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Thomas Kaminski (ur. 23 października 1992 w Dendermonde) – belgijski piłkarz polskiego pochodzenia, występujący na pozycji bramkarza. Syn Jacka Kamińskiego, reprezentanta Polski w siatkówce.
Karierę piłkarską Kaminski rozpoczynał w mieście Zellik, w lokalnych klubach Zellik Sport i Asse-Zellik. Następnie trafił do AFC Tubize i KAA Gent. W 2008 roku, w wieku 16 lat podpisał pierwszy profesjonalny kontrakt w Germinalu Beerschot. W ciągu trzech sezonów rozegrał w barwach zespołu z Antwerpii 37 spotkań.
Kariera piłkarska
Kariera klubowa
W sierpniu 2011 roku pomimo zainteresowania ze strony klubów holenderskich – VVV Venlo i Rody Kerkrade, Kaminski podpisał roczny kontrakt z Oud-Heverlee Leuven. Jednocześnie związał się obowiązującym od sezonu 2012/2013 pięcioletnim kontraktem z Anderlechtem, co wywołało nieprawdziwe informacje, jakoby młody Belg był wypożyczony do Leuven z drużyny Fiołków. W ekipie OHL Kaminski był podstawowym bramkarzem i zagrał w 25 spotkaniach Eerste klasse. Zgodnie z umową, latem 2012 Belg przeniósł się do Anderlechtu, w którego barwach zadebiutował 25 sierpnia 2012 roku w meczu ze swoim poprzednim klubem – OH Leuven. W 2014 roku został wypożyczony do Anorthosisu Famagusta, a w 2015 do FC København.
W 2016 przeszedł do KV Kortrijk. Barwy tego klubu nosił do roku 2018 i przeszedł do klubu KAA Gent w którym grał 2 lata. W roku 2020 przeniósł się na wyspy brytyjskie i rozpoczął grę w Angielskim klubie Blackburn Rovers.
Kariera reprezentacyjna
Kaminski jest reprezentantem Belgii w kolejnych kategoriach wiekowych: U-15, U-16, U-17, U-19 i U-21, w której zadebiutował 28 marca 2011 roku.
Przypisy
Bibliografia
Profil na stronach Belgijskiego Związku Piłki Nożnej
Belgijscy piłkarze polskiego pochodzenia
Piłkarze Germinalu Beerschot
Piłkarze Oud-Heverlee Leuven
Piłkarze RSC Anderlecht
Piłkarze Anorthosisu Famagusta
Piłkarze FC København
Piłkarze KV Kortrijk
Urodzeni w 1992
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Chinese Kali Mandir
In Kolkata's China Town, a temple to the Hindu goddess Kali.
Chinese Kali Mandir Courtesy Debmalya Das
Chinese Kali Mandir Deejayrocks2 / CC BY-SA 4.0
Courtesy Debmalya Das
Top Places in Kolkata
South Park Street Cemetery
Ronald Ross Memorial
Cheng Hi, a resident of Tangra in Kolkata, is a busy man at this time of the year. He is doing the last-minute preparations for Kali puja, an annual celebration of the Hindu goddess Kali, which will be organized at the Chinese Kali mandir on the night of Diwali, the biggest Indian festival. The temple is the only one of its kind created by the Chinese community of Kolkata.
Tangra is a region in east Kolkata which is also known as China Town. For many generations, it housed several tanneries owned by people of Hakka Chinese origin. The temple is visited throughout the year by the local Chinese and Bengali people, not just on special occasions like Kali puja.
"According to one of the popular local legends, some six decades ago when a Chinese boy fell sick and was not responding to any treatments his parents rushed him to an altar made of two black stones under a tree which was worshiped as an incarnation of goddess Kali by local Bengalis. After the parents offered puja at the altar the boy gradually recovered and as an expression of gratitude his parents built a Kali temple at the same location, which later came to be known as Chinese Kali mandir," said Ching, who has been involved with the temple management for the past 19 years. The black stones and the tree are still there inside the temple.
Kolkata's Chinese community was founded by immigrants who arrived in the late 18th century, taking up work as carpenters, tanners, and dock workers. In the 1960s, however, this community faced significant discrimination, as tensions rose between India and China during the Sino-Indian War. Considering ethnic Chinese residents as potential spies, the Indian government ordered many Chinese-Indians to leave the country and held thousands of those who did not comply in internment camps. As a result of these discriminatory practices, many Chinese-Indian residents of Kolkata and other cities emigrated to countries such as Canada, Australia, and the United States.
Every evening, a Bengali Hindu priest does the evening prayer rituals and chants Sanskrit mantras. But one may also find people from the Chinese community bowing down to pray in front of the goddess and lighting up Chinese incense sticks and burning handmade papers as a mark of veneration to the temple deity.
Entry is free. The temple ritual starts 5 p.m. every evening.
cultures hinduism gods immigration temples
diwashgahatraj
Matheswartala Road
Tangra
Kolkata, 700046
Nahoum and Sons
An eclectic mix of influences rot together in harmony in this India necropolis.
The world's largest secondhand book market is a haven for bibliophiles and bargain hunters.
This Jewish bakery, among the last in the region, sells legendary sweets from a century-old storefront.
Pen Hospital
For more than 70 years, this small shop has been a go-to for Kolkata's fountain pen connoisseurs and collectors.
Kumari Ghar
A brick building in Kathmandu's Durbar Square is home to Nepal's most prominent living goddess.
Visa Balaji Temple
A centuries-old shrine where Indians pray for U.S. visas.
Darjeeling, India
Mahakal Mandir
At this unique temple in Darjeeling, Hindu and Buddhist practices co-exist harmoniously.
Denden-gu
A small shrine dedicated to a god of lightning and electronics.
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{"url":"https:\/\/vulcanhammer.net\/2022\/07\/05\/constitutive-elasticity-equations-three-dimensional-formulation\/","text":"Posted in Academic Issues, Geotechnical Engineering\n\nConstitutive Elasticity Equations: Three-Dimensional Formulation\n\nThis is the beginning of what hopefully will be an enlightening series on elasticity and plasticity, primarily as it appears in finite element code. It is taken from a number of sources, including (but not limited to) Owen and Hinton (1980), Smith and Griffiths (1988) and Cook, Malkus and Plesha (1989.) Although finite element code has certainly advanced, the basics are still the same, especially with elasticity (plasticity is another matter.) More information on finite element analysis in geotechnics (with a link to Owen and Hinton) can be found here.\n\nThe basic elasticity relationship is familiar to engineers and can be stated as follows:\n\n$\\sigma = E \\epsilon$ (1)\n\nwhere\n\n\u2022 $\\sigma =$ stress\n\u2022 $E =$ Young\u2019s modulus, or modulus of elasticity\n\u2022 $\\epsilon =$ strain\n\nIt can be used in one-dimensional analyses such as tension and compression testing; we\u2019ll come back to that later. But real problems are always in three dimensions in one way or another. Let\u2019s start by considering an infinitesimal element (these are discussed in Verruijt) that looks like the one in Figure 69. There are many ways of looking at this element. For one thing, it is in static equilibrium, which means that the stresses can be changed along with the coordinate system and the results still be in static equilibrium. Related to this is the fact that there is one coordinate system at which point the shear stresses vanish and the three normal stresses that are left are the principal stresses; this is discussed here.\n\nThese, however, are not our main interest here. Our main interest is how the element deforms under stress. For now we will restrict our discussion to elastic, path-independent stresses. We will rewrite Equation (1) is a slightly different form:\n\n$D \\epsilon = \\sigma$ (2)\n\nNow both $\\sigma$ and $\\epsilon$ are vectors; you can see the multiple stresses in Figure 69. We use the notation $D$ because the modulus of elasticity $E$ is a material property and thus is a part of the elasticity matrix $D$.\n\nIn any case, when the actual vector and matrices are substituted into Equation (2), for three-dimensional problems such as shown in Figure 69 the equations look like this:\n\n\\left[\\begin{array}{cccccc} {\\frac{{\\it E}\\,\\left(1-\\nu\\right)}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & {\\frac{{\\it E}\\,\\nu}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & {\\frac{{\\it E}\\,\\nu}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & 0 & 0 & 0\\\\ \\noalign{\\medskip}{\\frac{{\\it E}\\,\\nu}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & {\\frac{{\\it E}\\,\\left(1-\\nu\\right)}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & {\\frac{{\\it E}\\,\\nu}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & 0 & 0 & 0\\\\ \\noalign{\\medskip}{\\frac{{\\it E}\\,\\nu}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & {\\frac{{\\it E}\\,\\nu}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & {\\frac{{\\it E}\\,\\left(1-\\nu\\right)}{\\left(1+\\nu\\right)\\left(1-2\\,\\nu\\right)}} & 0 & 0 & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & {\\frac{{\\it E}\\,\\left(1-\\nu\\right)}{\\left(1+\\nu\\right)\\left(2-2\\,\\nu\\right)}} & 0 & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & 0 & {\\frac{{\\it E}\\,\\left(1-\\nu\\right)}{\\left(1+\\nu\\right)\\left(2-2\\,\\nu\\right)}} & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & 0 & 0 & {\\frac{{\\it E}\\,\\left(1-\\nu\\right)}{\\left(1+\\nu\\right)\\left(2-2\\,\\nu\\right)}} \\end{array}\\right]\\left[\\begin{array}{c} \\epsilon_{{x}}\\\\ \\noalign{\\medskip}\\epsilon_{{y}}\\\\ \\noalign{\\medskip}\\epsilon_{{z}}\\\\ \\noalign{\\medskip}\\gamma{}_{{\\it xy}}\\\\ \\noalign{\\medskip}{\\it \\gamma}_{{\\it yz}}\\\\ \\noalign{\\medskip}{\\it \\gamma}_{{\\it zx}} \\end{array}\\right]=\\left[\\begin{array}{c} \\sigma_{{x}}\\\\ \\noalign{\\medskip}\\sigma_{{y}}\\\\ \\noalign{\\medskip}\\sigma_{{z}}\\\\ \\noalign{\\medskip}\\tau_{{\\it xy}}\\\\ \\noalign{\\medskip}\\tau_{{\\it yz}}\\\\ \\noalign{\\medskip}\\tau_{{\\it zx}} \\end{array}\\right] (3)\n\nIt should be immediately apparent that there are many repetitious quantities. There are many ways of dealing with the problem. For this analysis we will substitute Lam\u00e9\u2019s constants, which are\n\n$\\lambda = \\frac{\\nu E}{(1+\\nu)(1-2\\nu)}$ (4)\n\n$G = \\frac{E}{2(1+\\nu)}$ (5)\n\nwhere $\\nu$ is Poisson\u2019s Ratio and $G$ is the shear modulus of elasticity. The latter is very useful in soil mechanics (which can be seen, for example, here) and in fact Equation (4) can be restated as follows:\n\n$\\lambda = \\frac{2\\nu G}{1-2\\nu}$ (6)\n\nAt this point we need to observe one important thing: no matter whether you use Equation (4) or (6), the Lam\u00e9 constant $\\lambda$ blows up (or becomes singular, as we say in polite circles) when $\\nu = \\frac{1}{2}$. Materials where this is the case (such as very soft clays) are in reality fluids, the normal stresses become equal and the shear stresses only appear when the fluid moves. Avoiding this condition is one of the challenges of geotechnical finite element analysis.\n\nIt\u2019s also worth noting that this is the same Lam\u00e9 who came up with the theory behind shrink fits that is important with Vulcan hammers.\n\nIn any case, if we apply Equations (5) and (4) or (6), we obtain the following:\n\n\\left[\\begin{array}{cccccc} {\\frac{\\left(1-\\nu\\right)\\lambda}{\\nu}} & \\lambda & \\lambda & 0 & 0 & 0\\\\ \\noalign{\\medskip}\\lambda & {\\frac{\\left(1-\\nu\\right)\\lambda}{\\nu}} & \\lambda & 0 & 0 & 0\\\\ \\noalign{\\medskip}\\lambda & \\lambda & {\\frac{\\left(1-\\nu\\right)\\lambda}{\\nu}} & 0 & 0 & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & G & 0 & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & 0 & G & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & 0 & 0 & G \\end{array}\\right] \\left[\\begin{array}{c} \\epsilon_{{x}}\\\\ \\noalign{\\medskip}\\epsilon_{{y}}\\\\ \\noalign{\\medskip}\\epsilon_{{z}}\\\\ \\noalign{\\medskip}\\gamma{}_{{\\it xy}}\\\\ \\noalign{\\medskip}{\\it \\gamma}_{{\\it yz}}\\\\ \\noalign{\\medskip}{\\it \\gamma}_{{\\it zx}} \\end{array}\\right]=\\left[\\begin{array}{c} \\sigma_{{x}}\\\\ \\noalign{\\medskip}\\sigma_{{y}}\\\\ \\noalign{\\medskip}\\sigma_{{z}}\\\\ \\noalign{\\medskip}\\tau_{{\\it xy}}\\\\ \\noalign{\\medskip}\\tau_{{\\it yz}}\\\\ \\noalign{\\medskip}\\tau_{{\\it zx}} \\end{array}\\right] (7)\n\nWe see from this that we have simplified assembling the constitutive matrix considerably. From a coding standpoint, doing this eliminates not only the possibility of coding error but also computational cost by eliminating the sheer number of operations. This is especially useful if these matrices are reconstituted during the analysis (as one would expect if strain-softening is applied, for example.)\n\nIf we multiply the left hand side through, we have\n\n\\left[\\begin{array}{c} \\sigma_{{x}}\\\\ \\noalign{\\medskip}\\sigma_{{y}}\\\\ \\noalign{\\medskip}\\sigma_{{z}}\\\\ \\noalign{\\medskip}\\tau_{{\\it xy}}\\\\ \\noalign{\\medskip}\\tau_{{\\it yz}}\\\\ \\noalign{\\medskip}\\tau_{{\\it zx}} \\end{array}\\right] = \\left[\\begin{array}{c} {\\frac{\\left(1-\\nu\\right)\\lambda\\,\\epsilon_{{x}}}{\\nu}}+\\lambda\\,\\epsilon_{{y}}+\\lambda\\,\\epsilon_{{z}}\\\\ \\noalign{\\medskip}\\lambda\\,\\epsilon_{{x}}+{\\frac{\\left(1-\\nu\\right)\\lambda\\,\\epsilon_{{y}}}{\\nu}}+\\lambda\\,\\epsilon_{{z}}\\\\ \\noalign{\\medskip}\\lambda\\,\\epsilon_{{x}}+\\lambda\\,\\epsilon_{{y}}+{\\frac{\\left(1-\\nu\\right)\\lambda\\,\\epsilon_{{z}}}{\\nu}}\\\\ \\noalign{\\medskip}G{\\it \\gamma}_{{\\it xy}}\\\\ \\noalign{\\medskip}G{\\it \\gamma}_{{\\it yz}}\\\\ \\noalign{\\medskip}G{\\it \\gamma}_{{\\it zx}} \\end{array}\\right] (8)\n\nInversely, if we want to know the strains, we would compute these by the relationship\n\n$\\epsilon = D^{-1} \\sigma$ (9)\n\nInverting,\n\nD_{e}^{-1}=\\left[\\begin{array}{cccccc} -{\\frac{\\nu}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & {\\frac{{\\nu}^{2}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & {\\frac{{\\nu}^{2}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & 0 & 0 & 0\\\\ \\noalign{\\medskip}{\\frac{{\\nu}^{2}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & -{\\frac{\\nu}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & {\\frac{{\\nu}^{2}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & 0 & 0 & 0\\\\ \\noalign{\\medskip}{\\frac{{\\nu}^{2}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & {\\frac{{\\nu}^{2}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & -{\\frac{\\nu}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}} & 0 & 0 & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & {G}^{-1} & 0 & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & 0 & {G}^{-1} & 0\\\\ \\noalign{\\medskip}0 & 0 & 0 & 0 & 0 & {G}^{-1} \\end{array}\\right] (10)\n\nand the strains can be computed as follows\n\n\\left[\\begin{array}{c} \\epsilon_{{x}}\\\\ \\noalign{\\medskip}\\epsilon_{{y}}\\\\ \\noalign{\\medskip}\\epsilon_{{z}}\\\\ \\noalign{\\medskip}\\gamma{}_{{\\it xy}}\\\\ \\noalign{\\medskip}{\\it \\gamma}_{{\\it yz}}\\\\ \\noalign{\\medskip}{\\it \\gamma}_{{\\it zx}} \\end{array}\\right] = \\left[\\begin{array}{c} -{\\frac{\\nu\\,\\sigma_{{x}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}+{\\frac{{\\nu}^{2}\\sigma_{{y}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}+{\\frac{{\\nu}^{2}\\sigma_{{z}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}\\\\ \\noalign{\\medskip}{\\frac{{\\nu}^{2}\\sigma_{{x}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}-{\\frac{\\nu\\,\\sigma_{{y}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}+{\\frac{{\\nu}^{2}\\sigma_{{z}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}\\\\ \\noalign{\\medskip}{\\frac{{\\nu}^{2}\\sigma_{{x}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}+{\\frac{{\\nu}^{2}\\sigma_{{y}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}-{\\frac{\\nu\\,\\sigma_{{z}}}{\\lambda\\,\\left(-1+\\nu+2\\,{\\nu}^{2}\\right)}}\\\\ \\noalign{\\medskip}{\\frac{\\tau_{{\\it xy}}}{G}}\\\\ \\noalign{\\medskip}{\\frac{\\tau_{{\\it yz}}}{G}}\\\\ \\noalign{\\medskip}{\\frac{\\tau_{{\\it zx}}}{G}} \\end{array}\\right] (11)\n\nRelationships such as Equations (8) and (11) will be more useful when we consider two-dimensional problems.\n\nReferences\n\n\u2022 Cook, R.D., Malkus, D.S. and Plesha, M.E. (1989). Concepts and Applications of Finite Element Analysis, John Wiley & Sons, Inc., New York, NY\n\u2022 Owen, D.R.J., and Hinton, E. (1980). Finite Elements in Plasticity: Theory and Practice. Pineridge Press, Swansea, Wales.\n\u2022 Smith, I.M., and Griffiths, D.V. (1988). Programming the Finite Element Method. Second Edition. John Wiley & Sons, Chichester, England.\n\nNote: after this was first posted, some errors were discovered. These were corrected with the assistance of Jaeger, J.C. and Cook, N.G.W. (1979)\u00a0Fundamentals of Rock Mechanics. London: Chapman and Hall. This book has an excellent treatment of basic theoretical solid mechanics in general and elasticity in particular. Although they do not put the equations in matrix form, they do use the Lam\u00e9 constants in their formulation, albeit in a different way than is done here. We apologise for any inconvenience or confusion.\n\n10 thoughts on \u201cConstitutive Elasticity Equations: Three-Dimensional Formulation\u201d\n\n1. thomas mochizuki says:\n\nthe book by cook is excellent especially the section on geometric non-linearity. will you\nbe covering the programming of the finite element(s) to be used ?\n\nLike\n\n1. This series will focus on material elasticity and plasticity. Maybe the geometric part later. My PhD advisor (Dr. James C. Newman III) recommended Cook to help solve a problem with plasticity and dynamic analysis.\n\nLike\n\n1. thomas mochizuki says:\n\nI got interested in geometric non-linearity a long time ago because of fly rod design which is large scale deflections which Cook covered somewhat. There was a guy at MIT who wrote a\nprogram for such but wouldn\u2019t help me out. Anyway I\u2019d like to follow this\nand study along with it to better understand the math. I\u2019ve done this with advanced\nelasticity with the book by sokolnikov but admittedly not very well. will this email keep me up to date with what\u2019s happening?\n\nLike\n\n2. If you want to keep up with this blog, you need to subscribe. Scroll down until you see the \u201cSubscribe\u201d box (you may already have passed it.) I\u2019m doing this because I have written FEA code and my stats show FEA is a topic of interest.\n\nLike\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed.","date":"2023-03-25 04:43:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 23, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8371867537498474, \"perplexity\": 1684.1410484503785}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945315.31\/warc\/CC-MAIN-20230325033306-20230325063306-00132.warc.gz\"}"}
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{"url":"http:\/\/piccolboni.info\/2010\/07\/map-reduce-algorithm-for-connected.html","text":"In a recently published book about algorithms for the map reduce model of computation, a simple connected components algorithm based on lablel propagation is proposed, but its complexity depends on the diameter of the graph, which can be very large. It turns out we can get rid of that dependency with a completely different algorithm, ported from the PRAM model.\n\nThe authors observe the many if not most practically occurring web-sized graphs have small diameters (the \"small world\" phenomenon) and therefore their algorithm is of practical importance. At the same time, new types of large graphs become available to researchers and practitioners who want to data mine them and do not know a priori what the diameter of these graphs is. And if you don't buy this motivation for an algorithm with a diameter-independent upper bound, it's just \"because it's there\".\n\nResearching the literature for parallel algorithms for connected components, I found one called Random Mate (I am not sure if the correct attribution is to Hillel Gazit or John H. Reif, my access to older literature is limited, but I found a nice write up ) that seems amenable to a map reduce implementation. The basic idea is that, instead of extending components one or few nodes at a time, we should merge them and enjoy an exponential increase in component size for each iteration, on average. More specifically, components are represented with directed forests overlaid on top of the input graph, that is using the same nodes but a separate set of edges. The trees in the forest are kept very shallow, essentially stars or very close to stars, so that eventually each node will have as parent the root of a tree, providing a convenient representation for each component. The most technical bit is that at each step, nodes are randomly assigned to two sets, let's call them \"high\" and \"low\" (the above write up uses \"male\" and \"female\"), but what counts really is the type of the parent in the forest. Edges that hit two nodes with parents of the same type are out of the game for that specific iteration. Edges that hit nodes with one high and one low parent can only be used in one direction, that is to attach the forest of the low parent under the forest of the high parent. If we didn't do this, working in parallel we could end up creating loops, thus turning trees into general graphs (specifically, DAGs) and violating a key invariant of the algorithm. It is still possible that a low node could have multiple neighboring high nodes belonging to different trees, but only one wins based on the properties of the PRAM CRCW model of parallel computation which specifies a way for concurrent writes to be sorted out.\n\nThe following is a sketch of a map-reduce version of this PRAM algorithm. Its correctness and performance bounds follow from the fact that it closely emulates the original, step by step and therefore the original proof is still valid. In my pseudo code, we will consider tables as on disk data structures and consider join operations as primitive operations. Tables are implemented as distributed file system files. See the aforementioned book for the details on how to implement joins, or the Hive software. We will build a directed forest on top of the graph, so we will devote a table $F$ to the edge list of the forest and on to $E$, the input graph $G=(V,E)$ (undirected edges are represented as pairs of directed ones). $F$ is initialized as $(v,v) \\;\\forall v \\in V$ that is all the self-loops (this is a slight departure from the definition of a forest, but let me still call it a forest, it makes for simpler code with no special handling of the root nodes). The first operation is a join described in SQL-like language:\nselect E.u as u , E.v as v, F1.v as p1, F2.v as p2 from F F1 join E on E.u = F1.u join F F2 on E.v = F2.u\nThat is, we \"annotate\" each edge with the parents in the forest of each vertex hit by the edge. Now we describe a map phase with input $(u, v, p_1, p_2)$. $r$ is the map reduce round number.\ndef map(u, v, p1, p2): if p1 == p2: #same component return [] #do nothing else: h1 = hash(p1, r)%2 #randomly assign \"high\" and \"low\" h2 = hash(p2, r)%2 if h1 == h2: return [] #if same give up else: return [(p1,p2) if h1 else (p2,p1)] #otherwise merge one into the other\nThis phase, in the original formulation for the PRAM, requires a CRCW model, without preference for a specific variant, so we will use the arbitrary CRCW model and implement it in the reduce phase, the first node of the edge being designated as the key.\ndef reduce(list): return list[0]\n(this is amenable to be used as a combiner as well)\nThe output of this reducer becomes F', a table of updates to the forest that needs to be merged with the existing F.\nselect coalesce(F'.u, F.u), coalesce(F'.v, F.v) from F left outer join F' on F.u = F'.u\nwhich is a way of saying: keep all the edges in F unless there is one in F' with the same starting node, then replace it.\n\nFinally, there is a path shortening phase that turns trees in the forest into stars. This is implemented with a self join on the forest.\nselect F1.u, F2.v from F F1 join F F2 on F1.v = F2.u\nAgain this replaces F. The termination criterion is that if there are no edges in G with different parents, we are done.\n\nAs in the original algorithm, the number of iteration is $O(\\log(N))$, each of which requires sorting of all the edges, the graph's and the forest's. There are highly optimized sorting implementations for the map reduce model, but I haven't found a discussion of their asymptotic complexity. From a quick look at Terasort, I think it requires $O(N\\log(N))$ work and takes $O(\\log(N))$ time with $\\Omega(N)$ available processors. These are the dominating computational costs, as the mappers and reducers above all execute in constant time, for an overall $O(N(\\log(N))^2)$ work and $O((\\log(N))^2)$ time. There is one detail to take into account for the reducer that implements the effects of the arbitrary CRCW PRAM model: for a high degree node, one reducer might get a very large proportion of the input. This is not a problem, as the reducer can just return after reading the first line of input and we can use the reducer as a combiner for and additional optimization.\n\nAnother potential optimization could be to use, instead of a random bipartition of the nodes, a random priority. This way more mergers would happen at each step. The central part of the algorithm becomes:\ndef map(u, v, p1, p2): if p1 == p2: #same component return [] #do nothing else: h1 = hash(p1, r) #randomly assign priority h2 = hash(p2, r) if h1 == h2: return [] #if same give up else: return [(p1,p2) if h1 > h2 else (p2,p1)] #otherwise merge one into the other\nThat would require though a more powerful path shortening phase, since the correctness proof relies on the trees in the forest having depth one at the end of each iteration, and the current path shortening can only halve the depth. I expect that this modified algorithm would be faster for dense graphs for which the forest has many fewer edges.\n\nEven without this optimization, you might have noticed that the diameter of the graph was notably absent from the discussion, and in fact this algorithm works well even for paths, which was our original goal. It could require more iterations though than the label propagation algorithm when the diameter of the graph is $\\Omega(\\log(N))$ as far as I can tell from these bounds, but I don't have an example where it does. It might be possible to show that this algorithm works faster than $O(\\log(N))$ when the diameter is small.","date":"2018-01-19 05:26:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.49662572145462036, \"perplexity\": 804.2575329463656}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084887746.35\/warc\/CC-MAIN-20180119045937-20180119065937-00720.warc.gz\"}"}
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O Bell 212 Twin Huey é um helicóptero médio bimotor produzido pela Bell Helicopter. É uma versão atualizada e mais moderna do Bell UH-1 Iroquois.
O 212 é comercializado para operadores civis e tem uma configuração de quinze assentos, com um piloto e quatorze passageiros. Na configuração da carga o 212 tem uma capacidade interna de 220 pés cúbicos (6,23 metros cúbicos) e uma carga externa de até 5,000 lb (2,268 kg).
Desenvolvimento
Com base em uma fuselagem maior do Bell 205, o Bell 212 foi desenvolvido originalmente para as forças canadenses como o CUH-1N e, mais tarde, redesignado como o CH-135. As forças canadenses receberam 50 unidades a partir de maio de 1971. Ao mesmo tempo, as Forças Armadas dos Estados Unidos ordenaram 294 unidades sob a designação de UH-1N.
Em 1971 o 212 foi desenvolvido para aplicações comerciais. Entre os primeiros utilizadores do 212 na aviação civil está a CHC Helicopter da Noruega, que utilizou o 212 para ser utilizado como apoio à plataformas de petróleo.
Operadores governamentais
Alguns operadores do Bell 212:
Guarda Costeira do Canadá
Polícia Nacional da Colômbia
Guarda Costeira do Japão
Polícia Sérvia
Polícia Nacional Eslovênia
Polícia Real Thai
Departamento policial do Condado de San Bernardino
Departamento de bombeiros de San Diego
Departamento policial do Condado de Ventura
Helicópteros dos Estados Unidos
Aeronaves da Bell
Helicópteros da Bell
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{"url":"http:\/\/incommunity.it\/xmqx\/linear-transformation-matrix-calculator.html","text":"# Linear Transformation Matrix Calculator\n\nThe kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation). First, we need to find the inverse of the A matrix (assuming it exists!) Using the Matrix Calculator we get this: (I left the 1\/determinant outside the matrix to make the numbers simpler) Then multiply A-1 by B (we can use the Matrix Calculator again): And we are done! The solution is: x = 5, y = 3, z = \u22122. The important conclusion is that every linear transformation is associated with a matrix and vice versa. Linear transformations. Number of rows and columns decides the shape of matrix i. Kissinger Institute for Computing and Information Sciences Radboud University Nijmegen Version: autumn 2017 A. A point v in 2 can be transformed to a point v' in 3 with this equation: v' = B(A^-1)v where (A^-1) is the inverse of A. Proof: The linear transformation has an inverse function if and only if it is one-one and onto. , to get the kernel of. In physics related uses, they are used in the study of. Matrix calculator supports matrices with up to 40 rows and columns. It is helpful to sketch the graph and find the projections of i and j geometrically. In part (a), we computed that T(e 1) = 2 6 6 4 2 0 2 3 7 7 5, and part of our given information is that T(e 2) = 2 6 6 4 5 2 2 3 7 7 5. Matrix rank is calculated by reducing matrix to a row echelon form using elementary row operations. De\ufb01ne T : V \u2192 V as T(v) = v for all v \u2208 V. Linear Transformations However, what if the nonhomogeneous right\u2010hand term is discontinuous? There exists a method for solving such problems that can also be used to solve less frightening IVP's (that is, ones that do not involve discontinuous terms) and even some equations whose coefficients are not constants. Let's now define components. Elementary transformations of a matrix find a wide application in various mathematical problems. We can eliminate theta by squaring both sides and adding them (I have taken the liberty to transpose the first term on the right hand side of the equation, which is independent of theta, and corresponds to the average stress). Maths - Calculation of Matrix for 3D Rotation about a point In order to calculate the rotation about any arbitrary point we need to calculate its new rotation and translation. Be careful! Matrix multiplication is not commumative. As every vector space property derives from vector addition and scalar multiplication, so too, every property of a linear transformation derives from these two defining properties. 2x2 matrices are most commonly employed in describing basic geometric. The first property deals with addition. Then T is a linear transformation. In the latter case the matrix is invertible and the linear equation system it represents has a single unique solution. Suppose T : V \u2192. Linear transformations and matrices 94 4. Some interesting transformations to try: - enter this as - enter this as. Formally, the singular value decomposition of an m\u00d7n real or complex matrix M is a factorization of the form. The line L: y = 6\/5*x. Practice problems here: Note: Use CTRL-F to type in search term. Consider the case of a linear transformation from Rn to Rm given by ~y = A~x where A is an m \u00d7 n matrix, the transformation is invert-ible if the linear system A~x = ~y has a unique solution. The change of basis formula B = V 1AV suggests the following de nition. Click 'Show basis vectors' to see the effect of the transformation on the standard basis vectors , (also called ). A Matrix and a vector can be multiplied only if the number of columns of the matrix and the the dimension of the vector have the same size. The number of equations in the system: Change the names of the variables in the system. And, if you real don't understand, you can see this video link which show the same topic. Looking for a primer on how to solve matrix problems using a TI-89 graphing calculator? See how it's done with this free video algebra lesson. 3x4 Projection Matrix. Matrix methods represent multiple linear equations in a compact manner while using the. 5),(0,1)] is a linear transformation. This java applet is a simulation that demonstrates some properties of matrices and how they can be used to describe a linear transformation in two dimensions. Their inner product x\u22a4yis actually a 1\u00d71 matrix: x\u22a4y= [s] where s= Xm i=1 x iy i. Play around with different values in the matrix to see how the linear transformation it represents affects the image. Now that we have some good context on linear transformations, it's time to get to the main topic of this post - affine transformations. See also addrow and append. Another standard is book\u2019s audience: sophomores or juniors, usually with a background of at least one semester of calculus. Library: Inverse matrix. So add the two rows on your scratch paper:. Just like on the Systems of Linear. for any vectors and in , and. The red lattice illustrates how the entire plane is effected by multiplication with M. The table lists 2-D affine transformations with the transformation matrix used to define them. If is a linear transformation mapping to and \u2192 is a column vector with entries, then (\u2192) = \u2192for some \u00d7 matrix , called the transformation matrix of. f(kA)=kf(A). This means that applying the transformation T to a vector is the same as multiplying by this matrix. Applications of Matrix: A major application of matrices is to represent linear transformation. This page is not in its usual appearance because WIMS is unable to recognize your web browser. To deter-mine the least squares estimator, we write the sum of squares of the residuals (a function of b)as S(b) \u00bc X e2 i \u00bc e 0e \u00bc (y Xb)0(y Xb) \u00bc y0y y0Xb b0X0y \u00feb0X0Xb: (3:6) Derivation of least squares estimator. To find the image of a point, we multiply the transformation matrix by a column vector that represents the point's coordinate. 3: Matrix of a Linear Transformation If T : Rm \u2192 Rn is a linear transformation, then there is a matrix A such that T(x) = A(x) for every x in Rm. (TODO: implement these alternative methods). The Matrix Row Reducer will convert a matrix to reduced row echelon form for you, and show all steps in the process along the way. I would like to transform into a system in the form A*X = 0. You can input only integer numbers, decimals or fractions in this online calculator (-2. We de ne T Aby the rule T A(x)=Ax:If we express Ain terms of its columns as A=(a 1 a 2 a n), then T A(x)=Ax = Xn i=1 x ia i: Hence the value of T A at x is the linear combination of the columns of A which is the ith. Find the Kernel. The calculator below solves the quadratic equation of. Thank you so much, your explanation made it so much clearer! $\\endgroup$ \u2013 Kim Apr 20 '14 at 18:26. The above expositions of one-to-one and onto transformations were written to mirror each other. Type an integer or a simplified fraction. One of the homework assignments for MAT 119 is to reduce a matrix with a graphing calculator. Singular Value Decomposition (SVD) tutorial. , to get the kernel of. The following numbered formulas (M1,. The arrows denote eigenvectors corresponding to eigenvalues of the same color. For the calculation of a determinant, only the parameters are used. These are called eigenvectors (also known as characteristic vectors). In other words, di erent vector in V always map to di erent vectors in W. Exponential to linear transformation comparative analysis of student's achievement in algbraic simultaneous equations and word problem leading to simultaneous equations 5th grade math adding, subtracting, multiplying, dividing fractions work sheet. \"Reflection transformation matrix\" is the matrix which can be used to make reflection transformation of a figure. Matrix Multiplication for a Composition. That means you can combine rotations, and keep combining them, and as long as you occasionally correct for round-off error, you will always have a rotation matrix. The \ufb01rst column of the required matrix is P\u00a11 S TPBe1 = I2T(b1) = T(b1. In fact, every linear transformation (between finite dimensional vector spaces) can. So, for example, you could use this test to find out whether people. A matrix is said to be rank-deficient if it does not have full rank. For example, the linear equation x 1 - 7 x 2 - x 4 = 2. To deter-mine the least squares estimator, we write the sum of squares of the residuals (a function of b)as S(b) \u00bc X e2 i \u00bc e 0e \u00bc (y Xb)0(y Xb) \u00bc y0y y0Xb b0X0y \u00feb0X0Xb: (3:6) Derivation of least squares estimator. Above all, they are used to display linear transformations. Examples: (a) Prove that if T(x) = Axwhere A is an m \u00d7n matrix, then T is a linear transformation. Linear transformation De\ufb01nition. The change of basis formula B = V 1AV suggests the following de nition. form unrolled into an equation and above is just another way of representing it in linear algebra way. We proceed with the above example. The vector may change its length, or become zero (\"null\"). Suppose that \\begin {align*} T (\\mathbf {u})&=T\\left ( \\begin {bmatrix} 1 \\\\ [\u2026] Find an Orthonormal Basis of the Range of a Linear Transformation Let T: R2. Rref Calculator for the problem solvers. This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule. A 3x3 matrix maps 3d vectors into 3d vectors. The inverse is equivalent to subtracting. Example 3 The re\ufb02ection matrix R D 01 10 has eigenvalues1 and 1. They are most commonly used in linear algebra and computer graphics, since they can be easily represented, combined and computed. Graphing quadratic functions: General form VS. Transformation Matrix Properties Transformation matrices have several special properties that, while easily seen in this discussion of 2-D vectors, are equally applicable to 3-D applications as well. In physics related uses, they are used in the study of. where a,b,c,d are complex constants. The inverse of a linear transformation De nition If T : V !W is a linear transformation, its inverse (if it exists) is a linear transformation T 1: W !V such that T 1 T (v) = v and T T (w) = w for all v 2V and w 2W. In XYZ, any color is represented as a set of positive values. The transformation that works this magic is called the Cholesky transformation; it is represented by a matrix that is the \"square root\" of the covariance matrix. There are no hard and fast rules for making change of variables for multiple integrals. Linear algebra is a sub-field of mathematics concerned with vectors, matrices, and linear transforms. Discussed are the situations when a linear system has no solution or infinite solutions. Observability and controllability tests will be connected to the rank tests of ceratin matrices: the controllability and observability matrices. If a transformation satisfies two defining properties, it is a linear transformation. If the parent graph is made steeper or less steep (y = \u00bd x), the transformation is called a dilation. Linear transformations. Recall that if a set of vectors v 1;v 2;:::;v n is linearly independent, that means that the linear combination c. A linear transformation (multiplication by a 2\u00d72 matrix) followed by a translation (addition of a 1\u00d72 matrix) is called an affine transformation. For example, consider a rotation in the common world by a positive angle around the up-axis. Matrix Calculator Matrix Calculator computes all the important aspects of a matrix: determinant, inverse, trace , norm. Then an example of using this technique on a system of three equations with three unknowns. xla is an addin for Excel that contains useful functions for matrices and linear Algebra: Norm, Matrix multiplication, Similarity transformation, Determinant, Inverse, Power, Trace, Scalar Product, Vector Product, Eigenvalues and Eigenvectors of symmetric matrix with Jacobi algorithm, Jacobi's rotation matrix. We can call this the before transformation matrix: I1,I2,I3,In refer to the dimensions of the matrix (or number of rows and columns). Use the result matrix to declare the final solutions to the. transformations (or matrices), as well as the more di\ufb03cult question of how to invert a transformation (or matrix). pose of the residual vector e is the 1 n matrix e0 \u00bc (e 1, , e n). Could anyone help me out here? Thanks in. Leave extra cells empty to enter non-square matrices. A linear transformation T: R n \u2192 R m has an inverse function if and only if its kernel contains just the zero vector and its range is its whole codomain. It does not give only the inverse of a 2x2 matrix, and also it gives you the determinant and adjoint of the 2x2 matrix that you enter. Engineers use matrices to model physical systems and perform accurate calculations that are needed for complex mechanics to work. For math, science, nutrition, history. The arrows denote eigenvectors corresponding to eigenvalues of the same color. They are most commonly used in linear algebra and computer graphics, since they can be easily represented, combined and computed. A general matrix or linear transformation is di\ufb03cult to visualize directly, however one can under-. Visualizing linear transformations. Then the image can be used to perform the next linear transformation. In a square matrix the diagonal that starts in the upper left and ends in the lower right is often called the main diagonal. In that context, an eigenvector is a vector\u2014different from the null vector\u2014which does not change direction in the transformation (except if the transformation turns the vector to the opposite direction). y+2z-w = 0 2x+8y+2z-6w = 0 2x+7y-5w = 0 Step 2: Represent the system of linear equations in matrix form. r mp s 0 1 0 _2 0 0 0 0 1 s mp r 0 2 0 _1 0 0 0 0 1 This means we must be careful about the order of application of graphics transformations. com To create your new password, just click the link in the email we sent you. Such a repre-sentation is frequently called a canonical form. 0 x 3 + 3 x 1 = 3. In the computer graphics realm, they\u2019re also used for things like \u2014 surprise! \u2014 linear transformations and projecting 3D images onto a 2D screen. The \ufb01rst column of the required matrix is P\u00a11 S TPBe1 = I2T(b1) = T(b1. The Square Root Matrix Given a covariance matrix, \u03a3, it can be factored uniquely into a product \u03a3=U T U, where U is an upper triangular matrix with positive diagonal entries and the. Previous Post Next Post. Enter a matrix, and this calculator will show you step-by-step how to calculate a basis for the Column Space of that matrix. It is well known that a solvable system of linear algebraic equations has a solution if and only if the rank of the system matrix is full. 2 Null Spaces, Column Spaces, & Linear Transformations Definition The null space of an m n matrix A, written as Nul A,isthesetofallsolutionstothe homogeneous equation Ax 0. Describe the image of the linear transformation T from R2 to R2 given by the matrix A = \" 1 3 2 6 # Solution T \" x1 x2 # = A \" x1 x2 # = \" 1 3 2 6 #\" x1 x2. A linear transformation may or may not be injective or surjective. Account Details Login Options Account Management Settings Subscription Logout. Graphing Calculator. The numerals a, b, and c are coefficients of the equation, and they. Thus the system of equations. Furthermore, the kernel of T is the null space of A and the range of T is the column space of A. Stretch means we are look at the top half of the table, and then x-axis invariant means. Let L: R^3-->R^2 be a linear transformation such that L (x1, x2, x3) = (x1+x2+x3, x1-x2+x3) a) Determine the matrix representing the linear transformation. Formally, the singular value decomposition of an m\u00d7n real or complex matrix M is a factorization of the form. Specifically, if T: n m is a linear transformation, then there is a unique m n matrix, A, such that T x Ax for all x n. A linear transformation may or may not be injective or surjective. The previous three examples can be summarized as follows. The above expositions of one-to-one and onto transformations were written to mirror each other. That is, any vector or matrix multiplied by an identity matrix is simply the original vector or matrix. Introduction to Linear Algebra exam problems and solutions at the Ohio State University (Math 2568). This means that applying the transformation T to a vector is the same as multiplying by this matrix. The Attempt at a Solution I tried constructing a matrix using the vectors being applied to T and row reducing it. The next step is to get this into RREF. Image: (intrinsic\/internal camera parameters). The second eigenvector is. {1: ; 2: ; 3: } Fill the system of linear equations: Entering data into the inverse matrix method calculator. 1 Subspaces and Bases 0. We are allowed to perform the matrix multiplications of r and s before multiplying by square ,. The inverse of a linear transformation Theorem: Let A be an n x m matrix. Eigenvalues and Eigenvectors Projections have D 0 and 1. If your transformation matrix represents a rotation followed by a translation, then treat the components separately. Read the instructions. It combines a user-friendly presentation with straightforward, lucid language to clarify and organize the techniques and applications of linear algebra. To nd the image of a transformation, we need only to nd the linearly independent column vectors of the matrix of the transformation. THEOREM 2. For example, the linear equation x 1 - 7 x 2 - x 4 = 2. Note that q is the number of columns of B and is also the length of the rows of B, and that p is the number of rows of A and is also the length of the columns of A. This complex matrix calculator can perform matrix algebra, all the previously mentioned matrix operations and solving linear systems with complex matrices too. Furthermore, the kernel of T is the null space of A and the range of T is the column space of A. Perspective perspective(). Thus matrix multiplication provides a wealth of examples of linear transformations between real vector spaces. Write the standard matrix A for the transformation T. You may choose a shape to apply transformations to, and zoom and in out using the slider. Take the coordinate transformation example from above and this time apply a rigid body rotation of 50\u00b0 instead of a coordinate transformation. Graphing quadratic functions: General form VS. How to nd the matrix representing a linear transformation 95 5. In particular, A and B must be square and A;B;S all have the same dimensions n n. (Use a comma to separate answers as needed. Thus, the standard matrix. A matrix form of a linear system of equations obtained from the coefficient matrix as shown below. Matrix of Linear Xformations & Linear Models. have the same number of rows and columns) as the vector matrix of the figure it transforms, since this is a pre-requisite for matrix addition. Instead x 1, x 2, you can enter your names of variables. The linear transformation of primary interest in matrix theory is the transformation y =Ax. Initially, it was a sub-branch of linear algebra, but soon it grew to cover subjects related to graph theory, algebra, combinatorics and statistics as well. Just like on the Systems of Linear. More in-depth information read at these rules. Now we can define the linear. 2x\u22123y=8 4x+5y=1 2 x. There is an m n matrix A such that T has the formula T(v) = Av for v 2Rn. To compute the cumulative distribution of Y = g(X) in terms of the cumulative distribution of X, note that F. The arrows denote eigenvectors corresponding to eigenvalues of the same color. Error-correcting codes are used, e. Change of basis formula relates coordinates of one and the same vector in two different bases, whereas a linear transformation relates coordinates of two different vectors in the same basis. How could you find a standard matrix for a transformation T : R2 \u2192 R3 (a linear transformation) for which T([v1,v2]) = [v1,v2,v3] and T([v3,v4-10) = [v5,v6-10,v7] for a given v1,,v7? I have been thinking about using a function but do not think this is the most efficient way to solve this question. Matrix Multiplication, Addition and Subtraction Calculator; Matrix Inverse, Determinant and Adjoint Calculator. Matrix rank is calculated by reducing matrix to a row echelon form using elementary row operations. An augmented matrix is a combination of two matrices, and it is another way we can write our linear system. SheLovesMath. In fact, every linear transformation (between finite dimensional vector spaces) can. You can redifine the matrix. This matrix calculator allows you to enter your own 2\u00d72 matrices and it will add and subtract them, find the matrix multiplication (in both directions) and the inverses for you. Here you can calculate matrix rank with complex numbers online for free with a very detailed solution. Then T A: Rm \u2192 Rn is invertible if and only if n = m = rank(A). In other words rotation about a point is an 'proper' isometry transformation' which means that it has a linear and a rotational component. A basis of a vector space is a set of vectors in that is linearly independent and spans. Some of the techniques summarized. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Example 6: Find the loop currents in the D. A more formal understanding of functions. a system of linear equations with inequality constraints. The Linear Transformation Grapher. You can enter any number (not letters) between \u221299 and 99 into the matrix cells. It seems to be particularly fond of Flens and Seldon. English Espa\u00f1ol Portugu\u00eas \u4e2d\u6587 (\u7b80\u4f53) \u05e2\u05d1\u05e8\u05d9\u05ea \u0627\u0644\u0639\u0631\u0628\u064a\u0629. Write the standard matrix A for the transformation T. Transformation Matrix Main Concept A linear transformation on a vector space is an operation on the vector space satisfying two rules: , for all vectors , , and all scalars. The matrix A=[1,2;2,1;1,1] (three rows and two columns) induces a linear map from R 2 to R 3, with domain R 2. Before we define an elementary operation, recall that to an nxm matrix A, we can associate n rows and m columns. 1 T(~x + ~y) = T(~x) + T(~y)(preservation of addition) 2 T(a~x) = aT(~x)(preservation of scalar multiplication) Linear Transformations: Matrix of a Linear Transformation Linear Transformations Page 2\/13. 1 De nitions A subspace V of Rnis a subset of Rnthat contains the zero element and is closed under addition and scalar multiplication: (1) 0 2V (2) u;v 2V =)u+ v 2V (3) u 2V and k2R =)ku 2V. A linear transformation between two \ufb01nite dimensional vector spaces is uniquely determined once the images of an ordered basis for the domain are speci\ufb01ed. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Note that has rows and columns, whereas the transformation is from to. Enter a matrix, and this calculator will show you step-by-step how to calculate a basis for the Column Space of that matrix. Then T A: Rm \u2192 Rn is invertible if and only if n = m = rank(A). Note that q is the number of columns of B and is also the length of the rows of B, and that p is the number of rows of A and is also the length of the columns of A. This permits matrices to be used to perform translation. The two defining conditions in the definition of a linear transformation should \"feel linear,\" whatever that means. The calculator below will calculate the image of the points in two-dimensional space after applying the transformation. Matrix Calculator Matrix Calculator computes all the important aspects of a matrix: determinant, inverse, trace , norm. Matrix transformation matrix() Describes a homogeneous 2D transformation matrix. One-to-one transformations are also known as injective transformations. In physics related uses, they are used in the study of electrical circuits, quantum mechanics and optics. Multiply Two Matrices. If I use to denote the matrix of the linear transformation f, this result can be expressed more concisely as Proof. Linear Programming Calculator is a free online tool that displays the best optimal solution for the given constraints. situations: 1) as the set of solutions of a linear homogeneous system or 2) as the set of all linear combinations of a given set of vectors. \u201cMatrix decomposition refers to the transformation of a given matrix into a given canonical form. Matrix rank is calculated by reducing matrix to a row echelon form using elementary row operations. The table lists 2-D affine transformations with the transformation matrix used to define them. Vector transformations. In other words, the matrix (number) corresponding to the composition is the product of the matrices (numbers) corresponding to each of the \"factors\" and of. Read the instructions. {1: ; 2: ; 3: } Fill the system of linear equations: Entering data into the inverse matrix method calculator. By pre-multiplying both sides of these equations by the inverse of Q , Q 1 , one obtains the. If they are, prove it; if not, provide a counterexample to one of the properties: (a) T : R2!R2, with T x y = x+ y y Solution: This IS a linear transformation. Preimage of a set. This is also called reduced row echelon form (RREF). It is helpful to sketch the graph and find the projections of i and j geometrically. Another way, the one that will be used, it is multiplying the inverse of the basis matrix of B by the basis matrix. Model matrix. In the chart, A is an m \u00d7 n matrix, and T: R n \u2192 R m is the matrix transformation T (x)= Ax. The image of T is the x1\u00a1x2-plane in R3. When working with systems of linear equations, there were three operations you could perform which would not change the solution set. De nition: A matrix B is similar to a matrix A if there is an invertible matrix S such that B = S 1AS. In linear algebra, the Singular Value Decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics. Now we can define the linear. You do this with each number in the row and coloumn, then move to the next row and coloumn and do the same. Proof The conclusion says a certain matrix exists. Statistics: Linear Regression example. Step 1: To Begin, select the number of rows and columns in your Matrix, and press the \"Create Matrix\" button. De\ufb01ne T : V \u2192 V as T(v) = v for all v \u2208 V. 0 0 0 d n 3 7 7 7 5: The linear transformation de ned by Dhas the following e ect: Vectors are. Determine whether the following functions are linear transformations. Therefore. If the new transform is a roll, compute new local Y and X axes by rotating them \"roll\" degrees around the local Z axis. com is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus. Some interesting transformations to try: - enter this as - enter this as. What better way to prove something exists than to actually build it?. Show also that this map can be obtained by first rotating everything in the plane \u03c0 \/ 4 {\\displaystyle \\pi \/4} radians clockwise, then projecting onto the x {\\displaystyle x} -axis, and then rotating \u03c0 \/ 4 {\\displaystyle \\pi \/4} radians counterclockwise. 1;1\/ is unchanged by R. What is linear programming? What is a logarithm? StudyPug is a more interactive way of study math and offers students an easy access to stay on track in their math class. Specifically, if T: n m is a linear transformation, then there is a unique m n matrix, A, such that T x Ax for all x n. situations: 1) as the set of solutions of a linear homogeneous system or 2) as the set of all linear combinations of a given set of vectors. Theorem: linear transformations and matrix transformations. A matrix is said to be singular if its determinant is zero and non-singular otherwise. NOTE 1: A \" vector space \" is a set on which the operations vector addition and scalar multiplication are defined, and where they satisfy commutative, associative, additive. Note this also handles scaling even though you don't need it. Although we would almost always like to find a basis in which the matrix representation of an operator is. Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality. By using this website, you agree to our Cookie Policy. system, find a transformation M, that maps a representation in XYZ into a representation in the orthonormal system UVW, with the same origin \u2022The matrix M transforms the UVW vectors to the XYZ vectors y z x u=(u x,u y,u z) v=(v x,v y,v z) Change of Coordinates \u2022 Solution: M is rotation matrix whose rows are U,V, and W: \u2022 Note: the inverse. In the computer graphics realm, they\u2019re also used for things like \u2014 surprise! \u2014 linear transformations and projecting 3D images onto a 2D screen. Function: addcol (M, list_1, \u2026, list_n) Appends the column(s) given by the one or more lists (or matrices) onto the matrix M. G o t a d i f f e r e n t a n s w e r? C h e c k i f i t \u2032 s c o r r e c t. If our set of linear equations has constraints that are deterministic, we can represent the problem as matrices and apply matrix algebra. 2), and sketch both v and its image T(v). In fact, we will now show that every linear transformations fromFn to Fmis a matrix linear transformation. Thus we get that $x = \\begin{bmatrix} \\frac{2}{13} & \\frac{3}{26}\\\\ \\frac{3}{13} & -\\frac{1}{13} \\end{bmatrix}\\begin{bmatrix}w_1\\\\ w_2 \\end{bmatrix} = \\begin{bmatrix. Stationary Matrix Calculator. The inverse of a linear transformation Theorem: Let A be an n x m matrix. The red lattice illustrates how the entire plane is effected by multiplication with M. In part (a), we computed that T(e 1) = 2 6 6 4 2 0 2 3 7 7 5, and part of our given information is that T(e 2) = 2 6 6 4 5 2 2 3 7 7 5. Moreover, if P is the matrix with the columns C 1, C 2, , and C n the n eigenvectors of A, then the matrix P-1 AP is a diagonal matrix. For example, when using the calculator, \"Power of 2\" for a given matrix, A, means A 2. Linear transformations. In this section we consider the topic of Vectors, Matrices and Arrays and their application in solving Linear Equations and other linear algebra problems. Calculator for Matrices. This applet allows you to experiment with 2x2-matrices and linear transformations of the plane. This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule. I'll introduce the following terminology for the composite of a linear transformation and a translation. A linear transformation T from a n-dimensional space R n to a m-dimensional space R m is a function defined by a m by n matrix A such that: y = T(x) = A * x, for each x in R n. Number of rows and columns decides the shape of matrix i. It works over GF(q) for q = 2,3,4*,5,7,11. In this problem we consider a linear transformation that takes vectors from R3 and returns a vector in R3. Use the result matrix to declare the final solutions to the. 1 Matrix Linear Transformations Every m nmatrix Aover Fde nes linear transformationT A: Fn!Fmvia matrix multiplication. For example, the following matrix is diagonal: 2 6 6 4 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 3 7 7 5: An upper triangular matrix has zero entries everywhere below the diagonal (a ij = 0 for i>j). Essentially a \u201cpower\u201d regression is a transformation of variables to obtain an ordinary linear regression model. In other words, the matrix A is diagonalizable. Then T is a linear transformation. \u2013 Multiply the current matrix by the translation matri x \u2022 glRotate {fd }(TYPE angle, TYPE x, TYPE y, TYPE z) \u2013 Multiply the current matrix by the rotation matrix that rotates an object about the axis from (0,0,0) to (x, y, z) \u2022 glScale {fd }(TYPE x, TYPE y, TYPE z) \u2013 Multiply the current matrix by the scale matrix Examples. Library: Inverse matrix. The \ufb01rst column of the required matrix is P\u00a11 S TPBe1 = I2T(b1) = T(b1. Created by Sal Khan. LINEAR MODELS IN BUSINESS, SCIENCE, AND ENGINEERING. Functions and linear transformations. The same transformation can be used in using a Wiimote to make a low-cost interactive whiteboard or light pen (due to Johnny Chung Lee). First, we need a little terminology\/notation out of the way. That means you can combine rotations, and keep combining them, and as long as you occasionally correct for round-off error, you will always have a rotation matrix. There are three coordinate systems involved --- camera, image and world. Or you can type in the big output area and press \"to A\" or \"to B\" (the calculator will try its best to interpret your data). You can enter a new linear transformation by entering values in the matrix at top-left. NOTE 1: A \" vector space \" is a set on which the operations vector addition and scalar multiplication are defined, and where they satisfy commutative, associative, additive. Projection onto a subspace. G o t a d i f f e r e n t a n s w e r? C h e c k i f i t \u2032 s c o r r e c t. Just type matrix elements and click the button. For every two vectors A and B in R n. Now we can define the linear. (One of the requirements. STRETCH ANSWER. An Open Text by Ken Kuttler Linear Transformations: Matrix of a Linear Transformation Lecture Notes by Karen Sey arth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA) This license lets others remix, tweak, and build upon your work non-commercially, as long as they credit you and license their new creations. You can input only integer numbers, decimals or fractions in. Hence, aI = a, IX = X, etc. By using this website, you agree to our Cookie Policy. The inverse of a linear transformation Theorem: Let A be an n x m matrix. However, not every linear transformation has a basis of eigen vectors even in a space over the field of complex numbers. Show that$T_A$is one-to-one and define$T_{A^{-1}} : \\mathbb{R}^n \\to \\mathbb{R}^n$. Each of the above transformations is also a linear transformation. The first step is to create an augmented matrix having a column of zeros. Vector space) that is compatible with their linear structures. Understand the relationship between linear transformations and matrix transformations. In fact, matrices were originally invented for the study of linear transformations. In matrix form, these transformation equations can be written as 2 1 2 sin cos cos sin u u u u Figure 1. 1 De nitions A subspace V of Rnis a subset of Rnthat contains the zero element and is closed under addition and scalar multiplication: (1) 0 2V (2) u;v 2V =)u+ v 2V (3) u 2V and k2R =)ku 2V. Determine value of linear transformation from R^3 to R^2. Transforming a matrix to reduced row echelon form. This elegant matrix calculator deploys one single interface which can be used to enter multiple matrices including augmented matrices representing systems of linear equations!. You can also drag the images of the basis vectors to change. Singular value decomposition takes a rectangular matrix of gene expression data (defined as A, where A is a n x p matrix) in which the n rows represents the genes, and the p columns represents the experimental conditions. Get access to all the courses and over 150 HD videos with your subscription. Note that vector u is the left column of the matrix and v is the right column. [email protected] Here, it is calculated with matrix A and B, the result is given in the result matrix. im (T): Image of a transformation. What is the matrix of the identity transformation? Prove it! 2. Two Examples of Linear Transformations (1) Diagonal Matrices: A diagonal matrix is a matrix of the form D= 2 6 6 6 4 d 1 0 0 0 d 2 0. Vector and matrix algebra This appendix summarizes the elementary linear algebra used in this book. First, we can view matrix-matrix multiplication as a set of vector-vector products. 2 is a rotation, but other values for the elements of A. Linear Algebra - Transformation Matrix for Scaling 2D Objects - Duration: 19:17. Linear transformation T: R3 --> R2 Homework Statement Find the linear transformation T: R3 --> R2 such that: T(1,0,0) = (2,1) T(0,1,1) = (3,2) T(1,1,0) = (1,4) The Attempt at a Solution I've been doing some exercises about linear transformations (rotations and reflections. Even though students can get this stuff on internet, they do not understand exactly what has been explained. g) The linear transformation TA: Rn \u2192 Rn de\ufb01ned by A is onto. A square matrix is any matrix whose size (or dimension) is $$n \\times n$$. Visualizing linear transformations. De\ufb01nition 1 If B \u2208 M nq and A \u2208 M pm, the. To begin the process of row reduction, we create a matrix consisting of the numbers in our linear equation. The solve () method is the preferred way. The table lists 2-D affine transformations with the transformation matrix used to define them. inv () and linalg. This is important with respect to the topics discussed in this post. For example, the linear equation x 1 - 7 x 2 - x 4 = 2. To transform from XYZ to RGB (with D65 white point), the matrix transform used is : [ R ] [ 3. Such a linear transformation can be associated with an m\u00d7n matrix. Matrix of a linear transformation. The rotation is defined by one rotation angle ( a ) , and the scale change by one scale factor ( s ). NET Numerics is part of the Math. Covered topics include special functions, linear algebra, probability models, random numbers, interpolation, integration, regression, optimization problems and more. Step 1: System of linear equations associated to the implicit equations of the kernel, resulting from equalling to zero the components of the linear transformation formula. Re ections in R2 97 9. The leading entry in each row is the only non-zero entry in its column. 5),(0,1)] is a linear transformation. We need to prove two statements: 1) Every linear transformation from R n to R m satisfies these properties and 2) Every function from R n to R m satisfying these properties is a linear transformation. This elegant matrix calculator deploys one single interface which can be used to enter multiple matrices including augmented matrices representing systems of linear equations!. LINEAR MODELS IN BUSINESS, SCIENCE, AND ENGINEERING. What better way to prove something exists than to actually build it?. Theorem Let T be as above and let A be the matrix representation of T relative to bases B and C for V and W, respectively. The previous three examples can be summarized as follows. Here you can calculate a matrix transpose with complex numbers online for free. Notice how it\u2019s a matrix full of zeros with a 1 along the diagonal. h) The rank of A is n. 3) Skew - transformation along the X or Y axis 4) Translate - move element in XY direction linear transformations also can be represented by Matrix function. G o t a d i f f e r e n t a n s w e r? C h e c k i f i t \u2032 s c o r r e c t. A square matrix is any matrix whose size (or dimension) is $$n \\times n$$. \u201cMatrix decomposition refers to the transformation of a given matrix into a given canonical form. y+2z-w = 0 2x+8y+2z-6w = 0 2x+7y-5w = 0 Step 2: Represent the system of linear equations in matrix form. Discrete Probability Distributions. More on matrix addition and scalar multiplication. Matrix of a Linear Transformation. -coordinates and transform it into a region in uv. The Row Space Calculator will find a basis for the row space of a matrix for you, and show all steps in the process along the way. Drawing in the (u,v) window produces the preimage in the (x,y) window. The red lattice illustrates how the entire plane is effected by multiplication with M. The Square Root Matrix Given a covariance matrix, \u03a3, it can be factored uniquely into a product \u03a3=U T U, where U is an upper triangular matrix with positive diagonal entries and the. First, we need to find the inverse of the A matrix (assuming it exists!) Using the Matrix Calculator we get this: (I left the 1\/determinant outside the matrix to make the numbers simpler) Then multiply A-1 by B (we can use the Matrix Calculator again): And we are done! The solution is: x = 5, y = 3, z = \u22122. com To create your new password, just click the link in the email we sent you. all points in the x-y plane, into a new set of 2d vectors (or, equivalently, a new set of points). Dimension also changes to the opposite. Quick Quiz. If a linear transformation T: R n \u2192 R m has an inverse function, then m = n. Check that T is a linear transformation. This free app is a math calculator, which is able to calculate the determinant of a matrix. It can be expressed as $$Av=\\lambda v$$ where $$v$$ is an eigenvector of $$A$$ and $$\\lambda$$ is the corresponding eigenvalue. Practice problems here: Note: Use CTRL-F to type in search term. De nition: A matrix B is similar to a matrix A if there is an invertible matrix S such that B = S 1AS. So the skew transform represented by the matrix `bb(A)=[(1,-0. Moreover, if P is the matrix with the columns C 1, C 2, , and C n the n eigenvectors of A, then the matrix P-1 AP is a diagonal matrix. Video explanation on solving for a parameter in a linear equation. The transformation defines a map from \u211d3 to \u211d3. Also known as homogeneous transformation; linear. Translation is not a linear transformation, since all linear transformation must map the origin onto itself. A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space. For example, they lay in a basis of the known Gauss' method (method of exception of unknown values) for solution of system of linear equations [1]. Lecture 8: Examples of linear transformations While the space of linear transformations is large, there are few types of transformations which are typical. In part (a), we computed that T(e 1) = 2 6 6 4 2 0 2 3 7 7 5, and part of our given information is that T(e 2) = 2 6 6 4 5 2 2 3 7 7 5. 0 0 0 d n 3 7 7 7 5: The linear transformation de ned by Dhas the following e ect: Vectors are. Thank you so much, your explanation made it so much clearer!$\\endgroup\\$ \u2013 Kim Apr 20 '14 at 18:26. You see, it just clicks, and the whole point is that the inverse matrix gives the inverse to the linear transformation, that the product of two matrices gives the right matrix for the product of two transformations--matrix multiplication really came from. This project for my Linear Algebra class is about cryptography. Vector transformations. Such a linear transformation can be associated with an m\u00d7n matrix. After checking the residuals' normality, multicollinearity, homoscedasticity and priori power, the program interprets the results. a) Prove that a linear map T is 1-1 if and only if T sends linearly. \u2022 After the midterm, we will focus on matrices. Transformation Matrix Properties Transformation matrices have several special properties that, while easily seen in this discussion of 2-D vectors, are equally applicable to 3-D applications as well. Let\u2019s take a look at the following problem: x + y = 2 3x + 4y + z = 17 x + 2y + 3z = 11. The joint moment generating function of is Therefore, the joint moment generating function of is which is the moment generating function of a multivariate normal distribution with mean and covariance matrix. The inverse of a linear transformation De nition If T : V !W is a linear transformation, its inverse (if it exists) is a linear transformation T 1: W !V such that T 1 T (v) = v and T T (w) = w for all v 2V and w 2W. It's possible to observe a matrix as a particular linear transformation. Graphing Calculator. You can draw either lines, points, or rectangles, and vary the transformation as well. linear transformation. Many of the items contained in the Matrix & Vector menu work with a matrix that you must first define. Some interesting transformations to try: - enter this as - enter this as. Quick Quiz. Transforming a matrix to reduced row echelon form. Moreover, there are similar transformation rules for rotation about and. In XYZ, any color is represented as a set of positive values. Now we can define the linear. What better way to prove something exists than to actually build it?. Step 1: System of linear equations associated to the implicit equations of the kernel, resulting from equalling to zero the components of the linear transformation formula. The basis and vector components. Rref Calculator for the problem solvers. This section will simply cover operators and functions specifically suited to linear algebra. This process, called Gauss-Jordan elimination, saves time. Transformation Matrix Main Concept A linear transformation on a vector space is an operation on the vector space satisfying two rules: , for all vectors , , and all scalars. Elementary matrix transformations retain equivalence of matrices. This means that applying the transformation T to a vector is the same as multiplying by this matrix. Every linear transformation T: Fn!Fm is of the form T Afor a unique m nmatrix A. 2 Functions and Variables for Matrices and Linear Algebra. Matrix Algebra. A matrix is in reduced row echelon form (rref) when it satisfies the following conditions. An example is the linear transformation for a rotation. A square matrix is any matrix whose size (or dimension) is $$n \\times n$$. Rotations in the plane 96 8. Pick the 1st element in the 1st column and eliminate. This expression is the solution set for the system of equations. These are called eigenvectors (also known as characteristic vectors). Lastly, we will look at the Diagonal Matrix Representation and an overview of Similarity, and make connections between Eigenvalues (D-Matrix or B-Matrix) and Eigenvectors (P matrix or basis). To find the image of a point, we multiply the transformation matrix by a column vector that represents the point's coordinate. Determinants determine the solvability of a system of linear equations. Understand the relationship between linear transformations and matrix transformations. Final Answer: \u2022 2 \u00a14 5 0 \u00a11 3 \u201a Work: If S is the standard basis of R2 then P S = I2. Suppose and are linear transformations. There are no hard and fast rules for making change of variables for multiple integrals. It does not give only the inverse of a 2x2 matrix, and also it gives you the determinant and adjoint of the 2x2 matrix that you enter. Consider a linear transformation T from to and a basis of. By using this website, you agree to our Cookie Policy. Free matrix and vector calculator - solve matrix and vector operations step-by-step This website uses cookies to ensure you get the best experience. Showing that any matrix transformation is a linear transformation is overall a pretty simple proof (though we should be careful using the word \"simple\" when it comes to linear algebra!) But, this gives us the chance to really think about how the argument is structured and what is or isn't important to include - all of which are critical skills when it comes to proof writing. Nul A x: x is in Rn and Ax 0 (set notation) EXAMPLE Is w 2 3 1 in Nul A where A 2 1 1 4 31? Solution: Determine if Aw 0: 2 1 1 4 31 2 3 1 0 0 Hence w is in Nul A. Find more Widget Gallery widgets in Wolfram|Alpha. The transformation to this new basis (a. These are called eigenvectors (also known as characteristic vectors). Suppose T : R3!R2 is the linear transformation dened by T 0 @ 2 4 a b c 3 5 1 A = a b+c : If B is the ordered basis [b1;b2;b3] and C is the ordered basis [c1;c2]; where. The rotation is defined by one rotation angle ( a ) , and the scale change by one scale factor ( s ). If is a linear transformation mapping to and \u2192 is a column vector with entries, then (\u2192) = \u2192for some \u00d7 matrix , called the transformation matrix of. By using this website, you agree to our Cookie Policy. Write the standard matrix A for the transformation T. Matrix theory is a branch of mathematics which is focused on study of matrices. Step 1: System of linear equations associated to the implicit equations of the kernel, resulting from equalling to zero the components of the linear transformation formula. Linear transformations as matrix vector products. Generalized Linear Models Structure Generalized Linear Models (GLMs) A generalized linear model is made up of a linear predictor i = 0 + 1 x 1 i + :::+ p x pi and two functions I a link function that describes how the mean, E (Y i) = i, depends on the linear predictor g( i) = i I a variance function that describes how the variance, var( Y i. The Attempt at a Solution I tried constructing a matrix using the vectors being applied to T and row reducing it. The calculator will find the null space of the given matrix, with steps shown. It is created by adding an additional column for the constants on the right of the equal signs. In XYZ, any color is represented as a set of positive values. For every two vectors A and B in R n. Pick the 1st element in the 1st column and eliminate. Error-correcting codes are used, e. There are no hard and fast rules for making change of variables for multiple integrals. Time-saving video explanation and example problems on how to solve for a parameter in a simple linear equation. Or you can type in the big output area and press \"to A\" or \"to B\" (the calculator will try its best to interpret your data). We begin with an understanding of the Matrix of a Linear Transformation by associating a matrix T, with ordered bases B and C. The red lattice illustrates how the entire plane is effected by multiplication with M. If this is the case, its. Join 100 million happy users! Sign Up free of charge:. The model matrix transforms a position in a model to the position in the world. Translation is not a linear transformation, since all linear transformation must map the origin onto itself. Transformation Matrix Properties Transformation matrices have several special properties that, while easily seen in this discussion of 2-D vectors, are equally applicable to 3-D applications as well. Matrix of a Linear Transformation, Column Vectors Suppose that T : {\u2102}^{n} \u2192 {\u2102}^{m} is a linear transformation. However, linear algebra is mainly about matrix transformations, not solving large sets of equations (it\u2019d be like using Excel for your shopping list). You can enter a new linear transformation by entering values in the matrix at top-left. A system of an equation is a set of two or more equations, which have a shared set of unknowns and therefore a common solution. Eigenvectors and Linear Transformations Video. The leading entry in each row is the only non-zero entry in its column. Given a photo of a whiteboard taken at an angle, synthesize a perspective-free view of the whiteboard. Do similar calculations if the transform is a pitch or yaw. A matrix is in reduced row echelon form (rref) when it satisfies the following conditions. Ctrl + [scroll wheel] to zoom in and out. Instead x 1, x 2, you can enter your names of variables. This permits matrices to be used to perform translation. Suppose that T : V \u2192 W is a linear map of vector spaces. Image: (intrinsic\/internal camera parameters). The inverse of a linear transformation Theorem: Let A be an n x m matrix. 2x2 matrices are most commonly employed in describing basic geometric. Find the matrix for a stretch, factor 3, x-axis invariant. \u2022 If transformation of vertices are known, transformation of linear combination of vertices can be achieved \u2022 p and q are points or vectors in (n+1)x1 homogeneous coordinates \u2013 For 2D, 3x1 homogeneous coordinates \u2013 For 3D, 4x1 homogeneous coordinates \u2022 L is a (n+1)x(n+1) square matrix \u2013 For 2D, 3x3 matrix \u2013 For 3D, 4x4 matrix. A linear transformation may or may not be injective or surjective. 1 The Null Space of a Matrix. These last two examples are plane transformations that preserve areas of gures, but don\u2019t preserve distance. A Matrix and a vector can be multiplied only if the number of columns of the matrix and the the dimension of the vector have the same size. So, for example, you could use this test to find out whether people. 2 Null Spaces, Column Spaces, & Linear Transformations Definition The null space of an m n matrix A, written as Nul A,isthesetofallsolutionstothe homogeneous equation Ax 0. All linear transformations from Rn to Rm are of the form L(x) = Ax for some A. The image of a linear transformation or matrix is the span of the vectors of the linear transformation. Just like on the Systems of Linear. (b): Find the standard matrix for T, and brie y explain. Produce a matrix that describes the function's action. can be entered as: x 1 + x 2 + x 3 + x 4 = Additional features of inverse matrix method calculator. This tells us the following. Linear algebra calculator app designed for matrix operations Matrix Magus, by Asterism. If the stress tensor in a reference coordinate system is $$\\left[ \\matrix{1 & 2 \\\\ 2 & 3 } \\right]$$, then after rotating 50\u00b0, it would be. Now we can define the linear. The n n matrix B that transforms [x] B into [T(x)] B is called the -matrix of T for instance for all x in : [T(x)] B = B[(x)] B. Above all, they are used to display linear transformations. xla is an addin for Excel that contains useful functions for matrices and linear Algebra: Norm, Matrix multiplication, Similarity transformation, Determinant, Inverse, Power, Trace, Scalar Product, Vector Product, Eigenvalues and Eigenvectors of symmetric matrix with Jacobi algorithm, Jacobi's rotation matrix. Note that the transformation matrix for a translation must be the same size (i. The article explains how to solve a system of linear equations using Python's Numpy library. They are most commonly used in linear algebra and computer graphics, since they can be easily represented, combined and computed. This means that the null space of A is not the zero space. (b) W (c) Rank = 2, Nullity = 1 (Remark: Draw a picture. Explanation:. Final Answer: \u2022 2 \u00a14 5 0 \u00a11 3 \u201a Work: If S is the standard basis of R2 then P S = I2. For example: the coordinates of point A in those two coordinate systems are (i,j,k) and (x,y,z), separately. Linear Transformation Exercises Olena Bormashenko December 12, 2011 1. The change of basis formula B = V 1AV suggests the following de nition. Linear Transformation.\n4dzqjutfktyzn, vgg1uar62y4b7m8, bgszwxvbmk9o, ogy8avtxr5, r9cwvcq9448g4r, gq2j8z7654b44qw, rm8x5u3pmp8b, dzlq4hah5vt, ilmadmz2ae1, 3imwbc7cjp95m, y5aiiz7oz0ua50, brvebl1an0xwfak, s3crctcsirap9, z19061opp0, m9o7ylm7h7ij5n, p9untv5hpwy2cc, 5hepn8xdg9, onodwsjl0keh, fozcx1dngua3b, qiwcbi4ybn, g3b10ywvxvn, fgz2zulwqvno50, p5de7kjm5h2ikk, hwvu4z1k3d, d9iubht5o7ue, zpmdwsyevyjp, kabula9lewc5, s3evy6t7u1y, 3k7jr4m0w47","date":"2020-08-05 07:44:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8110021948814392, \"perplexity\": 447.12948749176496}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439735916.91\/warc\/CC-MAIN-20200805065524-20200805095524-00161.warc.gz\"}"}
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Q: how can i use `#function` symbol in a `inline` function? I'd like to use the name of function to resolve some problems, but #function seems not to work well with @inline(__always), here is my codes:
@inline(__always) func log() {
print(#function)
}
func a() { log() } // want 'a()', but got 'log()'
func b() { log() }
func c() { log() }
//...
Can anybody explain? or that's just a stupid idea.
A: If your intention is to print the name of the function which calls
log(), then you should pass it as default argument (which is evaluated
in the context of the caller), as demonstrated
in Building assert() in Swift, Part 2: __FILE__ and __LINE__ in the Swift blog.
Example:
@inline(__always) func log(_ message: String, callingFunction: String = #function) {
print("\(callingFunction): \(message)")
}
func a() { log("Hello world") }
func b() { log("Foo") }
func c() { log("Bar") }
a() // a(): Hello world
b() // b(): Foo
c() // c(): Bar
This works regardless of whether the log function is inlined or not.
(Inlining does not change the semantics of a program. In particular,
it does not mean that the source code of func log is included from
the source code of func a() and compiled as a single function.)
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Side Feature, Social System, The Khilafah
Let the Ummah Stand Up to these Malicious Institutions and Thwart their Plans
written by Official Spokeswoman of the Women's Section in Hizb ut Tahrir in Wilayah Sudan
A group of entities and bodies intends to conduct a procession at noon today, Thursday 02/01/2020, to demand the signing of the Convention on the Elimination of All Forms of Discrimination against Women (CEDAW), and said that the procession will meet the Prime Minister and the Minister of Justice to deliver the memo, and announced that the bodies participating in the procession include "The Strategic Initiative for Women in the Horn of Africa, "The Initiative of No to the Oppression of Women, the Women's Union, as well as the resistance committees and the Kendakat gathering in southern Khartoum. (At-Tayyar Newspaper)
The Women's Section in Hizb ut Tahrir Wilayah Sudan would like to clarify the following facts:
First: The feminist societies demanding the CEDAW agreement represent only themselves because the demand for the application of this agreement reflects a subordination and imitation of the Kaffir West who defies Islam by accusing its provisions of discriminating against women. So, they drafted the terms of this agreement targeting the Muslim woman to keep Muslim women from adhering to the provisions of Shariah by calling for the empowerment of women to dispose of their bodies and abolishing the legal guardianship of men over them in an explicit targeting of the Muslim family, inciting Muslim women against the provisions of Islam, and working to enact laws that rebel against the provisions of Islamic Shariah, in evasion from the noble values and high morals so that her life can go according to the West's liberal view, and the constant incitement to defend what are called freedoms, while constantly manifesting that the gardens of pleasure lie in the Kafir West's way of life, and that backwardness is to remain under the Shariah rules that restrict the freedom of the individual!
Second: These suspicious feminist societies complement the path of perverting women, and their fall in the quagmire of decadence and corruption by educating them the Western culture and concepts with glamorous slogans! Such as claiming to support women, informing them of their rights, helping them to prove themselves, and strengthening their personality in order to enhance their role in life! While the truth is that they want her to be a distorted image of Western women panting behind glamorous slogans and false fame, so she falls into constant misery.
Third: The activities of these associations have increased, as they disseminate their poisons with claims to rehabilitate rural women, spread the concepts of reproductive health, and use of condoms under the pretext of combating AIDS, as well as spreading gender culture, combating violence against women and eliminating all forms of discrimination against them, and other slogans that appear glamorous on the surface. The work of these associations has led ignorant women to rebel against the values of Islam, the family, the father, and the general custom influenced by Islam, which led to the spread of vice, prostitution, divorce, spinsterhood, and other problems that affect women primarily.
The people of Sudan must be cautious of these feminist societies, which do not need to ascertain that they are a Western commodity, whether in terms of the creed on which they are based (secularism), or the system that they follow, which is the capitalist system, which adopts freedoms. This appears in the programs they adopt, and the issues that they raise, and their Western sources of funding speak of treachery and mercenaries. The book, Spotlight on the Sudanese Feminist Movement, revealed this support in a statement by the U.S. Embassy, in which the Embassy announced a $35,000 monthly aid to one of Sudan's leading women's associations, as well as the project "Eradicating habits harmful to family health" adopted by Babiker Badri Scientific Association for Feminist Studies is sponsored by the American Public Welfare Association.
These feminist associations must raise their hand from our honours, for we are fully convinced that we have a great role in this life. A woman in Islam is a mother, a housewife, and an honour that must be protected. And Islam definitely dose not consider the mother, sister, daughter, and wife as slaves to men as it is in the culture of these associations belonging to the West. Rather, it described those who honour them with kindness and those who humiliate them with meanness, and obliged men to take care of them, because the Messenger ﷺ said: «وَالرَّجُلُ رَاعٍ عَلي أَهْلِ بَيْتِهِ وَهُوَ مَسْئُولٌ عَنْ رَعِيَّتِهِ» "a man is a guardian of his family and is responsible for them." And considered death for the sake of protecting woman and preserving her as a Shahadah (martyrdom) by the saying of the Messenger ﷺ: «وَمَنْ قُتِلَ دُونَ أَهْلِهِ فَهُوَ شَهِيدٌ» "And he who is killed while defending his family is a martyr."
O Muslims:
Stand up to these malicious institutions, and thwart their plans by preventing your women and daughters from visiting these associations, and attending their destructive malicious activities and events. Let us fortify them with the great Islamic culture, and wise Islamic concepts, especially the Shariah provisions related to women in their rights and duties, and their pivotal role in life. Let all men and women work with men and women workers to establish the second Khilafah Rashidah (righlty guided Caliphate) on the method of the Prophethood, which will protect women, families, society, and indeed all humanity from these institutions, their supporters, and their financiers.
Official Spokeswoman of the Women's Section in Hizb ut Tahrir in Wilayah Sudan
Thursday, 07th Jumada I 1441 AH
Filed under: Side Feature, Social System, The Khilafah
Previous PostThe Economic Hitmen of Pakistan – The Treachery Continues
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Idiot Nearly Crashes Tuned Porsche 911 GT2 RS While Drifting on Public Roads
He claimed it was okay to do this because it's Jan. 1 and "there are no people around."
By James GilboyJanuary 2, 2020
Todd Schleicher/Mimino Instagram
James GilboyView James Gilboy's Articles
twitter.com/_JamesGilboyinstagram.com/jamesgilboy
With gobs of power, rear-biased weight distribution, and tons of tire, the Porsche 911 GT2 RS is already one of the world's least forgiving cars when pushed to its limits. As this guy, this guy, and this guy demonstrate, it's just too much car for some people. Without lightning-quick, track-honed instincts, searching for the GT2 RS's limitations could mean spending your day in the shop—or worse, the hospital. Now imagine how much more of a handful a tuned GT2 RS is on a public road. Overestimating your grip on a less-refined version of a brutally fast Porsche risks dire consequences, as one GT2 RS owner discovered while caning his car on New Year's Day.
Instagram user miminogarage took his 991 GT2 RS out to play on the "empty" roads on Wednesday, claiming that with no police or "people around" he decided to break the rear tires loose a few times. When attempting a slide down a tight onramp, however, he rotated the car too much and found himself on course to kiss the wall.
Fortunately, his front tires found grip just inches from disaster, and after some correction, the Porsche came back under control. Though he avoided writing off his 911, we're not sure we can say the same of his pants. Then again, that just gives him an excuse to start the new decade with a new wardrobe rather than a nasty body shop bill and points on his license.
By the looks of his Instagram account, a yearly track membership or track-day outing is well within this person's financial reach, so perhaps he should stay off public roads. Yes, even on New Year's Day.
Someone Abandoned This Poor Porsche 911 GT2 RS Somewhere in Chile
Adopt a neglected and abused supercar this holiday season.
Man Crashes Brand-New Porsche 911 GT2 RS Just Minutes After Going on Test Drive
You break it, you buy it?
Watch a Porsche 911 GT2 RS Crash Into a $3.5M Pagani Huayra BC
Lack of skill = pretty big bill.
Watch a 1958 Maserati 250F Race Car Drift Through the Streets of Monaco
This is what dreams are made of.
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Projects and Campaigns
Yellow Pad
Dispatches from the Enchanted Kingdom
AER Buzz
Project and Campaigns
Sin Tax Reforms
Sin Tax Revenues for Universal Health Coverage
Clean Fuels for the Poor
Policy Reforms on Investments
Competition Reforms
Bangsamoro Basic Law (BBL)
About AER
Charting New Paths
Women and Men of Action for Economic Reforms
At what cost?
Posted by Action for Economic Reforms | Jan 27, 2009 | Dispatches from the Enchanted Kingdom
Buencamino is a fellow of Action for Economic Reforms. This piece was published in the January 28, 2009 edition of the Business Mirror, page A6.
"What one must lose in order to win is sometimes not worth the price of playing." – The Rude Pundit
Whether by design or not, Philippine Drug Enforcement Agency (PDEA) Director Dionisio Santiago's psy-war against Department of Justice (DOJ) prosecutors laid out the red carpet for brown shirts.
Legislators filed bills to reinstate the death penalty. A bishop from Infanta, Quezon urged Mrs. Arroyo "to use the iron hand of the government and smash all shabu laboratories, not only in Quezon but in every nook and corner of the country."
Mrs. Arroyo directed the Philippine National Police (PNP) and the PDEA to "endeavor to eliminate the number of drug cases dismissed due to mere technicalities."
There's a trial balloon for the return of Jovito Palparan, a soldier who never allowed mere technicalities to temper his war against suspected communist rebels and their sympathizers.
"We're studying what would be the immediate utilization for him. He's being considered for the DDB (Dangerous Drugs Board). Maybe, he'll be one of the board members,'' said Executive Secretary Eduardo Ermita.
The chair of the DDB, former senator and comedian Vicente "Tito" Sotto, said Palparan would be "a welcome addition to the committee. With his expertise, he can help us formulate strategies against drug pushers."
The comedian added that he considers Palparan's berdugo reputation an asset.
"If that is his image, then that will work well for us. It's the drug traffickers who should fear him, not the public,'' Sotto said with a straight face.
For his part Director Santiago told the Philippine Daily Inquirer, "You give me people, I'll utilize them and judge them according to how they will perform."
We are being told the Philippines will become a narco-state if we don't wage a total war on drugs. We are being told that drug traffickers are making humongous profits, billions of dollars, feeding the habit of a population that can barely afford three square meals a day. Where is all that money coming from?
But no one dares to ask. Nobody questions statistics. Instead, we have moral crusaders who say that if we must cut corners to defeat the drug menace then so be it.
That sentiment comes from the sort that spouted an inanity like, "we are prepared to lose our freedoms and our rights just to move this country forward."
Such silliness is music to the ears of Director Santiago who once admitted he sees nothing wrong with planting drugs on anyone "publicly known to be peddling drugs but always escapes arrest" and Jovito Palparan who reputedly sees nothing wrong with planting suspected rebels six feet under.
Our war on drugs may end up becoming like Bush's war on terror. We can lose our soul as America did when Bush trampled on the US Constitution and the Bill of Rights in the name of defeating terrorists.
Let's be cautious. Let's not become so overzealous we will need to be told what President Barack Obama told the American people:
"[w]e reject as false the choice between our safety and our ideals. Our Founding Fathers, faced with perils we can scarcely imagine, drafted a charter to assure the rule of law and the rights of man, a charter expanded by the blood of generations. Those ideals still light the world, and we will not give them up for expedience's sake."
PreviousPeace, Justice, and Obama
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Beyoncé for President: Popularity is not a free pass to the Presidency
Over 60 civil society organizations call on President Duterte to veto the Vape Bill
Uncertainty and polarization
The Marcos family and Lucio Tan
The relevance of Rizal's Indolence of the Filipino
Action for Economic Reforms (AER)
Action for Economic Reforms (AER) is a public interest organization that conducts policy analysis and advocacy on key economic issues. It was founded in 1996 by a group of progressive scholars as an independent, reform-oriented and activist policy group.
Unit 1403 West Trade Center, 132 West Avenue,
Quezon City, Philippines 1104
Phone: (63)(2) 426 5626
Fax: (63)(2) 426 5626
2019 Action for Economic Reforms. All rights reserved.
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Joeri Fransen (10 juli 1981) is een Vlaams zanger, die bekend werd als de winnaar van Idool 2004. Aanvankelijk bracht hij muziek uit enkel onder zijn voornaam Joeri, nadien onder zijn volledige naam.
Muzikale carrière
Nadat hij in 2004 Idool gewonnen had, verkocht zijn debuutsingle "Ya 'bout to find out" zeer goed: eind 2004 waren er reeds meer dan 25.000 exemplaren van verkocht, wat hem een gouden plaat opleverde. Het nummer verscheen ook meteen vanuit het niets op de eerste plaats in de Vlaamse Ultratop 50. In 2005 volgde zijn eerste en enige album "True lies". Het stond 21 weken in de Vlaamse Ultratop 100 Albums en behaalde als hoogste plaats de vierde positie. De singles "High and alive" en "We came this far" uit hetzelfde album behaalden eveneens de Ultratop 50, met respectievelijk als hoogste positie nummer 8 en 35. De vierde single uit het album, het titelnummer "True lies", haalde de top niet. Het album en de singles werden uitgebracht bij Sony BMG, zoals voorzien in het contract met Idool.
Na een zeer snel en massaal succes, daalde dit weer even snel. De ontgoocheling was groot. Eind 2005 brak hij met het managementsbureau Stageplan (van Bob Savenberg). Het contract met Sony BMG stopte eveneens.
In 2006 bracht hij een nieuwe single uit: "Different stars". Deze single werd uitgebracht bij het kleine managementbureau Jemmusic. De single kreeg echter nauwelijks aandacht in de media en flopte, wat Joeri tot uitspraken leidde dat zijn carrière "zo goed als dood" was.
In 2007 trad hij nog op met enkele covergroepen, doch zonder succes. Fransen weigerde sindsdien ieder contact met de pers.
In 2010 probeerde Joeri Fransen via fan-funding bij het Franse AKA Music € 35.000 te pakken te krijgen voor een nieuwe single. Dit leidde niet tot een nieuwe single.
In 2013 nam Katastroof samen met Joeri Fransen een nieuwe versie van het nummer "Tengels lieke" op. Het stond op het album Vrindjespolletiek. Joeri Fransen wordt ook vermeld op de website van Katastroof, omdat hij enkele jaren geleden samen met Katastrooflid Stef Bef de groep Home vormde.
Politiek
Bij de gemeenteraadsverkiezingen van 2006 was Joeri Fransen kandidaat voor de VLDplus in zijn woonplaats Herentals. Dit zorgde voor spanningen in de plaatselijke partijafdeling omdat de kandidatenlijsten niet correct zouden opgesteld zijn. De plaatselijke partijvoorzitter trad daarom af. Joeri Fransen behaalde slechts 121 stemmen, waarmee ook zijn politieke carrière meteen eindigde.
Idool optredens
Semi-finale (groep 1): "Lean On Me" (Bill Withers)
Top 10: "If I Had A Rocket Launcher" (Bruce Cockburn)
Top 9: "All Night Long" (Lionel Richie)
Top 8: "Geen Toeval" (Marco Borsato)
Top 7: "Everybody's Talking" (Harry Nilsson)
Top 6: "Lovin' Whiskey" (Rory Block)
Top 5: "Feeling Good" (Nina Simone)
Top 4: "Black" (Pearl Jam)
Top 4: "Clocks" (Coldplay)
Top 3: "Wicked Game" (Chris Isaak)
Top 3: "Out of Time" (Chris Farlowe)
Top 2: "Ya Bout To Find Out" (winnaarssingle)
Top 2: "Everybody's Talking" (reprise) (Harry Nilsson)
Top 2: "Georgia On My Mind" (Ray Charles)
Discografie
"Ya 'bout to find out" (debuutsingle - 2004)
"High and alive" (single - 2005)
True lies (debuutalbum - 2005)
"We came this far" (single - 2005)
"True lies" (single - 2005)
"Different stars" (single - 2006)
Samen met Katastroof:
"Tengels lieke" (albumtrack op Vrindjespolletiek - 2013)
Vlaams zanger
Idols
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\section{Introduction}
When small quantities of $^3$He are added to bulk superfluid $^4$He below $T{\sim}100{\rm\, mK}$ the atoms occupy so-called Andreev states \cite{ANDREEV} and form a degenerate two-dimensional fermion system. At low temperatures, and when the surface density of $^3$He $n_\mathrm{3S}$ is less than about half a monolayer, the $^3$He chemical potential can be described \cite{GUO} by
\begin{equation}
\mu_3=E^{(0)}_\mathrm{3S}+\left(\frac{\pi\hbar^2}{m_\mathrm{3S}}+\frac{ V_\mathrm{3S}}{2}\right)n_\mathrm{3S}
\label{mu3}
\end{equation}
where $E^{(0)}_\mathrm{3S}$ is the binding energy of a single $^3$He atom (effective mass $m_\mathrm{3S}$) to the $^4$He surface. $V_\mathrm{3S}$ parameterises the $^3$He--$^3$He interaction.
\par
Values of $m_\mathrm{3S}$ and $V_\mathrm{3S}$ inferred from measurements of thermodynamic properties of the surface, such as surface tension, are strongly covariant (see \cite{EDWARDS} for example). However, we have been able to obtain independent values for these quantities using a new method based on quantum evaporation \cite{BAIRD,CDHW}, as follows:
\begin{figure}[b]
\begin{center}\leavevmode
\includegraphics[width=1.0\linewidth]{CW990615-1.eps}
\caption{Schematic diagram of the quantum evaporation experiment} \label{EXPT}\end{center}\end{figure}
\par
A tightly collimated beam of high-energy phonons \cite{PHONONS} was directed at normal incidence to the free liquid surface (Fig.\ \ref{EXPT}a). The evaporated $^3$He atom beam was also collimated so that only atoms with a nearly zero component of momentum parallel to the surface were detected. An incident phonon of energy $E_\mathrm{p}$ imparts kinetic energy to the ejected atom, which has bare mass $m_3$, such that
\begin{equation}
{\frac{1}{2}} m_3v^2-E_\mathrm{p}=E^{(0)}_\mathrm{3S}+{\frac{1}{2}}V_\mathrm{3S} n_\mathrm{3S}.
\label{KE}
\end{equation}
The distribution of phonon energies arriving at the surface is unaffected by $n_\mathrm{3S}$ at the coverages used, and the value of $E^{(0)}_\mathrm{3S}/k_{\mathrm B}=-5.02\pm0.03\,\mathrm{K}$ \cite{EDWARDS} is constant. Therefore, the variation with $n_\mathrm{3S}$ of arrival times of $^3$He atoms at the bolometer is due simply to the term $\frac{1}{2}V_\mathrm{3S} n_\mathrm{3S}$ in Eqn\ \ref{KE}. Our measured first-arrival times were consistent with Eqn\ \ref{KE} for coverages up to $n_\mathrm{3S}=4\,\mathrm{nm}^{-2}$ (Fig.\ \ref{ENERGIES}) and we conclude that
$V_\mathrm{3S}/k_{\mathrm B}=(0.23\pm0.02)\,\mathrm{K\,nm^2}$. Knowledge of this value eliminates the uncertainty due to covariance (see above) in the value of $m_\mathrm{3S}$ inferred from measurements \cite{EDWARDS,EDWARDS2} of surface-sound velocity and surface tension. Hence, the best estimate of $m_\mathrm{3S}$ can refined from $(1.45\pm0.10)m_\mathrm{3}$ to $m_\mathrm{3S}=(1.53\pm0.02)m_\mathrm{3}$.
\begin{figure}[b]
\begin{center}\leavevmode
\includegraphics[width=1.0\linewidth]{CW990615-2.eps}
\caption{Values of $E^{(0)}_\mathrm{3S}+\frac{1}{2}V_\mathrm{3S} n_\mathrm{3S}$ deduced from measured first-arrival times for $^3$He atoms evaporated by high-energy phonons as a function of surface density $n_\mathrm{3S}$.} \label{ENERGIES}\end{center}\end{figure}
\par
We have also used a slight variant of the experiment (Fig.\ \ref{EXPT}b) to search for evidence of a second `excited' state predicted by Pavloff and Treiner \cite{PAVLOFF}. Atoms in this state have a smaller binding energy $E^{(1)}_\mathrm{3S}$ to the surface than those in the first surface state, and the signature of its occupation is therefore an additional faster component in the detected signal. This component appeared, albeit at a level not greatly above the detector noise, on all signals taken below $T=60\,\mathrm{mK}$ with surface coverages of, and above, about 0.6 monolayers, {\it i.e.} $n_\mathrm{3S}=4\,\mathrm{nm}^{-2}$ (Fig.\ \ref{SECONDSTATE}). From our preliminary measurements of first-arrival times, we find that $E^{(1)}_\mathrm{3S}=-3.4\pm0.4\,\mathrm{K}$, in agreement with the predictions \cite{PAVLOFF}.
\par
Although this paper has discussed exclusively the states of $^3$He above bulk $^4$He, we note that thin film and layered systems have some comparable properties and have been investigated by other groups using NMR, third-sound and heat capacity measurements \cite{ANDERSON,DANN,MORISHITA}.
\begin{figure}[btp]
\begin{center}\leavevmode
\includegraphics[width=1.0\linewidth]{CW990615-3.eps}
\caption{$^3$He atoms from the lowest surface-state start arriving at $85\,\mathrm{\mu s}$, the earlier component (shaded) is due to atoms from the excited surface state. The largest signal component (inset) is due to evaporated $^4$He atoms.} \label{SECONDSTATE}\end{center}\end{figure}
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Richard Foster Flint (March 1, 1902 - June 6, 1976) was an American geologist.
Biography
He was born in Chicago on March 1, 1902.
Flint graduated from the University of Chicago and earned his Ph.D. in geology at the University of California graduating in 1925. He then joined Yale as a member of the faculty, becoming a full professor in 1945.
Flint was recognized for his leadership role in Quaternary period geology with extensive work on effects of glaciations in northeastern America.
He also performed research in Washington State to understand the last ice age's impact on the Northwest, gaining some notoriety for his opposition to the Missoula Floods hypothesis, which was posed by J Harlen Bretz. He presented a detailed and thoughtful argument against the possibility of catastrophic floods; a position which has subsequently fallen into disfavor based on a wide collection of evidence.
He died on June 6, 1976, in New Haven, Connecticut.
Major publications include
Outlines of Physical Geology, 1941
Introduction to Geology, 1962
Radiocarbon measurements, 1967
Glacial Geology and the Pleistocene Epoch (Glacial and Pleistocene Geology), 1957
Glacial and Quaternary Geology, 1971
Honors
In 1972 he was awarded the Prestwich Medal, a medal awarded by the Geological Society of London, for significant contributions in the science of Geology.
References
20th-century American geologists
Quaternary geologists
1902 births
1976 deaths
University of Chicago alumni
University of California, Berkeley alumni
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Q: convert spanish characters in HTML doc I have a HTML file and it has some information in spanish. I am using a third party control to convert this HTML file into RTF document. The third party software I am using is Subsystems HTML Addon.
The HTML file has
<META http-equiv="Content-Type" content="text/html; charset=utf-8">
I think the subsystems software is not able to recognize the characters greater than 127.
I tried replacing the characters with ASCII>127 to their HTML entity code For ex, ò with • and then sending the document to converter but that didn't help.
Any one has any ideas?
A: The magic word is encoding. The question is what encoding your HTML file is in, and what encoding you need in your RTF file.
Here is some very good basic reading on the issue if you're interested. Otherwise, you'll have to determine the HTML files actual, and the RTF file's actual encodings.
A: Did you try just asking their technical support? The product page says it supports Arabic/Hebrew and Asian languages, so it sounds like they definitely support values above ASCII 127. It's probably an issue with getting the right encoding.
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}
| 6,213
|
But America needs to fix its addiction to easy money and the current crisis could have been the best thing happening to the country in a long while for its curative properties. Its going to be bluntet by the shale gas boom as this puts new easy money at the fingertips of politicians.
America is the new energy El Dorado – what about Europe now?
America rapidly transforms from the energy problem child to become the new el Dorado. The economy is pimped by all the investment going the other side of the Atlantic. The population is blessed with much cheaper fuel and lets not forget – the environment does better than any time after World War 2. Why cant we have that in Europe?
The late 1970ies saw the rise of various green political movements in Europe. Today they are a fundamental feature of any European parliament. Ever since, they were known as those who would fight for Mother Nature and on the side for progressive politics. Do they?
I have seen many of those plans and they have never worked. Not because they are African. This stuff cannot work if you put it in Europe or North America. Because grand Master plans generally are impossible to realize. They inherently lack flexibility, which is necessary to deal with unforeseen events or twists and turns in the market or the international energy scene.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,048
|
Russia in Crimea—Not 'Munich 1938' Yet Again
By Ivan Eland
Also published in The Huffington Post Tue. April 8, 2014 « Show Fewer
Western hysteria surrounding Russia's seizing of Crimea is rooted in a larger problem with U.S. foreign policy—"the Munich 1938 syndrome." Ever since British Prime Minister Neville Chamberlain allowed Adolf Hitler to take over German-speaking portions of Czechoslovakia at a Munich meeting in 1938, appeasement has gone out of favor as a traditional, respectable tool in foreign policy. Since that time, what had been an acceptable way of sometimes conciliating or buying off an opponent has taken on a universally cowardly connotation. (As U.S. General David Petraeus skillfully and quietly demonstrated in extricating the United States from Iraq by buying off most enemy Sunni guerrillas and getting them to fight against al Qaeda, the policy is still used occasionally, but we just don't call it the a-word.)
However, a good case can be made that the now-vilified Chamberlain saved Britain's bacon during World War II and that, despite his pledge to the contrary, Winston Churchill was the one who sold the British Empire down the river, even forfeiting Britain's standing as a world power. In fact, Chamberlain had little choice but to accommodate Hitler's Germany at Munich; the British were behind the Germans in their military build up and needed time to catch up. Furthermore, Chamberlain ordered the Royal Air Force—which like many other military organizations, then and since, was enamored with the macho "cult of the offensive"—to transfer resources from offensive strategic bombing into fighter air defense. In 1940, augmented air defenses helped save Britain by defeating the attacking German Luftwaffe in the Battle of Britain.
In contrast, Churchill, now the hero of World War II for being prime minister at the time of that titanic battle and during most of the war, benefitted greatly from Chamberlain's buying of time through earlier appeasement and his using that time to augment the British military, especially its air defenses. Moreover, the always-belligerent Churchill helped pave the way for the rise of Hitler in the first place (as did America's Woodrow Wilson) by beating the drums to enter World War I, which tipped the balance in a largely European continental conflict to the allies and led to Hitler's rise out of allied post-war humiliation of the Germans. "Winning" the massive conflicts of World Wars I and II financially ruined Britain, causing it to ultimately lose its empire and even its status as a world power.
After Munich in 1938, instead of learning that appeasement or conciliation is sometimes the best policy, or even a necessary one, great powers took away a mistaken lesson—that any perceived threat, no matter how small, had to be met immediately with forceful action in order to deter an opponent from further snow-balling aggression. The United States was especially prone to this false lesson during World War II and thereafter. President Franklin Delano Roosevelt, taking the wrong message from Munich 1938, refused to meet with the Japanese prime minister to conciliate U.S.-Japanese differences over Japan's empire building in East Asia and retaliatory American attempts to strangle the Japanese military and economy with an oil embargo (the United States was then the world's largest producer of oil). FDR's refusal led to the desperate Japanese attack on Pearl Harbor and U.S. involvement in World War II. During the Cold War, springing directly out of the "Munich 1938 syndrome" were fears that communism would spread like falling dominoes in country after country, if it wasn't stopped in backwater countries such as Korea, Vietnam, Afghanistan, Nicaragua, Angola, Grenada, etc.
And now, after Russia has invaded and annexed Crimea, calls have arisen in the West to impose harsh economic penalties on Russia, deploy U.S. military equipment in NATO countries near Russia, give Ukraine substantial financial aid, and spend more on NATO defenses—all to deter Russia from further aggression. Make no mistake about it, Vladimir Putin and Russia were wrong in using force to annex a Crimea that probably wanted to be a part of the Russian Federation anyway and could very well do the same with the equally Russophilic eastern Ukraine.
Congratulating Russia for violating the sovereignty of a nation-state isn't a good idea, but overreacting isn't either. Some demonstration of displeasure on the part of Western governments to deter further potential Russian actions probably isn't a bad idea. However, Ukraine owes a lot of money to Russia, and huge amounts of Western financial aid given to Ukraine will merely be sent to Russia to pay it off, thus helping Russia. Spending more money on NATO defenses wouldn't be a bad idea if it were the nearby European countries that were going to do it; but they will not. For decades, wealthy European countries have relied on the United States to provide their security, and the U.S. military-industrial-congressional complex (MICC) has found it profitable to do so. The MICC is now hyping the allegedly renewed Russian threat to attempt to reverse declining defense spending in the United States.
In the worst case, Russia could also try to incorporate the Russian-speaking peoples of eastern Ukraine into Russia, as it did with Crimea. While not a good development, Russia's military is probably incapable of going any farther to conquer and occupy the vast stretches of the remainder of Ukraine. Despite glowing accounts of a much-improved Russian military in Crimea in The New York Times, the improvement lies mainly with elite Russian Special Forces units. Militarily, today's Russia is certainly no Soviet Union. And despite likely presidential candidate Hillary Clinton's assertion that Putin is another Hitler, he is not. He is not trying to take over Europe and is merely trying to salvage what he can of Ukraine after a democratically elected pro-Russian government was overthrown by pro-Western street mobs.
Whenever U.S. politicians want to hype a threat, they compare the minor villain of the day to Hitler—Slobodan Milosevic in Serbia, Saddam Hussein in Iraq, Muammar Gaddafi in Libya, choose your ruler of North Korea from the family Kim, and now Putin in Russia—and solemnly imply that the new bad buy must be stopped or we'll have Munich 1938 all over again. Hardly.
Ivan Eland is Senior Fellow at the Independent Institute and Director of the Institute's Center on Peace & Liberty.
Defense and Foreign PolicyDiplomacy and Foreign AidEuropeGovernment and PoliticsInternational Economics and DevelopmentPolitical History
FROM Ivan Eland
Restoring the Republic after Congressional Failure
Eleven Presidents
Promises vs. Results in Achieving Limited Government
Recarving Rushmore
Ranking the Presidents on Peace, Prosperity, and Liberty
No War for Oil
U.S. Dependency and the Middle East
Partitioning for Peace
An Exit Strategy for Iraq
Recarving Rushmore (First Edition)
The Empire Has No Clothes (Updated Edition)
U.S. Foreign Policy Exposed
The Empire Has No Clothes (First Edition)
Putting "Defense" Back into U.S. Defense Policy
Rethinking U.S. Security in the Post-Cold War World
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,336
|
package com.intellij.ide.ui.customization;
import com.intellij.icons.AllIcons;
import com.intellij.ide.IdeBundle;
import com.intellij.openapi.actionSystem.ActionManager;
import com.intellij.openapi.actionSystem.AnAction;
import com.intellij.openapi.actionSystem.AnSeparator;
import com.intellij.openapi.actionSystem.ex.QuickList;
import com.intellij.openapi.actionSystem.ex.QuickListsManager;
import com.intellij.openapi.diagnostic.Logger;
import com.intellij.openapi.fileChooser.FileChooserDescriptor;
import com.intellij.openapi.keymap.impl.ui.ActionsTree;
import com.intellij.openapi.keymap.impl.ui.ActionsTreeUtil;
import com.intellij.openapi.keymap.impl.ui.Group;
import com.intellij.openapi.options.ConfigurationException;
import com.intellij.openapi.project.Project;
import com.intellij.openapi.project.ProjectManager;
import com.intellij.openapi.ui.DialogWrapper;
import com.intellij.openapi.ui.Messages;
import com.intellij.openapi.ui.TextFieldWithBrowseButton;
import com.intellij.openapi.util.IconLoader;
import com.intellij.openapi.util.Pair;
import com.intellij.openapi.util.io.FileUtil;
import com.intellij.openapi.util.text.StringUtil;
import com.intellij.openapi.vfs.VfsUtil;
import com.intellij.openapi.vfs.VirtualFile;
import com.intellij.openapi.wm.ex.WindowManagerEx;
import com.intellij.openapi.wm.impl.IdeFrameImpl;
import com.intellij.packageDependencies.ui.TreeExpansionMonitor;
import com.intellij.ui.DocumentAdapter;
import com.intellij.ui.InsertPathAction;
import com.intellij.ui.ScrollPaneFactory;
import com.intellij.ui.treeStructure.Tree;
import com.intellij.util.ImageLoader;
import com.intellij.util.ObjectUtils;
import com.intellij.util.ui.EmptyIcon;
import com.intellij.util.ui.UIUtil;
import com.intellij.util.ui.tree.TreeUtil;
import org.jetbrains.annotations.Nullable;
import javax.swing.*;
import javax.swing.event.DocumentEvent;
import javax.swing.event.TreeSelectionEvent;
import javax.swing.event.TreeSelectionListener;
import javax.swing.tree.*;
import java.awt.*;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.io.File;
import java.io.IOException;
import java.util.*;
import java.util.List;
/**
* User: anna
* Date: Mar 17, 2005
*/
public class CustomizableActionsPanel {
private static final Logger LOG = Logger.getInstance("#com.intellij.ide.ui.customization.CustomizableActionsPanel");
private JButton myEditIconButton;
private JButton myRemoveActionButton;
private JButton myAddActionButton;
private JButton myMoveActionDownButton;
private JButton myMoveActionUpButton;
private JPanel myPanel;
private JTree myActionsTree;
private JButton myAddSeparatorButton;
private final TreeExpansionMonitor myTreeExpansionMonitor;
private CustomActionsSchema mySelectedSchema;
private JButton myRestoreAllDefaultButton;
private JButton myRestoreDefaultButton;
public CustomizableActionsPanel() {
//noinspection HardCodedStringLiteral
Group rootGroup = new Group("root", null, null);
final DefaultMutableTreeNode root = new DefaultMutableTreeNode(rootGroup);
DefaultTreeModel model = new DefaultTreeModel(root);
myActionsTree.setModel(model);
myActionsTree.setRootVisible(false);
myActionsTree.setShowsRootHandles(true);
UIUtil.setLineStyleAngled(myActionsTree);
myActionsTree.setCellRenderer(new MyTreeCellRenderer());
setButtonsDisabled();
final ActionManager actionManager = ActionManager.getInstance();
myActionsTree.getSelectionModel().addTreeSelectionListener(new TreeSelectionListener() {
public void valueChanged(TreeSelectionEvent e) {
final TreePath[] selectionPaths = myActionsTree.getSelectionPaths();
final boolean isSingleSelection = selectionPaths != null && selectionPaths.length == 1;
myAddActionButton.setEnabled(isSingleSelection);
if (isSingleSelection) {
final DefaultMutableTreeNode node = (DefaultMutableTreeNode)selectionPaths[0].getLastPathComponent();
String actionId = getActionId(node);
if (actionId != null) {
final AnAction action = actionManager.getAction(actionId);
myEditIconButton.setEnabled(action != null &&
action.getTemplatePresentation() != null);
}
else {
myEditIconButton.setEnabled(false);
}
}
else {
myEditIconButton.setEnabled(false);
}
myAddSeparatorButton.setEnabled(isSingleSelection);
myRemoveActionButton.setEnabled(selectionPaths != null);
if (selectionPaths != null) {
for (TreePath selectionPath : selectionPaths) {
if (selectionPath.getPath() != null && selectionPath.getPath().length <= 2) {
setButtonsDisabled();
return;
}
}
}
myMoveActionUpButton.setEnabled(isMoveSupported(myActionsTree, -1));
myMoveActionDownButton.setEnabled(isMoveSupported(myActionsTree, 1));
myRestoreDefaultButton.setEnabled(!findActionsUnderSelection().isEmpty());
}
});
myAddActionButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
final List<TreePath> expandedPaths = TreeUtil.collectExpandedPaths(myActionsTree);
final TreePath selectionPath = myActionsTree.getLeadSelectionPath();
if (selectionPath != null) {
DefaultMutableTreeNode node = (DefaultMutableTreeNode)selectionPath.getLastPathComponent();
final FindAvailableActionsDialog dlg = new FindAvailableActionsDialog();
dlg.show();
if (dlg.isOK()) {
final Set<Object> toAdd = dlg.getTreeSelectedActionIds();
if (toAdd == null) return;
for (final Object o : toAdd) {
final ActionUrl url = new ActionUrl(ActionUrl.getGroupPath(new TreePath(node.getPath())), o, ActionUrl.ADDED,
node.getParent().getIndex(node) + 1);
addCustomizedAction(url);
ActionUrl.changePathInActionsTree(myActionsTree, url);
if (o instanceof String) {
DefaultMutableTreeNode current = new DefaultMutableTreeNode(url.getComponent());
current.setParent((DefaultMutableTreeNode)node.getParent());
editToolbarIcon((String)o, current);
}
}
((DefaultTreeModel)myActionsTree.getModel()).reload();
}
}
TreeUtil.restoreExpandedPaths(myActionsTree, expandedPaths);
}
});
myEditIconButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
myRestoreAllDefaultButton.setEnabled(true);
final List<TreePath> expandedPaths = TreeUtil.collectExpandedPaths(myActionsTree);
final TreePath selectionPath = myActionsTree.getLeadSelectionPath();
if (selectionPath != null) {
EditIconDialog dlg = new EditIconDialog((DefaultMutableTreeNode)selectionPath.getLastPathComponent());
dlg.show();
if (dlg.isOK()) {
myActionsTree.repaint();
}
}
TreeUtil.restoreExpandedPaths(myActionsTree, expandedPaths);
}
});
myAddSeparatorButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
final List<TreePath> expandedPaths = TreeUtil.collectExpandedPaths(myActionsTree);
final TreePath selectionPath = myActionsTree.getLeadSelectionPath();
if (selectionPath != null) {
DefaultMutableTreeNode node = (DefaultMutableTreeNode)selectionPath.getLastPathComponent();
final ActionUrl url = new ActionUrl(ActionUrl.getGroupPath(selectionPath), AnSeparator.getInstance(), ActionUrl.ADDED,
node.getParent().getIndex(node) + 1);
ActionUrl.changePathInActionsTree(myActionsTree, url);
addCustomizedAction(url);
((DefaultTreeModel)myActionsTree.getModel()).reload();
}
TreeUtil.restoreExpandedPaths(myActionsTree, expandedPaths);
}
});
myRemoveActionButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
final List<TreePath> expandedPaths = TreeUtil.collectExpandedPaths(myActionsTree);
final TreePath[] selectionPath = myActionsTree.getSelectionPaths();
if (selectionPath != null) {
for (TreePath treePath : selectionPath) {
final ActionUrl url = CustomizationUtil.getActionUrl(treePath, ActionUrl.DELETED);
ActionUrl.changePathInActionsTree(myActionsTree, url);
addCustomizedAction(url);
}
((DefaultTreeModel)myActionsTree.getModel()).reload();
}
TreeUtil.restoreExpandedPaths(myActionsTree, expandedPaths);
}
});
myMoveActionUpButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
final List<TreePath> expandedPaths = TreeUtil.collectExpandedPaths(myActionsTree);
final TreePath[] selectionPath = myActionsTree.getSelectionPaths();
if (selectionPath != null) {
for (TreePath treePath : selectionPath) {
final ActionUrl url = CustomizationUtil.getActionUrl(treePath, ActionUrl.MOVE);
final int absolutePosition = url.getAbsolutePosition();
url.setInitialPosition(absolutePosition);
url.setAbsolutePosition(absolutePosition - 1);
ActionUrl.changePathInActionsTree(myActionsTree, url);
addCustomizedAction(url);
}
((DefaultTreeModel)myActionsTree.getModel()).reload();
TreeUtil.restoreExpandedPaths(myActionsTree, expandedPaths);
for (TreePath path : selectionPath) {
myActionsTree.addSelectionPath(path);
}
}
}
});
myMoveActionDownButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
final List<TreePath> expandedPaths = TreeUtil.collectExpandedPaths(myActionsTree);
final TreePath[] selectionPath = myActionsTree.getSelectionPaths();
if (selectionPath != null) {
for (int i = selectionPath.length - 1; i >= 0; i--) {
TreePath treePath = selectionPath[i];
final ActionUrl url = CustomizationUtil.getActionUrl(treePath, ActionUrl.MOVE);
final int absolutePosition = url.getAbsolutePosition();
url.setInitialPosition(absolutePosition);
url.setAbsolutePosition(absolutePosition + 1);
ActionUrl.changePathInActionsTree(myActionsTree, url);
addCustomizedAction(url);
}
((DefaultTreeModel)myActionsTree.getModel()).reload();
TreeUtil.restoreExpandedPaths(myActionsTree, expandedPaths);
for (TreePath path : selectionPath) {
myActionsTree.addSelectionPath(path);
}
}
}
});
myRestoreAllDefaultButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
mySelectedSchema.copyFrom(new CustomActionsSchema());
patchActionsTreeCorrespondingToSchema(root);
myRestoreAllDefaultButton.setEnabled(false);
}
});
myRestoreDefaultButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
final List<ActionUrl> otherActions = new ArrayList<ActionUrl>(mySelectedSchema.getActions());
otherActions.removeAll(findActionsUnderSelection());
mySelectedSchema.copyFrom(new CustomActionsSchema());
for (ActionUrl otherAction : otherActions) {
mySelectedSchema.addAction(otherAction);
}
final List<TreePath> treePaths = TreeUtil.collectExpandedPaths(myActionsTree);
patchActionsTreeCorrespondingToSchema(root);
restorePathsAfterTreeOptimization(treePaths);
myRestoreDefaultButton.setEnabled(false);
}
});
patchActionsTreeCorrespondingToSchema(root);
myTreeExpansionMonitor = TreeExpansionMonitor.install(myActionsTree);
}
private List<ActionUrl> findActionsUnderSelection() {
final ArrayList<ActionUrl> actions = new ArrayList<ActionUrl>();
final TreePath[] selectionPaths = myActionsTree.getSelectionPaths();
if (selectionPaths != null) {
for (TreePath path : selectionPaths) {
final ActionUrl selectedUrl = CustomizationUtil.getActionUrl(path, ActionUrl.MOVE);
final ArrayList<String> selectedGroupPath = new ArrayList<String>(selectedUrl.getGroupPath());
final Object component = selectedUrl.getComponent();
if (component instanceof Group) {
selectedGroupPath.add(((Group)component).getName());
for (ActionUrl action : mySelectedSchema.getActions()) {
final ArrayList<String> groupPath = action.getGroupPath();
final int idx = Collections.indexOfSubList(groupPath, selectedGroupPath);
if (idx > -1) {
actions.add(action);
}
}
}
}
}
return actions;
}
private void addCustomizedAction(ActionUrl url) {
mySelectedSchema.addAction(url);
myRestoreAllDefaultButton.setEnabled(true);
}
private void editToolbarIcon(String actionId, DefaultMutableTreeNode node) {
final AnAction anAction = ActionManager.getInstance().getAction(actionId);
if (isToolbarAction(node) &&
anAction.getTemplatePresentation() != null &&
anAction.getTemplatePresentation().getIcon() == null) {
final int exitCode = Messages.showOkCancelDialog(IdeBundle.message("error.adding.action.without.icon.to.toolbar"),
IdeBundle.message("title.unable.to.add.action.without.icon.to.toolbar"),
Messages.getInformationIcon());
if (exitCode == Messages.OK) {
mySelectedSchema.addIconCustomization(actionId, null);
anAction.getTemplatePresentation().setIcon(AllIcons.Toolbar.Unknown);
anAction.setDefaultIcon(false);
node.setUserObject(Pair.create(actionId, AllIcons.Toolbar.Unknown));
myActionsTree.repaint();
setCustomizationSchemaForCurrentProjects();
}
}
}
private void setButtonsDisabled() {
myRemoveActionButton.setEnabled(false);
myAddActionButton.setEnabled(false);
myEditIconButton.setEnabled(false);
myAddSeparatorButton.setEnabled(false);
myMoveActionDownButton.setEnabled(false);
myMoveActionUpButton.setEnabled(false);
}
private static boolean isMoveSupported(JTree tree, int dir) {
final TreePath[] selectionPaths = tree.getSelectionPaths();
if (selectionPaths != null) {
DefaultMutableTreeNode parent = null;
for (TreePath treePath : selectionPaths)
if (treePath.getLastPathComponent() != null) {
final DefaultMutableTreeNode node = (DefaultMutableTreeNode)treePath.getLastPathComponent();
if (parent == null) {
parent = (DefaultMutableTreeNode)node.getParent();
}
if (parent != node.getParent()) {
return false;
}
if (dir > 0) {
if (parent.getIndex(node) == parent.getChildCount() - 1) {
return false;
}
}
else {
if (parent.getIndex(node) == 0) {
return false;
}
}
}
return true;
}
return false;
}
public JPanel getPanel() {
return myPanel;
}
private static void setCustomizationSchemaForCurrentProjects() {
final Project[] openProjects = ProjectManager.getInstance().getOpenProjects();
for (Project project : openProjects) {
final IdeFrameImpl frame = WindowManagerEx.getInstanceEx().getFrame(project);
if (frame != null) {
frame.updateView();
}
//final FavoritesManager favoritesView = FavoritesManager.getInstance(project);
//final String[] availableFavoritesLists = favoritesView.getAvailableFavoritesLists();
//for (String favoritesList : availableFavoritesLists) {
// favoritesView.getFavoritesTreeViewPanel(favoritesList).updateTreePopupHandler();
//}
}
final IdeFrameImpl frame = WindowManagerEx.getInstanceEx().getFrame(null);
if (frame != null) {
frame.updateView();
}
}
public void apply() throws ConfigurationException {
final List<TreePath> treePaths = TreeUtil.collectExpandedPaths(myActionsTree);
if (mySelectedSchema != null) {
CustomizationUtil.optimizeSchema(myActionsTree, mySelectedSchema);
}
restorePathsAfterTreeOptimization(treePaths);
CustomActionsSchema.getInstance().copyFrom(mySelectedSchema);
setCustomizationSchemaForCurrentProjects();
}
private void restorePathsAfterTreeOptimization(final List<TreePath> treePaths) {
for (final TreePath treePath : treePaths) {
myActionsTree.expandPath(CustomizationUtil.getPathByUserObjects(myActionsTree, treePath));
}
}
public void reset() {
mySelectedSchema = new CustomActionsSchema();
mySelectedSchema.copyFrom(CustomActionsSchema.getInstance());
patchActionsTreeCorrespondingToSchema((DefaultMutableTreeNode)myActionsTree.getModel().getRoot());
myRestoreAllDefaultButton.setEnabled(mySelectedSchema.isModified(new CustomActionsSchema()));
}
public boolean isModified() {
CustomizationUtil.optimizeSchema(myActionsTree, mySelectedSchema);
return CustomActionsSchema.getInstance().isModified(mySelectedSchema);
}
private void patchActionsTreeCorrespondingToSchema(DefaultMutableTreeNode root) {
root.removeAllChildren();
if (mySelectedSchema != null) {
mySelectedSchema.fillActionGroups(root);
for (final ActionUrl actionUrl : mySelectedSchema.getActions()) {
ActionUrl.changePathInActionsTree(myActionsTree, actionUrl);
}
}
((DefaultTreeModel)myActionsTree.getModel()).reload();
}
private static class MyTreeCellRenderer extends DefaultTreeCellRenderer {
public Component getTreeCellRendererComponent(JTree tree,
Object value,
boolean sel,
boolean expanded,
boolean leaf,
int row,
boolean hasFocus) {
super.getTreeCellRendererComponent(tree, value, sel, expanded, leaf, row, hasFocus);
if (value instanceof DefaultMutableTreeNode) {
Object userObject = ((DefaultMutableTreeNode)value).getUserObject();
Icon icon = null;
if (userObject instanceof Group) {
Group group = (Group)userObject;
String name = group.getName();
setText(name != null ? name : group.getId());
icon = ObjectUtils.notNull(group.getIcon(), AllIcons.Nodes.Folder);
}
else if (userObject instanceof String) {
String actionId = (String)userObject;
AnAction action = ActionManager.getInstance().getAction(actionId);
String name = action != null ? action.getTemplatePresentation().getText() : null;
setText(!StringUtil.isEmptyOrSpaces(name) ? name : actionId);
if (action != null) {
Icon actionIcon = action.getTemplatePresentation().getIcon();
if (actionIcon != null) {
icon = actionIcon;
}
}
}
else if (userObject instanceof Pair) {
String actionId = (String)((Pair)userObject).first;
AnAction action = ActionManager.getInstance().getAction(actionId);
setText(action != null ? action.getTemplatePresentation().getText() : actionId);
icon = (Icon)((Pair)userObject).second;
}
else if (userObject instanceof AnSeparator) {
setText("-------------");
}
else if (userObject instanceof QuickList) {
setText(((QuickList)userObject).getDisplayName());
icon = AllIcons.Actions.QuickList;
}
else {
throw new IllegalArgumentException("unknown userObject: " + userObject);
}
setIcon(ActionsTree.getEvenIcon(icon));
if (sel) {
setForeground(UIUtil.getTreeSelectionForeground());
}
else {
setForeground(UIUtil.getTreeForeground());
}
}
return this;
}
}
private static boolean isToolbarAction(DefaultMutableTreeNode node) {
return node.getParent() != null && ((DefaultMutableTreeNode)node.getParent()).getUserObject() instanceof Group &&
((Group)((DefaultMutableTreeNode)node.getParent()).getUserObject()).getName().equals(ActionsTreeUtil.MAIN_TOOLBAR);
}
@Nullable
private static String getActionId(DefaultMutableTreeNode node) {
return (String)(node.getUserObject() instanceof String ? node.getUserObject() :
node.getUserObject() instanceof Pair ? ((Pair)node.getUserObject()).first : null);
}
protected boolean doSetIcon(DefaultMutableTreeNode node, @Nullable String path, Component component) {
if (StringUtil.isNotEmpty(path) && !new File(path).isFile()) {
Messages
.showErrorDialog(component, IdeBundle.message("error.file.not.found.message", path), IdeBundle.message("title.choose.action.icon"));
return false;
}
String actionId = getActionId(node);
if (actionId == null) return false;
final AnAction action = ActionManager.getInstance().getAction(actionId);
if (action != null && action.getTemplatePresentation() != null) {
if (StringUtil.isNotEmpty(path)) {
Image image = null;
try {
image = ImageLoader.loadFromStream(VfsUtil.convertToURL(VfsUtil.pathToUrl(path.replace(File.separatorChar,
'/'))).openStream());
}
catch (IOException e) {
LOG.debug(e);
}
Icon icon = new File(path).exists() ? IconLoader.getIcon(image) : null;
if (icon != null) {
if (icon.getIconWidth() > EmptyIcon.ICON_18.getIconWidth() || icon.getIconHeight() > EmptyIcon.ICON_18.getIconHeight()) {
Messages.showErrorDialog(component, IdeBundle.message("custom.icon.validation.message"), IdeBundle.message("title.choose.action.icon"));
return false;
}
node.setUserObject(Pair.create(actionId, icon));
mySelectedSchema.addIconCustomization(actionId, path);
}
}
else {
node.setUserObject(Pair.create(actionId, null));
mySelectedSchema.removeIconCustomization(actionId);
final DefaultMutableTreeNode nodeOnToolbar = findNodeOnToolbar(actionId);
if (nodeOnToolbar != null){
editToolbarIcon(actionId, nodeOnToolbar);
node.setUserObject(nodeOnToolbar.getUserObject());
}
}
return true;
}
return false;
}
private static TextFieldWithBrowseButton createBrowseField(){
TextFieldWithBrowseButton textField = new TextFieldWithBrowseButton();
textField.setPreferredSize(new Dimension(200, textField.getPreferredSize().height));
textField.setMinimumSize(new Dimension(200, textField.getPreferredSize().height));
final FileChooserDescriptor fileChooserDescriptor = new FileChooserDescriptor(true, false, false, false, false, false) {
public boolean isFileSelectable(VirtualFile file) {
//noinspection HardCodedStringLiteral
return file.getName().endsWith(".png");
}
};
textField.addBrowseFolderListener(IdeBundle.message("title.browse.icon"), IdeBundle.message("prompt.browse.icon.for.selected.action"), null,
fileChooserDescriptor);
InsertPathAction.addTo(textField.getTextField(), fileChooserDescriptor);
return textField;
}
private class EditIconDialog extends DialogWrapper {
private final DefaultMutableTreeNode myNode;
protected TextFieldWithBrowseButton myTextField;
protected EditIconDialog(DefaultMutableTreeNode node) {
super(false);
setTitle(IdeBundle.message("title.choose.action.icon"));
init();
myNode = node;
final String actionId = getActionId(node);
if (actionId != null) {
final String iconPath = mySelectedSchema.getIconPath(actionId);
myTextField.setText(FileUtil.toSystemDependentName(iconPath));
}
}
@Override
public JComponent getPreferredFocusedComponent() {
return myTextField.getChildComponent();
}
protected String getDimensionServiceKey() {
return getClass().getName();
}
protected JComponent createCenterPanel() {
myTextField = createBrowseField();
JPanel northPanel = new JPanel(new BorderLayout());
northPanel.add(myTextField, BorderLayout.NORTH);
return northPanel;
}
protected void doOKAction() {
if (myNode != null) {
if (!doSetIcon(myNode, myTextField.getText(), getContentPane())) {
return;
}
final Object userObject = myNode.getUserObject();
if (userObject instanceof Pair) {
String actionId = (String)((Pair)userObject).first;
final AnAction action = ActionManager.getInstance().getAction(actionId);
final Icon icon = (Icon)((Pair)userObject).second;
action.getTemplatePresentation().setIcon(icon);
action.setDefaultIcon(icon == null);
editToolbarIcon(actionId, myNode);
}
myActionsTree.repaint();
}
setCustomizationSchemaForCurrentProjects();
super.doOKAction();
}
}
@Nullable
private DefaultMutableTreeNode findNodeOnToolbar(String actionId){
final TreeNode toolbar = ((DefaultMutableTreeNode)myActionsTree.getModel().getRoot()).getChildAt(1);
for(int i = 0; i < toolbar.getChildCount(); i++){
final DefaultMutableTreeNode child = (DefaultMutableTreeNode)toolbar.getChildAt(i);
final String childId = getActionId(child);
if (childId != null && childId.equals(actionId)){
return child;
}
}
return null;
}
private class FindAvailableActionsDialog extends DialogWrapper{
private JTree myTree;
private JButton mySetIconButton;
private TextFieldWithBrowseButton myTextField;
FindAvailableActionsDialog() {
super(false);
setTitle(IdeBundle.message("action.choose.actions.to.add"));
init();
}
protected JComponent createCenterPanel() {
Group rootGroup = ActionsTreeUtil.createMainGroup(null, null, QuickListsManager.getInstance().getAllQuickLists());
DefaultMutableTreeNode root = ActionsTreeUtil.createNode(rootGroup);
DefaultTreeModel model = new DefaultTreeModel(root);
myTree = new Tree();
myTree.setModel(model);
myTree.setCellRenderer(new MyTreeCellRenderer());
final ActionManager actionManager = ActionManager.getInstance();
mySetIconButton = new JButton(IdeBundle.message("button.set.icon"));
mySetIconButton.setEnabled(false);
mySetIconButton.addActionListener(new ActionListener() {
public void actionPerformed(ActionEvent e) {
final TreePath selectionPath = myTree.getSelectionPath();
if (selectionPath != null) {
doSetIcon((DefaultMutableTreeNode)selectionPath.getLastPathComponent(), myTextField.getText(), getContentPane());
myTree.repaint();
}
}
});
myTextField = createBrowseField();
myTextField.getTextField().getDocument().addDocumentListener(new DocumentAdapter() {
protected void textChanged(DocumentEvent e) {
enableSetIconButton(actionManager);
}
});
JPanel northPanel = new JPanel(new BorderLayout());
northPanel.add(myTextField, BorderLayout.CENTER);
final JLabel label = new JLabel(IdeBundle.message("label.icon.path"));
label.setLabelFor(myTextField.getChildComponent());
northPanel.add(label, BorderLayout.WEST);
northPanel.add(mySetIconButton, BorderLayout.EAST);
northPanel.setBorder(BorderFactory.createEmptyBorder(0, 0, 5, 0));
JPanel panel = new JPanel(new BorderLayout());
panel.add(northPanel, BorderLayout.NORTH);
panel.add(ScrollPaneFactory.createScrollPane(myTree), BorderLayout.CENTER);
myTree.getSelectionModel().addTreeSelectionListener(new TreeSelectionListener() {
public void valueChanged(TreeSelectionEvent e) {
enableSetIconButton(actionManager);
final TreePath selectionPath = myTree.getSelectionPath();
if (selectionPath != null) {
final DefaultMutableTreeNode node = (DefaultMutableTreeNode)selectionPath.getLastPathComponent();
final String actionId = getActionId(node);
if (actionId != null) {
final String iconPath = mySelectedSchema.getIconPath(actionId);
myTextField.setText(FileUtil.toSystemDependentName(iconPath));
}
}
}
});
return panel;
}
protected void doOKAction() {
final ActionManager actionManager = ActionManager.getInstance();
TreeUtil.traverseDepth((TreeNode)myTree.getModel().getRoot(), new TreeUtil.Traverse() {
public boolean accept(Object node) {
if (node instanceof DefaultMutableTreeNode) {
final DefaultMutableTreeNode mutableNode = (DefaultMutableTreeNode)node;
final Object userObject = mutableNode.getUserObject();
if (userObject instanceof Pair) {
String actionId = (String)((Pair)userObject).first;
final AnAction action = actionManager.getAction(actionId);
Icon icon = (Icon)((Pair)userObject).second;
action.getTemplatePresentation().setIcon(icon);
action.setDefaultIcon(icon == null);
editToolbarIcon(actionId, mutableNode);
}
}
return true;
}
});
super.doOKAction();
setCustomizationSchemaForCurrentProjects();
}
protected void enableSetIconButton(ActionManager actionManager) {
final TreePath selectionPath = myTree.getSelectionPath();
Object userObject = null;
if (selectionPath != null) {
userObject = ((DefaultMutableTreeNode)selectionPath.getLastPathComponent()).getUserObject();
if (userObject instanceof String) {
final AnAction action = actionManager.getAction((String)userObject);
if (action != null &&
action.getTemplatePresentation() != null &&
action.getTemplatePresentation().getIcon() != null) {
mySetIconButton.setEnabled(true);
return;
}
}
}
mySetIconButton.setEnabled(myTextField.getText().length() != 0 &&
selectionPath != null &&
new DefaultMutableTreeNode(selectionPath).isLeaf() &&
!(userObject instanceof AnSeparator));
}
@Nullable
public Set<Object> getTreeSelectedActionIds() {
TreePath[] paths = myTree.getSelectionPaths();
if (paths == null) return null;
Set<Object> actions = new HashSet<Object>();
for (TreePath path : paths) {
Object node = path.getLastPathComponent();
if (node instanceof DefaultMutableTreeNode) {
DefaultMutableTreeNode defNode = (DefaultMutableTreeNode)node;
Object userObject = defNode.getUserObject();
actions.add(userObject);
}
}
return actions;
}
protected String getDimensionServiceKey() {
return "#com.intellij.ide.ui.customization.CustomizableActionsPanel.FindAvailableActionsDialog";
}
}
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,864
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Ріміні (, ) — місто та муніципалітет в Італії на узбережжі Адріатичного моря, у регіоні Емілія-Романья, столиця провінції Ріміні.
Ріміні розташоване на відстані близько 240 км на північ від Рима, 110 км на південний схід від Болоньї.
Населення — (2014).
Щорічний фестиваль відбувається 14 жовтня. Покровитель — San Gaudenzio di Rimini.
Українці в Ріміні
Після Другої світової війни в окрузі Ріміні були розташовані табори полонених, куди прибули також вояки 1 Української Дивізії Української-Національної Армії (див. Дивізія «Галичина») кількістю понад 10 000, що здали зброю британцям в Австрії у травні 1945. Від початку червня до 16 жовтня дивізійники перебували у таборі ч. 5-ц б. Беллярії, згодом до половини червня 1947 у таборі ч. 1 б. Мірамаре. Життя у таборі 5-ц б. Беллярії мало познаки тимчасовости; тоді внаслідок дій радянської репатріаційної місії зголосилося до повернення на рідні землі бл. 900 осіб (8,5 %). У таборі ч. 1 були кращі життєві умови, внутрішня організація табору мала військову структуру. Табір очолював комендант (генерал-хорунжий М. Краг, підполковник Р. Долинський, майор С. Яськевич) з штабом, з 7 відділами та 8 референтурами. Таборяни були поділені на 8 полків і частини окремого призначення. У Рімінському таборі, станом на 1946 р., було 9310 осіб, у тому ч. 288 старшин, 822 підстаршин та 8200 вояків.
Релігійну опіку таборяни одержували від 4 укр.-католицьких і одного православного душпастирів-капелянів Дивізії. Охорона здоров'я була під опікою 16 лікарів і досить великої кількості санітарів. Шкільництво охоплювало ряд шкіл: початкову (для малограмотних), гімназію, торг., учительську семінарію, сер. техн., однорічну рільничу, лісництва, драматичну, дяківську та ремісничу, з численними відділами; діяли різні курси і народний університет.
Культурно-освітня праця охоплювала: таборовий театр, хори «Бурлака», «Славута» і православний церковний хор, мистецьку спілку «Веселка», три оркестри, бандуристів і ревелерсів. Виходили: щоденник «Життя в Таборі», тижневик «Батьківщина», двотижневик «Юнацький Зрив», гумористично-сатиричний двотижневик «Оса», літературно-мистецькі й популярно-наукові журнали: «Наш Шлях» і «Гроно» та неперіодичні вид. Було організоване спортивне життя (серед ін. масове змагання за Відзнаку Фізичної Вправности), і діяли різні об'єднання (правників, учителів, інженерів, ремісничий цех з підвідділами, студентська громада, літературно-мистецький клуб, товариство філателістів, пластові гуртки, читальні «Просвіти»).
Особливу опіку над табором у Ріміні розгорнув єпископ І. Бучко як голова римського Українського Допомогового Комітету.
Демографія
Уродженці
Джованні Авреліо Авгурелло (1456—1524) — італійський поет
Федеріко Фелліні (1920—1993) — італійський кінорежисер та сценарист.
Сусідні муніципалітети
Белларія-Іджеа-Марина
Коріано
Риччоне
Сан-Мауро-Пасколі
Сантарканджело-ді-Романья
Серравалле
Веруккіо
Каттоліка
Галерея
Література
Див. також
Список муніципалітетів провінції Ріміні
Примітки
Міста Італії
Муніципалітети провінції Ріміні
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,137
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Home » Streaming Service » Netflix » Haunted season 3, episode 4 recap – "The Witch Behind the Wall"
Haunted season 3, episode 4 recap – "The Witch Behind the Wall"
Jonathon Wilson 1
Netflix, TV Recaps
"The Witch Behind the Wall" has an intriguing concept that it handles in, predictably, the absolute silliest way possible.
This recap of Haunted season 3, episode 4, "The Witch Behind the Wall", contains spoilers.
I'll tell you what's an eerie sound — sawing. Although I suppose any noise is a bit weird when it's created by a scruffy-looking old woman building a model of a house complete with little dummies, their faces made of cut-outs from photographs. Brandy claims to have been tormented by a woman who lived in her house behind an adjoining wall and judging by the state of her in that cold open, this might be a pretty believable episode.
Brandy's mother was single, worked two or three jobs to make ends meet, and thus was never at home. The kids apparently stayed away from the mean old lady's house, which makes sense, since when their ball goes over her fence she punctures it with a big pair of scissors, like that dude from the Clock Tower games. But old ladies do old lady things. When the family moves into her rambling old house, they just assume she died. It's a big, exciting new place with room for everyone, so you've got to look at the silver lining.
Predictably, the "feeling" and the "atmosphere" in the room upstairs changed, and we creepily see the old woman on the other side of the wall. Then we see her telekinetically choke Brandy using her little voodoo doll, so scratch what I said at the top about Haunted season 3, episode 4 possibly being believable. (Imagine how much scarier this would be if there was no supernaturalism here at all and old girl just lived there, in the wall).
The old woman continues to torment the family by playing with her facsimile of the crib, noosing the dolls, rattling the walls, and so on. It's not a bad horror concept, this, but the idea that it's true is just silly. I've said before and I'll say again that this show would work so much better if it was just short horror vignettes and not "real-life accounts". The re-enactment stuff is so fun in "The Witch Behind the Wall" that I kept groaning every time we returned to the present-day roundtable.
Eventually, the kids follow an open door to the old woman's creepy modeling room and find their faces stuck on the dolls. When she returns they have to hide under the table, inches from her dirty, sandalled trotters. When they make a noisy escape she blows smoke into the model of the house and lo and behold, her withered old face appears in a column of smoke, barking at them to get out. Apparently, the final straw for them to move was a minister coming to bless the house and claiming that the place was too evil for a cleanse. Bit lazy, on his part, right?
Locke and Key season 3, episode 7 recap – "Curtain"
Devil in Ohio episode 4 recap – "Rely-upon"
1 thought on "Haunted season 3, episode 4 recap – "The Witch Behind the Wall""
Brandy Q Lane
Brandy's full name is Brandy Q Lane and she writes about true paranormal stories on reddit under the name Crazyturtlemama. The real story behind this witch story is called "The Haunted House on Alexandria." Her, nor her siblings full names weren't used for some reason.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,071
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Q: Preload Solution while blazor wasm starts The first time a blazor wasm app is executed it downloads the necessary data into the client.
This may take a while.
So i want to show the user something(plain html or somthing quick) instead of this preloading page.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,574
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Woodstock 50: Florida readers remember the mud, the drugs, the magical music
By Larry Aydlette
Aug 9, 2019 at 9:41 AM Aug 9, 2019 at 9:45 AM
Florida men and women look back on the memories, drugs and magical music of Woodstock.
That older gentleman and lady standing next to you at the Publix checkout line?
Fifty years ago, they might have been rolling in the mud at Woodstock.
Related: What was happening in Palm Beach County 50 years ago?
Quite a few of our readers attended the iconic festival in August 1969 at Max Yasgur's farm in Bethel, N.Y.
There's an old phrase, often attributed to hippie prankster Wavy Gravy, that says, "If you can remember the '60s, you weren't there."
But our readers were there -- and they remember.
Here are some of their stories.
Related: A Lake Worth summer of '69 love story
'Sitting, eating and sleeping in mud'
It was to be a first adventure after high school graduation and before starting college. I was going to Woodstock with two friends, Barbara and Donna, equipped with a borrowed sleeping bag, rubberized vinyl rain coat and a bag of non-perishable foods (a spray can of cheese was one of the items I remember).
The three of us splurged, purchasing the three-part ticket for "three days of Peace and Music" and had gotten seats on a chartered school bus for the two-and-a-half-hour ride to the festival. I remember that the bus driver said he had big plans for a family-outing back on Long Island that afternoon. He would not make it.
Even before we got close to Bethel, traffic slowed to a crawl. The two-lane road became three lanes, then four as cars took over the oncoming lanes. The bus was barely inching along. Impatient passengers got off. Walking was faster. I doubt the bus driver got home until very late that evening.
We followed the crowd, excitement rising, as we heard Richie Havens singing off in the distance. I never saw a ticket booth. The concert field was like a crowded beach on a hot summer day. Finding a pretty good spot to lay out our stuff, just to the left of the stage, we settled in to watch the crew setting up and adjusting equipment.
Tired of sitting, I took a walk. That was a mistake. It was almost impossible to find my friends in the sea of people. Between sets, announcements paging lost people (and telling people to get off the sound towers) were repeated over and over.
After sunset, on cue from the stage, the crowd would ignite their lighters. Like thousands of fireflies, or a city seen from the air, the lights flickered all around us in the dark. It was quite beautiful.
Then, of course, it rained and rained and rained. We were sitting, eating and sleeping in mud. Saturday morning, we awoke to a sea of brown. It was impossible for us to stay another night. I had a drenched sleeping bag, several inches of water in the pockets of my raincoat and my blue leather shoes were filled with water.
Tired, soaked, and dirty, Donna hunkered down and started to cry. She couldn't go any further. But we had to keep going, there wasn't a choice. One mud sucking step after another, we headed to the road hoping to hitch a ride toward home. Our Woodstock exodus was a '60s hippie version of Gone With the Wind's scene of the wounded and dying in the aftermath of the battle of Atlanta.
There was a woman in the throes of some drug-induced hysteria tearing her clothes off. Locals offering sandwiches. Cars full of people, inching along, bumper to bumper, with additional people sitting on the cars' trunks. We were dumped off our ride when the car lurched forward. I tore up my knee, earning my Woodstock purple heart.
If only we had cell phones at that time. We found a pay phone and called home to find we were media stars. Our worried parents told us radio, TV and newspapers were full of accounts of the thousands attending Woodstock.
We returned home relatively intact. My sleeping bag was a total loss, and, for days after, my feet were stained blue by the dye from my sodden shoes.
-- Carol Erenrich, West Palm Beach
'Love and innocence was in the air'
I had a subscription to the Village Voice and saw an ad for Woodstock. My friend Bud and I decided to go. Bud took his Volkswagen Beetle four-speed without his mother's approval and we left Monroeville, PA, outside of Pittsburgh for Woodstock on Wednesday. We took turns driving. Bud was 17 and I was 16. This was the first time I ever drove a stick shift and we had problems.
The car did not make it and we were stuck in Hershey, PA and had to junk the car. We continued the journey and hitchhiked to New York and slept in a bug-infested field with our suitcases. The next morning we arrived in NYC, just in time to see the ticker-tape parade for the astronauts that had landed on the Moon.
Afterwards we continued our journey to Woodstock, with no real knowledge of where we were going. When we entered Connecticut, we knew that was the wrong direction and we got out of our ride and switched sides of the highway and got our first ride. It was a single guy with a Plymouth, push button car. He said he was going to Woodstock. His name was Elliott and he was attending Harvard.
After a while we started noticing the backup of traffic. The radio announcements were saying to turn back. We said no way and proceeded to find a parking spot about 3-5 miles outside of Max Yasgur's farm site. We did not have tickets. It was Friday in the late morning and Elliot parked his car on the side of the street where others were parking and camping.
We were going to part ways, when we asked him if we could leave our luggage in his car until Monday morning, as we felt like idiots going to a festival with our suitcases. He said OK. Trust, love and innocence was in the air.
When we walked in the direction of the music, we noticed that the fences were already torn down, and I believe I could hear the group Canned Heat. It looked amazing and people were offering speed and more. For us, the party was just getting started.
Yes, we did sleep in the mud. Yes, we did put in an announcement that turned up on the (Woodstock) album and, yes, we did find Elliott again.
-- Sheldon Klasfeld, Boca Raton
'My tent filled with folks I didn't even know'
In 1969, I was 23 years old and living in central New Jersey. I was the lead singer of the rock band, Men Working. We had just been signed to Richie Havens' new MGM recording company, Stormy Forest, (which) gave us complimentary tickets to an upcoming event: Woodstock. We had never received free tickets to anything and thought it might be fun.
Of course, we had no idea what Woodstock would become, literally and figuratively, to my generation and those that would follow.
I packed up a six-person tent and traveled with a couple of the guys in my group the Wednesday prior to the show's start that Friday. My wife could not get time off from work to go up early, so we planned for her to arrive with our manager's family that Friday evening. I had packed enough food to last until she would arrive with the rest.
We arrived to find a humongous swarm of people looking for parking, for a place to pitch their tent, for a space just to be. I could see right away that this was not going to be a well-controlled event.
We pitched our tent and checked out the staging area. Then we settled in for the night. Sometime during the night, it began to rain. And rain. My tent quickly filled with folks I didn't even know just trying to stay dry. I counted 14 people at one point!
I went back to my car and unlocked it to give shelter to more people. (Nothing was taken, and my car was left as clean as they found it). I tried to make a phone call to my wife, Maurene, to let her know that connecting would be extremely difficult. I was unable to make that call because the pay phones had filled up with coins and no calls could be made from them.
We enjoyed the concerts from the hillside, it was fun to watch Richie's opening and be part of the crowd. I think it was at that moment we began to realize just how special and unique this whole thing was. People were amazingly well behaved. But, like many others, we were out of food, and especially because of the rain, it was a mess.
I never did connect with my wife, my manager or his family. It was only after returning home that I found out that when they headed for Woodstock, they were turned away by the New York State Police some 25 miles away from the site. They decided to go to Grossinger's Resort, which had become a last-minute staging area for the performers. So, while I made my way through the mud, hungry, tired and wet, my wife and manager hung out with the performers and the press and enjoyed wonderful food at the resort.
By the way, music is as important to me today as it was back then. Despite having a career for 36 years with Lucent Technologies, my passion has always been music. Today I am the leader of Mystery Lane, a well-known band on the Treasure Coast.
-- Albert Miller, Stuart
'Sludge, excitement and my bloody eye-patch'
The summer of '69. We just graduated. The lucky seven of us were going to Woodstock. We needed a tent. I had a clever idea (I thought at the time). We'll take our 20-foot round pool cover and toss it over the picnic-table umbrella. Add some strings and wooden spikes and voila!, we had our coverage.
We piled into Tony's dad's boat-like Cadillac and left for Yasgur's farm around noon. About 2 miles out, traffic came to a crawl. Long-hairs abandoned their cars. Hippies hopped on and off the boat-car for a free ride. Doobies being passed as we sang, and laughed toward the promised land.
We finally arrive at midnight, it's raining, we hurriedly pitched our pool-tent. We crash.
Morning comes. We're gasping for breath from zero ventilation from the plastic 'dome-tent'.
We began to put away the umbrella tent, then, WHACK!, a single umbrella spoke snapped, zapped me in the eye. A direct hit. Dripping blood. It hurt. Expecting the worst, I ran to the side-view mirror. At first I thought my eye ball exploded into a mangled mess, but, luckily, my eyelid got in the way. I made an eye-patch. Thousands of freaks everywhere, sludge, excitement, bloody eye-patch. I fit right in.
To this day, 50 years later, I'll peek at the still-present eyelid scar as a reminder and say, "Hey, I went to Woodstock and made it back in one piece."
-- Al Manning, West Palm Beach
'As the sun set, the music world changed forever'
I had gotten a call from a friend in high school that there was going to be this concert, in New York, with a ton of bands. It would be for three days so we would have to camp out. We picked a spot about three hundred yards from the stage to set up our tent. It consisted of poles and plastic sheeting. We brought some food with us, canned ravioli. I guess our judgement was "clouded" when we thought of that as being our only food. Needless to say, the food was gone in a very short time and later that night, a guy on a bad acid trip, ran through our tent and destroyed it. And then it rained.
The next morning, we woke up and found ourselves surrounded by a mass of people. Where the hell did they all come from? There were tens of thousands! More important, where were the bathrooms? During that day we took turns "observing" the sights. Talk about an out of body experience! We couldn't wait for the music to begin. And as the sun began to set, the music world changed forever.
First up, Richie Havens. I still tear up, fifty years later, when I hear his music. That night we also heard Arlo Guthrie, and fell asleep as Joan Baez finished "We Shall Overcome."
Saturday, we heard Country Joe McDonald, Santana, John B. Sebastian, Canned Heat, Mountain, Grateful Dead, Creedence Clearwater Revival. And then it really got going around 2 in the morning. Janis Joplin strutted on to the stage. When she finished one of her greatest performances, up steps Sly and the Family Stone. Can I take anymore? Yup. The next up were the pinball wizards, The Who. At around 6 in the morning the music finally stopped. But not for long.
Since it was now Sunday, I figured to get some rest before the afternoon music started. In my sleep I heard the words, "Good morning people" shouted over the sound system. I'd heard that voice before. Could it be her or the dream of a seventeen year old? It was her: Grace Slick and Jefferson Airplane. At 8 in the morning! That saw us through brunch -- saltine crackers and water -- and set us up for the afternoon.
First up was Joe Cocker. Are you kidding me? Then Country Joe and the Fish. Ten Years After, The Band, Johnny Winter, Blood, Sweat, and Tears, Crosby, Stills, and Nash and Young. We finally crashed around 4 am.
When we woke on Monday morning, there were more abandoned sandals than people. We moved our position to within fifty yards of the stage. It was the size of a "normal" concert and there was one group I had to hear. At around 7:30, Sha Na Na stepped onto the stage. No, they were not who I stayed for, it was the guy that came on next to close out Woodstock. For me, the greatest guitar player of our time: Jimi Hendrix.
You will never hear the Star-Spangled Banner played like that ever again. My long weekend was complete.
-- Jeff Eagle, West Palm Beach
'Bartering our marijuana cigarettes for sandwiches'
It sounded like the perfect way to spend a hot August weekend when my friend, Alex, called to tell me he had bought two tickets for Woodstock.
Before we left late on a Friday afternoon, I booked a pedicure, using my favorite color, "Hot Coral," so my feet would look pretty in the grass. I dressed in my bell bottom jeans, tie-dyed T-shirt, sandals, leather belt, and lots of turquoise and beaded jewelry. And I packed similar outfits in my back pack, including a flowered head band for my long blonde hair.
When we finally reached the festival, no one was able to collect tickets as a sea of naked and half- naked, mud covered bodies tried to push forward. It felt exactly like being in a stalled, jam-packed, non-air-conditioned subway car, and both of us had the same immediate response - "Let's get out of here!"
But there was no way to turn back.
And then, quite suddenly, we began to breathe more slowly and marvel at the enormity of the crowd. Surprisingly, our first reactions of discomfort and annoyance started to morph into excitement and exhilaration, as we realized we were part of a huge happening in which Love and Acceptance were key! This discovery completely changed the way both of us adapted during the next three days.
Eventually, we found a small flat area where we could set up our tent. However, we hadn't been able to carry our plastic sheeting, pillows or other amenities on the long walk from our car and we had also become quite hungry. I had assumed there would be food stands readily available, but there were none in sight, and although Alex had brought along some perfectly rolled joints and plenty of water, neither of us had thought to bring anything to eat.
So I was deployed to forage for food. That first night I went from tent to tent bartering our wonderful marijuana cigarettes for other peoples' sandwiches. Everyone was laid back, loving, and rather stoned and we were well fed but quite boringly sober. This unfortunate state was rectified by the generosity of other tent occupants and the prevailing atmosphere of Peace and Love. And now all we needed was to reach the area where we could hear the Music!
During the next two days, it took considerable effort to finally get through the crowds to listen to a few of the many artists including Joan Baez and Janis Joplin. Then on the third day, our departure was halted because the cars we'd parked for miles along the road, had slid down an incline in the rain, and all of us had to wait for AAA to come to the rescue. That delay, however, turned out to be a bonus, enabling us to hear Jimi Hendrix who had arrived late due to traffic.
I don't remember much about the drive back to New York or the heat, the rain, the mud, the crowds, or the inconvenience. What I do remember is the Peace, the Love, and the Music – and how much the world needs all of them now.
Woodstock was a wonderful once in a lifetime experience that forever changed the way I look at and react to life. I am so grateful and so glad that I was there!
-- Kristi Witker-Coons, West Palm Beach
Florida Time archives: Get caught up on the stories you've missed
This story originally published to palmbeachpost.com, and was shared to other Florida newspapers in the GateHouse Media network via the Florida Wire. The Florida Wire, which runs across digital, print and video platforms, curates and distributes Florida-focused stories. For more Florida stories, visit here, and to support local media throughout the state of Florida, consider subscribing to your local paper.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,089
|
Tour de France Grand Départ survey
West Yorkshire Combined Authority
This dataset contains the individual responses to a survey conducted by West Yorkshire Combined Authority which sought to understand how people visiting the opening stages of the Tour de France in Yorkshire on the weekend of 5-6 July 2014 made their travel choices, including the sources of information they consulted, and what level of disruption they experienced on the roads and on public transport.
The feedback survey took the form of an online survey and was designed by the Institute for Transport Studies, University of Leeds. The survey went live on 24 July 2014 and was open for 19 days, closing on 11 August 2014 and attracting a total of 1,881 respondents.
Many of the survey questions were multiple choice and had several parts – the questions are described in 2 separate data files.
Please note, this is a one-off dataset and will not be updated.
transportTour de Francecyclingsporthack my routecyclebicycle
Survey questions (12.06 kB)
The survey questions
Answer options (6.15 kB)
List of answer options
Responses (1.47 MB)
Responses to survey questions
Request Access to Tour de France Grand Départ survey
WYCA
Tour de France Grand Départ Survey (c) West Yorkshire Combined Authority, 2014. This information is licensed under the terms of the Open Government Licence.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,918
|
\section{Introduction}
Collective decisions can produce conflict when no outcome is acceptable to all agents involved. Tensions may arise while options are being considered and discussed, but they may also appear or worsen once a decision has been made, particularly if preference strengths have not been accounted for or if some agents are disenfranchised entirely.
In many situations there is an implicit recourse for the frustrated and the downtrodden---they can leave. An agent, or a set of agents, may leave a community if they are sufficiently dissatisfied with the outcome of a decision, or a bundle of decisions.
When group cohesiveness is valued, it is sensible for decision makers to consider how each potential decision might effect camaraderie. This is commonly done informally through discussion when group sizes are small enough to deliberate, and through polling when communities are large. Here we propose a formal mechanism to take such considerations into account.
In typical voting scenarios, agents express preferences over alternatives and a single alternative is elected, which must then be universally accepted as the outcome by all agents in the community. In our setting, agents do not have to accept a particular alternative, and can fork instead. Consequently, the set of possible outcomes for every decision is not just the set of alternatives; rather, an outcome is a partition of the agents into one or more coalitions, with each coalition selecting an alternative that appeals to its members.
By designing voting rules that account for forking preferences, we empower aggrieved minorities to leverage the threat of leaving in order to pressure the majority into concessions. Importantly, we enable minorities to coordinate
with minimal overhead, by eliciting additional information during the voting process.
In our model, voters individually indicate conditions under which they prefer to leave,
and the mechanism then identifies a group that can benefit from forking, thereby
eliminating the need for campaigning or coordination among the disgruntled minority.
The value of stability is inherent in digital and analogue communities alike.
In most current blockchain protocols (i.e., proof-of-work and proof-of-stake) a fork can only be initiated by a majority or a powerful minority.
When a fork does occur, enacted by only a subset of the agents, all agents must determine what ``side" of the fork they want to be on.\footnote{In blockchain forks, currency owners may be able to run both protocols, but miners must choose how to allocate their computing resources, and programmers must choose how to allocate their personal time.}
In this sense, the forks we study are also relevant to version control systems such as Git, where anyone can initiate a fork. In our setting this is an edge case where a voter is willing to split from the group by themselves and others may follow, but we also allow groups of voters to fork together when none of them may be willing to do so on their own.
Since forks can be tumultuous, we wish to design democratic systems that enable communities to efficiently find states that are stable, in the sense that no further forks will occur. To this end, we put forward a formal model that approaches this challenge from the perspective of computational social choice.
\subsubsection*{Related Work}
Our paper is positioned at the interface of the research on blockchain technology and computational social choice.
\subsubsection*{Blockchain}
A blockchain is a replicated data structure designed to guarantee the integrity of data (e.g., monetary transactions in cryptocurrency applications such as Bitcoin~\cite{bitcoin}) and computations on them, combined with consensus protocols, which allow peers to agree on their content (e.g., who has been paid) and to ensure that no double spending of currency has occurred (see \cite{narayanan16handbook,wattenhofer17distributed} for recent overviews). By now, blockchain is an established technology, and cryptocurrency applications are attracting considerable attention~\cite{cryptocurrency,cryptoimpact}.
When the community of a specific blockchain protocol---such as the Bitcoin protocol---is not satisfied with it, it may break into several subcommunities. The community typically consists of developers, who build the software, miners, who operate the protocol, and users, e.g., account holders. When some of the developers of the current protocol decide to modify it, they create an alternative branch that obeys their new protocol, and if it attracts a sufficient number of miners and users, the result is a so-called \emph{hard fork}. Several such hard forks have been documented, including among the most influential cryptocurrencies. Bitcoin (cf.~\cite{hardfork}), despite its relatively short history, has already undergone seven hard forks.
At the moment, a key feature of these hard forks is that they happen through an informal social process, and, crucially, in ways that are completely exogenous to the protocol underpinning the blockchain. In blockchain terminology, they are said to happen `off-chain'. This points to a lack of governance in most current blockchain systems, for better or worse.
Against this backdrop there have been attempts at incorporating protocol amendment procedures within blockchain protocols themselves (so-called `on-chain' governance, cf. \cite{tezos}); more generally, the issue of governance is attracting increasing attention~\cite{beck2018governance,reijers2018now}.
We are not aware, however, of research that approaches forking as a social choice problem, and aims for an algorithmic solution. We lay the foundations for this approach here.
\subsubsection*{Computational social choice}
Social choice theory studies preference aggregation methods for various settings~\cite{brandt2016handbook}. Our social choice setting is closely related to assignment problems, as the result of a fork is that each agent is ``assigned'' to a community.
In this context, we mention works on judgement aggregation~\cite{Grossi_2014},~\cite[ch. 17]{brandt2016handbook} (which, formally, can capture assignments as well) and on partition aggregation~\cite{bulteaupartition}.
To the best of our knowledge, the specific social choice setting we consider is novel.
In a broader context, we mention work on stability in coalition formation games~\cite{cgt-book},~\cite[ch. 15]{brandt2016handbook} as well as the recent paper on deliberative majorities~\cite{deliberativemajorities}, which studies coalition formation in a general voting setting.
Our model is also related to the group activity selection problem with (increasing) ordinal preferences ({\sf o-GASP}) \cite{darmann2015group} (see also \cite{darmann2018group,darmann2018social}), where our notion of stability corresponds to core
stability in {\sf o-GASP}. However, due to our focus on strategy-proofness, and the fact that our setting does not admit a no-choice option (void activity in {\sf o-GASP}), most of the existing
results for {\sf o-GASP} are not directly relevant to our study, so we chose not to use
the {\sf o-GASP} formalism.
\subsubsection*{Version control}
Forking is not limited to the cryptocurrency setting; in particular, forking is relevant to projects of open-source code, in which a community jointly writes a piece of code and may experience different opinions regarding the code that is being written.
Indeed, there is some work on using social choice mechanisms (and, in particular liquid democracy) for revision control systems~\cite{liquidgit}. Others have been studying the phenomena of forking in open source projects; see, e.g., the work of Zhou et al.~\cite{zhou2020has}.
\subsubsection*{Outline and Contributions}
We describe a formal model of social choice for community forking, in which agents report their preferences over alternatives relative to possible forks. Throughout the paper, we focus on the setting where the number of available alternatives is two; towards the end of the paper, we discuss the challenges in extending our approach to three or more alternatives.
The paper is structured as follows.
Section~\ref{section:model} describes the formal model.
Section~\ref{section:stability} examines whether stable solutions always exist and whether they can be found efficiently in terms of computation and elicitation.
In Section~\ref{section:strategicbehavior}, we consider strategic agent behavior.
In Section~\ref{sec:three} we discuss the extension of our framework to more than two alternatives. We conclude in Section~\ref{sec:conclusions}.
The main contributions of our paper are as follows:
\begin{itemize}
\item
We devise a polynomial time algorithm (Algorithm~\ref{algorithm:general}) for our setting that finds a stable assignment for a very broad and natural domain restriction.
\item
We propose a modification of Algorithm~\ref{algorithm:general} (Algorithm~\ref{algorithm:generaliterative}) that allows for efficient iterative preference elicitation.
\item
We prove an impossibility result (Theorem~\ref{theorem:notstrategyproof}), showing that there is no algorithm that is strategyproof for profiles with more than one stable assignment.
\item
We establish that Algorithm~\ref{algorithm:general} is strategyproof for profiles that admit a unique stable solution; the impossibility result mentioned above then implies that it is optimal in that sense.
\end{itemize}
\section{Formal Model}\label{section:model}
\subsubsection*{Setting}
We have a set of agents $V = \{v_1, \ldots, v_n\}$. This community will vote on a set of two alternatives $\{A,B\}$ (say, cryptocurrency protocols or locations).
However, unlike in most voting scenarios, the agents are not all bound to accept the same winning alternative.
Agents have the ability to \emph{fork}, or forge a new community centered around the ``losing" alternative.
Ultimately, either all of the agents will remain in a single community or they will split into two communities that have accepted opposite alternatives.
\subsubsection*{Agent Preferences}
Agents care about what alternative their community adopts and how many people are in their community, but not the identities of the other agents in their community.
We can represent agent preferences as total orders over the possible tuples $(S,j)$, where $S \in \{A,B\}$ is the alternative to which they are assigned and $j \in [1,n]$ is the number of agents in their community (including themselves and $j-1$ other agents). We denote the set of all such tuples by $\mathcal{S}$
The preference relation $(A,j) \succ_i (B,k)$ means that agent $v_i$ would prefer to be in a community of size $j$ that accepts alternative $A$ rather than a community of size $k$ that accepts alternative $B$.
Agent preferences are \emph{monotonic} in the size of their community, so given a fixed alternative, they would always prefer to be in a larger community.
Formally, for each agent $v_i \in V$ we have $(S,j) \succ_i (S,k)$ for all $1 \leq k < j \leq n$ and $S \in \{A,B\}$.
We denote the set of all monotonic total orders over $\mathcal{S}$ by $\ensuremath{\mathcal{T}} \xspace$.
For $S\in\{A, B\}$, let $V_S^*$ denote the set of agents who prefer $(S,n)$ to $(S', n)$.
We will overload notation and use $v_i$ to represent both an agent $v_i \in V$ and their preference ordering $v_i \in \ensuremath{\mathcal{T}} \xspace$. In a similar fashion, $V$ is the set of agents and also the preference {\em profile}, or collection of the voters' total orders, $V \in \ensuremath{\mathcal{T}} \xspace^n$.
We refer to the pair $(V, \{A, B \})$, with $V \in \ensuremath{\mathcal{T}} \xspace^n$, as a {\em forking problem}.
\begin{example} \label{ex_three_agents}
Suppose we have $n = 3$ agents, $V = \{v_1, v_2, v_3\}$. Consider the preferences of a single agent $v_i$. By monotonicity, $(A, 3) \succ_i (A, 2) \succ_i (A, 1)$ and $(B, 3) \succ_i (B, 2) \succ_i (B, 1)$ must hold for all agents $v_i \in V$. However, these two orders may be interleaved differently for different agents.
\end{example}
\subsubsection*{Assignments}
We refer to the community that accepts alternative $A$ (resp., $B$) as community $A$ (resp., $B$).
An assignment $f: V \rightarrow \{A,B\}$ assigns agents to one of the two communities, and we denote by $f(v_i) \in \{A,B\}$ the community into which agent $v_i$ is placed.
The set $\ensuremath{\mathcal{F}} \xspace$ is the set of all $2^n$ possible assignments, or partitions, of the agents.
Voters' preferences over $\mathcal{S}$ induce preferences over assignments in $\ensuremath{\mathcal{F}} \xspace$: a voter
$v_i$ prefers an assignment $f$ to an assignment $g$ if
$(f(v_i), |f^{-1}(f(v_i)|) \succ_i (g(v_i), |g^{-1}(g(v_i)|)$.
Given a forking problem $(V, \{A, B \})$ a voting rule $R : \ensuremath{\mathcal{T}} \xspace^n \rightarrow \ensuremath{\mathcal{F}} \xspace$ selects an assignment $R(V) = f \in \ensuremath{\mathcal{F}} \xspace$.
We let $a = |f^{-1}(A)|$ be the size of community $A$, and similarly for $b = |f^{-1}(B)|$.
\section{Stability}\label{section:stability}
Our primary goal is to construct stable assignments.
An assignment is stable if no subset of agents has an incentive to move simultaneously to a new community.
\begin{definition}[$k$-Stability]
An assignment $f : V \to \{A, B\}$ is
{\em stable} if there is no assignment $f' : V \to \{A, B\}$
such that each voter $v_i$ with $f'(v_i)\neq f(v_i)$ prefers $f'$ over $f$.
\end{definition}
A voting rule $R$ is {\em stable} if it returns a stable assignment whenever one exists.
\begin{example}\label{example:stable}
Consider two agents, $V = \{v_1, v_2\}$, where $v_1: (A,2) \succ_1 (A,1) \succ_1 (B,2) \succ_1 (B,1)$ and $v_2: (B,2) \succ_2 (B,1) \succ_2 (A,2) \succ_2 (A,1)$. Each agent would prefer to be alone at their preferred alternative to being together with the other agent at their less preferred alternative.
Thus, the only stable assignment $f$ has $f(v_1) = A$ and $f(v_2) = B$.
\end{example}
\subsection{Finding Stable Solutions}
When preferences are monotonic, there must be at least one stable assignment, and it can be computed in polynomial time.
\begin{algorithm}[t]
\caption{General Stable Assignment Rule}
\label{algorithm:general}
\begin{algorithmic}
\STATE $V_A = V$, $V_B = \emptyset$, $a \gets |V_A|$, $b \gets |V_B|$
\WHILE{{\tt true}}
\STATE $k \gets \max\{j: 0\le j\le a, |\{v_i \in V : (B, b+j) \succ_i (A,a)\}| \geq j\}$
\IF{$k = 0$}
\STATE return $\{V_A,V_B\}$
\ELSE
\STATE Let $X = \{v_i \in V : (B, b+k) \succ_i (A,a)\}$
\STATE $V_B \gets V_B \cup X$, $V_A \gets V_A \setminus X$
\STATE $a \gets |V_A|$, $b \gets |V_B|$
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\begin{theorem} \label{theorem:generalstable}
There is a polynomial time assignment rule (Algorithm \ref{algorithm:general}) that finds a stable assignment for any monotonic profile.
\end{theorem}
\begin{proof}
Consider Algorithm~\ref{algorithm:general}. Let $a = |V_A|$ and $b = |V_B|$. Initially, we place all agents in $V_A$, so $a = n$ and $b = 0$.
If this assignment is not stable, then there exists a subset $X$ of some $k > 0$ agents that all prefer $(B,k)$ to $(A,n)$. We move all these agents to $V_B$.
Monotonicity implies that moving additional agents from $V_A$ to $V_B$ will never cause agents in $V_B$ to want to move back to $V_A$; thus, as long as we are not in a stable state, there must be a subset of agents at $V_A$ who would prefer to move together to $V_B$. As long as such a set of agents exists, we continue to move them over together.
This procedure halts in at most $n$ steps, and when it halts, the result must be stable, as there is no subset of agents who will move together.
A naive implementation of the algorithm loops at most $n$ times, and each computation of the set $X$ of agents to move takes $O(n^2)$ time.
\qed\end{proof}
\subsection{Elicitation}
Algorithm~1 does not use all of the information in agents' preferences. An iterative version of the algorithm can ask only for the information it needs.
Instead of assuming that the total orders of all agents are given explicitly in the input, we place all of the agents at $A$ and at each iteration we ask the $a$ remaining agents at $A$ for the minimum value $j$ such that if $j$ agents could be moved to $B$, they would now prefer the new community $(B, b+j)$ over their current community $(A,a)$.
Once an agent has been moved to $B$ there is no need to ask them for any more information.
In Algorithm~\ref{algorithm:generaliterative} we repeatedly query agents about the conditions under which they are willing to leave their current community.
Agents indicate their preferences with a single integer that says how many agents would have to move with them for them to prefer leaving over the status quo.
\begin{algorithm}[t]
\caption{General Stable Assignment Rule with Iterative Elicitation}
\label{algorithm:generaliterative}
\begin{algorithmic}
\STATE $V_A = V$, $V_B = \emptyset$, $a \gets |V_A|$, $b \gets |V_B|$
\WHILE{{\tt true}}
\STATE Ask each agent $v_i$ in $V_A$ for the smallest value ${j \in [0,a]}$ such that $(B, b+j) \succ_i (A,a)$
\STATE $k \gets \min\{j: j \in [0,a], |\{v_i \in V : (B, b+j) \succ_i (A,a)\}| \geq j\}$
\IF{$k = 0$}
\STATE return $\{V_A,V_B\}$
\ELSE
\STATE Let $X = \{v_i \in V : (B, b+k) \succ_i (A,a)\}$
\STATE $V_B \gets V_B \cup X$, $V_A \gets V_A \setminus X$
\STATE $a \gets |V_A|$, $b \gets |V_B|$
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}
Ideally, we would like to only have to query each agent a small number of times.
If agents' preferences are structured, it becomes possible to compute stable assignments with little information.
To capture this intuition, we introduce the concept of non-critically-interleaving preferences.
\begin{definition}[Non-critically-interleaving]
A preference is \emph{non-critically-interleaving} if it is monotonic and $(A,j) \succ (B,n) \succ (B, n-j) \succ (A, j-1)$ or $(B,j) \succ (A,n) \succ (A, n-j) \succ (B, j-1)$ for some $j \in [1,n]$.
A profile is \emph{non-critically-interleaving} if it contains only non-critically-interleaving preferences.
\end{definition}
When preferences are non-critically-interleaving, we only need to ask each agent whether they prefer $(A,n)$ or $(B,n)$, and the minimum value of $j$ such that they would rather be at their preferred alternative in a coalition of size $j$ than at the other alternative in a coalition of size $n$.
From this information the relevant part of the preference order of each agent can be inferred, and so Algorithm \ref{algorithm:generaliterative} will compute a stable assignment.
\subsection{Uniqueness}
While at least one stable assignment must exist for all monotonic profiles (Theorem~\ref{theorem:generalstable}), it is not necessarily unique.
\begin{example}\label{example:manystable}
Let $V = \{v_1, v_2, v_3, v_4\}$ be a set of four agents with preferences that contain the following prefixes, respectively:
\begin{itemize}
%
\item $v_1:(B,4) \succ_1 (B,3) \succ_1 (A,4) \succ_1 (B,2) \succ_1 (A,3) \succ_1 \cdots$
%
\item $v_2: (B,4) \succ_2 (B,3) \succ_2 (B,2) \succ_2 (A,4) \succ_2 (B,1) \succ_2 \cdots$
%
\item $v_3: (A,4) \succ_3 (A,3) \succ_3 (A,2) \succ_3 (B,4) \succ_3 (A,1) \succ_3 \cdots$
%
\item $v_4: (A,4) \succ_4 (A,3) \succ_4 (B,4) \succ_4 (A,2) \succ_4 (B,3) \succ_4 \cdots$
%
\end{itemize}
Regardless of how the remainder of the preference profile is filled, as long as monotonicity is maintained, there are at least three stable assignments: (1) all agents at $A$; (2) all agents at $B$; or (3) $v_1$ and $v_2$ at $B$ and $v_3$ and $v_4$ at $A$.
\end{example}
We would like to identify conditions under which a profile admits a unique stable assignment.
One extreme case is when preferences are non-interleaving.
\begin{definition}[Non-interleaving]
A preference order is \emph{non-interleaving} if it is monotonic and either $(A,1) \succ (B,n)$ or $(B,1) \succ (A,n)$.
A profile is \emph{non-interleaving} if it contains only non-interleaving preference orders.
\end{definition}
The profile in Example \ref{example:stable} is an instance of a non-interleaving profile.
If an agent's preference is non-interleaving, then their choice of community is independent of the other agents: they would rather be alone at their preferred alternative than with everyone else at the other alternative. Thus, their preference is described by a single bit of information: it suffices
to know whether they are in $V_A^*$ or in $V_B^*$.
Non-interleaving preferences can be viewed as a degenerate case of non-critically-interleaving preferences when $j = 1$.
\begin{observation}
When preferences are non-interleaving, there is a unique stable assignment.
\end{observation}
\begin{proof}
The only stable assignment assigns to all agents in $V_A^*$ to $A$ and all agents in $V_B^*$ to $B$.
Otherwise, an agent assigned to the opposite community will wish to move, even if on their own.
\qed\end{proof}
Non-interleaving preferences can be generalized to domains of preferences that guarantee unique stable assignments. Informally, we say that an agent is $k$-loyal to an alternative $S$ if
they prefer to be at $S$ with $k$ other agents to being at
the other alternative in a coalition of size $n$.
\begin{definition}[$k$-Loyalty]
An agent $v_i \in V$ is {\em $k$-loyal} to alternative $S$, $k\in [n]$, if $v_i\in V_S^*$ and $(S,k) \succ_i (S',n)$ for $S' \neq S$.
\end{definition}
When all agents are sufficiently loyal to their preferred alternatives, there is a unique stable assignment.
\begin{proposition} \label{theorem:allforced}
Suppose there exist some $k_1$, $k_2$ such that $k_1\leq |V_A^*|$, $k_2\leq |V_B^*|$,
every agent in $V_A^*$ is $k_1$-loyal, and every agent in $V_B^*$ is $k_2$-loyal.
Then there is a unique stable assignment.
\end{proposition}
\begin{proof}
By construction, any stable assignment must have all agents in $V_A^*$ assigned to $A$, because otherwise those assigned to $B$ would prefer to move together to $A$, forming a coalition of size $|V_A^*|\ge k_1$ at $A$.
Symmetrically, any stable assignment must have all agents in $V_B^*$ assigned to $B$, as otherwise those assigned to $A$ would prefer to move together to $B$, forming a coalition of size at least $|V_B^*|\ge k_2$ at $B$.
\qed\end{proof}
Proposition~\ref{theorem:allforced} holds because all agents must necessarily be assigned to their preferred alternative.
We now examine a sub-domain of non-critically-interleaving preferences in which there is always a unique stable assignment, but not all agents are necessarily assigned to their preferred alternative.
\begin{proposition} \label{theorem:tworounds}
Suppose agents' preferences are non-critically-interleaving. Let
\begin{align*}
V_A' &= \argmax\limits_{U \subseteq V} |\{v_i \in U : (A, |U|) \succ (B,n)\}|\ , \\
V_B' &= \argmax\limits_{U \subseteq V} |\{v_i \in U : (B, |U|) \succ (A,n)\}|\ .
\end{align*}
If none of the agents in $V_A^*\setminus V_A'$ are $(n - |V_B'|)$-loyal and none of the agents in $V_B^*\setminus V_B'$ are $(n - |V_A'|)$-loyal, then there is a unique stable assignment.
\end{proposition}
\begin{proof}
Note first that, by monotonicity, the set $\argmax$ in the definition of $V_A'$ and $V_B'$ is a singleton, so $V_A'$ and $V_B'$ are well-defined.
As with Proposition~\ref{theorem:allforced}, for any assignment to be stable it must assign all agents in $V_A'$ to $A$ and those in $V_B'$ to $B$.
For the remaining agents, they must necessarily be assigned to the opposite alternative, because there cannot be enough agents at their most preferred alternative for them to stay there.
\qed\end{proof}
The maximal class of profiles for which there is a unique stable solution is still more general than those we describe above.
We can use Algorithm~\ref{algorithm:general} to characterize the set of profiles that admit a unique stable assignment.
Let $R_A$ be the assignment rule given by Algorithm~\ref{algorithm:general}, and let $R_B$ be the complementary assignment rule that starts with all agents at $B$ and iteratively moves them to $A$ in the same manner.
\begin{theorem}
Algorithm~\ref{algorithm:general} $(R_A)$ and the reverse assignment rule $(R_B)$ return the same assignment if and only if the profile admits a unique stable assignment.
\end{theorem}
\begin{proof}
If the stable assignment is unique, then both $R_A$ and $R_B$ must return this assignment.
We now show that if $R_A$ and $R_B$ return the same stable assignment, then it must be the unique stable assignment.
Let $V_A^1$ and $V_B^1$ be the communities in the stable assignment $f_1 = R_A(V)$.
Let $V_A^2$ and $V_B^2$ be the communities according to a different stable assignment $f_2$.
By monotonicity and the properties of Algorithm~1
we have $V_A^1\subsetneq V_A^2$.
Consider the set of agents $V_A^2 \setminus V_A^1$, and in particular, the agent(s) in this set that were the first to be moved to~$B$ by $R_A$. At the beginning of the iteration in which they were moved, the number of agents at $A$ had to be at least $|V_A^2|$ (before moving). This contradicts the claim that $f_2$ is stable, as there are agents in $V_A^2$ preferring to move together to $B$.
\qed\end{proof}
\subsection{Cohesiveness}
Not all stable assignments may be equally attractive. In real life, forking comes at a cost, such as the need to replicate infrastructure and to carve out or abandon intellectual property or goodwill, as well as the social and emotional cost of separation.
The cost of forking within our framework is implicit in the preferences of the agents. In line with the monotonicity of preferences, it is natural that the community may want to avoid forks when possible.
When it is desirable to avoid forking, we prefer stable assignments that place all agents at the same alternative over those that fork. We call these non-forking assignments.
A profile is said to be \emph{cohesive} if it admits at least one non-forking stable assignment;
otherwise, we say that a profile is {\em forking}. The profile in Example~\ref{example:manystable} is cohesive, although it also permits a forked stable assignment.
The assignment in Example \ref{example:stable} is stable, but the input profile is forking, because no stable assignment exists with all agents in one community.
For a profile to be cohesive there must be at least one alternative (w.l.o.g, $A$) such that for all $j \leq n$, there are fewer than $j$ agents who prefer $(B,j)$ over $(A,n)$.
The following example shows a cohesive profile with no stable forked assignments.
\begin{example}\label{example:nonforkingstable}
Let $V = \{v_1, v_2\}$, where $(A,2) \succ_1 (B,2) \succ_1 (A,1) \succ_1 (B,1)$ and $(B,2) \succ_2 (A,2) \succ_2 (B,1) \succ_2 (A,1)$. Each agent would prefer to be together with the other agent at their less preferred alternative rather than alone at their preferred alternative. This profile is cohesive, and only admits stable assignments that are non-forking.
\end{example}
\section{Strategyproofness}\label{section:strategicbehavior}
So far we have considered the existence and the possibility of efficiently computing stable assignments when agents report their preferences truthfully. Another important question is whether there exist strategyproof stable assignment rules, i.e., rules that output stable assignments and do not incentivize the agents to misreport their true preferences.
\begin{definition}[Strategyproofness]
A rule $R$ is {\em strategyproof over domain $D \subseteq \ensuremath{\mathcal{T}} \xspace^n$} if for all profiles $V \in D$ and assignments $f = R(V)$, there is no agent $v_i \in V$ that can unilaterally change her preference order to $v_i'$, creating a new profile $V'$ such that she prefers $f'(v_i)$ over $f(v_i)$,
where $f' = R(V')$.
\end{definition}
Similarly, a rule is $k$-strategyproof in our setting if no subset of agents of size $k$ can simultaneously report false preferences to yield an assignment they all prefer. Naturally, $k$-strategyproofness implies $(k-1)$-strategyproofness.
\begin{definition}[$k$-Strategyproofness]
A rule $R$ is {\em $k$-strategyproof over domain $D \subseteq \ensuremath{\mathcal{T}} \xspace^n$} if for all profiles $V \in D$ and assignments $f = R(V)$, there is no subset of agents $U \subseteq V$ of size $|U| \leq k$ that can simultaneously change their preferences, creating a new profile $V'$ such that each agent $v_i \in U$ prefers $f'(v_i)$ over $f(v_i)$, where $f' = R(V')$.
\end{definition}
For the domain of all monotonic profiles, no strategyproof stable rules exist.
This can be seen from Example \ref{example:nonforkingstable}.
In this example there are two stable assignments, one creating $(A,2)$ and the other creating $(B,2)$. Suppose the agents are both assigned to $B$. If $v_1$ were to change their reported preferences to $(A,2) \succ_1 (A,1) \succ_1 (B,2) \succ_1 (B,1)$, then $(A,2)$ would become the only stable assignment for the new profile, which $v_1$ clearly prefers over $(B,2)$. This profile is symmetric, so if the agents were to be assigned to $(A,2)$ (by some tie-breaking mechanism) then $v_2$ has the opportunity to be strategic.
In fact, no strategyproof stable assignment rule can exist for any domain containing a profile that admits two or more stable assignments.
\begin{theorem} \label{theorem:notstrategyproof}
No assignment rule can be strategyproof over a domain that includes a profile that admits more than one stable assignment.
\end{theorem}
\begin{proof}
Suppose profile $V$ permits at least two stable assignments, and our assignment rule~$R$ picks one of them, $f_1 = R(V)$.
Let $f_2$ be the closest stable assignment to $f_1$, in the sense that there is no other assignment $f_3$ such that $|f_3^{-1}(A)|$ is between $|f_1^{-1}(A)|$ and $|f_2^{-1}(A)|$, and consequently no $|f_3^{-1}(B)|$ between $|f_1^{-1}(B)|$ and $|f_2^{-1}(B)|$ (indeed, $f^{-1}(A) + f^{-1}(B) = n$ for any $f$).
For brevity, let $V_A^1 = f_1^{-1}(A)$, $V_A^2 = f_2^{-1}(A)$,
$V_B^1 = f_1^{-1}(B)$, $V_B^2 = f_2^{-1}(B)$.
Assume that $|V_A^1| > |V_A^2|$ and $|V_B^2| > |V_B^1|$; later we will see that this assumption is without loss of generality.
From monotonicity, we know that $V_A^2 \subset V_A^1$ and $V_B^1 \subset V_B^2$.
Consider an agent $v_i \in V_A^1$ with $(B, |V_B^2|) \succ_i (A, |V_A^1|) \succ_i (B, |V_B^1|+1)$. At least one such agent must exist because otherwise $f_2$ could not be stable, as all the agents in $V_A^1 \cap V_B^2$ would prefer to move together to $A$ rather than stay in $(B, |V_B^2|)$.
If $v_i$ commits to $B$ by falsely reporting that they prefer $(B, 1)$ to $(A,n)$, then they must be assigned to~$B$ (as otherwise the assignment would not be stable). By construction, however, no intermediate stable state can exist between $f_1$ and $f_2$, so agents will prefer to move from $V_A^1$ to $B$ until it is of size $|V_B^2|$, which is what $v_i$ preferred.
By symmetry, if our rule $R$ had picked assignment $f_2$ instead of $f_1$ then we would have the same result; thus, our assumption that $|V_A^1| > |V_A^2|$ and $|V_B^2| > |V_B^1|$ is indeed
without loss of generality.
\qed\end{proof}
The above theorem means that the domain consisting of those profiles that admit a unique stable assignment is the maximal domain for which a stable assignment rule can be strategyproof.
Next, we show that, whenever a profile admits a unique stable assignment, Algorithm~\ref{algorithm:general} is strategyproof.
\begin{lemma} \label{lemma:strategyproof}
Algorithm~\ref{algorithm:general} is strategyproof over the domain of all profiles that admit a unique stable assignment.
\end{lemma}
\begin{proof}
Consider a run of Algorithm~1 in which it assigns some agent $v_i$ to~$B$ and outputs the assignment $f$.
First, observe that $v_i$ cannot misreport their preferences so that they would end up in a larger community at $B$ that they prefer. If the agents assigned to $A$ do not move at any iteration, then $v_i$ moving at an earlier to later iteration has no effect on them. And, due to monotonicity, $v_i$ staying at $A$ would not further entice anyone to move to $B$.
Second, we want to show that agent $v_i$ cannot manipulate the outcome so that it will be assigned to a larger community at $A$ that it prefers.
Suppose that at the beginning of the iteration when $v_i$ is moved from $A$ to $B$, the size of the community at $A$ is $a$.
The agents who were moved from $A$ to $B$ at an iteration before the iteration at which $v_i$ is moved will be assigned to $B$ regardless of what $v_i$ reports.
In general, agents moved to $B$ together at one iteration must end up at $B$ regardless of the preferences of those moved to $B$ at later iterations and those who stay at $A$.
As a consequence, $v_i$ can never induce an assignment with a community at $A$ of size greater than~$a$.
Since $v_i$ was moved at the iteration when the size of $A$ was $a$, it must be to a community $B$ that they prefer over $(A,a)$.
Therefore no agent $v_i \in B$ can deviate profitably from their true preferences.
It remains to show that no agent assigned to $A$ can benefit from strategic behavior. By symmetry, if there is a unique stable assignment, then if Algorithm~\ref{algorithm:general} starts with all agents at $A$ and moves them in batches to $B$, or starts at $B$ and moves them in batches to $A$, then it must return the same assignment.
We can therefore use the same argument as above for agents assigned to $A$ according to the algorithm that initializes $A$ and $B$ in the opposite way.
\qed\end{proof}
The result extends to $n$-strategyproofness, or group-strategyproofness, since, if we consider any coalition of agents assigned to $B$ by Algorithm~\ref{algorithm:general}, and look at only those who were moved first (in the same iteration as one another, but before everyone else in the coalition who was moved), then they have no incentive to misreport their preferences for the same reason as the agent in the proof of Lemma~\ref{lemma:strategyproof}.
By combining this with Theorem~\ref{theorem:notstrategyproof}, we arrive at our main result.
\begin{theorem}
Algorithm~\ref{algorithm:general} is group-strategyproof over the domain of all profiles that admit a unique stable assignment.
\end{theorem}
Just how common are profiles that admit strategyproof stable assignment rules?
One specific domain restriction that implies group-strategyproofness of Algorithm~\ref{algorithm:general} is the domain of non-interleaving profiles (recall that, for this domain restriction, placing all agents at their preferred alternative is stable).
Non-interleaving preferences are indeed very extreme, in that agents ignore each other completely.
However, if we relax this extreme constraint on preferences even the slightest bit, we can lose strategyproofness.
\begin{definition}[Minimally-interleaving]
A preference order is \emph{minimally-interleaving} if it is monotonic and $(A,2) \succ (B,n) \succ (A,1) \succ (B, n-1)$ or $(B,2) \succ (A,n) \succ (B,1) \succ (A, n-1)$.
A profile is a \emph{minimally-interleaving} if it contains only non-interleaving and minimally-interleaving preferences.
\end{definition}
Minimally-interleaving preferences can be interpreted as just barely extending non-interleaving preferences to allow that agents may be willing to go with their less preferred alternative if they would otherwise be alone with their more preferred alternative. Note that the minimally-interleaving domain is still a severely restricted domain. In particular, it allows each agent to specify only one of four possible orders.
However, it turns out that if we allow just minimal interleaving, then there is no assignment rule that is both stable and strategyproof.
\begin{observation}\label{theorem:1Inter_strategy}
There is no assignment rule that is both stable and strategyproof for all minimally-interleaving preference profiles.
\end{observation}
\begin{proof}
Let $R$ be a stable assignment rule and consider the profile from Example~\ref{example:nonforkingstable}:
$v_1: (A,2) \succ_1 (B,2) \succ_1 (A,1) \succ_1 (B,1)$;
$v_2: (B,2) \succ_2 (A,2) \succ_2 (B,1) \succ_2 (A,1)$.
Note that, indeed, this profile is minimally-interleaving.
Observe that the only stable assignments are the two that place both agents in the same community. Since there are two stable assignments, Theorem~\ref{theorem:notstrategyproof} implies that $R$ cannot be strategyproof for this profile.
For illustrative purposes, consider the case in which $v_1$ votes strategically by reporting $v_1': (A,2) \succ_1 (A,1) \succ_1 (B,2) \succ_1 (B,1)$.
The only stable assignment places both agents at $A$, creating $(A,2)$, which $v_1$ prefers to $(B,2)$.
So if $R$ placed both agents at $(B,2)$, it cannot be strategyproof.
\qed\end{proof}
We say that a profile is {\em $k$-interleaving} if it may contain preference orders in which $(S',n) \succ \dots \succ (S,n) \succ (S',k)$, but not in which $(S',n) \succ \dots \succ (S,n) \succ (S',k+1)$, where $S' \neq S$.
Hence, non-interleaving preferences are equivalent to $0$-interleaving; minimally-interleaving preferences are the same as $1$-interleaving; and $n$-interleaving is the domain of all monotonic preferences. Naturally, the set of all $k$-interleaving preferences encompasses all $(k-1)$-interleaving preferences, so no strategyproof stable assignment rule can exist for $k \geq 1$.
While non-interleaving is a sufficient condition for strategyproofness, the next example demonstrates that it is not a necessary condition.
\begin{example}\label{example:mixed}
%
Consider two agents, $V = \{v_1, v_2\}$, where $v_1: (A,2) \succ_1 (A,1) \succ_1 (B,2) \succ_1 (B,1)$ and $v_2: (B,2) \succ_2 (A,2) \succ_2 (B,1) \succ_2 (A,1)$.
%
Any stable assignment must have $v_1$ at $A$, independent of the preferences of $v_2$. Agent $v_2$ would prefer to be at $A$ with $v_1$ to being alone at $B$, so the only stable assignment has both agents at~$A$, and neither agent has an incentive to be strategic.
\end{example}
Notice that our interleaving conditions apply to the preference order of each agent individually. Example \ref{example:mixed} suggests that we should instead consider restrictions on the profile as a whole.
%
While we know that the necessary and sufficient conditions for stable strategyproof assignment rules to exist is that there be a unique stable assignment, characterizing the profiles for which this occurs is an interesting challenge.
\section{Forking with More than Two Alternatives} \label{sec:three}
So far, we focused on the case of two alternatives. We conclude the paper with two observations about the general case:
(1) stable assignments are no longer guaranteed to exist;
(2) deciding whether an assignment is stable is NP-complete.
\begin{proposition}
There exist monotonic profiles with no stable assignment.
\end{proposition}
\begin{proof}
Consider the problem with three agents $V = \{v_1, v_2, v_3\}$, three alternatives $\{A,B,C\}$, and a following profile:
\begin{itemize}[topsep=0pt]
\item
$v_1: \cdots \succ_1 (B,2) \succ_1 (A,2) \succ_1 (A,1) \succ_1 (B,1) \succ_1 (C,3) \succ_1 \cdots$
\item
$v_2: \cdots \succ_2 (C,2) \succ_2 (B,2) \succ_2 (B,1) \succ_2 (C,1) \succ_2 (A,3) \succ_2 \cdots$
\item
$v_3: \cdots \succ_3 (A,2) \succ_3 (C,2) \succ_3 (C,1) \succ_3 (A,1) \succ_3 (B,3) \succ_3 \cdots$
\end{itemize}
Assume for contradiction that this profile admits a stable assignment $f$.
As $v_2$ prefers $(B,1)$ to $(A,3)$, the assignment $f$ cannot assign $v_2$ to $A$.
By considering $v_1$ and $v_3$, we conclude that $|f^{-1}(s)|<3$
for every $S\in\{A, B, C\}$.
Suppose $f(v_1)=A$, $f(v_3)=A$. Then we have $f(v_2)=B$
because $(B,1) \succ_2 (C,1)$. But in this case, $v_1$ would prefer to move to $B$ since $(B,2) \succ_1 (A,2)$.
By the same reasoning $|f^{-1}(S)|\neq 2$ for each $S\in\{A, B, C\}$.
The only remaining option is to have one voter at each alternative. Let $v$
be the voter at $A$. If $v=v_1$, then $v_1$ prefers to move to $B$
and if $v=v_3$ then $v_3$ would prefer to move to $C$. Finally, if $v=v_2$
then $v_1$ prefers to join $v_2$ at $A$.
\qed\end{proof}
From a complexity-theoretic perspective, it is then natural to ask if there are efficient algorithms
for (a) checking whether a given assignment is stable, and (b) deciding if a given profile
admits a stable assignment. It turns out that, while the answer to the first question is `yes',
the answer to the second question is likely to be `no'.
\begin{proposition}\label{prop:in-np}
We can decide in polynomial time whether a given assignment for a forking problem is stable.
\end{proposition}
\begin{proof}
Note first that if an assignment $f$ is not stable, then this can be witnessed
by a deviation in which all deviating agents move to the same alternative (say, $A$).
Indeed, the agents who deviate from $f$ by moving to $A$ would find this move beneficial
even if other agents did not move (in particular, due to monotonicity, they benefit
from other agents not moving away from $A$). Thus, to decide if a given assignment $f$ is stable,
it suffices to consider deviations that can be described by a pair $(S, n_S)$,
where $S$ is an alternative and $n_S>f^{-1}(S)$. For each such pair, we need
to check if there are $n_S-f^{-1}(S)$ agents who are currently not assigned to $S$,
but prefer $(S, n_S)$ to their current circumstances.
\qed\end{proof}
\begin{proposition}\label{prop:nphard}
Deciding whether a forking problem admits a stable assignment is NP-complete.
\end{proposition}
\begin{proof}[Sketch]
By Proposition~\ref{prop:in-np}, our problem is in NP.
For hardness we adapt the reduction argument of Darmann~\cite[Th. 3]{darmann2015group}, establishing NP-hardness for the core stability problem in {\sf o-GASP} with increasing preferences.
That construction makes use of so-called void activities, which are available in {\sf o-GASP}, but not in forking problems. In the profile constructed for~\cite[Th. 3]{darmann2015group}, the occurrence of void activities in each agent's preference needs to be replaced by $(S^*, 1)$, where $S^*$ denotes the top alternative in the agent's preference.
\qed\end{proof}
Our hardness reduction produces an instance where the number of alternatives
is linear in the number of voters. The complexity of finding a stable assignment for a fixed number of
alternatives (e.g., $m=3$) remains open.
\section{Conclusions and Future Work} \label{sec:conclusions}
In the real world, communities sometimes fracture, or fork. This can generally be seen as a consequence of the decisions the community has made.
If agents associate freely, with the ever-present option of leaving, then we can account for this possibility within collective decision making procedures. This enables minorities to threaten a fork in protest against the tyranny of the majority while giving the majority an opportunity to concede to prevent a fork.
Such a forking process also facilitates the emergence of new communities, as it may be easier to sprout a community from an existing one rather than to build one from scratch.
We have shown that, while it may not be difficult to find stable partitions of a set of agents, constructing strategyproof rules is only possible in restricted domains. While the necessary and sufficient conditions for strategyproofness remain an interesting open question, we have identified a range of circumstances that are sufficient for strategyproofness.
Lastly, we have shown that efficient preference elicitation is possible and desirable.
The social choice setting we considered is, to the best of our knowledge, novel and our work has only made the first steps towards its analysis. Several directions for future research present themselves:
(1) first, as mentioned above, settling the question about the domain restrictions that are necessary and sufficient for the existence of stable and strategy-proof assignment rules is a priority;
(2) second, natural generalizations of the setting we propose will be worth investigating---e.g., settings with several alternatives (similarly to how, e.g., large miners can be present in several forks), or settings in which the identities of the agents matter (as agents may wish to fork with other specific agents);
(3) third, studying mechanisms for the converse problem, in which several communities could merge into a new one;
and
(4) fourth, enabling a majority to remove troublesome or faulty agents (e.g. Sybils) by forcing a fork.
\section*{Acknowledgements}
Ehud Shapiro is the Incumbent of The Harry Weinrebe Professorial Chair of Computer
Science and Biology. We thank the generous support of the Braginsky Center for the
Interface between Science and the Humanities. Nimrod Talmon was supported by the
Israel Science Foundation (ISF; Grant No. 630/19).
Ben Abramowitz was supported in part by NSF award CCF-1527497.
\bibliographystyle{plain}
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Martha J. Farah
Martha J. Farah is Walter H. Annenberg Professor of Natural Sciences in the Department of Psychology at the University of Pennsylvania, where she directs the Center for Neuroscience & Society. She has worked on many topics within neuroscience, including vision, prefrontal function, emotion, and development. In her three decades of research she has witnessed the advent of functional neuroimaging, the burgeoning of cognitive neuroscience, and its expansion into the study of social and affective processes. She is now focusing her attention on the ethical, legal and social implications of these developments.
An Introduction with Readings
Martha J. Farah 2010
Explores the ethical, legal, and societal issues arising from brain imaging, psychopharmacology, and other new developments in neuroscience.
Neuroscience increasingly allows us to explain, predict, and even control aspects of human behavior. The ethical issues that arise from these developments extend beyond the boundaries of conventional bioethics into philosophy of mind, psychology, theology, public policy, and the law. This broader set of concerns is the subject matter of neuroethics. In this book, leading neuroscientist Martha Farah introduces the reader to the key issues of neuroethics, placing them in scientific and cultural context and presenting a carefully chosen set of essays, articles, and excerpts from longer works that explore specific problems in neuroethics from the perspectives of a diverse set of authors. Included are writings by such leading scientists, philosophers, and legal scholars as Carl Elliot, Joshua Greene, Steven Hyman, Peter Kramer, and Elizabeth Phelps. Topics include the ethical dilemmas of cognitive enhancement; issues of personality, memory and identity; the ability of brain imaging to both persuade and reveal; the legal implications of neuroscience; and the many ways in which neuroscience challenges our conception of what it means to be a person.
Neuroethics is an essential guide to the most intellectually challenging and socially significant issues at the interface of neuroscience and society. Farah's clear writing and well-chosen readings will be appreciated by scientist and humanist alike, and the inclusion of questions for discussion in each section makes the book suitable for classroom use.
Contributors Zenab Amin, Ofek Bar-Ilan, Richard G. Boire, Philip Campbell, Turhan Canli, Jonathan Cohen, Robert Cook-Degan, Lawrence H. Diller, Carl Elliott, Martha J. Farah, Rod Flower, Kenneth R. Foster, Howard Gardner, Michael Gazzaniga, Jeremy R. Gray, Henry Greely, Joshua Greene, John Harris, Andrea S. Heberlein, Steven E. Hyman, Judy Iles, Eric Kandel, Ronald C. Kessler, Patricia King, Adam J. Kolber, Peter D. Kramer, Daniel D. Langleben, Steven Laureys, Stephen J. Morse, Nancey Murphy, Eric Parens, Sidney Perkowitz, Elizabeth A. Phelps, President's Council on Bioethics, Eric Racine, Barbara Sahakian, Laura A. Thomas, Paul M. Thompson, Stacey A. Tovino, Paul Root Wolpe
Patient-Based Approaches to Cognitive Neuroscience, Second Edition
Martha J. Farah and Todd E. Feinberg 2005
A review of current developments in cognitive neuroscience that integrates data from behavioral, imaging, and genetic studies of patients with research-oriented cognitive theories.
The cognitive disorders that follow brain damage are an important source of insights into the neural bases of human thought. This second edition of the widely acclaimed Patient-Based Approaches to Cognitive Neuroscience offers state-of-the-art reviews of the patient-based approach to central issues in cognitive neuroscience by leaders in the field.
The second edition has been thoroughly updated, with new coverage of methods from imaging to transcranial magnetic stimulation to genetics and topics from plasticity to executive function to mathematical thought. Part I, on the history and methods of cognitive neuroscience and behavioral neurology, includes two new chapters on imaging, one covering the basics of fMRI in normal humans and the other on the functional imaging of brain-damaged patients, as well as updated chapters on electrophysiological methods and computer modeling. Part II, on perception and attention, includes new chapters on visual perception and spatial cognition as well as attention, visual, tactile, and auditory recognition, music perception, body concept, and delusions. Part III, on language, covers many aspects of language processing in adults and children, including reading. Part IV discusses memory and prefrontal function, including semantic memory and executive functions. Part V covers dementias and developmental disorders, among them Alzheimer's and Parkinson's diseases, mental retardation, ADHD, and autism, and includes a chapter on the molecular genetics of cognitive disorders.
Visual Agnosia, Second Edition
The cognitive neuroscience of human vision draws on two kinds of evidence: functional imaging of normal subjects and the study of neurological patients with visual disorders. Martha Farah's landmark 1990 book Visual Agnosia presented the first comprehensive analysis of disorders of visual recognition within the framework of cognitive neuroscience, and remains the authoritative work on the subject. This long-awaited second edition provides a reorganized and updated review of the visual agnosias, incorporating the latest research on patients with insights from the functional neuroimaging literature. Visual agnosia refers to a multitude of different disorders and syndromes, fascinating in their own right and valuable for what they can tell us about normal human vision. Some patients cannot recognize faces but can still recognize other objects, while others retain only face recognition. Some see only one object at a time; others can see multiple objects but recognize only one at a time. Some do not consciously perceive the orientation of an object but nevertheless reach for it with perfected oriented grasp; others do not consciously recognize a face as familiar but nevertheless respond to it autonomically. Each disorder is illustrated with a clinical vignette, followed by a thorough review of the case report literature and a discussion of the theoretical implications of the disorder for cognitive neuroscience.
The second edition extends the range of disorders covered to include disorders of topographic recognition, and both general and selective disorders of semantic memory, as well as expanded coverage of face recognition impairments. Also included are a discussion of the complementary roles of imaging and patient-based research in cognitive neuroscience, and a final integrative chapter presenting the "big picture" of object recognition as illuminated by agnosia research.
Patient-Based Approaches to Cognitive Neuroscience
The cognitive disorders that follow brain damage are an important source of insight into the neural bases of human thought. Although cognitive neuroscience is sometimes equated with cognitive neuroimaging, the patient-based approach to cognitive neuroscience is responsible for most of what we now know about the brain systems underlying perception, attention, memory, language, and higher-order forms of thought including consciousness. This volume brings together state-of-the-art reviews of the patient-based approach to these and other central issues in cognitive neuroscience, written by leading authorities.
Part I covers the history, principles, and methods of patient-based neuroscience: lesion method, imaging, computational modeling, and anatomy. Part II covers perception and vision: sensory agnosias, disorders of body perception, attention and neglect, disorders of perception and awareness, and misidentification syndromes. Part III covers language: aphasia, language disorders in children, specific language impairments, developmental dyslexia, acquired reading disorders, and agraphia. Part IV covers memory: amnesia and semantic memory impairments. Part V covers higher cognitive functions: frontal lobes, callosal disconnection (split brain), skilled movement disorders, acalculia, dementia, delirium, and degenerative conditions including Alzheimer's disease, Parkinson's disease, and Huntington's disease.
ContributorsMichael P. Alexander, Russell M. Bauer, Kathleen Baynes, D. Frank Benson, H. Branch Coslett, Jeffrey L. Cummings, Tim Curran, Antonio R. Damasio, Hanna Damasio, Ennio De Renzi, Maureen Dennis, Mark D'Esposito, Martha J. Farah, Todd E. Feinberg, Michael S. Gazzaniga, Georg Goldenberg, Jordan Grafman, Kenneth M. Heilman, Diane M. Jacobs, Daniel I. Kaufer, Daniel Y. Kimberg, Maureen W. Lovett, Richard Mayeux, M.-Marsel Mesulam, Bruce L. Miller, Robert D. Nebes, Robert D. Rafal, Marcus E. Raichle, Timothy Rickard, David M. Roane, David J. Roeltgen, Leslie J. Gonzalez Rothi, Eleanor M. Saffran, Daniel L. Schacter, Karin Stromswold, Edward Valenstein, Robert T. Watson, Tricia Zawacki, Stuart Zola
Visual Agnosia
Disorders of Object Recognition and What They Tell Us about Normal Vision
Visual Agnosia is a comprehensive and up-to-date review of disorders of higher vision that relates these disorders to current conceptions of higher vision from cognitive science, illuminating both the neuropsychological disorders and the nature of normal visual object recognition.
Brain damage can lead to selective problems with visual perception, including visual agnosia the inability to recognize objects even though elementary visual functions remain unimpaired. Such disorders are relatively rare, yet they provide a window onto how the normal brain might accomplish the complex task of vision. Visual Agnosia reviews a century of case studies of higher-level visual deficits following brain damage, places them in the general context of current neuroscience, and draws relevant conclusions about the organization of normal visual processing. It is unique in drawing on research in cognitive psychology, computational vision, visual neurophysiology, and neuropsychology to interpret the agnosias and draw inferences from them about visual object recognition.
Following a historical account of agnosia research, Visual Agnosia offers a taxonomy of a wide range of agnosia syndromes, describing and interpreting the syndromes in terms of the latest theoretical models of visual processing and ultimately bringing them to bear as evidence on a variety of questions in the study of higher vision.
Visual Agnosia is included in the Issues in Biology of Language and Cognition series, edited by John Marshall.
An Invitation to Cognitive Science, Second Edition, Volume 2
Visual Cognition
Stephen M. Kosslyn and Daniel N. Osherson 1995
An Invitation to Cognitive Science provides a point of entry into the vast realm of cognitive science, offering selected examples of issues and theories from many of its subfields. All of the volumes in the second edition contain substantially revised and as well as entirely new chapters.
Rather than surveying theories and data in the manner characteristic of many introductory textbooks in the field, An Invitation to Cognitive Science employs a unique case study approach, presenting a focused research topic in some depth and relying on suggested readings to convey the breadth of views and results. Each chapter tells a coherent scientific story, whether developing themes and ideas or describing a particular model and exploring its implications.
The volumes are self contained and can be used individually in upper-level undergraduate and graduate courses ranging from introductory psychology, linguistics, cognitive science, and decision sciences, to social psychology, philosophy of mind, rationality, language, and vision science.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,273
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Q: Pandas, loop through each row of a column and append string to next column previous row I've done this task through Excel, but it has taken a hell of a long time to run across the 300,000 rows of data so I was hoping I could get it done a lot faster using python.
What I have is like the following data frame,
PartID Notes
0 1 Fiv
1 2 Six
2 3 Pot
3 4 Lep
4 Date is New
The issue is I have been given a file where PartID has strings in it so index 4 has 'Date is' which should be in the Notes section. In Excel, what I did was use the value function to change everything to values so that the numbers became values while text were changed to be empty. I then used a macro to say if the row below is empty, then append the data to the Notes column in the row above and the output would look like below
PartID Notes
0 1 Fiv
1 2 Six
2 3 Pot
3 4 Lep Date is
4 5 New
Is there a way to do the same thing in Python using pandas?
Thanks!
A: I think you need, pd.to_numeric and pd.Series.shift
a=df['PartID'].shift(-1).fillna('')
b=df['Notes']+a.loc[pd.to_numeric(a,errors='coerce').isnull()]
df['Notes']=b.combine_first(df['Notes'])
df['PartID']=np.arange(1,len(df['Notes'])+1)
print(df)
PartID Notes
1 Fiv
2 Six
3 Pot
4 Lep Date is
5 New
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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A two-part project published in Behance shows us Google's flexible, yet solid set of guidelines that strengthen the company's visual identity. Spearheaded by Art Director Christopher Bettig and Senior Graphic Designer Christopher Bettig together with fellow designers Alex Griendling, Jefferson Cheng, Yan Yan and Zachary Gibson, the summary focuses on iconography, divided into two parts which include product icons, logo mockups, user interface and illustrations.
The creation of this sweet guidelines which started in January 2012 is based on Google's design style credited to ECD Chris Wiggins along with designers Jesse Kaczmarek, Nicholas Jitkoff and Jonathan Lee. Check out the full set of guidelines here and here.
Share us your thoughts by leaving a message in the comment box below. Find You The Designer on Facebook, Twitter, Pinterest and Google Plus for more updates. Also, don't forget to subscribe to our blog for the latest design inspirations, stories and freebies. Speaking of freebies, check out our free print templates page for your print design needs. Stay awesome everyone!
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{
"redpajama_set_name": "RedPajamaC4"
}
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Home » Consequences » Invisible consequences » Hyperacusis
The denominator "sound hypersensitivity"
Many people think that overstimulation for sound and hyperacusis are the same. That is not true. A mixed picture of hyperacusis and overstimulation by sound may arise after brain injury. They may also occur separately. The two concepts are summarized under the unclear denominator "sound hypersensitivity".
Short explanation
With hyperacusis the dynamic range of hearing has decreased, ears cannot adjust to changes in loudness. Sounds are perceived as too loud, sharp or painful. The sound tolerance limit has dropped. Hyperacusis can be measured by a UCL test, uncomfortable loudness test. Above a limit, sound becomes intolerable.
Over-stimulation for sound means that people cannot separate sounds properly, background noise immediately imposes itself as foreground sound, but people are not necessarily bothered by volume. In the UCL test (uncomfortable loudness test), people with only sound stimulation do not appear to have a different score. The sound tolerance limit has not dropped.
Difference with overstimulation by sound
In case of over-stimulation for sound, a person cannot stand multiple sound sources. The brain cannot filter the sounds. This person cannot tolerate background noise or may experience all sound as a mixture of sounds. Following a conversation in noise becomes impossible. It is impossible to understand what is being said when someone else is talking in the room at the same time.
When this person is tired or very busy, sound is no longer tolerated. A stacking effect occurs between other forms of sensory and cognitive stimulation. This is a common invalidating and socially isolating consequence of brain injury. Rest and stopping background noise are the only remedies.
With hyperacusis, the sound hurts or sounds as intolerably loud, or sharply experienced, and the ear cannot adjust quickly to varying volume. Living with an inner volume control that is constantly on too loud.
Dynamic range lost and pain
With hyperacusis the ears have lost their so-called "dynamic range". This is the ability of our ears to quickly adapt to changing sound levels. The sound tolerance limit has dropped. Normally someone can still tolerate 80-90 dB well.
Sound can hurt and can be experienced as unbearable and too loud or sharp. It seems as if the volume control is constantly too loud. The hearing care professional can do a UCL test, an Uncomfortable Loudness test (UnComfortable Level).
A person with hyperacusis does not have to have a hearing loss. If someone with hyperacusis does have hearing loss, it is called 'recruitment'. Recruitment involves a damaged inner ear.
Characteristics hyperacusis
Feeling uncomfortable with sound,
People who suffer from this cover their ears or try to get away from the noise,
People who suffer from this can be angry, tense, sad or anxious about noise
People experience sound as painful, sharp or unbearably loud
Sound tolerance limit has dropped
Dynamic range of hearing has decreased, ears cannot adjust to changes in volume.
Research is still being done into causes, but not everything is known
Exposure, in the present and the past, to loud noise, for example in the workplace, but also to exposure to loud music
Head and brain injuries, including whiplash
Use of certain medicines
Squirting the ear which was done careless
Problems with the jaw joint (eg Cranio Mandibular Joint (CMD)
CPRS - Complex Regional Pain Syndrome
Heredity,
Lyme disease (tick bite)
Furthermore, it can be a symptom of:
Ménière's disease (attacks of vertigo, poor hearing and ringing in the ears)
Bell's Palsy (acute paralysis of the facial nerve)
William's Syndrome / Williams Beuren syndrome (delayed development, heart defect and behavioral problems)
Tay-Sachs disease (metabolic disease)
Fibromyalgia (non-joint-related chronic muscle connective tissue pain and pressure point pain / soft tissue rheumatism)
Autism (According to American research, 40% of people with autism spectrum disorder are hypersensitive to loud sounds)
Some cry babies seem to be sensitive to sounds, whether or not in combination with other factors
A third problem: tinnitus
Hyperacusis, literally translated from Greek for 'hearing too much', often occurs in combination with ringing in the ears.
Tinnitus, or ringing in the ears, is derived from the Latin word for 'ringing'. It is hearing damage and includes various sounds that cannot be heard outside the head: whistling sounds, buzzing, hissing, ringing or grumbling up to and including waterfall sounds. If the cilia in the ear are damaged, you will hear worse or they will pass on the wrong information, such as a beep, hum or a noise.
There are various causes for tinnitus.
Side effect of medicines
Exposure to noise
Please note that terms are confused when it comes to sound problems with brain injury. We hope to have highlighted on this page the differences between:
Sound sensitivity
Overstimulation for sound
Auditory processing problems
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,487
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#ifndef _EXT_H_
#define _EXT_H_
#define C74_MAX_SDK_VERSION 0x0517
#if C74_NO_CONST == 0
#define C74_CONST const
#else
#define C74_CONST
#endif
#include "ext_prefix.h" /* this header must always be first */
BEGIN_USING_C_LINKAGE
#include "ext_mess.h"
typedef short (*fretint)(void *, ...); /* kludge to cast to function returning int */
typedef short (*eachdomethod)(void *, ...);
typedef long (*exprmethod)(void *, ...);
typedef long (*fptr)(void *, ...);
typedef void *(*methodptr)(void *, ...);
#define clock_free freeobject
#define binbuf_free freeobject
#define wind_free freeobject
#define ASSIST_INLET 1
#define ASSIST_OUTLET 2
/**
This macro being defined means that getbytes and sysmem APIs for memory management are unified.
This is correct for Max 5, but should be commented out when compiling for old max targets.
@ingroup memory
*/
#define MM_UNIFIED
#include "ext_types.h"
#include "ext_maxtypes.h"
#include "ext_byteorder.h"
#include "ext_proto.h"
END_USING_C_LINKAGE
#endif /* _EXT_H_ */
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,771
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Q: Blocks retaining the class they just got passed to? We have a class that wraps NSURLConnection. It accepts a block that it calls back when it finishes loading. To give you an idea, see below. When you send a request, it saves the callback on the instance. Assume my class is named Request
// from Request.h
@property (nonatomic, copy) void(^callback)(Request*);
- (void) sendWithCallback:(void(^)(Request*))callback;
My code to use one looks something like this:
Request * request = [Request requestWithURL:url];
[request sendWithCallback:^(Request * request) {
// do some stuff
}]
My question is: what does the block do to the retain count of request? Does it copy/retain it? Notice that I didn't put __block in front of the definition.
I just changed something major in Request (switched from a synchronous NSURLConnection to async ASIHTTPRequest), and it started deallocing almost immediately after sending (causing delegate methods to call a dealloced object). With the sync NSURLConnection, that never happened.
I guess it makes sense that it would get dealloced with async, but how would I retain request appropriately? If I retained it right after I created it, I'd have to release it in the callback, but the callback doesn't get called if the request is cancelled, and would create a memory leak.
A:
what does the block do to the retain count of request? Does it copy/retain it?
No, it doesn't.
Request * request = [Request requestWithURL:url];
[request sendWithCallback:^(Request * request) {
// The request argument shadows the request local variable,
// this block doesn't retain the request instance.
}]
If the block doesn't have the request argument,
Request * request = [Request requestWithURL:url];
[request sendWithCallback:^{
// If you use the request local variable in this block,
// this block automatically retains the request instance.
}]
In this case, it would cause retain cycles (the request retains the block, the block retains the request).
*
*AsyncURLConnection.h
*AsyncURLConnection.m
Please take a look at my AsyncURLConnection class. NSURLConnection retains AsyncURLConnection instance, so you don't own AsyncURLConnection stuff by yourself.
How to use
[AsyncURLConnection request:url completeBlock:^(NSData *data) {
// Do success stuff
} errorBlock:^(NSError *error) {
// Do error stuff
}];
A: Blocks won't automatically retain or copy object arguments. It's the same semantics as passing object arguments to methods or functions — the block, method, or function should retain its arguments if there's potential for the current autorelease pool to drain before the block, method, or function has finished using the arguments.
Note the workflow in your scenario. This code:
Request * request = [Request requestWithURL:url];
[request sendWithCallback:^(Request * request) {
// do some stuff
}];
does not execute the block yet, nor does it pass any argument to the block. It creates a block and passes it as an argument to -sendWithCallback:. The block has a parameter called request of type Request * but the actual argument hasn't been passed yet.
At some point later in your code, and assuming you've stored the block in callback, that block will be called as:
callback(someRequest); // or callback(self);
or
self.callback(someRequest); // or self.callback(self);
or
aRequest.callback(someRequest); // or someRequest.callback(someRequest);
depending on who's responsible for calling it. At this point, whoever calls the callback should have a reference to a valid request (someRequest), and that request is the argument passed to the block.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,392
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package com.github.lite2073.emailvalidator.impl;
import com.github.lite2073.emailvalidator.EmailValidationResult;
import com.github.lite2073.emailvalidator.EmailValidationResult.State;
import com.github.lite2073.emailvalidator.EmailValidator;
public class CommonsValidatorEmailValidator implements EmailValidator {
private org.apache.commons.validator.routines.EmailValidator emailValidator = org.apache.commons.validator.routines.EmailValidator
.getInstance();
@Override
public EmailValidationResult validate(String email) {
if (email == null) {
return new EmailValidationResult(State.ERROR, "Null email");
}
boolean isValid = emailValidator.isValid(email);
return new EmailValidationResult(isValid ? State.OK : State.ERROR,
"Email failed to pass the hibernate validator");
}
@Override
public EmailValidationResult validate(String email, boolean checkDns) {
if (checkDns) {
throw new UnsupportedOperationException();
} else {
return validate(email);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 450
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It has been a good year for new fossil taxa on this blog. I'm pleased to present a fauna of Early Silurian crinoids from the Hilliste Formation (Rhuddanian) exposed on Hiiumaa Island, western Estonia. They are described in a paper that has just appeared in the Journal of Paleontology (early view) written by that master of Silurian crinoids, Bill Ausich of Ohio State University, and me, his apprentice.
Here's the simplified caption for the above composite image: Rhuddanian crinoids from western Estonia: (1) Bedding surface comprised primarily of crinoid columnals and pluricolumnals; (2) Radial circlet of an unrecognizable calceocrinid; (3) Basal circlet of an unrecognizable calceocrinid; (4) Holdfast A: Virgate radices anchored in coarse skeletal debris; (5) Holdfast D: Simple discoidal holdfast cemented to a bryozoan; (6, 7, 8) Hiiumaacrinus vinni n. gen. and n. sp.: 6, D-ray lateral view of calyx, 7, E-ray lateral view of calyx, 8, basal view of calyx; (9) Holdfast B: Dendritic holdfast in coarse skeletal debris; (10) Eomyelodactylus sp. columnal; (11) Holdfast C: Simple discoidal holdfast cemented to a tabulate coral; (12) Two examples of Holdfast E: Stoloniferous holdfasts cemented to a tabulate coral; (13) Protaxocrinus estoniensis n. sp. lateral view of partial crown, top of radial plate indicated by line.
No crinoid paper is complete without camera lucida drawings (scale bar for all figures is one mm): (1) Hiiumaacrinus vinni n. gen. and n. sp.; (2) Radial circlet of an unrecognizable calceocrinid; (3) Basal circlet of an unrecognizable calceocrinid; (4) Protaxocrinus estoniensis n. sp.
There are two new species and one new genus here. Hiiumaacrinus vinni is named first after the lovely Estonian island where the species is found, and then after our good friend and colleague Olev Vinn (above) at the University of Tartu. Olev first introduced me to the Ordovician and Silurian of Estonia, and then was an excellent field companion for Bill and me on our Estonian field trips.
A reminder where Hiiumaa Island is, and for that matter, the nation of Estonia.
Here is Hilliste Quarry on Hiiumaa Island. Still one of my favorite places to work. Very, very quiet.
Here is Bill Ausich in the quarry during our 2012 expedition. The pose is known among paleontologists as "the Walcott".
Here is one of the specimens collected by Bill in July of 2012. You may recognize this field scene as figure 12 in the top image of this post. These are two examples of crinoid holdfasts on a tabulate coral.
Please welcome Hiiumaacrinus vinni and Protaxocrinus estoniensis to the paleontological world!
Ausich, W.I. and Wilson, M.A. 2016. Llandovery (Early Silurian) crinoids from Hiiumaa Island, Estonia. Journal of Paleontology (early view).
Ausich, W.I., Wilson, M.A. and Vinn, O. 2012. Crinoids from the Silurian of Western Estonia (Phylum Echinodermata). Acta Palaeontologica Polonica 57: 613‒631.
Ausich, W.I., Wilson, M.A. and Vinn, O. 2015. Wenlock and Pridoli (Silurian) crinoids from Saaremaa, western Estonia (Phylum Echinodermata). Journal of Paleontology 89: 72‒81.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Explores issues, topics, and tasks related to personal, educational, and career choices. Addresses educational and career planning, personal characteristics and individual preferences, life and work values and interests, decision making, goal setting, and job/career search preparations. Credit/no-credit only.
Introduces a theoretical and experiential understanding of team development, consensus decision-making, sharing values, diversity, facilitation, conflict resolution, and dialogue. Theory is based on emerging views of teams and organizations as self-organizing systems.
Independent fieldwork in community agencies, apprenticeships, internships, as approved for College of Arts and Sciences credit. Faculty sponsor and internship supervisor are required. Credit/no-credit only. Offered: AWSpS.
Independent study conducted in organizations in our communities, complementing a designated course.
Using a research project from another course students refine writing skills and expand skills in accessing, identifying, and critically evaluating information. Must be concurrently enrolled in another IAS course. Credit/no-credit only.
Strengthens quantitative reasoning and develops problem solving and critical thinking skills through studying mathematics that can be used in everyday lives and careers.
Students nominated by faculty may participate on the editorial board of the Policy Journal. Board members are responsible for managing the content and production of the Policy Journal which is produced at least once per year, with the possibility of additional volumes if sufficient numbers of quality submissions are received. Credit/no-credit only.
Provides direct experience in teaching and facilitation. Students gain in-depth background on subject material along with training in teaching techniques and facilitation approaches. Credit/no-credit only.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 2,968
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2. 指代处命题(常见指示代词比如this, that, it, the+n等)。比如In December 2010 America's Federal Trade Commission (FTC) proposed adding a "do not track"(DNT) option to internet browsers, so that users could tell advertisers that they did not want to be followed. Microsoft's Internet Explorer and Apple's Safari both offer DNT; Google's Chrome is due to do so this year. In February the FTC and Digital Advertising Alliance(DAA) agreed that the industry would get cracking on responding to DNT requests. 这段话中最后一句的the industry属于阅读中比较特别的指代,此处如果命题的话,需要还原在前面句子找到他指代的对象,不难发现,应该是Microsoft's Internet Explorer,Apple's Safari,Google's Chrome这三个的总称internet browser developers.
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{
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
For many years, the study of the electric and magnetic
polarizabilities of hadronic systems has deserved a significant
interest towards the understanding of strong interactions.
The polarizabilities account for the distortion of a system (induced
dipole moments) in the presence of external quasistatic electric
and magnetic fields, so that they determine the lowest-order
response of the system's internal degrees of freedom to the
electromagnetic interactions \cite{review}. In particular, in the case of
a real Compton scattering (RCS) process, it is well known from low-energy
theorems that the corresponding spin-averaged forward scattering amplitude
is determined up to second order in the photon energy by the sum of both
the electric and magnetic polarizabilities, $\alpha+\beta$ \cite{low}.
One has indeed
\begin{equation}
f(\omega^2)= -\frac{Z^2 e^2}{4\pi M} + (\alpha+\beta)\, \omega^2 +
O (\omega^4) \,,
\label{fw}
\end{equation}
where $\omega$ is the energy of the (real) photon, and the first term on
the right hand side corresponds to the Thomson limit, which depends only
on global properties of the system: its mass and its electric charge.
The use of a forward dispersion relation allows to find a connection
between the sum $\alpha+\beta$ in (\ref{fw}) and the total
photo-absorption cross section for the system, $\sigma_{tot}(\omega)$.
This relation can be expressed in the form of a sum rule \cite{suma},
\begin{equation}
\alpha+\beta=\frac{1}{2\pi^2}\int^\infty_{\omega_{th}}
d\omega\;\frac{\sigma_{tot}(\omega)}{\omega^2} \,,
\end{equation}
where $\omega_{th}$ stands for the threshold inelastic excitation of the
system. Thus the sum of the structure-dependent polarizabilities can be
estimated by performing experimental measurements of $\sigma_{tot}
(\omega)$ up to large enough photon energies.
Unfortunately, this approach cannot be used to determine both
parameters $\alpha$ and $\beta$ independently, as it is clear from the fact
that only the sum $\alpha+\beta$ appears in (\ref{fw}). In order to get
separate sum rules from RCS processes, it is necessary to take into account
also nonforward amplitudes, which in general lead to the introduction of
additional difficulties. This is e.g.\ the case if one considers the
backward amplitude, which is related to the difference $\alpha-\beta$; it can
be seen that the corresponding dispersion relation needs
information not only on the s-channel photo-absorption cross sections, but
also on the t-channel two-photon processes \cite{bef}. As a consequence,
the analysis turns out to be more involved and includes some model dependence.
There is, however, an alternative possibility. As it was
shown in Ref.~\cite{bt} some time ago, it is possible to obtain a sum rule
for $\alpha$ that involves the longitudinal part of the {\em virtual} photon
cross section $\sigma_L(\omega,q^2)$ in the region of quasi-real photons.
In fact, although the longitudinal cross section itself vanishes in the
on-shell photon limit, it is found that the integral of the slope of
$\sigma_L(\omega,q^2)$ at $q^2=0$ is directly related to the electric
polarizability. The sum rule can be
derived by considering an unsubtracted dispersion relation for the
longitudinal forward virtual Compton amplitude $T_L(\omega,q^2)$, with the
only ingredients of gauge invariance, analiticity and unitarity. The result
in Ref.~\cite{bt} explicitly reads
\begin{equation}
\alpha - (\frac{e^2}{4\pi}) \frac{\mu^2}{4 M^3} =
\frac{1}{2\pi^2}\int^\infty_{\omega_{th}} d\omega\;
\lim_{q^2\rightarrow 0} \frac{\sigma_L(\omega,q^2)}{-q^2} \,,
\label{regla}
\end{equation}
where $\mu$ is the corresponding anomalous magnetic moment. We stress
that the presence or not of a subtraction for $T_L$ cannot be inferred
from the Thomson limit, therefore Eq.\ (\ref{regla})
is valid under the assumption that the integrand in the right hand side
has an adequate high-energy behaviour.
In this paper, we make use of the above longitudinal sum rule to derive
a theoretical expression for the electric polarizability of nuclear
systems. In order to deal with the nuclear photo-absorption cross section,
we proceed by performing a conventional nonrelativistic treatment of the
nuclear interactions, keeping only leading terms in powers of the inverse
mass of the system. In this regard, however, there is an essential
difference with respect to previous approaches: since our starting point
for the evaluation of the electric polarizability is relation
(\ref{regla}), the nonrelativistic approximations are introduced here only
{\em after} having taken into account the properties of gauge invariance,
analyticity and unitarity of the $S$ matrix, which have already been
invoked to obtain the sum rule. In contrast, these properties are not
exactly satisfied from the very beginning when the nonrelativistic limit
is taken directly on the real part of the Compton amplitude. This
last procedure has been followed e.g.\ in Refs.~\cite{otros,friar}.
We find that our treatment allows to improve the result for $\alpha$ obtained
in these previous analyses, leading to the presence of additional
contributions that depend on the nuclear interaction.
To evaluate the significance of these contributions, we
consider the simple case of a proton-neutron system interacting via a
separable Yamaguchi potential.
The calculation of the electric polarizability from the sum rule
(\ref{regla}) is performed in Section II. In Section III we present
the application to the Yamaguchi-like interacting system, while
Section IV contains our conclusions.
\section{Nuclear electric polarizability}
Starting from the sum rule in Eq.\ (\ref{regla}), we derive in this
section the leading contributions to the nuclear electric polarizability,
up to first order in the inverse nuclear mass $M_A^{-1}$. To this end, it
is convenient to write the integrand on the right hand side of
(\ref{regla}) in terms of the so-called longitudinal response, which is
measured in many nuclei by means of inelastic electron scattering.
This function is defined as
\begin{equation}
R_L(|{\bf q}|,\omega)=\sum_{n>0}|\langle n|\rho({\bf q})|0\rangle |^2
\,\delta(\omega-\omega_{rec}-E_n+E_0)\,,
\label{rl}
\end{equation}
where $\rho({\bf q})$ represents the charge operator, $\omega_{rec}=
|{\bf q}|^2/(2M_A)$ is the recoil energy of the nuclear system, and $E_0$
($|0\rangle$) and $E_n$ ($|n\rangle$) are the eigenvalues (eigenstates) of
the nuclear Hamiltonian corresponding to the ground and excited states,
respectively. For a spinless nucleus, it is easy to see that
Eq.\ (\ref{regla}) can be written as
\begin{equation}
\alpha={1\over 2\pi}\int_{\omega_{th}}^\infty\,d\omega\,
\left.{R_L(|{\bf q}|,\omega)
\over \omega \,|{\bf q}|^2}\right|_{|{\bf q}|\to\omega}\,.
\label{alfrl}
\end{equation}
From the analysis of electron scattering, it is seen that the longitudinal
response displays a strong collectivity at small momenta, whereas for
large momenta a single particle character is observed. In particular,
it has been shown \cite{ELO} that the proton-neutron dynamical correlations
generate a high-energy tail that can be relevant for the Coulomb sum rule.
We notice that, although this effect could also be significant in the
quasi-real photon limit, the high-energy contributions to the longitudinal
sum rule for $\alpha$ will be in general suppressed in view of the inverse
powers of $\omega$ entering the integrand in Eq.\ (\ref{alfrl}).
Let us work out the right hand side of (\ref{alfrl}). We begin by considering
the charge operator in (\ref{rl}), which at lowest relativistic order
is given by
\begin{equation}
\rho({\bf q})=e\sum_{j=1}^Z\exp{(i {\bf q}\cdot {\bf r}_j)}\,,
\end{equation}
where $Z$ is the number of protons and ${\bf r}_j$ are their coordinates
with respect to the nuclear center of mass. Then, performing an expansion in
powers of $|{\bf q}|$, one has
\begin{eqnarray}
\alpha&=&{e^2\over 2\pi}\int_{\omega_{th}}^\infty\,{d\omega\over \omega}
\,\lim_{|{\bf q}|\to\omega} \left\{
\delta\left(\omega-{|{\bf q}|^2\over 2M_A }-E_n+E_0\right) \,\sum_{n>0}
\,\Biggl[\,|\langle 0|C_1|n\rangle |^2 \right.\Biggr. \nonumber \\
& & \left.\left. + \, {|{\bf q}|^2\over 4}
\left(|\langle 0|C_2|n\rangle |^2-{4\over 3}
\langle 0|C_1|n\rangle\langle n|C_3|0\rangle \right) + O(|{\bf q}|^4)
\,\right]\,\right\}\,,
\label{siete}
\end{eqnarray}
where $C_n\equiv\sum_{i=1}^Z z_i^n$. In this expression, it is worth to
notice the presence of the electric dipole operator $C_1$ at leading order
in $|{\bf q}|$, while higher multipoles are also relevant to the
next-to-leading order (their importance will be seen later). Taking
explicitly the limit in the right hand side, and performing the integration
in $\omega$, the expression for $\alpha$ reads
\begin{eqnarray}
\alpha & = & {e^2\over 2\pi} \sum_{n>0}
\left(1-{\epsilon_n\over M_A}\right)^{-1}
\Biggl[\epsilon_n^{-1} |\langle 0|C_1|n\rangle |^2 \Biggr. \nonumber \\
& & \Biggl. + {\epsilon_n \over 4}
\left(|\langle 0|C_2|n\rangle |^2-{4\over 3}
\langle 0|C_1|n\rangle \langle n|C_3|0\rangle\right) +
O(\epsilon_n^2)\Biggr] \,,
\end{eqnarray}
where $\epsilon_n=M_A\left(1-\sqrt{1-2\,(E_n-E_0)/M_A}\,\right)$.
Finally, expanding up to first order in $(E_n-E_0)/M_A$, one gets
\begin{eqnarray}
\alpha & = & \frac{e^2}{2\pi} \sum_{n>0} \left[
\frac{|\langle 0|C_1|n\rangle|^2}{E_n-E_0} + \frac{1}{2M_A}
|\langle 0|C_1|n\rangle|^2 \right. \nonumber \\
& & + \frac{1}{4}(E_n-E_0) \left( |\langle 0|C_2|n\rangle|^2 -\frac{4}{3}
\langle 0|C_1|n\rangle \langle n|C_3|0\rangle \right) \nonumber \\
& & + \Biggl. \frac{3}{8M_A} (E_n-E_0)^2
\left(|\langle 0|C_2|n\rangle|^2 -\frac{4}{3}
\langle 0|C_1|n\rangle\langle n|C_3|0\rangle\right) +
O(M_A^{-2})\Biggr]\,.
\label{alf1}
\end{eqnarray}
Eq.\ (\ref{alf1}) gives $\alpha$ in terms of different energy-weighted
sums involving nuclear matrix elements of the $C_n$ operators $(n=1,2,3)$.
It is also possible to express this result by means of sum
rules, i.e.\ of average values of commutators and anticommutators of the
$C_n$ operators and the total nuclear Hamiltonian \cite{OTrep}. In order to
do this, let us first rewrite the nuclear polarizability in terms
of the moments $m_p$ of the longitudinal response function, defined by
\begin{equation}
m_p(|{\bf q}|)\equiv \int_{0^+}^\infty d\tilde\omega \;
(\tilde\omega)^p\, R_L(|{\bf q}|,\omega) \,,
\end{equation}
where $\tilde\omega=\omega-|{\bf q}|^2/(2 M_A)$. It can be easily verified
that the following relations between the derivatives of the moments and the
energy-weighted sums hold:
\begin{mathletters}
\label{mom}
\begin{eqnarray}
m'_{-1}(0)&\equiv&\left.\frac{dm_{-1}(|{\bf q}|)}{d|{\bf q}|^2}
\right|_{|{\bf q}|^2=0} =
\sum_{n> 0} \frac{|\langle 0|C_1|n\rangle|^2}{E_n-E_0} \\
m'_{0}(0)&\equiv&\left.\frac{dm_0(|{\bf q}|)}{d|{\bf q}|^2}
\right|_{|{\bf q}|^2=0} = \sum_n |\langle 0|C_1|n\rangle|^2
\label{momb} \\
m''_{1}(0)&\equiv &\left.\frac{d^2m_1(|{\bf q}|)}{(d|{\bf q}|^2)^2}
\right|_{|{\bf q}|^2=0}
= {1\over2}\sum_n (E_n-E_0) \left(|\langle 0|C_2|n\rangle|^2
-\frac{4}{3} \langle 0|C_1|n\rangle\langle n|C_3|0\rangle \right) \\
m''_{2}(0)&\equiv & \left.\frac{d^2m_2(|{\bf q}|)}{(d|{\bf q}|^2)^2}
\right|_{|{\bf q}|^2=0} =
{1\over 2}\sum_n (E_n-E_0)^2 \left( |\langle 0|C_2|n\rangle|^2
-\frac{4}{3} \langle 0|C_1|n\rangle\langle n|C_3|0\rangle \right)
\label{momd}
\end{eqnarray}
\end{mathletters}
\hspace{-.27cm}
thus $\alpha$ can be written as
\begin{equation}
\alpha=\left(\frac{e^2}{4\pi}\right) \left[ 2\, m'_{-1}(0) +
\frac{1}{M_A} m'_0(0) + m{''}_1(0)+\frac{3}{2M_A} m{''}_2(0) +
O\left(M_A^{-2}\right)\right]\,.
\label{alfyam}
\end{equation}
Now, by using the closure property, it can be shown that the derivatives
of the nonnegative moments in Eqs.\ (\ref{mom}) satisfy the following sum
rules\cite{OTrep}:
\begin{mathletters}
\label{momvac}
\begin{eqnarray}
m'_{0}(0)&=& {1\over 2}\langle 0|\{C_1,C_1\}|0\rangle \\
m''_{1}(0)&=& {1\over 4}\langle 0|\left[C_2,[H,C_2]\right]|0\rangle
-{1\over 3}\langle 0|\left[C_1,[H,C_3]\right]|0\rangle \label{mom1} \\
m''_{2}(0)&=&
{1\over 4}\langle 0|\{[C_2,H],[H,C_2]\}|0\rangle
-\frac{1}{3} \langle 0|\{[C_1,H],[H,C_3]\}|0\rangle \, . \label{mom2}
\end{eqnarray}
\end{mathletters}
\hspace{-.27cm}
Moreover, some parts of the commutators and anticommutators can be
explicitly evaluated; in particular, the double commutators containing
the kinetic part $T$ of the Hamiltonian ($H=T+V$) give
rise to terms proportional to the proton radius, leading to
\begin{mathletters}
\label{conm}
\begin{eqnarray}
\langle 0|\left[C_2,[H,C_2]\right]|0\rangle & = & {4\over m}
\left({1\over 3}\langle 0|\sum_{i=1}^Z {\bf r}_i^2|0\rangle - {1\over A}
\langle 0|C_1\,C_1|0\rangle \right)+\langle 0|\left[C_2,[V,C_2]
\right]|0\rangle
\label{conm1} \\
\langle0|\left[C_1,[H,C_3]\right]|0\rangle & = & {1\over m}\left(1-
{Z\over A}\right) \langle0|\sum_{i=1}^Z {\bf r}_i^2|0\rangle +
\langle 0|\left[C_1,[V,C_3]\right]|0\rangle\,,
\label{conm2}
\end{eqnarray}
\end{mathletters}
\hspace{-.27cm}
where we have approximated $M_A\simeq m\,A$, being $m$ the nucleon mass.
By making use of relations (\ref{mom}-\ref{conm}), the nuclear electric
polarizability can finally be written as
\begin{eqnarray}
\alpha&=& \left( \frac{e^2}{4\pi} \right) \left[ 2 \sum_{n > 0}
\frac{|\langle 0|C_1|n\rangle |^2}{E_n-E_0} + \frac{Z}{3 M_A} \langle 0|
\sum_{i=1}^Z {\bf r}_i^2|0\rangle +\frac{1}{4} \langle 0|[C_2,[V,C_2]]|0
\rangle\right. -\frac{1}{3} \langle 0|[C_1,[V,C_3]]|0\rangle \nonumber \\
& & \left. + \frac{3}{8 M_A}
\langle 0|\{[C_2,V],[V,C_2]\}|0\rangle -\frac{1}{2 M_A}
\langle 0|\{[C_1,V],[V,C_3]\}|0\rangle + O(M_A^{-2})
\right]\,.
\label{main}
\end{eqnarray}
Eq.\ (\ref{main}) represents the main result of this work. It is seen
that, beyond the leading dipole term, there is a contribution to $\alpha$
proportional to $\langle {\bf r}^2\rangle$, plus potential-dependent
terms of order $V$ and $V^2/M_A$. Let us remark that previous nonrelativistic
analyses \cite{otros,friar} lead only to the first two terms in
(\ref{main}). The significance of the potential-dependent contributions
obtained from our calculation will be illustrated in the next section,
where we consider the case of a simple solvable proton-neutron system.
The study of Eq.\ (\ref{main}) for realistic
nuclear potentials leading to exchange-current contributions will be the
subject of analysis in future work.
As a final comment, let us point out that the above result includes both
the contributions coming from collective nuclear effects and
those arising from the polarizability of the individual nucleons inside the
nucleus. In fact, it can be seen that the photoabsorption cross section
appears to be dominated by the nucleonic excitations for energies above
the pion mass. Below this threshold, there is a region
where the cross section still shows a volume character, which can be
described in terms of the interaction of the photon with quasi-deuteron
systems. Thus, for these energies, the contribution to the polarizability
can also be understood as a nucleonic one, being the nucleons modified by
the surrounding medium \cite{ERC}. Within our formulation, the identification
of the volume contributions is not trivial, since
the result in Eq.\ (\ref{main}) is obtained after performing an integration
over the whole spectrum. We recall, however, that the sum rule for $\alpha$
considered here weights the longitudinal response inversely with
$\omega$, hence the effects coming from the high energy region are
in general expected to be suppressed.
\section{Proton--neutron system with a Yamaguchi potential}
As an application of the above result, let us
evaluate the contributions to $\alpha$ for a simple case,
namely a proton-neutron system interacting via a separable potential
\begin{equation}
V=\lambda\, |g\rangle \langle g|\;,\hspace{1cm} \lambda < 0 \,.
\end{equation}
Denoting by $\psi_0({\bf p})$ and $\psi_{\bf k}({\bf p})$ the
bound state wave function and the scattering solution for this potential
respectively, the longitudinal structure function $R_L$ will be given by
\begin{equation}
R_L(|{\bf q}|,\omega)= \int d^3k\;
\delta(\omega-\frac{{\bf k}^2}{2M}-\epsilon_B)\,
\left|\int d^3p\;\psi_{\bf k}({\bf p})\,\psi_0({\bf p}-{\bf q})\right|^2 \,,
\label{sl}
\end{equation}
where $M$ is the total mass of the system and $\epsilon_B$ stands for the
binding energy corresponding to the $\psi_0$ state. We consider for
simplicity the particular case of a Yamaguchi potential \cite{yam},
\begin{equation}
\langle {\bf p}|g\rangle =\frac{\sqrt{\beta}}{\pi}\frac{1}{p^2+\beta^2}\;,
\hspace{1cm} \epsilon_B=\frac{\beta^2}{2M} \;,
\label{yam}
\end{equation}
which allows to perform the integrals in (\ref{sl}) analytically. After
some algebra, it is found \cite{rosen} that the longitudinal structure
function can be written in terms of the variables $x\equiv
\sqrt{\omega/\epsilon_B-1}$ and $y\equiv |{\bf q}|/\beta$ as
\begin{equation}
R_L(y,x)=\frac{16}{\pi \epsilon_B} \frac{x}{D^3} \left[
(1+x^2+y^2)^2+\frac{4}{3} x^2 y^2 - D\, F_0(y) \left( 1+\frac{y^2}{2}
-\frac{y^4}{2(1+x^2)}\right) \right] \,,
\label{slxy}
\end{equation}
where $D\equiv (1-x^2+y^2)^2+4x^2$, and $F_0(y)$ is the elastic form
factor,
\begin{equation}
F_0(y)=\frac{1}{Z} \langle\psi_0|\rho({\bf q})|\psi_0\rangle=
\left(1+\frac{y^2}{4}\right)^{-2} \,.
\end{equation}
In the case under consideration, the integrals corresponding to the moments
in (\ref{mom}) are convergent, so that the terms in the right
hand side of (\ref{alfyam}) can be evaluated using the expression for $R_L$
in (\ref{slxy}). Considering as before the terms in $\alpha$ up to
order $M^{-1}$, we have
\begin{equation}
\alpha=\left(\frac{e^2}{4\pi}\right) \left[ \frac{7}{12 M \epsilon_B^2}
+ \frac{1}{2M^2\epsilon_B} - \frac{3}{16 M^2 \epsilon_B} +
\frac{9}{8 M^3} \right] \,,
\label{alfin}
\end{equation}
where the four terms correspond to those in the right hand side of Eq.\
(\ref{alfyam}), respectively. Now, it is instructive to compare this
expression for $\alpha$ with the main result in Eq.\ (\ref{main}),
in order to identify the contribution of the potential-dependent terms for
this simple case. From (\ref{mom2}), we see that the $V^2$ part in
(\ref{main}) corresponds to the $m{''}_2(0)$
contribution, which yields the $M^{-3}$ term in (\ref{alfin}). This is
consistent with the nonrelativistic approximation, which assumes
implicitly that the nuclear interactions are relatively ``weak'', and both
the potential and kinetic energies should be considered to be order $(1/M)$
\cite{friar}. Furthermore, it is found that the remaining
potential contributions to (\ref{main}) correspond to the $m{''}_1(0)$
term. In fact, in the $Z=1$ case the kinetic contributions cancel out,
and we end up with
\begin{equation}
\frac{1}{4} \langle 0|[C_2,[V,C_2]]|0\rangle
-\frac{1}{3} \langle 0|[C_1,[V,C_3]]|0\rangle = m{''}_1(0)=
-\frac{3}{16M^2\epsilon_B}
\label{pot}
\end{equation}
Finally, we see that the contributions of $m'_{-1}(0)$ and $m'_0(0)$
(first two terms in (\ref{alfin})) correspond to the first two terms in
(\ref{main}), respectively. Then we conclude, at least for this model,
that {\em the lowest order potential-dependent contribution to $\alpha$
(i.e.\ that in (\ref{pot})) has the same order of magnitude as the
$\langle {\bf r}^2 \rangle$ term}. This is once again consistent with
treating the potential as order $1/M$. On the other hand, we recall that
previous nonrelativistic calculations for $\alpha$, such as those in
Refs.~\cite{otros,friar} led only to the $\langle{\bf r}^2\rangle$
correction. In this simple example we find that the additional
potential-dependent terms provided by the longitudinal sum rule approach
contribute significantly to the electric polarizability and therefore
should also be taken into account.
\section{Conclusions}
We have presented here a novel approach to analyze the electric
polarizability of nuclear systems. The distinctive feature of our analysis is
the use of the sum rule displayed in Eq.\ (\ref{regla}), which arises from
the assumption of an unsubtracted dispersion relation for the longitudinal
forward virtual Compton amplitude. This sum rule involves the virtual
photo-absorption cross section of the system in the region of quasi-real
photons.
We have calculated the leading terms contributing to the electric
polarizability for a spinless nucleus, up to first order in the inverse
nuclear mass. To do this, we have assumed that the photo-absorption
longitudinal cross section shows an adequate high-energy behaviour, so
that the treatment of the nucleus as a nonrelativistic system is consistent
with the use of the sum rule.
The main result of our calculation is shown in Eq.\ (\ref{main}). As expected,
it is found that the lowest order contribution to $\alpha$ is given by the
inverse energy-weighted sum of the strengths of inelastic dipole
excitations. Beyond this leading order, we obtain
a contribution proportional to $\langle{\bf r}^2\rangle$, plus
potential-dependent terms that appear
to be also significant. In particular, if the
kinetic and potential energies are both considered to be order $M^{-1}$
(which is consistent with the nonrelativistic approximation), the
terms proportional to $V$ in (\ref{main}) are shown to be of the same
order of magnitude as the $\langle{\bf r}^2\rangle$ one.
This is e.g.\ the case for a proton-neutron system interacting through
a separable Yamaguchi potential. As stated above, the presence of the
$\langle{\bf r}^2 \rangle$ contribution at the order $M^{-1}$ had been
derived many years ago by explicit nonrelativistic calculations of the
Compton amplitude for a bound system. The origin of the new
potential-dependent corrections in our analysis can be traced back to the
properties of gauge invariance, analiticity and unitarity of the $S$ matrix,
which are implicitly taken into account once the sum rule for $\alpha$ has
been invoked.
It is amazing that the requirement of completely general conditions on
the virtual Compton scattering amplitude automatically leads to the
presence of exchange current contributions to the nuclear polarizability.
The analysis of such contributions for realistic nuclear potentials
deserves a detailed study that will be the subject of future work.
\acknowledgements
G.\ O.\ would like to thank the Departament de F\'{\i}sica Te\`orica
of the University of Valencia for warm hospitality. D.\ G.\ D.\ has
been supported by a grant from the Commission
of the European Communities, under the TMR programme (Contract
N$^\circ$ ERBFMBICT961548). This work has been funded by CICYT,
Spain, under Grant AEN-96/1718.
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{"url":"https:\/\/www.lmfdb.org\/ModularForm\/GL2\/Q\/holomorphic\/3024\/2\/k\/c\/","text":"# Properties\n\n Label 3024.2.k.c Level 3024 Weight 2 Character orbit 3024.k Analytic conductor 24.147 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4\n\n# Related objects\n\n## Newspace parameters\n\n Level: $$N$$ $$=$$ $$3024 = 2^{4} \\cdot 3^{3} \\cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\\chi]$$ $$=$$ 3024.k (of order $$2$$, degree $$1$$, not minimal)\n\n## Newform invariants\n\n Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\\Q(\\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\\Z[a_1, \\ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 756) Sato-Tate group: $\\mathrm{U}(1)[D_{2}]$\n\n## $q$-expansion\n\nCoefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.\n\n $$f(q)$$ $$=$$ $$q + ( 2 + \\zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 + \\zeta_{6} ) q^{7} + ( 1 - 2 \\zeta_{6} ) q^{13} + ( -5 + 10 \\zeta_{6} ) q^{19} -5 q^{25} + ( -6 + 12 \\zeta_{6} ) q^{31} + q^{37} + 8 q^{43} + ( 3 + 5 \\zeta_{6} ) q^{49} + ( 5 - 10 \\zeta_{6} ) q^{61} -11 q^{67} + ( -1 + 2 \\zeta_{6} ) q^{73} + 13 q^{79} + ( 4 - 5 \\zeta_{6} ) q^{91} + ( -11 + 22 \\zeta_{6} ) q^{97} +O(q^{100})$$ $$\\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{7} + O(q^{10})$$ $$2q + 5q^{7} - 10q^{25} + 2q^{37} + 16q^{43} + 11q^{49} - 22q^{67} + 26q^{79} + 3q^{91} + O(q^{100})$$\n\n## Character values\n\nWe give the values of $$\\chi$$ on generators for $$\\left(\\mathbb{Z}\/3024\\mathbb{Z}\\right)^\\times$$.\n\n $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$\n\n## Embeddings\n\nFor each embedding $$\\iota_m$$ of the coefficient field, the values $$\\iota_m(a_n)$$ are shown below.\n\nFor more information on an embedded modular form you can click on its label.\n\nLabel $$\\iota_m(\\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$\n1889.1\n 0.5 \u2212 0.866025i 0.5 + 0.866025i\n0 0 0 0 0 2.50000 0.866025i 0 0 0\n1889.2 0 0 0 0 0 2.50000 + 0.866025i 0 0 0\n $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles\n\n## Inner twists\n\nChar Parity Ord Mult Type\n1.a even 1 1 trivial\n3.b odd 2 1 CM by $$\\Q(\\sqrt{-3})$$\n7.b odd 2 1 inner\n21.c even 2 1 inner\n\n## Twists\n\nBy twisting character orbit\nChar Parity Ord Mult Type Twist Min Dim\n1.a even 1 1 trivial 3024.2.k.c 2\n3.b odd 2 1 CM 3024.2.k.c 2\n4.b odd 2 1 756.2.f.b 2\n7.b odd 2 1 inner 3024.2.k.c 2\n12.b even 2 1 756.2.f.b 2\n21.c even 2 1 inner 3024.2.k.c 2\n28.d even 2 1 756.2.f.b 2\n36.f odd 6 1 2268.2.x.d 2\n36.f odd 6 1 2268.2.x.f 2\n36.h even 6 1 2268.2.x.d 2\n36.h even 6 1 2268.2.x.f 2\n84.h odd 2 1 756.2.f.b 2\n252.s odd 6 1 2268.2.x.d 2\n252.s odd 6 1 2268.2.x.f 2\n252.bi even 6 1 2268.2.x.d 2\n252.bi even 6 1 2268.2.x.f 2\n\nBy twisted newform orbit\nTwist Min Dim Char Parity Ord Mult Type\n756.2.f.b 2 4.b odd 2 1\n756.2.f.b 2 12.b even 2 1\n756.2.f.b 2 28.d even 2 1\n756.2.f.b 2 84.h odd 2 1\n2268.2.x.d 2 36.f odd 6 1\n2268.2.x.d 2 36.h even 6 1\n2268.2.x.d 2 252.s odd 6 1\n2268.2.x.d 2 252.bi even 6 1\n2268.2.x.f 2 36.f odd 6 1\n2268.2.x.f 2 36.h even 6 1\n2268.2.x.f 2 252.s odd 6 1\n2268.2.x.f 2 252.bi even 6 1\n3024.2.k.c 2 1.a even 1 1 trivial\n3024.2.k.c 2 3.b odd 2 1 CM\n3024.2.k.c 2 7.b odd 2 1 inner\n3024.2.k.c 2 21.c even 2 1 inner\n\n## Hecke kernels\n\nThis newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\\mathrm{new}}(3024, [\\chi])$$:\n\n $$T_{5}$$ $$T_{11}$$ $$T_{13}^{2} + 3$$\n\n## Hecke characteristic polynomials\n\n$p$ $F_p(T)$\n$2$ 1\n$3$ 1\n$5$ $$( 1 + 5 T^{2} )^{2}$$\n$7$ $$1 - 5 T + 7 T^{2}$$\n$11$ $$( 1 - 11 T^{2} )^{2}$$\n$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$\n$17$ $$( 1 + 17 T^{2} )^{2}$$\n$19$ $$( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} )$$\n$23$ $$( 1 - 23 T^{2} )^{2}$$\n$29$ $$( 1 - 29 T^{2} )^{2}$$\n$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$\n$37$ $$( 1 - T + 37 T^{2} )^{2}$$\n$41$ $$( 1 + 41 T^{2} )^{2}$$\n$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$\n$47$ $$( 1 + 47 T^{2} )^{2}$$\n$53$ $$( 1 - 53 T^{2} )^{2}$$\n$59$ $$( 1 + 59 T^{2} )^{2}$$\n$61$ $$( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$\n$67$ $$( 1 + 11 T + 67 T^{2} )^{2}$$\n$71$ $$( 1 - 71 T^{2} )^{2}$$\n$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$\n$79$ $$( 1 - 13 T + 79 T^{2} )^{2}$$\n$83$ $$( 1 + 83 T^{2} )^{2}$$\n$89$ $$( 1 + 89 T^{2} )^{2}$$\n$97$ $$( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} )$$","date":"2020-10-02 00:03:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.950946033000946, \"perplexity\": 7571.713713414825}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600402132335.99\/warc\/CC-MAIN-20201001210429-20201002000429-00008.warc.gz\"}"}
| null | null |
Photograph of Lt. Col. Franklin H. Clack, CSA, 1828-1864
Identifier: WLU-Coll-PP-0091
Washington and Lee University, University Library Special Collections and Archives
This collection consists of an 8 x 11 photograph on board of Lt. Col. Franklin H. Clack of Louisiana in his Confederate captains uniform. The image is a 1945 copy of a retouched original photograph of Clack. It is a 3/4 standing view. The photograph was given to Washington and Lee University in 1945 and was accompanied by a letter which details Clack's service records. The letter mentions that the original photograph was taken on March 30, 1862. Clack died as a result of a wound on April 24, 1864.
Clack, Franklin H., Lieutenant Colonel
United States -- Confederate States of America
WLU Coll PP. Print and Photographic Collections
Nicholson, Robert Livingston (Person)
Part of the Washington and Lee University, University Library Special Collections and Archives Repository
http://library.wlu.edu/specialcollections
Lexington VA 24450 USA
specialcollections@wlu.edu
Photograph of Lt. Col. Franklin H. Clack, CSA, 1828-1864, WLU-Coll-PP-0091. Washington and Lee University, University Library Special Collections and Archives. http://localhost:8081/repositories/5/resources/786 Accessed January 28, 2023.
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| 8,946
|
{"url":"https:\/\/brilliant.org\/problems\/what-is-the-universal-number\/","text":"# What is the universal number\n\nLevel pending\n\nThe universal number is a number that you get when break down any word...it has to break down any word.\n\n....So whats the universal number??\n\n\u00d7","date":"2016-10-23 22:16:21","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8725805878639221, \"perplexity\": 2811.3050927146724}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988719437.30\/warc\/CC-MAIN-20161020183839-00087-ip-10-171-6-4.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/60958\/how-do-i-format-compile-correctly","text":"# How do I format Compile[] correctly?\n\nMy Mathematica skill is still rusty, so kindly bear with me.\n\nI'm having problem formatting expression to compile the function correctly:\n\n$\\sum _{g=1}^G \\sum _{n=1}^{\\text{Ns}} -\\frac{e^{\\frac{\\text{Kg}}{P \\gamma _{g,n}}} \\text{Kg} \\beta _{g,n}}{\\text{Log}[2]} \\text{ExpIntegralEi}\\left[-\\frac{\\text{Kg}}{P \\gamma _{g,n}}\\right]$\n\nAll variables are known except $\\beta_{g,n}$ which is an optimization variable. Here's what I've done to express it usingcompile[] function.\n\ncostFxn =\nCompile[{{P, _Real}, {Ns, _Integer}, {gh, _Real}, {Kg, _Integer}, {G, _Integer},\n{\\beta_{g, n}, _Integer}},\n\nSum[-Exp[Kg\/(P gh[[g,n]])](Kg \\beta_{g,n})\/Log[2] ExpIntegalEi[-Kg\/(P gh[[g,n]])], {g,1,G},{n,1,Ns}]\n]\n\n\nwhen I try executing this snippet, I get part spec error.\n\n'Compile::part: \"Part specification gh[[1,1]] cannot be compiled since the argument is not a tensor of sufficient rank. Evaluation will use the uncompiled function.\"'\n\n\nI've been rummaging through the help file but not quite sure of how to correct this error.\n\nSince gh is a tensor, you need to say what rank it is, so replace {gh, _Real} with {gh, _Real, 2} to fix the error.\n\ncostFxn = Compile[\n{{P, _Real}, {Ns, _Integer}, {gh, _Real, 2},\n{Kg, _Integer}, {G, _Integer}, {betaGN, _Integer}},\nSum[\n-Exp[Kg\/(P gh[[g, n]])] (Kg * betaGN)\/\nLog[2] ExpIntegralEi[-Kg\/(P gh[[g, n]])],\n{g, 1, G},\n{n, 1, Ns}]]\n\n\nHowever, note that ExpIntegralEi is not a compilable function, so will leave a call to MainEvaluate inside the compiled function.\n\nThat said, comparing the compiled and uncompiled versions, we find:\n\nDo[costFxn[5, 2, {{5, 1}, {2, 3}}, 3, 2, 1], {1000}] \/\/ AbsoluteTiming\n(* 0.009018 seconds *)\n\nmyCostFxn[P_, Ns_, gh_, Kg_, G_, betaGN_] :=\nN[Sum[-Exp[Kg\/(P gh[[g, n]])] (Kg * betaGN)\/\nLog[2] ExpIntegralEi[-Kg\/(P gh[[g, n]])], {g, 1, G}, {n, 1, Ns}]]\n\nDo[myCostFxn[5, 2, {{5, 1}, {2, 3}}, 3, 2, 1], {1000}] \/\/ AbsoluteTiming\n(* 0.067047 seconds *)\n\ncostFxn[5, 2, {{5, 1}, {2, 3}}, 3, 2, 1]\n(* = 23.4363 *)\nmyCostFxn[5, 2, {{5, 1}, {2, 3}}, 3, 2, 1]\n(* = 23.4363 *)\n\n\nSo there is a benefit to compiling.\n\n## Edit\n\nIf you mean for $\\beta_{g,n}$ to be a variable that depends on the value of g and n, then you need to pass it as a tensor too.\n\ncostFxn2 = Compile[\n{{P, _Real}, {Ns, _Integer}, {gh, _Real, 2},\n{Kg, _Integer}, {G, _Integer}, {betaGN, _Integer, 2}},\nSum[\n-Exp[Kg\/(P gh[[g, n]])] (Kg * betaGN[[g,n]])\/\nLog[2] ExpIntegralEi[-Kg\/(P gh[[g, n]])],\n{g, 1, G},\n{n, 1, Ns}]]\n\ncostFxn2[5, 10, RandomReal[{1, 3}, {3, 10}], 4, 3, RandomInteger[{1, 3}, {3, 10}]]\n(* = 355.973 *)\n\n\u2022 Using artguments $P = 5; Ns = 10; G = 3; Kg = 4; gh = RandomReal[\\{1, 3\\}, {G, Ns}]; \\beta_{g,n}$ gives argument error: Argument <>...<> at position 3 should be a machine-size real number \u2013\u00a0Afloz Sep 30 '14 at 7:35\n\u2022 That's probably because RandomReal[1,3,G,Ns] is incorrect syntax. If you want a matrix that has dimensions G x Ns than try RandomReal[{1, 3}, {G, Ns}]. \u2013\u00a0dr.blochwave Sep 30 '14 at 7:37\n\u2022 Try using costFxn[5, 10, RandomReal[{1, 3}, {3, 10}], 4, 3, 1] and you'll see that works. \u2013\u00a0dr.blochwave Sep 30 '14 at 7:40\n\u2022 Oh, I see the problem. Actually, I did $RandomReal[\\{1,3\\},\\{G,Ns\\}]$. The copy pasting removed the formatting. Problem is, since I'll be using the resulting equation as cost function, the $\\beta_{g,n}$ should remain in the compiled function. If we replace last arg $1$ with $\\beta_{g,n}$, same error occurs. \u2013\u00a0Afloz Sep 30 '14 at 7:47\n\u2022 Ok, so there is a bigger reason for that - Compiled functions only accept Integers, Reals, Complex Numbers and Booleans. The documentation is here: reference.wolfram.com\/language\/ref\/Compile.html - look under the \"Details & Options\" part. What you want to do is symbolic, which won't work. \u2013\u00a0dr.blochwave Sep 30 '14 at 7:49","date":"2020-01-29 18:52:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2379167675971985, \"perplexity\": 7789.8925530731385}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251801423.98\/warc\/CC-MAIN-20200129164403-20200129193403-00438.warc.gz\"}"}
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Q: how securing a public nodejs api to only requests from the frontend I have a node frontend express server and a node api express server.
How can I best ensure that only requests that are made to the api are made from the frontend express server?
There is no user authentication so the user will not be sending a jwt with each request.
A: The easiest way would be to set the Content Security Policy using Helmet.js And you can easily add other security features using Helmet.
const helmet = require('helmet')
app.use(helmet.contentSecurityPolicy({
directives: {
defaultSrc: ["'self'"],
// styleSrc probably not needed but you can set those too
styleSrc: ["'self'", 'maxcdn.bootstrapcdn.com']
}
}))
This effectively tells the browser "only load things that are from my own domain"
https://helmetjs.github.io/docs/csp/
https://github.com/helmetjs/helmet
A: in my opinions, you should
*
*configure the firewall of api server to accept only ip address of the frontend express server with only port 443 also.
*please add basic authentication to the header in every APIs, then the front-end must attach some username/password or secret to the header of all APIs calls.
*in your server APIs, please include your security library, e.g., helmet.js also. it can help you secure your APIs server.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,184
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{"url":"https:\/\/www.physicsforums.com\/threads\/obtain-the-resistance-of-a-ntc-thermistor.98167\/","text":"Obtain the resistance of a ntc thermistor\n\n1. Nov 3, 2005\n\nSparky2020\n\nHI.\nIm really struggling to work out how to obtain the resistance of a ntc thermistor from just having the temperature. Ive been given the characteristic temp. of 3900K, labelled B for some reason. the only equation i can find to use is R = A.e to the power of B\/T. I think this is somwthing to do with the base of natural logarithms, but when i asked my lecturer, he said no knowledge of logs was needed.\n\n2. Nov 3, 2005\n\n3. Nov 3, 2005\n\nSparky2020\n\nThe question asks that if the ntc thermistor has a resistance of 47 ohms at 20 degrees celsius, and its characteristic temp is 3900K then what is its resistance at 100 degrees celsius?\nI cant even find a suitable equation which relates its resistance to its temperature.\n\n4. Nov 3, 2005\n\nverty\n\nI am going to guess here because I don't actually know. I'm guessing the resistance is 0 at the characteristic temperature. Perhaps you know about that. I will also assume a linear relationship. If it isn't linear, you need more information anyhow.\n\nAssuming this, it's a ratio. Actually, I'll let you work it out rather. Try with these assumptions.","date":"2017-11-19 00:13:09","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8082869052886963, \"perplexity\": 647.590344500779}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934805114.42\/warc\/CC-MAIN-20171118225302-20171119005302-00343.warc.gz\"}"}
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\section*{Introduction}
Quantum processors and quantum algorithms promise substantial advantages in computational tasks \cite{Harrow2017_supremacy}. Recent experiments have shown significant improvements towards the realization of large scale quantum devices operating in the regime in which classical computers cannot reproduce the output of the calculation \cite{Harrow2017_supremacy,Arute2019, Wu_2021_supremacy, Zhong_GBS_supremacy, zhong2021phaseprogrammable}.
An intriguing computational problem concerning quantum photonic processors is Boson Sampling (BS) and, more recently, its variant Gaussian Boson Sampling (GBS). The BS paradigm corresponds to sampling from the output distribution of a Fock state with $n$ indistinguishable photons after the evolution through a linear optical $m$-port interferometer \cite{AA, Brod19review}. This problem turns out to be intractable for a classical computer, while a dedicated quantum device can tackle such a task towards unequivocal demonstration of quantum computational advantage. The GBS variant replaces the quantum resource of the BS, i.e the Fock state, with single-mode squeezed vacuum states (SMSV). This change to the original problem enhances the samples generation rate with respect to BS performed with probabilistic sources, and preserves the hardness of sampling from a quantum state \cite{Lund_SBS,wcqoscct,Hamilton2017, Deshpande_GBS_th_supremacy}.
The GBS problem has drawn attention for the practical chance to achieve the quantum advantage regime. After the small scale experiments \cite{Zhong19, Paesani2019, thekkadath2022experimental},
the latest GBS instances have just reached the condition where the quantum device has solved the task faster than current state-of-the-art classical strategies \cite{Zhong_GBS_supremacy, zhong2021phaseprogrammable}. The interest in GBS also concerns applications for sampling for gaussian states beyond the original computational advantage. The probability of counting $n$-photon in the output ports of a GBS is proportional to the squared \emph{hafnians} of an appropriately constructed matrix, that takes into account the unitary transformation $U$ representing the optical circuit and the covariance matrix of the input state. Computing Hafnians of a matrix is as hard as computing \emph{permanents} that describe the amplitude of the BS output states. The hafnians have a precise interpretation in graph theory since their calculation corresponds to counting the perfect matchings in a graph. The adjacency matrix of a graph can be encoded in a GBS, and then the collected samples are informative about the graph properties. Recently, GBS-based algorithms for solving well-known problems in graph theory have been formulated \cite{Arrazzola_densesubgraph, Shuld_GBS_graphsimilarity, Bradler_2021} and tested in a first proof-of-principle experiment of the GBS within a reconfigurable photonic nano-chip \cite{Arrazola2021}.
These results on the BS framework are thus bringing back photonic platforms as a promising approach to implement quantum algorithms. In parallel, this development is currently accompanied by research efforts aimed at identifying suitable and efficient strategies for system certification. This is indeed a crucial requirement, both for benchmarking quantum devices reaching the quantum advantage regime, as well as validating the operation of such systems whenever they are employed to solve specific computations. While several methodologies have been developed and reported, certification of quantum processors is still an open problem. In the case of BS and GBS, direct calculation or sampling from the output distribution cannot be performed efficiently by classical means, and are thus not viable for large-scale implementations with many photons and ports of the optical circuit \cite{clifford2017classical,Neville2017,Quesada_exact_simulation,Quesada_exact_simulation_speedup, Bulmer_Markov_GBS}. Then, it is preferable to switch the problem towards a validation approach, i.e to exclude that the samples could be reproduced by specifically chosen classical models. The validation tests first developed for the BS problem focus on ruling out the uniform sampler, the distinguishable particle sampler and the mean-field sampler hypotheses \cite{Aaronson14, Tichy, Spagnolo2, Carolan15, Crespi16,Viggianiello18, Walschaers16, Giordani18, agresti2019pattern, FlaminiTSNE}. Recent efforts have been also dedicated to addressing partial photon distinguishability \cite{Viggianiello17optimal, Renema_partial_2020}, which is a crucial requirement that can spoil the complexity of the computation \cite{Renema_2018_classical, Moylett_2019}. This validation approach, originally conceived for the BS problem and based on defining suitable alternative hypotheses, has been subsequently extended to the GBS variant (see Fig.\ref{fig:validation}). In various experiments, the samples from GBSs have been validated against alternative classically-simulable hypotheses, such as the thermal, coherent and distinguishable SMSV states \cite{Zhong19, Paesani2019, Zhong_GBS_supremacy, zhong2021phaseprogrammable}.
These GBS validation examples include variations of Bayesian approaches \cite{Zhong_GBS_supremacy, zhong2021phaseprogrammable} or algorithms based on the statistical properties of two-point correlation functions \cite{Walschaers16, Giordani18} that can be used also for GBS to exclude thermal and distinguishable SMSV samplers \cite{hanbury}. In~\cite{zhong2021phaseprogrammable} a more refined analysis investigates the possibility of describing the experimental results as lower order interference processes, thus not involving all the generated photons. This approach is strictly related to sampling algorithms based on low-order interference approximations \cite{popova2021cracking} or low-order marginal probabilities
\cite{AA, Renema_partial_2020, renema2020marginal, villalonga2021efficient}.
In parallel, studies regarding the classical simulability of BS and GBS in terms of photon losses have also been carried out \cite{Oszmaniec_2018,GarciaPatron2019simulatingboson,Brod2020classicalsimulation, Qi_lossyGBS}.
Besides these examples of GBS validations, there is a lack of tailored algorithms for GBS that could be efficient in the regime of quantum advantage. In this work, we propose a validation protocol based on the deep connection between GBS and graph theory. We consider the features of the graph extracted from the GBS samples as a signature of the correct sampling from indistinguishable SMSV states. Within this framework, we present two approaches. The first method considers the space spanned by the feature vectors extracted from photon counting samples obtained from different gaussian states. Then, a classifier, such as a neural network, can be trained to identify an optimal hyper-surface to distinguish a true GBS and the mock-up hypotheses in this space. The second method investigates the properties of the kernel generated by the feature vectors of each class of gaussian state.
Both approaches exploit macroscopic quantities that can be retrieved in a reasonable time from the measured GBS samples.
This work is organized as follows. First, we review the concept of sampling from gaussian state of light and the relationship with counting graph perfect matchings. Then, we present the validation methods based on the properties of graph feature vectors and kernels. We conclude by providing insights on the effectiveness of the proposed approach to discriminate genuine GBS from different alternative hypotheses.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{GBS_validation.pdf}
\caption{\textbf{Gaussian Boson Sampling validation}. In the GBS paradigm $n$-photon configurations are sampled at the outputs of an optical circuit with $m$ ports. The aim of a validation algorithm is to exclude that the obtained samples could have been generated by classically-simulable models. Previous experiments mainly focused in techniques capable to rule out the uniform sampler, the thermal, coherent and distinguishable SMSV states hypotheses.}
\label{fig:validation}
\end{figure}
\section*{Gaussian Boson Sampling and its connection with graphs}
\begin{figure*}[ht]
\centering
\includegraphics[width = \textwidth]{fig2.pdf}
\caption{\textbf{Gaussian Boson Sampling and graph perfect matching.} a) Structure of the $2m\times 2m$ sampling matrix for $m$ independent gaussian states injected in a $m$-port interferometer. b) The hafnian as the operation to count the perfect matchings in a simple undirected graph with $n$ nodes. c) The permament as the same operation for a bipartite graph. d) The adjacency matrix of undirected graphs, even in the bipartite case, can be encoded in GBS devices. }
\label{fig:haf_perm}
\end{figure*}
\vspace{1ex}\noindent\textbf{Background.}
Here we briefly review the general theory of the probability to obtain $n$-photon configurations from a set of indistinguishable gaussian input states $\rho_i$ such that $\rho_{in}=\otimes_{i=1}^m \rho_i$, and distributed in $m$ optical modes, after the evolution in a multi-port interferometer.
Given the $2m\times2m$ covariance matrix $\sigma$ that identifies the gaussian state, and the output configuration $\vec{n}= (n_1, n_2, \dots, n_m)$, where $n_i$ is the number of photons detected in the output port $i$ such that $\sum_{i=1}^m n_i = n$, we have
\begin{equation}
\text{Pr}(\vec{n})=|\sigma_Q|^{-\frac{1}{2}}\frac{\text{Haf}\,(A_{\vec{n}})}{\prod_{\vec{n}}{n_i}!} \,.
\label{eq:hafn_gbs}
\end{equation}
The quantity $\sigma_Q$ is $\sigma+\frac{1}{2}\mathbb{I}_{2m}$ where $\mathbb{I}_{2m}$ is the $2m\times2m$ identity operator; $A_{\vec{n}}$ is a sub-matrix of the overall matrix $A$ that contains the information about the optical circuit represented by the transformation $U$ and the covariance of the input state, while $\text{Haf}$ stands for the hafnian of the matrix. More precisely, $A_{\vec{n}}$ is the $n \times n$ sub-matrix obtained by taking $n_i$ times the $i$-th row and the $i$-th column of $A$ \cite{Hamilton2017, DetailedstudyGBS}. The hafnian of $A_{\vec{n}}$ corresponds to the summation over the possible perfect matching permutations, i.e, the ways to partition the index set $\{1,\cdots ,n\}$ into $n$/2 pairs such that each index appears only in one pair (see also~\cite{Caianiello1953}).
The hafnian is in the \#P-complete complexity class, and is a generalization of the permanent of a matrix $M$ according to the following expression:
\begin{equation}
\text{Per}(M)=\text{Haf}\begin{pmatrix}
0 & M\\
M^t & 0
\end{pmatrix}\,.
\label{eq:permanent}
\end{equation}
The above description has been used to define a classically-hard sampling algorithm, using indistinguishable SMSV states with photon-counting measurements
\cite{wcqoscct, Hamilton2017, DetailedstudyGBS}. More specifically, in Fig. \ref{fig:haf_perm}a we report the structure of the sampling matrix $A$ for an input state $\rho$ that has zero displacement. In the language of quantum optics the displacement is the operation that generates a coherent state from the vacuum. Then, the blocks $B$ and $C$ highlighted in the Fig. \ref{fig:haf_perm}a correspond to the contribution of squeezed and thermal light respectively in the input state. Pure, indistinguishable, SMSV states display a $C=0$ and $B=U \text{diag} (\tanh{s_1}, \dots, \tanh{s_m}) U^t$, where $s_i$ are the squeezing parameters of each $\rho_i$ \cite{DetailedstudyGBS}. According to this representation the expression in Eq. \eqref{eq:hafn_gbs} becomes
\begin{equation}
\text{Pr}(\vec{n})_{\text{SMSV}} = |\sigma_Q|^{-\frac{1}{2}}\frac{|\text{Haf}\,({B}_{\vec{n}})|^2}{\prod_{\vec{n}}{n_i}!}\,,
\label{eq:smsv_states}
\end{equation}
where ${B}_{\vec{n}}$ is the submatrix of $B$ obtained from the string $\vec{n}$ as described at the beginning of the section.
\vspace{1ex}\noindent\textbf{Connection to graph theory.}
Recently, several works have identified a connection
between the GBS apparatus and graph theory \cite{ArrazolaQOpt,Arrazzola_densesubgraph, Bradler_2021}. These studies take advantage of such a relationship to formulate GBS-based algorithms in the context of graph-similarity and graph kernels.
The algorithms exploit the fact that the vectors extracted from GBS samples can be considered a feature space for a graph encoded inside the apparatus. In particular, they are strictly correlated to a class of classical graph kernels that count the number of $n$-matchings, i.e., the perfect matchings of the sub-graph with $n$ links in the original graph encoded inside the GBS. Given $A$ the adjacency matrix of the graph, the number of perfect matchings
is proportional to the hafnian of the matrix, thus corresponding to the output probabilities in Eqs. \eqref{eq:hafn_gbs} and \eqref{eq:smsv_states}.
Indeed, any symmetric matrix, such as the graph adjacency matrices, can be decomposed accordingly to the Takagi-Autonne factorization as $A = U \text{diag}(c\lambda_1,\dots, c\lambda_m) U^{t}$, where $\lambda_i$ are real parameters in the range $[0,1]$, $c$ is a scaling factor and $U$ is a unitary matrix. This decomposition matches with the expression of the sampling matrix $B$ of SMSV states when $\lambda_i = \tanh{s_i}$. Also squeezed states with very small displacement have a $n$-photon probability distribution that can be expressed through hafnians.
For example, displaced squeezed states have been investigated in the context of graph similarity, where a small amount of displacement has been employed as a hyper-parameter to enhance the graphs' classification accuracy \cite{Shuld_GBS_graphsimilarity}.
Regarding the sub-matrix selected by the sampling process, the configuration $\vec{n}$ identifies the elements of the sub-matrix $A_{\vec{n}}$ that represent an induced sub-graph (see Fig \ref{fig:haf_perm}b). The nodes of the original graph $A$ corresponding to detectors with zero counting are deleted, together with any edges connecting these nodes to the others.
If some elements $n_i$ of $\vec{n}$ are larger than one, i.e. these detectors count more than one photon, $A_{\vec{n}}$ describes what we call an \emph{extended induced sub-graph} in which the corresponding nodes and all their connections are duplicated $n_i$ times.
It is worth noting that also the permanent has a precise meaning in the context of graphs. Indeed, the matrix on the right-hand side of Eq. \eqref{eq:permanent} corresponds to the adjacency matrix of a bipartite graph. In other words, the permanent calculation provides the number of perfect matchings for this class of graphs (see Fig. \ref{fig:haf_perm}c). One may ask whether other sampling processes regulated by permanent calculations, such as the BS and the thermal samplers (see Appendix \ref{app:sampling}), could have a relationship with bipartite graphs.
The BS output distribution is defined by the permanent of the sub-matrix from the unitary transformation $U$ representing the circuit. It is clear that not all graphs can be represented by a unitary adjacency matrix. Furthermore, in the BS paradigm, the sub-matrix selected by the sampling process depends also on the input state. This implies that the resulting sub-graph could not have the same symmetries and properties as the original encoded in the $U$ matrix. The latter issue can be overcome by using thermal light, where only the output configuration $\vec{n}$ determines the sub-matrix. However, also for thermal light, the sampling matrix $C$ does not in general represent an adjacency matrix, thus preventing the possibility of encoding any bipartite graphs.
In conclusion, the GBS devices with squeezed states are the only ones that have a direct connection with graphs (see Fig. \ref{fig:haf_perm}d).
\section*{Feature vector-based validation algorithm}
In the following, we illustrate two validation algorithms tailored for GBS. The idea behind our protocols is to exploit the connection between the samples of a genuine GBS and the graph properties encoded in the device.
According to Eq. \eqref{eq:smsv_states} the most likely outcomes from the GBS are those with the highest hafnians, i.e. the output configurations that identify the sub-graph $A_{\vec{n}}$ with the largest number of perfect matchings.
However, we remind that the calculation of a single hafnian is a \#P-complete problem as the counting of the perfect matchings in a graph. Furthermore, estimation of the output probabilities from the quantum devices becomes unfeasible for large system sizes, and thus any protocol should not rely on this ingredient. Then, it is necessary for a successful validation algorithm to exploit quantities that do not depend
on the evaluation of the probability of a single $\vec{n}$, which would require exponential time for its estimation.
\vspace{1ex}\noindent\textbf{Feature vectors.}
It is possible to extract properties from a graph summarized in the so called feature vectors. In the GBS-based algorithms the features of the graph are extracted from a coarse-graining of the output configuration states. For instance, the probability to detect configurations $\{\vec{n}\}$ with $n_i = \{0,1\}$ is linked to the number of perfect matchings of the sub-graphs $\{A_{\vec{n}}\}$ of $A$ which do not have repetition of nodes and edges. Accordingly, the probability of the set of $\{\vec{n}\}$ with two photons in the same output will be connected to the perfect matching in sub-graphs with one repetition of a pair of nodes and edges. The collections of output configurations that identify a family of sub-graphs with a certain number of nodes and edges repetitions are called \emph{orbits} \cite{Shuld_GBS_graphsimilarity}. Given $n$ the total number of post-selected photons in the output, the orbit $O_{\vec{n}}$ is defined as the set of the possible index permutations of $\vec{n}$. In this work we consider the orbit $O_{[1,1,\dots,1,0 \dots 0]}$ that corresponds to output states with one or zero photon per mode; the orbit $O_{[2,1,\dots,1,0 \dots, 0]}$ that is the collection of the outputs with one mode occupied by two photons and $O_{[2,2,1\dots,1,0 \dots, 0]}$ with two distinct outputs hosting two photons. The graph feature vector components are identified by the probability of each orbit, defined as $Pr(O_{\vec{n}})=\sum_{\vec{n} \in O_{\vec{n}}}Pr(\vec{n})$. In the rest of this work we will refer to the probabilities of the orbits $O_{[1,1,\dots,1,0 \dots 0]}$, $O_{[2,1,\dots,1,0 \dots, 0]}$ and $O_{[2,2,1\dots,1,0 \dots, 0]}$ as $[1, \dots, 1]$, $[2,1, \dots1]$ and $[2,2,1,\dots,1]$ respectively.
The orbit probabilities can be estimated directly from photon counting measurements. This method can be applied in GBS experiments. In numerical simulation, direct sampling of photon counting is a viable approach for deriving orbit probabilities of gaussian states that can be sampled classically, such as distinguishable SMSV, thermal and coherent states (see Appendix \ref{app:sampling}). These states reproduce the scenarios that could occur in the experimental realizations of GBS devices. For example, photon losses turn the squeezed light into thermal radiation, while mode-mismatch, such as spectral and temporal distinguishability, breaks the symmetry of boson statistics. Exact estimation of the orbits for indistinguishable SMVS states can be performed by directly calculating all the hafnians, thus requiring evaluation of a large number of complex quantities. A different approach can be employed, based on approximating the orbits probability by a Monte Carlo simulation \cite{Killoran2019strawberryfields, Bromley_2020}. The outputs $\vec{n}$ within an orbit are selected uniformly at random and their exact probabilities are calculated. Then, the probability of the whole orbit after $N$ extractions can be approximated by $Pr(O_{\vec{n}}) \approx \frac{|O_{\vec{n}}|}{N}\sum_{i=1}^N Pr(\vec{n}_i)$, where $|O_{\vec{n}}|$ is the number of elements in the orbit. The adopted strategies reproduce the experimental conditions in which the orbits probabilities are estimated on a finite number $N$ of samples. The code for generating GBS data included routines from Strawberry Fields \cite{Killoran2019strawberryfields} and The Walrus \cite{Gupt2019} Python libraries.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{leaves.pdf}
\caption{\textbf{Orbits probabilities distribution.} In blue we report the feature vectors for $100$ genuine GBS devices with $m=400$. The squeezers parameters in each GBS were tuned to obtain a photon number distribution centered around $n \sim 16 \ll m$. Each cloud corresponds to the post-selection of different number of photons $n$ in the outputs. This is equivalent to look at the features of $n$-node sub-graphs. In yellow we report the thermal sampler case, in pink the distinguishable sampler, in red the distinguishable thermal sampler and in green the coherent light one.
GBS data were generated numerically via Monte Carlo approximation of the orbits probabilities. The maximum size achieved for the simulation corresponds to $n = 22$ for computation time reasons. The data of the other models were extracted from direct sampling of the photon counting. Thermal, coherent and distinguishable thermal samplers display also a non-zero probability to generate odd number of photons.}
\label{fig:leaves}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width = \textwidth]{new_dolphins.pdf}
\caption{\textbf{Orbits probabilities for different sizes of the GBS}. a) Orbits probabilities $[1, \dots, 1]$, $[2, 1, \dots, 1]$, $[2, 2, \dots, 1]$ for different samplers with $n$-photon $\in [4, 6, \dots, 22]$ and $m=n^2$ optical modes. In the blue-scale samples from a genuine GBS device, in yellow data from indistinguishable thermal states, in green the coherent states, in pink the distinguishable SMVS states and in red the distinguishable thermal light. For each $n$ and class of states we sampled $100$ sets of $U$ and $\{s_i\}$. Two orbits are not enough to discriminate the data, while in the space spanned by three orbits the various hypotheses are very well separated. b) Results of the classification accuracy of genuine GBS data by means of a neural network classifier. The network trained with trusted GBS data of smaller sizes indicated along the x axis is able to correctly classify larger GBS devices. }
\label{fig:fv_scaling}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{k_hist_all.pdf}
\caption{\textbf{Graph kernels distributions.} a) Kernels distributions for the graphs encoded in GBSs with $m$=400 and photon-number distribution centered around $n=16$. The feature vectors have been normalized to a given $n$ and to the space of the three orbits. We report the distributions for the GBS in blue, coherent light sources in green, distinguishable SMSV in pink and distinguishable thermal states in red. The four histograms display different features for any $n$. b-c) Kernels mean and standard deviation for the case $n=16$ and for increasing values of measured graphs. We observe that a small amount of experiments is enough to discriminate genuine GBS kernels. The uncertainties reported in the plots correspond to $3$ standard deviations.}
\label{fig:kernels}
\end{figure*}
\vspace{1ex}\noindent\textbf{Validation by classification.}
As a first method for validation, we propose the classification of these different samplers in the space spanned by the three feature vector components identified by the orbits $[1, \dots, 1]$, $[2,1,\dots,1]$ and $[2,2,1,\dots,1]$. In Fig.~\ref{fig:leaves} we give an insight of our intuition by reporting an example of the distribution of feature vectors for different graphs and sampler types. The colors underline samples from different models such as genuine GBS, distinguishable SMSV states, coherent light, indistinguishable thermal light emitters and distinguishable thermal states. In this simulation we consider $100$ optical random circuits with $m=400$ modes and $m$ sources set to produce a photon-number distribution centered in $n\ll m$. In this condition we are in the dilute regime where the orbits with low number of photons in the same output have the highest probability to occur. It is worth noting that in this estimation it is necessary to take into account the occurrence of the orbits in the whole space of the GBS, i.e the Hilbert space associated to the all possible $n$-photon states that can be generated by the squeezers.
Experimentally, such a method requires the knowledge of the photon-number distribution of the sources. Such requirement is not demanding since the characterization of the gaussian sources is a standard preliminary procedure in GBS experiments \cite{Zhong19,Paesani2019, Zhong_GBS_supremacy, Arrazola2021, zhong2021phaseprogrammable}. Alternatively, the orbits probability can be estimated by post-selecting samples with different total number $n$ and dividing the occurrence of photon counting belonging to the orbit with a given $n$ by the total number of samples. The data of the classical models were retrieved with such an approach while the GBS orbits were calculated via the Monte Carlo approximation.
These simulations show that three orbits are informative to discriminate among different gaussian samplers until the photon-number distribution is centered in the dilute regime. On the one hand, it is worth noting that the thermal light curve lies in the same plane of the GBS data but with somehow a smaller radius. The reason is that thermal radiation displays a non-zero probability to generate an odd number of photons. On the other hand, the distinguishability moves the two curves towards another plane that exhibits higher values of the probability of the orbit $[1,1,\dots 1]$. The physical intuition behind this behavior is that distinguishable particles do not interfere and, consequently, they have a lower probability of bunching.
To prove the effectiveness of feature vectors to validate a genuine GBS device of any size, we train a classifier such as feed-forward neural network with the data reported in Fig. \ref{fig:fv_scaling}. Experimental details are provided in Appendix \ref{app:3}. Here the samples correspond to different experiment layouts with number of modes $m = n^2$, and the number of post-selected photons varying in $n \in [4, 6, \dots 22]$. The size of the collected samples was $\sim 10^5$ for the classical gaussian states that generate a fraction of $\sim 10^3-10^4$ output configurations in the orbits under investigation. For the GBS data, we performed $\sim 10^4$ Monte Carlo extractions for the orbits probability estimation. The classifier reaches high level of accuracy, greater than 99\%. We performed a further study reported in Fig. \ref{fig:fv_scaling}b to check the ability of the network to generalize
to GBSs sizes not included in the training stage. To this aim we have trained the network with the data of Fig.~\ref{fig:fv_scaling}a up to $n = 12, 14, 16$, and subsequently computed the classification accuracy for the data with $n = 18, 20, 22$. The latter has been estimated on $100$ set of GBS for each $n$ and on $10$ independent training.
\vspace{1ex}\noindent\textbf{Validation via graph kernels.}
Other interesting quantities linked to feature vectors are the graph kernels, which can be employed to define a second method for validation. Here we study the linear kernels defined as the scalar product between pairs of feature vectors.
This method is less demanding in terms of number of measurements since it works even in the case where only samples from a given number $n$ of photons are post-selected at the output. In Fig.~\ref{fig:kernels}a we report the distributions of kernels for feature vectors normalized to the $3$-dimensional orbits space for a given number $n$ of post-selected photons. We note that kernels from distinguishable SMVS and distinguishable thermal states (Fig. \ref{fig:kernels}a) display the same gaussian distribution of the indistinguishable case, but they are centered at different kernel values for any $n$. Indeed, each histogram in the figure corresponds to the data of Fig. \ref{fig:leaves} for the $100$ sub-graph identified by $n=14$ and $n=16$. The coherent light data display the same average but show a larger variance. These differences highlighted in Fig.\ref{fig:kernels}a can be exploited to discriminate the coherent and distinguishable particles hypotheses.
To do this, we only require for the optical circuit to be reconfigurable, and perform enough experiments to retrieve the kernel distributions. Note that the number of kernels scales exponentially with the number of experiments, i.e. the number of sampled different graphs. More precisely, the number of kernels after $N$ experiments is $\begin{pmatrix}
N \\
2
\end{pmatrix}$. Thus, the kernels average and variance can be retrieved in a reasonable number of measurements as investigated in Fig.~\ref{fig:kernels}b-c.
The distributions of kernels from thermal samplers (not shown in the figure) are centered at the same values of genuine GBS with the same gaussian distribution. Thus, the discrimination of data from thermal indistinguishable emitters still requires the measurement of different $n$ number of photons in the outputs. This is not surprising if we consider the distribution of the feature vectors in Fig.~\ref{fig:leaves}. They display the same dispersion of the GBS data and, since we are now considering only the space of configurations with a given number of photons, the clouds collapse on each other.
\section*{Discussion}
In this work, we have presented a new approach to GBS validation that exploits the intrinsic connection between photon counting from specific classes of gaussian states of light and counting of perfect matchings in undirected graphs. Despite GBS-based algorithms in graph theory still need further studies to clarify their actual effectiveness and advantage with respect to the classical counterparts, the tools introduced in this context turn out to be informative in the framework of GBS experiments verification. We have seen how the feature vectors together with the graph kernels extracted from photon counting indicate the quantum nature of the sampling process. In fact, these quantities are very sensitive to imperfections that could occur in actual experiments, such as photon losses and distinguishability \cite{Shuld_GBS_graphsimilarity}. These two effects drive the device to act more similarly to thermal and distinguishable particles samplers that can be simulated efficiently by classical means.
The methods based on graph feature vectors and kernel distributions require a reasonable number of samples due to the coarse-graining of the output space of GBSs.
The method based on graph kernels requires fewer experiments with different graphs, in turn requiring the capability to tune the optical circuit $U$ and the squeezing parameters $s_i$. Nowadays, recent experimental results on integrated reconfigurable circuits \cite{Arrazola2021, Taballione_2021, hoch2021boson} enable large tunability and dimension of the matrix $U$. In addition, squeezing parameters can be tuned by changing the power of the pump laser that generates squeezed light from nonlinear crystals, and by tuning the relative squeezing parameters phases as recently demonstrated in~\cite{zhong2021phaseprogrammable}.
Further improvements to the approach adopted in this work can be foreseen. For instance, these include exploiting a more extensive orbit set or larger coarse-graining. These modifications could help in the validation of larger-scale instances of GBS. For example, it is possible to observe from Fig. \ref{fig:leaves} and Fig. \ref{fig:fv_scaling} that the orbits probabilities tend to zero with larger size due to the increasing dimension of the GBS Hilbert space.
A future perspective of such investigation may be the extension in the regime that exploits threshold detectors. This configuration has been adopted to prove quantum advantage, but its connection with graph feature vectors has not been investigated yet.
\section*{Acknowledgments}
This work is supported by the ERC Advanced grant QU-BOSS (Grant Agreement No. 884676) and ERC Starting grant SPECGEO (no. 802554). The authors wish to acknowledge financial support also by MIUR (Ministero dell'Istruzione, dell'Università e della Ricerca) via project PRIN 2017 "Taming complexity via QUantum Strategies: a Hybrid Integrated Photonic approach" (QUSHIP - Id. 2017SRNBRK). N.S. acknowledges funding from Sapienza Universit\`a di Roma via Bando Ricerca 2020: Progetti di Ricerca Piccoli, project "Validation of Boson Sampling via Machine Learning".
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\section{Introduction}\label{sec1}
Supersymmetry is a bold idea which arose in the seventies in string
and f\/ield theory. It was immediately realized that mechanisms of
spontaneous supersymmetry breaking should be investigated searching
for explanations of the apparent lack of supersymmetry in nature. In
a series of papers \cite{Wit1,Wit2,Wit3} Witten proposed to analyze this phenomenon in the simplest
possible setting: supersymmetric quantum mechanics. A new area of
research in quantum mechanics was born with far-reaching
consequences both in mathematics and physics.
Of course, there were antecedents in ordinary quantum mechanics
({\it nihil novum sub sole}), and indeed even before. The track can
be followed back to some work by Clif\/ford on the Laplacian operator,
see~\cite{Cliff}, quoted in~\cite{Hit}. Recast in modern SUSY
language, the Clif\/ford supercharge is:
\[
Q=\left(\begin{array}{lc} 0 & i\nabla_1+j\nabla_2+k\nabla_3
\\ 0 & 0 \end{array}\right), \qquad i^2=j^2=k^2=-1,
\qquad ij=-ji=k \,\, \, {\rm ciclyc},
\]
where $i$, $j$, $k$ are the imaginary unit quaternions and
\[
\nabla_1=\frac{\partial}{\partial x_1}+A_1(\vec{x}), \qquad
\nabla_2=\frac{\partial}{\partial x_2}+A_2(\vec{x}), \qquad
\nabla_3=\frac{\partial}{\partial x_3}+A_3(\vec{x})
\]
are the components of the gradient modif\/ied by the components of the
electromagnetic vector potential. The SUSY Hamiltonian is:
\[
Q^\dagger Q+QQ^\dagger=\left(\!\begin{array}{l@{}c} -\bigtriangleup +i
B_1(\vec{x})+j B_2(\vec{x})+ k B_3(\vec{x}) & 0
\\ 0 & -\bigtriangleup -i
B_1(\vec{x})-j B_2(\vec{x})- k B_3(\vec{x})\end{array}\!\right)\! ,
\]
the Laplacian plus Pauli terms. Needless to say, an identical
construction relates the Dirac operator in electromagnetic and/or
gravitational f\/ields backgrounds with the Klein--Gordon operator. The
factorization method of identifying the spectra of Schrodinger
operators, see~\cite{Inf} for a review, is another antecedent of
supersymmetric quantum mechanics that can also be traced back to the
19th century through the Darboux theorem.
In its modern version, supersymmetric quantum mechanics prompted the
study of many one-dimensional systems from a physical point of view.
A good deal of this work can be found in~\cite{Casal,
Ber,Khare,Junker}. A frequent starting point
in this framework is the following problem: given a non-SUSY
one-dimensional quantum Hamiltonian, is it possible to build a
supersymmetric extension? The answer to this question is positive
when one f\/inds a solution to the Riccati equation
\[
{1\over 2}\frac{dW}{dx} \frac{dW}{dx}+{\hbar\over
2}\frac{d^2W}{dx^2}=V(x),
\]
that identif\/ies the~-- a priori unknown~-- ``superpotential'' $W(x)$
from the~--~given~-- poten\-tial~$V(x)$. Several examples of this strategy
have been worked out in~\cite{Car}.
The formalism of physical supersymmetric systems with more than one
degree of freedom was f\/irst developed by Andrianov, Iof\/fe and
coworkers in a series of papers~\cite{Ioffe,Ioffe2},
published in the eighties. The same authors, almost simultaneously,
considered higher than one-dimensional SUSY quantum mechanics from
the point of view of the factorization of $N$-dimensional quantum
systems~\cite{Andr,Andr1}. Factorability, even though
essential in $N$-dimensional SUSY quantum mechanics, is not so
ef\/fective as compared with the one-dimensional situation. Some
degree of separability is also necessary to achieve analytical
results. For this reason we started a program of research in the
two-dimensional supersymmetric classical mechanics of Liouville
systems~\cite{Perelomov}; i.e., those separable in elliptic, polar,
parabolic, or Cartesian coordinates, see the papers \cite{AM} and
\cite{AoP}. We followed this path in the quantum domain for Type I
Liouville models in \cite{JPA}, whereas Iof\/fe et al.\ also studied
the interplay between supersymmetry and integrability in quantum and
classical settings in other type of models in~\cite{Andr2,Can}. In these papers, a new structure was
introduced: second-order supercharges provided intertwined scalar
Hamiltonians even in the two-dimensional case, see~\cite{Ioffe3} for
a review. This higher-order SUSY algebra allows for new forms of
non-conventional separability in two dimensions. There are two
possibilities: (1)~a~similarity transformation performs separation
of variables in the supercharges and some eigenfunctions (partial
solvability) can be found, see~\cite{Ioffe4,Ioffe5}. (2)~One
of the two intertwined Hamiltonian allows for exact separability:
the spectrum of the other is consequently known~\cite{Ioffe6,Ioffe7}.
The second dif\/f\/iculty with the jump in dimensions is the
identif\/ication of the superpotential. Instead of the Riccati
equation one must solve the PDE:
\[
{1\over
2}\vec{\nabla}W(\vec{x})\cdot\vec{\nabla}W(\vec{x})+{\hbar\over
2}\nabla^2W(\vec{x})=V(\vec{x}) .
\]
In our case, we look for solutions of this PDE when $V(\vec{x})$ is
the potential energy of the two Coulombian centers. We do not know
how to solve it in general, but two dif\/ferent strategies should help
us. First, following the work in \cite{KLPW} and \cite{KLPW1} on the
supersymmetric Coulomb problem, we shall choose the superpotential
as the solution of the Poisson equation:
\[
{\hbar\over 2}\nabla^2W(\vec{x})=V(\vec{x}) .
\]
The superpotential will be the solution of {\it another}
Riccati-like PDE where a classical piece must be added to the
potential of the two centers. Second, as in~\cite{Manton,Heumann} the selection of superpotential requires the solution
of the Hamilton--Jacobi equation:
\[
{1\over
2}\vec{\nabla}W(\vec{x})\cdot\vec{\nabla}W(\vec{x})=V(\vec{x})
.
\]
Again the superpotential must solve a third Riccati-like PDE, where
now a quantum piece must be added to the potential of the two
centers.
The organization of the paper is as follows: We start by brief\/ly
recalling the non-SUSY classical, Newtonian~\cite{landau}, and
quantal, Coulombian~\cite{Pauling,Greiner}, two-center
problem. We shall constrain the particle to move in one plane
containing the two centers. The third coordinate is cyclic and it
would be easy to extend our results to three dimensions. In Section~\ref{sec2} the formalism of two-dimensional SUSY quantum mechanics is
developed, and the superpotential of the f\/irst Type is identif\/ied
for two Coulombian centers. Bosonic zero-energy ground states are
also found. Sections~\ref{sec3} and~\ref{sec4} are devoted to formulating the SUSY
system in elliptic coordinates where the problem is separable in
order to f\/ind fermionic zero-energy ground states. It is also shown
that the spectral problem is tantamount to families of two ODE's of
Razavy~\cite{Razavy1}, and Whittaker--Hill type~\cite{Razavy2}, see
also~\cite{Bondar}. Since these systems are quasi-exactly solvable,
several eigenvalues are found following the work in~\cite{FGR}. In
Section~\ref{sec5} two centers of the same strength are studied and some
eigenfunctions are also found. Section~\ref{sec6} is fully devoted to the
analysis of the Manton--Heumann approach applied to the two
Coulombian centers. Finally, a summary is of\/fered in Section~\ref{sec7}.
\subsection{The classical problem of two Newtonian/Coulombian
centers}\label{sec1.1}
\begin{figure}[h]\centering
\includegraphics[height=3.5cm]{Guilarte-fig01}
\caption{Location of the two centers and distances
to the particle from the centers.}
\end{figure}
The classical action for a system of a light particle moving in a
plane around two heavy bodies which are sources of static
Newtonian/Coulombian forces is:
\[ \tilde{S} =\int d t \left\{ {1\over 2}\, m \left( {d x_1 \over d
t}{d x_1 \over d t} + {d x_2 \over d t}{d x_2 \over d t} \right) -
{\alpha_1\over r_1} - {\alpha_2 \over r_2} \right\}.
\]
The centers are located at the points $(x_1=-d$, $x_2=0)$,
$(x_1=d$, $x_2=0)$, their strengths are $\alpha_1=\alpha \geq
\alpha_2=\delta\alpha
> 0 $, $\delta \in (0,1]$, and
\[
r_1=\sqrt{(x_1-d)^2 + x_2^2} , \qquad
r_2=\sqrt{(x_1+d)^2+x_2^2}
\]
are the distances from the particle to the centers. In the following
formulas we show the dimensions of the coupling constants and
parameters and def\/ine non-dimensional variables:
\begin{gather*}
[\alpha_1] = [\alpha_2]=[\alpha] = M L^3 T^{-2}, \qquad
[d]=L, \qquad [\delta]=1,\\
x_1\rightarrow d\, x_1, \qquad x_2\rightarrow d\, x_2
, \qquad t\rightarrow \sqrt{\frac{d^3m}{\alpha}}\, t,
\\
r_1\rightarrow d\, r_1=d\sqrt{(x_1-1)^2 + x_2^2}, \qquad
r_2\rightarrow d\, r_2=d\sqrt{(x_1+1)^2+x_2^2}.
\end{gather*}
In the rest of the paper we shall use non-dimensional variables.
From the non-dimensional action
\[ \tilde{S} = \sqrt{md\alpha}\,
S=\sqrt{md\alpha} \int \, dt \, \left\{ {1\over 2} \left( {d x_1
\over dt}{d x_1 \over dt} + {d x_2 \over dt}{d x_2 \over dt} \right)
- {1\over r_1} - {\delta \over r_2} \right\},
\]
the linear momenta and Hamiltonian are def\/ined:
\begin{gather*}
p_1=\frac{\partial L}{\partial \dot{x_1}}=\frac{dx_1}{dt} , \qquad p_2=\frac{\partial L}{\partial
\dot{x_2}}=\frac{dx_2}{dt},
\\
\tilde{H}=\frac{\alpha}{d} H , \qquad H=
{1\over 2} (p_1^2+ p_2^2) +{1\over r_1} + {\delta \over r_2} .
\end{gather*}
This system is completely integrable because there exists a ``second invariant'' in involution with the Hamiltonian:
\[ \tilde{I}_2
= (m d \alpha) I_2 , \qquad I_2={1\over 2} (l^2 - p_2^2)+x_1
\left( {\delta \over r_2} -{1\over r_1} \right), \qquad l^2 =
(x_1 p_2-x_2 p_1)^2.
\]
\subsection{The quantum problem of two Coulombian centers of force}\label{sec1.2}
If $\sqrt{m\alpha d}$ is of the order of the Planck constant
$\hbar$,
the system is of quantum nature. Canonical quantization in terms of
the non-dimensional $\bar\hbar$ constant,
\begin{gather*}
p_i \rightarrow \hat{p}_i=-i\bar{\hbar}\frac{\partial}{\partial x_i}
, \qquad x_i \rightarrow \hat{x}_i=x_i,
\\
[\hat{x}_i,\hat{p_j}]=i\bar{\hbar}\delta_{ij} ,
\qquad \bar{\hbar}=\frac{\hbar}{\sqrt{md\alpha }},
\end{gather*}
converts the dynamical variables into operators. The quantum
Hamiltonian, $\hat{\tilde{H}}=\frac{\alpha}{d}\hat{H}$, and the
quantum symmetry operator, $\hat{\tilde{I}}_2=(m d \alpha)
\hat{I}_2$, are mutually commuting operators:
\begin{gather*} \hat{H}=
-{\bar{\hbar}\over 2} \left( \frac{\partial^2}{\partial x_1^2}+
\frac{\partial^2}{\partial x_2^2}\right) + {1\over r_1} + {\delta
\over r_2}, \qquad [\hat{H},
\hat{I}_2]=\hat{H}\hat{I}_2-\hat{I}_2\hat{H}=0 ,
\\ \hat{I}_2= -{\bar{\hbar}^2\over 2} \left( (x_1^2-1)
\frac{\partial^2}{\partial x_2^2} + x_2^2 \frac{\partial^2}{\partial
x_1^2} - 2 x_1 x_2 \frac{\partial^2}{\partial x_1 \partial x_2} -
x_1 \frac{\partial}{\partial x_1} - x_2 \frac{\partial}{\partial
x_2} \right) + x_1 \left( {\delta \over r_2} - {1 \over r_1} \right)
.
\end{gather*}
\section[Two-dimensional ${\cal N}=2$ SUSY quantum mechanics]{Two-dimensional $\boldsymbol{{\cal N}=2}$ SUSY quantum mechanics}\label{sec2}
We now describe how to build a non-specif\/ic system in
two-dimensional ${\cal N}=2$ SUSY quantum mechanics. Besides
commuting -- non-commuting -- operators there are anti-commuting~-- non-anti-commuting operators to be referred respectively as
``bosonic'' and ``fermionic'' by analogy with QFT. The
Fermi operators are represented on Euclidean spinors in ${\mathbb
R}^4$ by the Hermitian $4\times 4$ gamma matrices:
\begin{gather*} \psi^j_1=\frac{i}{\sqrt{2}}\gamma^j ,
\qquad \psi^j_2=-\frac{i}{\sqrt{2}}\gamma^{2+j}, \qquad
(\gamma^j)^\dagger=\gamma^j , \qquad (\gamma^{2+j})^\dagger=\gamma^{2+j},\\
\{\gamma^j,\gamma^k\}=2\delta^{jk}=\{\gamma^{2+j},\gamma^{2+k}\}
, \qquad \{\gamma^j,\gamma^{2+k}\}=0, \qquad
j,k=1,2.
\end{gather*}
The building blocks of the SUSY system are the two (${\cal N}=2$)
quantum Hermitian supercharges: $\hat{Q}_1^\dagger=\hat{Q}_1$, $\hat{Q}_2^\dagger=\hat{Q}_2$,
\[ \hat{Q}_1=\sqrt{{\bar{\hbar}}} \sum_{j=1}^2
\left(-i{\bar{\hbar}}{\partial\over\partial
x_j} \psi_1^j-\frac{\partial W}{\partial
x_j} \psi_2^j\right), \qquad
\hat{Q}_2=\sqrt{{\bar{\hbar}}} \sum_{j=1}^2
\left(-i{\bar{\hbar}}{\partial\over\partial
x_j} \psi_2^j+\frac{\partial W}{\partial
x_j} \psi_1^j\right).
\]
It is convenient to def\/ine the non-Hermitian supercharges
$\hat{Q}_\pm=\hat{Q}_1\pm i\hat{Q}_2$,
\begin{gather*}
\hat{Q}_+=i\sqrt{\bar{\hbar}} \left(\begin{array}{cccc}
0 & 0 & 0 & 0 \vspace{1mm}\\
\bar{\hbar}{\partial \over \partial x_1}-{\partial W \over
\partial x_1} & 0 & 0 & 0 \vspace{1mm}\\ \bar{\hbar}{\partial \over\partial
x_2}-{\partial W \over
\partial x_2} & 0 & 0 & 0
\vspace{1mm}\\ 0 & -\bar{\hbar}{\partial \over\partial x_2}+{\partial W \over\partial x_2}
& \bar{\hbar}{\partial \over\partial x_1}-{\partial W \over\partial
x_1} & 0
\end{array}\right) ,
\\
\hat{Q}_-=i\sqrt{\bar{\hbar}} \left(\begin{array}{cccc} 0 &
\bar{\hbar}{\partial \over \partial
x_1}+{\partial W \over \partial x_1} & \bar{\hbar} {\partial \over
\partial x_2}+{\partial W \over \partial x_2} & 0
\vspace{1mm}\\ 0 & 0 & 0 & -\bar{\hbar}{\partial \over \partial x_2}-{\partial W \over \partial x_2}
\vspace{1mm}\\ 0 & 0 & 0 & \bar{\hbar}{\partial \over \partial x_1}+{\partial W \over \partial x_1}
\vspace{1mm}\\ 0 & 0 & 0 & 0
\end{array}\right)
\end{gather*}
because their anti-commutator determines the Hamiltonian $\hat{H}_S$
of the supersymmetric system:
\[ \{\hat{Q}_+ , \hat{Q}_- \}=2 \bar{\hbar} \hat{H}_S,
\qquad [\hat{Q}_+,\hat{H}_S]=[\hat{Q}_-,\hat{H}_S]=0.
\]
The explicit form of the quantum SUSY Hamiltonian is enlightened by
the ``Fermi'' number $F= \sum\limits_{j=1}^2
\psi_+^j\psi_-^j$ operator:
\[
\hat{H}_S=\left(\begin{array}{cccc} \hat{h}^{(0)} & 0 & 0 &0 \\
0 & \hat{h}^{(1)}_{11} & \hat{h}^{(1)}_{12} & 0\\ 0 &
\hat{h}^{(1)}_{21}& \hat{h}^{(1)}_{22} & 0
\\ 0 & 0 & 0 & \hat{h}^{(2)}
\end{array}\right) , \qquad F={\displaystyle \sum_{j=1}^2}
\psi_+^j\psi_-^j=\left(\begin{array}{cccc} 0 & 0 & 0 & 0
\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &
2\end{array}\right).
\]
It has a block diagonal structure acting on the sub-spaces of the
Hilbert space of Fermi numbers 0, 1, and 2. In the sub-spaces of
Fermi numbers even ${\hat H}_S$ acts as ordinary dif\/ferential
Schr\"odinger operators. The scalar Hamiltonians are:
\begin{gather*}
2\hat{h}^{(f=0)}=-\bar{\hbar}^2\nabla^2+\vec{\nabla}W\vec{\nabla}W+\bar{\hbar}
\nabla^2W = -\bar{\hbar}^2\nabla^2 + 2 {\hat V}^{(0)},
\\ 2\hat{h}^{(f=2)}=
-\bar{\hbar}^2\nabla^2+\vec{\nabla}W\vec{\nabla}W-\bar{\hbar}
\nabla^2W =-\bar{\hbar}^2\nabla^2 + 2 {\hat V}^{(2)}.
\end{gather*}
In the sub-space of Fermi number 1, however, ${\hat H}_S$ is a
matrix of dif\/ferential operators, the $2\times 2$ matrix
Hamiltonian:
\begin{gather*} 2\hat{h}^{(f=1)}=\left(\begin{array}{cc}
-\bar{\hbar}^2\nabla^2+\vec{\nabla}W\vec{\nabla}W-\bar{\hbar}
\Box^2W &
-2\bar{\hbar} {\partial^2 W \over\partial x_1\partial x_2} \vspace{2mm}\\
-2\bar{\hbar}{\partial^2 W \over\partial x_1\partial x_2} &
-\bar{\hbar}^2\nabla^2+\vec{\nabla}W\vec{\nabla}W+\bar{\hbar}\Box^2W
\end{array}\right) ,
\\ \vec{\nabla}=\frac{\partial}{\partial x_1}\cdot
\vec{e}_1+\frac{\partial}{\partial x_2}\cdot \vec{e}_2 , \qquad
\nabla^2={\partial^2\over\partial x_1\partial
x_1}+{\partial^2\over\partial x_2\partial x_2} , \qquad
\Box^2={\partial^2\over\partial x_1\partial
x_1}-{\partial^2\over\partial x_2\partial x_2}.
\end{gather*}
This is exactly the structure unveiled in \cite{Ioffe,Ioffe2}. All the interactions expressed in ${\hat H}_S$ come
from the as yet unspecif\/ied function $W(x_1,x_2)$, which is thus
called the superpotential.
\subsection{The superpotential I for the two-center
problem}\label{sec2.1}
To build a supersymmetric system containing the interactions due to
two Coulombian centers of force, we must start by identifying the
superpotential. One possible choice, inspired by \cite{KLPW}, is
having the two-center potential energy in $\hat{h}^{(0)}$ in the
term proportional to $\bar\hbar$. We must therefore solve the
Poisson equation to f\/ind the superpotential I:
\begin{equation} {\bar{\hbar} \over 2} \nabla^2 \hat{W} = -{1\over
r_1} -{\delta\over r_2} , \qquad \hat{W}(x_1,x_2)=-\frac{2
r_1}{\bar{\hbar}} - \frac{2 \delta r_2}{\bar{\hbar}} .
\label{eq:sups}
\end{equation}
Note that the anticommutator between the supercharges induces a
$\bar{\hbar}$ factor in front of the Laplacian. This fact, in turns,
forces the singularity of the superpotential (henceforth, also of
the potential) at the classical limit $\bar{\hbar}=0$. The same
singularity arises in the bound state spectra of atoms, e.g., in the
energy levels of the hydrogen atom. The potential energies in the
scalar sectors are accordingly:
\[
{\hat V}_I^{(0)\choose (2)}=\frac{2}{\bar{\hbar}^2 } \left[ 1 +
\delta^2 + \delta \left( {r_1\over r_2} +{r_2\over r_1} -{4 \over
r_1 r_2} \right) \right] \mp \left({ 1\over r_1} +{\delta \over
r_2} \right).
\]
\begin{figure}[h]
\centering \includegraphics[height=4.5cm]{Guilarte-fig02}
\includegraphics[height=4.5cm]{Guilarte-fig03}
\caption{Cross section ($x_2=0$) and 3D
graphics of the quantum potential $\hat{V}^{(0)}$ for $\delta=1/2$.
Cases: Upper row: (a) $\bar{\hbar}=0.2$, (b) $\bar{\hbar}=0.4$.
Lower row: (a) $\bar{\hbar}=1$ and (b) $\bar{\hbar}=10$. Increasing
$\bar{\hbar}$ the centers become more and more attractive.}
\end{figure}
We stress that the superpotential I is a solution of the
Riccati-like PDE's:
\[ \vec{\nabla}\hat{W}\vec{\nabla}\hat{W}\pm\bar{\hbar}
\nabla^2{\hat W} =2{\hat V}_I^{(0)\choose (2)}
\]
and the scalar Hamiltonians for two SUSY Coulombian centers read:
\[ \hat{h}^{(0)\choose(2)}=-{\bar{\hbar}^2\over 2}\nabla^2+
\frac{2}{\bar{\hbar}^2 } \left[ 1 + \delta^2 + \delta \left(
{r_1\over r_2} +{r_2\over r_1} -{4 \over r_1 r_2} \right) \right]\mp
\left({ 1\over r_1} +{\delta \over r_2} \right) .
\]
\subsection{Bosonic zero modes I}\label{sec2.2}
The bosonic zero modes
\[ \hat{Q}_\pm\Psi_0^{(0)}(x_1,x_2)=0 , \qquad
\hat{Q}_\mp\Psi_0^{(2)}(x_1,x_2)=0 ,
\]
if normalizable, are the bosonic ground states of the system:
\[
\Psi_0^{(0)}(x_1,x_2)= \left(
\begin{array}{c} {\rm exp}[{(-2 r_1-2 \delta r_2) \over \bar{\hbar}^2}]
\\ 0 \\ 0 \\ 0
\end{array} \right), \qquad
\Psi_0^{(2)}(x_1,x_2)= \left(
\begin{array}{c} 0
\\ 0 \\ 0 \\ {\rm exp}[{(2 r_1+2 \delta r_2) \over
\bar{\hbar}^2}]\end{array}\right) .
\]
The norm of the true bosonic ground state $\Psi_0^{(0)}$ is f\/inite
and given in terms of Bessel functions:
\begin{gather*}
N(\bar{\hbar})=\int_{-\infty}^\infty \int_{-\infty}^\infty dx_1dx_2
\, e^{\frac{2}{\bar{\hbar}} \hat{W}(x_1,x_2)}= 2 \int_{0}^\infty
dx_1 \int_{0}^\infty dx_2 \, e^{\frac{2}{\bar{\hbar}^2} (-2
r_1-2\delta r_2)},\\
N(\bar{\hbar})=2\pi \left[
\frac{\bar{\hbar}^2}{4(1+\delta)}
I_0\left(\frac{4}{\bar{\hbar}^2}(1-\delta)\right)
K_1\left(\frac{4}{\bar{\hbar}^2}(1+\delta)\right)\right.\\
\left. \phantom{N(\bar{\hbar})=}{} +\frac{\bar{\hbar}^2}{4(1-\delta)}
I_1\left(\frac{4}{\bar{\hbar}^2}(1-\delta)\right)
K_0\left(\frac{4}{\bar{\hbar}^2}(1+\delta)\right)\right] .
\end{gather*}
In Fig.~3 plots of the zero energy bosonic ground state
probability density of f\/inding the particle in some area of the
plane are shown for several values of $\bar\hbar$. The drawings
reveal the physical meaning of the $\bar\hbar=0$ singularity: for
$\bar\hbar=0.2$ we see the particle probability density peaked
around the center on the right with a very small probability.
Exactly at $\bar\hbar=0$, $e^{-{4 r_1\over\bar\hbar^2}}$ is f\/inite
only for $r_1=0$ and zero otherwise, giving probability of f\/inding
the particle exactly in the center. $e^{-{4 \delta
r_2\over\bar\hbar^2}}$, however, is zero $\forall\, r_2$ meaning that
the probability of this state is zero at the classical limit; in
classical mechanics there are no isolated discrete states.
\begin{figure}[h]\centering
\includegraphics[height=2.5cm]{Guilarte-fig04}
\includegraphics[height=3.cm]{Guilarte-fig05}
\includegraphics[height=3.cm]{Guilarte-fig06}
\caption{Graphics of the norm as a
function of $\bar\hbar$ and the ground state probability density,
$|\Psi_0^{(0)}(x_1,x_2)|^2$, for $\delta=1/2$ and the values of
$\bar{\hbar} = 0.2, \, 0.4,\, 1$ and $10$. Note the extreme smallness
for $\bar{\hbar}=0.2$. The norms for these four cases are:
$N(0.2)= 3.5806 \cdot 10^{-47}$, $N(0.4)= 2.0576 \cdot
10^{-13}$, $N(1)=0.00942$ and $N(10)=1743.94$.}
\end{figure}
\section[Two-dimensional ${\cal N}=2$ SUSY quantum mechanics in elliptic coordinates]{Two-dimensional $\boldsymbol{{\cal N}=2}$ SUSY quantum mechanics\\ in elliptic coordinates}\label{sec3}
The search for more eigenfunctions of the SUSY Hamiltonian requires
the use of the separability in elliptic coordinates of the problem
at hand. This, in turn, needs the translation of our two-dimensional
${\cal N}=2$ SUSY system to elliptic coordinates. A general
reference (in Russian) where SUSY quantum mechanics is formulated in
curvilinear coordinates is \cite{Ioffe8}, see also \cite{Art} to
f\/ind a~more geometric version of SUSY QM on Riemannian manifolds.
The change from Cartesian to elliptic coordinates,
\begin{gather*} x_1=u v \in (-\infty,+\infty) ,\qquad
x_2=\pm \sqrt{(u^2-1) (1-v^2)} \in (-\infty,+\infty),
\\
u={1\over 2}(r_1+r_2)\in (1,+\infty) , \qquad
v={1\over 2} (r_2-r_1)\in (-1,1),
\end{gather*}
induces a map from the plane to the inf\/inite elliptic strip: $
{\mathbb R}^2\equiv (-\infty,+\infty)\times (-\infty,+\infty)
\Longrightarrow {\mathbb E}^2\equiv (-1,1)\times (1,+\infty)$.
This map also induces a non-Euclidean (but f\/lat) metric in ${\mathbb
E}^2$:
\[ g(u,v)=\left(
\begin{array}{cc} g_{uu}=\frac{u^2-v^2}{u^2-1} &
g_{uv}=0 \vspace{1mm}\\ g_{vu}=0 &
g_{vv}=\frac{u^2-v^2}{1-v^2}\end{array} \right) ,
\]
with Christof\/fel symbols:
\begin{gather*}
\Gamma_{uu}^u=\displaystyle\frac{-u (1-v^2)}{(u^2-v^2)(u^2-1)} ,\qquad
\Gamma_{vv}^v=\displaystyle\frac{v (u^2-1)}{(u^2-v^2)(1-v^2)} ,\qquad
\Gamma_{uv}^u=\Gamma_{vu}^u=\displaystyle\frac{-v}{u^2-v^2},
\\ \Gamma_{uu}^v=\displaystyle\frac{v
(1-v^2)}{(u^2-v^2)(u^2-1)} ,\qquad \Gamma_{vv}^u=\displaystyle\frac{-u
(u^2-1)}{(u^2-v^2)(1-v^2)} ,\qquad
\Gamma_{uv}^v=\Gamma_{vu}^v=\displaystyle\frac{u }{u^2-v^2} .
\end{gather*}
Using the zweig-bein chosen in this form,
\begin{gather*}
g^{uu}(u,v)=\sum_{j=1}^2e^u_j(u,v)e^u_j(u,v) , \qquad
g^{vv}(u,v)=\sum_{j=1}^2e^v_j(u,v)e^v_j(u,v) ,
\\
e^u_1(u,v)=\left(u^2-1\over u^2-v^2\right)^{{1\over 2}},
\qquad e^v_2(u,v)=\left(1-v^2\over u^2-v^2\right)^{{1\over 2}}
\end{gather*}
we now def\/ine ``elliptic'' spinors, ``elliptic'' Fermi
operators, and ``elliptic'' supercharges:
\begin{gather*}
\psi_{\pm}^{u}(u,v)=e^u_1(u,v)\psi^1_\pm , \qquad
\psi_\pm^v(u,v)=e^v_2(u,v)\psi_\pm^2 ,
\\
\hat{C}_+=-
i \sqrt{\bar{\hbar}} \left(\!\!\begin{array}{cccc} 0 & 0 & 0 & 0 \\
e^u_1\nabla_u^- & 0 & 0 & 0
\\ e^v_2\nabla_v^- & 0 & 0 & 0
\\ 0 & -e^v_2 \left( \nabla_v^- -{\bar\hbar v\over u^2-v^2} \right)
& e^u_1 \left( \nabla_u^- +{\bar\hbar u\over u^2-v^2} \right) & 0
\end{array}\!\!\right) \!, \qquad \nabla_u^\mp= \bar{\hbar} {\partial\over\partial u}\mp
{d\hat{F}\over du},
\\
\hat{C}_-=-i \sqrt{\bar{\hbar}} \left(\!\!\begin{array}{cccc} 0 & e^u_1
\left( \nabla_u^+ +{\bar\hbar u\over u^2-v^2} \right) & e^v_2 \left(
\nabla_v^+ -{\bar\hbar v\over u^2-v^2} \right) & 0
\\ 0 & 0 & 0 & -e^v_2\nabla_v^+
\\ 0 & 0 & 0 & e^u_1\nabla_u^+
\\ 0 & 0 & 0 & 0
\end{array}\!\!\right)\! , \qquad \nabla_v^\mp = \bar{\hbar}{\partial\over\partial v}\mp {d\hat{G}\over
dv} ,
\end{gather*}
where $ \hat{W}(u,v)=\hat{F}(u)+\hat{G}(v)$.
To obtain the supercharges in Cartesian coordinates from the
supercharges in elliptic coordinates, besides expressing $u$ and $v$
in terms of $x_1$ and $x_2$, one needs to act by conjugation with
the idempotent, Hermitian matrix:
\[
{\cal S} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\
0 & - v e^u_1(u,v) & - u e^v_2(u,v) & 0 \\ 0 & - u e^v_2(u,v) & v
e^u_1(u,v) & 0 \\ 0 & 0 & 0 & -1 \end{array} \right), \qquad
{\cal S} \hat{C}_+ {\cal S} = \hat{Q}_+ , \qquad {\cal S}
\hat{C}_- {\cal S} = \hat{Q}_- .
\]
Equation (\ref{eq:sups}) in elliptic coordinates,
\begin{gather} {\bar{\hbar} \over 2} \left[ {u^2 -1 \over
u^2-v^2} \left( {d^2 \hat{F} \over d u^2} + {u\over u^2-1 }{d
\hat{F} \over d u} \right) + {1 -v^2 \over u^2-v^2} \left( {d^2
\hat{G} \over d v^2}-{v\over 1-v^2} {d \hat{G} \over d v} \right)
\right]\nonumber\\
\qquad{} = -{(1+\delta) u \over u^2-v^2} + {(\delta-1) v\over
u^2-v^2} \label{eq:poisse}
\end{gather}
is separable:
\[ (u^2-1) {d^2 \hat{F} \over d u^2} + u {d \hat{F}
\over d u} + \frac{2 (1+\delta) u}{\bar{\hbar}} =\kappa =- (1-v^2)
{d^2 \hat{G}\over d v^2} + v {d \hat{G} \over d v} + \frac{2 (\delta
-1) v}{\bar{\hbar}} ,
\]
with separation constant $\kappa$. The general solution of equation
(\ref{eq:poisse}) depends on two integration constants (besides an
unimportant additive constant):
\begin{gather*}
\hat{W}(u,v;\kappa, C_1,C_2)= - 2 \frac{(1+\delta)}{\bar{\hbar}} u
+ {C_1\over \bar{\hbar}} \ln \big(u+\sqrt{u^2-1}\big) +{\kappa \over
2 \bar{\hbar}} \left( \ln \big(u+\sqrt{u^2-1}\big) \right)^2 \\
\phantom{\hat{W}(u,v;\kappa, C_1,C_2)=}{} + 2
\frac{(1-\delta)}{\bar{\hbar}} v+ {C_2 \over \bar{\hbar}} \arcsin v -{\kappa \over 2 \bar{\hbar}} \left( \arcsin v
\right)^2.
\end{gather*}
We f\/ind thus a three-parametric family of supersymmetric models for
which the potentials in the scalar sectors are:
\begin{gather*} {\hat V}_I^{(0)\choose
(2)}(x_1, x_2; \kappa, C_1,C_2)=\frac{2}{\bar{\hbar}^2 } \left[ 1
+ \delta^2 + \delta \left( {r_1\over r_2} +{r_2\over r_1} -{4 \over
r_1 r_2} \right) \right] \\
\qquad{} +\frac{1}{2 \bar{\hbar}^2 r_1 r_2} \Bigg[ \left( C_1+ \kappa \ln
\frac{r_1+r_2+\sqrt{(r_1 + r_2)^2 - 4}}{2}\right)^2 + \left( C_2 -
\kappa \arcsin \frac{r_2-r_1}{2}\right)^2 \\
\qquad{} - 2 (1 +
\delta) \sqrt{(r_1 + r_2)^2- 4 } \left( C_1 + \kappa \ln
\frac{r_1 + r_2 + \sqrt{(r_1 + r_2)^2 - 4}}{2}\right)\\
\qquad{}+ 2(1-\delta) \sqrt{4-(r_2-r_1)^2} \left( C_2 - \kappa
\arcsin \frac{r_2-r_1}{2 }\right) \Bigg] \mp \left({ 1\over r_1}
+{\delta \over r_2} \right) .
\end{gather*}
We shall restrict ourselves in the sequel (as before) to the
simplest choice $\kappa=C_1=C_2=0$ such that we shall work with the
``elliptic'' superpotential I
\[
\hat{W}(u,v) = - \frac{2 (1+\delta) u}{\bar{\hbar}} +\frac{ 2
(1-\delta) v}{\bar{\hbar}} , \qquad \hat{W}(x_1,x_2)=-\frac{2
r_1}{\bar{\hbar}} - \frac{2 \delta r_2}{\bar{\hbar}} ,
\]
because this election is signif\/icative and contains enough
complexity.
Considering families of superpotentials related to the same physical
system in our 2D framework dif\/fers from a similar analysis on the 1D
SUSY oscillator, see e.g.~\cite{Mielnik}, in two aspects: (a)~Because we solve the Poisson equation, not the Riccati equation, the
family of superpotentials induces dif\/ferent families of potentials in
both $\hat{h}^{(0)}$ and $\hat{h}^{(2)}$. (b) $\hat{h}^{(0)}$ and
$\hat{h}^{(2)}$ are not iso-spectral because are not directly
intertwined. The spectrum of $\hat{h}^{(1)}$ is the union of the
spectra of $\hat{h}^{(0)}$ and $\hat{h}^{(2)}$.
\subsection{Fermionic zero modes I}\label{sec3.1}
The fermionic zero modes
\begin{gather*} \hat{C}_+ \Psi_0^{(1)}(u,v)=0 \quad , \quad
\hat{C}_-\Psi_0^{(1)}(u,v)=0 , \qquad
\Psi_0^{(1)}(u,v)= \left(\begin{array}{c} 0 \\
\psi^{(1)1}_0(u,v) \\ \psi^{(1)2}_0(u,v) \\ 0
\end{array}\right),
\\
\Psi_0^{(1)}(u,v)= \left(\begin{array}{c} 0 \\
\psi^{(1)1}_0(u,v) \\ \psi^{(1)2}_0(u,v) \\ 0
\end{array}\right)=\frac{A_1}{\sqrt{u^2-v^2}}\left(
\begin{array}{c} 0\\
e^{\frac{-\hat{F}(u)+\hat{G}(v)}{\bar{\hbar}}}\\ 0 \\ 0
\end{array} \right)\ +\ \frac{A_2}{\sqrt{u^2-v^2}}\left(
\begin{array}{c} 0\\0\\
e^{\frac{\hat{F}(u)-\hat{G}(v)}{\bar{\hbar}}}\\ 0
\end{array} \right)
\end{gather*}
are fermionic ground states if normalizable. Because the norm is:
\begin{gather*} N(\bar{\hbar})=2\int_{-1}^1 \!\! dv\int_{1}^{\infty} \!\! du \! \left(
\frac{A_1^2}{\sqrt{(u^2-1)(1-v^2)}}\,
e^{-2\frac{\hat{F}(u)-\hat{G}(v)}{\bar{\hbar}}}\!+\frac{A_2^2}{\sqrt{(u^2-1)(1-v^2)}}\,
e^{2\frac{\hat{F}(u)-\hat{G}(v)}{\bar{\hbar}}}\right) ,
\end{gather*}
it is f\/inite if either $A_1=0$ or $A_2=0$. With our choice of sign
in $F(u)$ the fermionic ground state is:
\[
\Psi_0^{(1)}(u,v)= \frac{1}{\sqrt{u^2-v^2}}\left(
\begin{array}{c} 0\\ 0
\\e^{- \frac{ 2 (1+\delta) u + 2 (1-\delta) v}{\bar{\hbar}^2}} \\ 0
\end{array} \right)
\]
and the norm is also given in terms of Bessel functions:
\[
N(\bar{\hbar}) = 2 \int_1^{\infty} d u \int_{-1}^1 d \bar v
{e^{-\frac{4}{\bar{\hbar}^2} (1+\delta) u} \over \sqrt{ u^2 -1} }
{e^{-\frac{4}{\bar{\hbar}^2} (1 - \delta) v} \over \sqrt{1-v^2}}=2
\pi K_0 \left( {4 \over \bar{\hbar}^2} (1+\delta) \right) I_0
\left( {4 c\over \bar{\hbar}^2} (1-\delta) \right) .
\]
It is also possible to give the fermionic ground state in ${\mathbb
R}^2$ using the ${\mathbb S}$ matrix:
\[ \Psi_0^{(1)}(x_1,x_2)={\cal S}\Psi^{(1)}_0(r_1,r_2)=\left(
\begin{array}{c} 0\vspace{1mm}\\
-\frac{(r_1+r_2)}{4 \sqrt{r_1r_2}} \sqrt{\frac{4}{r_1 r_2}
-\frac{r_1}{r_2}-\frac{r_2}{r_1}+2}
\vspace{1mm}\\ i\frac{(r_2-r_1)}{4 \sqrt{r_1r_2}} \sqrt{\frac{4}{r_1 r_2}
-\frac{r_1}{r_2}-\frac{r_2}{r_1}-2} \vspace{1mm}\\ 0
\end{array} \right) e^{-\frac{2 ( \delta r_1
+r_2)}{\bar{\hbar}^2}} .
\]
\begin{figure}[h]\centering
\includegraphics[height=2.5cm]{Guilarte-fig07}
\includegraphics[height=3.0cm]{Guilarte-fig08}
\includegraphics[height=3.cm]{Guilarte-fig09}
\caption{Graphics of $N(\bar{\hbar})$ for:
$\delta=1/2$, $\kappa=0$. $|\Psi_0^{(1)}(x_1,x_2)|^2$ for
$\delta=1/2$, $\bar{\hbar} = 0.2, \, 1, \, 4$ and $10$. Norms:
$N(0.2)= 1.3518\cdot 10^{-45}$, $N(1)=0.0178$, $N(4)=7.3881$,
$N(10)=18.4297$.}
\end{figure}
\section{The bosonic spectral problem I}\label{sec4}
The spectral problem for the scalar Hamiltonians is also separable
in elliptic coordinates. Plugging in the separation ansatz
\[
\hat{h}^{(0)\choose
(2)}\psi^{(0)\choose(2)}_E(u,v)=E\psi^{(0)\choose(2)}_E(u,v) , \qquad
\psi^{(0)\choose(2)}_E(u,v)=\eta^{(0)\choose(2)}_E(u)\xi^{(0)\choose(2)}_E(v)
\]
in the above spectral equation we f\/ind:
\begin{gather*}
\left[-\bar\hbar^2(u^2-1)\frac{d^2}{du^2}-\bar\hbar^2u\frac{d}{du}+
\left(4 \frac{(1+\delta)^2}{\bar{\hbar}^2} (u^2-1) \mp 2 (1+\delta)
u - 2 E
u^2 \right)
\right] \eta^{(0)\choose(2)}_E(u)\\
\qquad{} =I \eta^{(0)\choose(2)}_E(u),
\\
\left[-\bar\hbar^2(1-v^2)\frac{d^2}{dv^2}+\bar\hbar^2v\frac{d}{dv}+
\left(4 \frac{(1-\delta)^2}{\bar{\hbar}^2} (1-v^2) \mp 2 (1-\delta)
v +2 E v^2 \right) \right] \xi^{(0)\choose(2)}_E(v)\\
\qquad{} = - I
\xi^{(0)\choose(2)}_E(v),
\end{gather*}
where $I$ is the eigenvalue of the symmetry operator
$\hat{I}=-\{\hat{h}^{(0)\choose (2)}+\hat{I}_2^{(0)\choose (2)}\}$.
Research on the solution of these two ODE's by power series
expansions will be published elsewhere. Here, we shall describe how
another change of variables transmutes the f\/irst ODE into Razavy
equation~\cite{Razavy1,Bondar}:
\[
-\frac{d^2\eta_{\pm}(x)}{dx^2}+ \left( \zeta_{\pm} \cosh 2x- M_\pm
\right) ^2 \eta_{\pm}(x)= \lambda_{\pm}\ \eta_{\pm}(x) , \qquad
x={1\over 2} {\rm arccosh}\, u
\]
with parameters:
\begin{gather*}
\zeta_{\pm}= \pm {2\over \bar{\hbar} } \sqrt{\frac{4}{\bar{\hbar}^2} (1+\delta)^2 - 2
E_\pm} , \qquad \lambda_{\pm}= M_\pm^2 + {4\over \bar{\hbar}^2}
\left(I_{\pm} + 4 \frac{(1+\delta)^2}{\bar\hbar^2}\right) ,\\
M_\pm^2 ={2
(1+\delta)^2 \over 2 (1+\delta)^2 -\bar\hbar^2 E_\pm}
\end{gather*}
Simili modo, another change of variables leads from the second ODE
to the Whittaker--Hill or Razavy trigonometric~\cite{Razavy2,Bondar}, equation
\[
{ d^2 \xi_{\pm}(y) \over d
y^2} + ( \beta_{\pm} {\rm cos} 2 y - N_\pm)^2 \ \xi_{\pm}(y) =
\mu_{\pm}\ \xi_{\pm}(y) , \qquad y={1\over 2} \arccos v\in
\big[0,\tfrac{\pi}{2}\big]
\]
with parameters:
\begin{gather*}
\beta_{\pm} = \mp{2\over \bar{\hbar}} \sqrt{\frac{4}{\bar\hbar^2}
(1-\delta)^2 - 2 E_\pm} , \qquad N^2_\pm = {2 (1-\delta)^2 \over 2
(1-\delta)^2 - \bar{\hbar}^2 E_\pm} , \\ \mu_{\pm} = N^2_\pm +
{4\over \bar{\hbar}^2} \left(I_{\pm} + \frac{4}{\bar{\hbar}^2}
(1-\delta)^2\right) .
\end{gather*}
Therefore, the spectral problem in the scalar sectors is tantamount
to the solving of two entangled sets~-- one per each pair~(E,I)~-- of
Razavy and Whittaker--Hill equations. If $M_\pm=n_1^\pm+1$,
$n_1^\pm\in {\mathbb N}^+$, the Razavy equation is QES; i.e., there
are known $n+1$ f\/inite eigenfunctions with an eigenvalue, see~\cite{FGR}:
\[E_\pm=E_{n_1^\pm} = 2
\frac{(1+\delta)^2}{\bar\hbar^2} \left(
1-\frac{1}{(n_1^\pm+1)^2}\right). \]
This means that for those values of $E_\pm$ one expects bound
eigenstates of the SUSY Hamiltonian, although the $v$-dependence
cannot be identif\/ied. If $N_\pm=n_2^\pm+1$, $n_2^\pm\in {\mathbb
N}^+$, there exist f\/inite eigenfunctions of the Whittaker--Hill
equation, with eigenvalues:
\[E_\pm=E_{n_2^\pm} = 2
\frac{(1-\delta)^2}{\bar\hbar^2} \left(
1-\frac{1}{(n_2^\pm+1)^2}\right).\]
Again, one expects these values of $E_\pm$ to be eigenvalues of the
SUSY Hamiltonian, although their eigenfunctions are expected to be
non-normalizable (except the $n_2^+=0$ case) because the
$u$-dependent part of the eigenfunction is non f\/inite and $u$ is a
non-compact variable. In any case, $n_1^+=n_2^+=0$ gives the bosonic
zero mode.
\subsection{The fermionic spectrum I}\label{sec4.1}
The eigenfunctions of the matrix Hamiltonian,
\begin{gather*} \hat{h}_{11}^{(1)}= -{1\over 2} \bar\hbar^2
\left( {\partial^2 \over
\partial x_1^2} + {\partial^2 \over \partial x_2^2} \right) +
{2\over\bar\hbar^2} \left[ 1 + \delta^2 + \delta \left(
{r_1\over r_2} +{r_2\over r_1} -{4 \over r_1 r_2} \right)
\right]\\
\phantom{\hat{h}_{11}^{(1)}=}{} - \left(\frac{(x_1-1)^2-x^2_2}{r_1^3} +
\delta\frac{(x_1+1)^2-x_2^2}{r_2^3} \right), \\
\hat{h}_{12}^{(1)} = \hat{h}_{21}^{(1)} = -2 \left(
\frac{x_2(x_1-1)}{r_1^3} + \delta\frac{x_2(x_1+1)}{r_2^3} \right),
\\
\hat{h}_{22}^{(1)} = -{1\over 2} \bar\hbar^2 \left( {\partial^2
\over \partial x_1^2} + {\partial^2 \over \partial x_2^2} \right) +
{2\over\bar\hbar^2} \left[ 1 + \delta^2 + \delta \left( {r_1\over
r_2} +{r_2\over r_1} -{4 \over r_1 r_2} \right) \right]
\\ \phantom{\hat{h}_{22}^{(1)} =}{}+ \left(\frac{(x_1-1)^2-x^2_2}{r_1^3} +
\delta\frac{(x_1+1)^2-x_2^2}{r_2^3} \right) ,
\end{gather*}
except the fermionic ground states, follows easily from the SUSY
algebra. The fermionic eigenfunctions in elliptic coordinates,
\begin{gather*}
\Psi^{(1)}_{E_+}(u,v)=\hat{C}_+\Psi^{(0)}_{E_+}(u,v)=\!\left(\!\!\!\begin{array}{c} 0\\\psi_{E_+}^{(1)1}(u,v)\\
\psi_{E_+}^{(1)2}(u,v)\\ 0
\end{array}\!\!\!\right)\!=-i\sqrt{\bar{\hbar}}\left(\!\!\!\begin{array}{c} 0\\e^u_1\nabla^-_u\psi_{E_+}^{(0)}(u,v)\\
e^v_2\nabla^-_v\psi_{E_+}^{(0)}(u,v)\\ 0
\end{array}\!\!\!\right) \! , \quad
E_+=\left\{\!\!\begin{array}{c}E_{n_1^+},\\ E_{n_2^+}, \end{array}\right.\!\!\!
\\
\Psi^{(1)}_{E_-}(u,v)=\hat{C}_-\Psi^{(2)}_{E_-}(u,v)
=\!\left(\!\!\!\begin{array}{c} 0\\\psi_{E_-}^{(1)1}(u,v)\\
\psi_{E_-}^{(1)2}(u,v)\\ 0
\end{array}\!\!\!\right)\!=-i\sqrt{\bar{\hbar}}\left(\!\!\!\begin{array}{c} 0\\-e^v_2\nabla^+_v\psi_{E_-}^{(2)}(u,v)\\
e^u_1\nabla^+_u\psi_{E_-}^{(2)}(u,v)\\ 0
\end{array}\!\!\!\right)\! , \quad
E_-=\left\{\!\!\begin{array}{c}E_{n_1^-},\\ E_{n_2^-},\end{array}\right.\!\!\!
\end{gather*}
in Cartesian coordinates $\Psi_{E_\pm}^{(1)}(x_1,x_2)={\cal
S}\Psi^{(1)}_{E_\pm}(u,v)$ read:
\begin{gather*}
\Psi^{(1)}_{E_+}(x_1,x_2)=\hat{Q}_+\Psi^{(0)}_{E_+}(x_1,x_2)=\!\left(\!\!\!\begin{array}{c} 0\vspace{1mm}\\ \psi_{E_+}^{(1)1}(x_1,x_2)\vspace{1mm}\\
\psi_{E_+}^{(1)2}(x_1,x_2)\vspace{1mm}\\ 0
\end{array}\!\!\!\right)\!=-i\sqrt{\bar{\hbar}}\left(\!\!\!\begin{array}{c} 0 \vspace{1mm} \\(\bar\hbar\frac{\partial}{\partial x_1}
-\frac{\partial W}{\partial x_1})\psi_{E_+}^{(0)}(x_1,x_2)\vspace{1mm}\\
(\bar\hbar\frac{\partial}{\partial x_2}-\frac{\partial W}{\partial
x_2})\psi_{E_+}^{(0)}(x_1,x_2)\vspace{1mm}\\ 0
\end{array}\!\!\!\right)\!,
\\
\Psi^{(1)}_{E_-}(x_1,x_2)=\hat{Q}_-\Psi^{(2)}_{E_-}(x_1,x_2)=\!\left(\!\!\!\begin{array}{c} 0\vspace{1mm}\\ \psi_{E_-}^{(1)1}(x_1,x_2)\vspace{1mm}\\
\psi_{E_-}^{(1)2}(x_1,x_2)\vspace{1mm}\\ 0
\end{array}\!\!\!\right)=-i\sqrt{\bar{\hbar}}\left(\!\!\!\begin{array}{c} 0\vspace{1mm}\\ (-\bar\hbar\frac{\partial}{\partial x_2}
-\frac{\partial W}{\partial x_2})\psi_{E_-}^{(2)}(x_1,x_2)\vspace{1mm}\\
(\bar\hbar\frac{\partial}{\partial x_1}+\frac{\partial W}{\partial
x_1})\psi_{E_-}^{(2)}(x_1,x_2)\vspace{1mm}\\ 0
\end{array}\!\!\!\right)\! .
\end{gather*}
\section{Two centers of the same strength}\label{sec5}
If the two centers have the same strength, $\delta=1$, the spectral
problem in the scalar sectors becomes tantamount to the two ODE's:
\begin{gather*}
\left[-\bar\hbar^2(u^2-1)\frac{d^2}{du^2}-\bar\hbar^2u\frac{d}{du}+
\left( \frac{16}{\bar{\hbar}^2} (u^2-1) \mp 4 u - 2 E
u^2 \right)
\right] \eta^{(0)\choose(2)}_E(u)=I \eta^{(0)\choose(2)}_E(u),
\\
\left[-\bar\hbar^2(1-v^2)\frac{d^2}{dv^2}+\bar\hbar^2v\frac{d}{dv}+
2 E v^2 \right] \xi^{(0)\choose(2)}_E(v) = - I
\xi^{(0)\choose(2)}_E(v).
\end{gather*}
Identical changes of variables as those performed in the
$0\leq\delta\leq 1$ cases now lead to the Razavy and Mathieu
equations \cite{Ince}, with parameters:
\begin{gather*}
-\frac{d^2\eta_{\pm}(x)}{dx^2}+ \left( \zeta_{\pm} \cosh 2x- M_\pm
\right) ^2 \eta_{\pm}(x)= \lambda_{\pm}\ \eta_{\pm}(x) ,
\\
\zeta_{\pm}= \pm {2\over \bar{\hbar} } \sqrt{\frac{16}{\bar{\hbar}^2} - 2 E_\pm}
,
\qquad M_\pm^2 ={8\over 8 -\bar\hbar^2 E_\pm} , \qquad
\lambda_{\pm}= M_\pm^2 + {4\over \bar{\hbar}^2} \left(I_{\pm} +
\frac{16}{\bar\hbar^2}\right) ,
\\ - { d^2 \xi_{\pm}(y) \over d y^2} + ( \alpha_\pm \cos 4 y
+ \sigma_{\pm} )\ \xi_{\pm}(y) = 0,
\qquad
\alpha_\pm = {4 E_\pm \over \bar{\hbar}^2} ,
\qquad \sigma_{\pm} = {4\over \bar{\hbar}^2} (I_{\pm} + E ) .
\end{gather*}
We now select the three lowest energy levels from f\/inite solutions of the Razavy
equation:
\[
E_0=0 , \qquad E_1={6\over \bar\hbar^2} , \qquad
E_2={64\over 9\bar\hbar^2}. \]
The corresponding eigenfunctions of the Razavy Hamiltonians for
$n_1^\pm=0,1,2$ are:
\begin{gather*}
\eta_{\pm}^{01}(u) = e^{\mp {4 u \over \bar{\hbar}^2}} , \qquad
\eta_{\pm}^{11} (u) = e^{\mp { 2 u \over \bar{\hbar}^2}}
\sqrt{2 (u+1)},\qquad
\eta_{\pm}^{12}(u) =- e^{\mp { 2 u \over \bar{\hbar}^2}} \sqrt{2
(u-1)},
\\ \eta_{\pm}^{21} (u) = - 2 e^{\mp {4 u \over 3
\bar{\hbar}^2}} \sqrt{ u^2 - 1} , \qquad
\eta_{\pm}^{22}(u) =\pm {3 \bar{\hbar}^2 \over 8} e^{\mp {4 u \over
3 \bar{\hbar}^2}} \left[ \pm {16 \over 3 \bar{\hbar}^2} u - 1 +
\sqrt{ 1 + { 256 \over 9
\bar{\hbar}^4}} \right], \\
\eta_{\pm}^{23}(u) =\pm {3 \bar{\hbar}^2 \over 8} e^{\mp {4 u \over
3 \bar{\hbar}^2}} \left[ \pm {16 \over 3 \bar{\hbar}^2} u - 1 -
\sqrt{ 1 + { 256 \over 9 \bar{\hbar}^4}} \right].
\end{gather*}
The energy degeneracy is labeled by the eigenvalues of the symmetry
operator
\begin{gather*}
I^{nm}_{\pm} = \frac{\bar{\hbar}^2}{4} (
\lambda^{nm}_{\pm} - (n+1)^2) - {16\over \bar{\hbar}^2} ,
\qquad m=1, 2, \dots, n+1,
\\
I_{\pm}^{01} = 0 , \qquad
I_{\pm}^{11} = -{\bar{\hbar}^2 \over 4} - \frac{12}{\bar{\hbar}^2} \mp
2 ,\qquad
I_{\pm}^{12} = -{\bar{\hbar}^2 \over 4} - \frac{12}{\bar\hbar^2} \pm
2,\\
I_{\pm}^{21} = - \bar{\hbar}^2 - {128 \over 9 \bar{\hbar}^2}
, \qquad
I_{\pm}^{22} = -{\bar{\hbar}^2 \over 2} - {128\over 9 \bar{\hbar}^2
} - {1\over 6} \sqrt{ 256 + 9 \bar{\hbar}^4},\\
I_{\pm}^{23} = -{\bar{\hbar}^2 \over 2} - {128\over 9
\bar{\hbar}^2} + {1\over 6} \sqrt{ 256 + 9 \bar{\hbar}^4}.
\end{gather*}
In this rotationally non-invariant system the symmetry operator
$\hat{I}$ replaces the orbital angular momentum in providing a basis
of common eigenfunctions with $\bar{H}$ in each degenerate in energy
sub-space of the Hilbert space in such a way that a quantum number
reminiscent of the orbital angular momentum arises.
Next we next consider even/odd in $v$ solutions of the Mathieu equations
\begin{gather*}
\xi^{nm}_{\pm{\rm even}}(v) = \frac{c_1}{2} \left( C\left[ a^{nm}_{\pm},
q_n, \arccos( v ) \right] + C\left[ a^{nm}_{\pm}, q_n, \arccos(-
v ) \right] \right) \\
\phantom{\xi^{nm}_{\pm{\rm even}}(v) =}{} + \frac{c_2}{2} \left( S\left[ a^{nm}_{\pm}, q_n, \arccos( v )
\right] + S\left[ a^{nm}_{\pm}, q_n, \arccos(- v ) \right] \right),
\\
\xi^{nm}_{\pm{\rm odd}}(v) = \frac{d_1}{2} \left( C\left[ a^{nm}_{\pm},
q_n, \arccos( v ) \right] - C\left[ a^{nm}_{\pm}, q_n, \arccos(-
v ) \right] \right) \\
\phantom{\xi^{nm}_{\pm{\rm odd}}(v) =}{} + \frac{d_2}{2} \left( S\left[ a^{nm}_{\pm}, q_n, \arccos( v )
\right] - S\left[ a^{nm}_{\pm}, q_n, \arccos(- v ) \right] \right).
\end{gather*}
The reason is that the invariance of the problem under
$r_1\leftrightarrow r_2\equiv v\leftrightarrow -v$ -- the exchange
between the two centers~-- forces even or odd eigenfunctions in $v$.
The parameters of the Mathieu equations determined by the spectral
problem are:
\[
a^{nm}_{\pm} = - \frac{\sigma^{nm}_{\pm}}{4} = -\frac{E_n +
I^{nm}_{\pm}}{\bar{\hbar}^2} , \qquad q_n =
\frac{\alpha_n}{8}= \frac{E_n} {2\bar{\hbar}^2} .
\]
To f\/it in with the parameters of the Razavy Hamiltonians set by
$n_1^\pm=0,1,2$ we must choose: \begin{gather*}
q_0 = 0 , \qquad a^{01}_{\pm} = 0, \\
q_1 = \frac{3}{\bar{\hbar}^4} , \qquad
a^{11}_{\pm} = {6 \over \bar{\hbar}^4} \pm\frac{
2}{\bar{\hbar}^2} + \frac{1}{4}, \qquad a^{12}_{\pm} = {6 \over
\bar{\hbar}^4} \mp \frac{ 2}{\bar{\hbar}^2} + \frac{1}{4}, \\
q_2 = \frac{32}{9 \bar{\hbar}^4} , \qquad
a^{21}_{\pm} = 1 + {64 \over 9 \bar{\hbar}^4} , \qquad
a^{22}_{\pm} =\frac{1}{2} + {64 \over 9\bar{\hbar}^4} +
{1\over 6 \bar{\hbar}^2 } \sqrt{256 + 9 \bar{\hbar}^4}, \\
a^{23}_{\pm} = {1 \over 2} + {64 \over 9\bar{\hbar}^4} - {1\over
6\bar{\hbar}^2} \sqrt{ 256 + 9 \bar{\hbar}^4} .
\end{gather*}
Therefore,
\[ \psi^{(0)}_{nm\,\,{\rm even/odd}}(u,v) = \eta^{nm}_+
(u) \xi^{nm}_{+{\rm even/odd}}(v) , \qquad n\geq 0, \qquad
m=1,2, \dots ,m+1 ,
\]
is a set of bound states of non-zero energy of the scalar
Hamiltonian of two SUSY Coulombian centers of the same strength. The
paired fermionic eigenstates are obtained through the action of
the appropriate supercharge. Since every non-zero-energy state come
in bosonic-fermionic pairs, the criterion for spontaneous
supersymmetry breaking is the existence of a~Fermi--Bose pair ground
state of positive energy connected one with each other by one of the
supercharges. Our SUSY Hamiltonian has both bosonic and fermionic
zero modes as single ground states; consequently, supersymmetry is
not spontaneously broken in this system.
In the next f\/igures we show several graphics of bosonic SUSY
eigenfunctions for the choice: $c_2=d_2=0$, $c_1=d_1=1$.
\begin{figure}[h]\centering
\includegraphics[height=3cm]{Guilarte-fig10} \
\includegraphics[height=3cm]{Guilarte-fig11}
\includegraphics[height=3cm]{Guilarte-fig12} \
\includegraphics[height=3cm]{Guilarte-fig13} \
\includegraphics[height=3cm]{Guilarte-fig14}
\caption{Graphics with ($\bar{\hbar}=1$):
$|\psi^{01}_{+{\rm even}}(x_1,x_2)|^2$; $|\psi^{11}_{+{\rm
even}}(x_1,x_2)|^2$ and $|\psi^{11}_{+{\rm odd}}(x_1,x_2)|^2$;
$|\psi^{21}_{+{\rm even}}(x_1,x_2)|^2$ and $|\psi^{21}_{+{\rm
odd}}(x_1,x_2)|^2$.}
\end{figure}
\subsection{Bosonic and fermionic ground states for two centers of the same strength}\label{sec5.1}
For comparison, we also plot the bosonic and fermionic ground states
for several values of $\bar\hbar$.
\begin{figure}[h]\centering
\includegraphics[height=2.75cm]{Guilarte-fig15}
\includegraphics[height=2.75cm]{Guilarte-fig16}
\includegraphics[height=2.75cm]{Guilarte-fig17}
\includegraphics[height=2.75cm]{Guilarte-fig18}
\caption{Graphics of the probability
density $|\Psi_0^{(0)}(x_1,x_2)|^2$ for $\delta=1$, and the values
of $\bar{\hbar} = 0.2, \, 1, \, 4$ and $10$.}
\end{figure}
\begin{figure}[h]
\includegraphics[height=2.7cm]{Guilarte-fig19}
\includegraphics[height=2.7cm]{Guilarte-fig20}
\caption{Graphics of
$|\Psi_0^{(1)}(x_1,x_2)|^2$ for $\delta=1$, and $\bar{\hbar} = 0.2,
\, 1, \, 4$ and $10$. The norms are: $N(0.2)= 7.7012\cdot 10^{-88}$,
$N(1)=0.0009$, $N(4)=5.8083$, $N(10)=16.6347$.}
\end{figure}
\section{The superpotential II for the two-center problem}\label{sec6}
There is another possibility to f\/ind the potential energy of two
Coulombian centers in a 2D ${\cal N}=2$ SUSY Hamiltonian. The
superpotential must satisfy the Hamilton--Jacobi equation rather than
the Poisson equation (\ref{eq:sups}):
\begin{equation} \frac{ 1}{r_1}+\frac{\delta}{r_2}=\frac{1}{2}
\left( \frac{\partial W}{\partial x_1}\right)^2+\frac{1}{2} \left(
\frac{\partial W}{\partial x_2}\right)^2.\label{eqHJ}
\end{equation}
Note that in this case the two centers must be repulsive to
guarantee a real $W$. This point of view, which follows the path
shown in \cite{Manton} and \cite{Heumann} for the Coulomb problem,
amounts to the quantization of a classical supersymmetric system,
only semi-positive def\/inite for repulsive potentials.
Again using elliptic coordinates, the Hamilton--Jacobi equation
separates
\[ \kappa =-(u^2-1) \left({dF\over d u}\right)^2 + 2 (1+\delta) u
, \qquad \kappa = (1-v^2) \left({dG\over d v}\right)^2 +2
(\delta-1) v \label{kappa}
\]
by plugging in (\ref{eqHJ}) the ansatz:
\[
W(u,v;\kappa)=F_a(u;\kappa) + G_b(v;\kappa) , \qquad a,b=0,1
.
\]
The quadratures
\[
F_a(u;\kappa)= (-1)^a\, \int_{1}^u \frac{\sqrt{2 (1+\delta)
u-\kappa}}{\sqrt{u^2-1}}\, du , \qquad G_b(v;\kappa)= (-1)^b\,
\int_{-1}^v \frac{\sqrt{2 (1-\delta) v+\kappa}}{\sqrt{1-v^2}}\, dv
\]
show that the separation constant $\kappa$ is constrained in order
to f\/ind real $F_a(u)$ and $G_b(v)$: $2(1-\delta) \leq \kappa \leq 2
(1+\delta)$. Note that there are two dif\/ferent possibilities: $a=b$
and $a\neq b$. A~dif\/ferent global sign only exchanges the $(0)$ with
the $(2)$ and the $(1)1$ with the $(1)2$ sectors. The superpotential
II is thus given in terms of incomplete and complete elliptic
integrals of the f\/irst and second type, see \cite{Abramowitz,Byrd}:
\begin{gather*} F_a(u;\kappa) = (-1)^a 2 i
\sqrt{\kappa+2(1{+}\delta) } \left( E\left[ \sin^{-1}
\sqrt{\frac{\kappa-2(1{+}\delta)u}{\kappa-2(1{+}\delta)
}},\frac{\kappa-2(1{+}\delta)}{\kappa+2(1{+}\delta) }\right]\right.\\ \left.
{} -E\left[
{\pi\over 2},\frac{\kappa-2(1{+}\delta)}{\kappa+2(1{+}\delta)
}\right]-F\left[ \sin^{-1}
\sqrt{\frac{\kappa-2(1{+}\delta)u}{\kappa-2(1{+}\delta)}},\frac{\kappa-2(1{+}\delta)}{\kappa+2(1{+}\delta)
}\right]+F\left[{\pi\over
2},\frac{\kappa-2(1{+}\delta)}{\kappa+2(1{+}\delta)}\right]\right),
\\ G_b(v;\kappa) = (-1)^b 2 i
\sqrt{\kappa-2(1{-}\delta) }\\
{}\times\left(- E\left[ \sin^{-1}
\sqrt{\frac{\kappa+2(1{-}\delta)v}{\kappa+2(1{-}\delta)
}},\frac{\kappa+2(1{-}\delta)}{\kappa-2(1{-}\delta)}\right]+E\left[ {\rm
sin}^{-1}\sqrt{\frac{\kappa-2(1{-}\delta)}{\kappa+2(1{-}\delta)}},\frac{\kappa+2(1{-}\delta)}{\kappa-2(1{-}\delta)
}\right]\right. \\ {} +\left. F\left[ \sin^{-1}
\sqrt{\frac{\kappa+2(1{-}\delta)v}{\kappa+2(1{-}\delta)}},\frac{\kappa+2(1{-}\delta)}{\kappa-2(1{-}\delta)
}\right] - F\left[{\rm
sin}^{-1}\sqrt{\frac{\kappa-2(1{-}\delta)}{\kappa+2(1{-}\delta)}},\frac{\kappa+2(1{-}\delta)}{\kappa-2(1{-}\delta)
}\right]\right).
\end{gather*}
The scalar Hamiltonians, both in elliptic and Cartesian coordinates,
read:
\begin{gather*} \hat{h}^{(0)\choose(2)}={1\over
2(u^2-v^2)}\Bigg\{-\bar\hbar^2\left((u^2-1)\frac{d^2}{du^2}+u\frac{d}{du}+
(1-v^2)\frac{d^2}{dv^2}- v\frac{d}{dv}\right) \\
\phantom{\hat{h}^{(0)\choose(2)}=}{}+
2(1+\delta)u+2(1-\delta)v \\
\phantom{\hat{h}^{(0)\choose(2)}=}{}
\pm\bar\hbar\left[(-1)^a(1+\delta)\sqrt{\frac{u^2-1}{2(1+\delta)u-\kappa}}+
(-1)^b(1-\delta)\sqrt{\frac{1-v^2}{2(1-\delta)v+\kappa}}
\right] \Bigg\},
\\
\hat{h}^{(0)\choose(2)}=-{\bar\hbar^2\over
2}\nabla^2 +\frac{1}{r_1}+\frac{\delta}{r_2}\\
\phantom{\hat{h}^{(0)\choose(2)}=}{} \pm\frac{\bar{\hbar}}{
4 r_1 r_2} \left\{ \frac{(-1)^a (1+\delta) \sqrt{(r_1+r_2)^2\!-4
}}{\sqrt{(1+\delta) (r_1+r_2)-\kappa}} + \frac{(-1)^b (1-\delta)
\sqrt{4-(r_1-r_2)^2}}{\sqrt{ \kappa-(1-\delta) (r_1-r_2)}} \right\} .
\end{gather*}
\subsection{Type IIa and Type IIb two-center SUSY quantum mechanics}\label{sec6.1}
We now present the graphics of the scalar potential for $a=b$, a
system that we shall call Type~IIa ${\cal N}=2$ SUSY two Coulombian
centers. By the same token, the system arising from the
superpotential I will be called Type I ${\cal N}=2$ SUSY two Coulombian
centers.
\begin{figure}[h]\centering
\includegraphics[height=4cm]{Guilarte-fig21}\
\includegraphics[height=4cm]{Guilarte-fig22}
\caption{3D graphics of the quantum
potential $\hat{V}^{(0)}$ for ${\bf a}={\bf b}={\bf 1}$ (or $\hat{V}^{(2)}$ for
${\bf a}={\bf b}={\bf 0}$). We choose $\delta=1/2$, $\kappa=3$. Cases: (a)
$\bar{\hbar}=0.2$, $\bar{\hbar}= 2$. Observe that when $\bar\hbar$
is increased this provides a reduction of the strength of the
repulsive centers and the left center becomes attractive whereas the
right center is still repulsive. (b) $\bar{\hbar}=4$ and
$\bar{\hbar}=10$, both centers are attractive. With increasing
$\bar{\hbar}$ the centers become more and more attractive.}
\end{figure}
If $a\neq b$, we shall call the system Type IIb ${\cal N}=2$ SUSY
two Coulombian centers. The scalar potential is drawn in the
graphics below for several values of $\bar\hbar$.
\begin{figure}[h]\centering
\includegraphics[height=4.8cm]{Guilarte-fig23} \
\includegraphics[height=4.8cm]{Guilarte-fig24}
\caption{3D graphics of the quantum
potential $\hat{V}^{(0)}$ for ${\bf a}={\bf 1}$, ${\bf b}={\bf 0}$ (or
$\hat{V}^{(2)}$ for ${\bf a}={\bf 0}$, ${\bf b}={\bf 1}$). We choose $\delta=1/2$,
$\kappa=3$. Cases: (a) $\bar{\hbar}=0.2$, $\bar{\hbar}= 2$. Observe
that once again increasing $\bar{\hbar}$ provides a~reduction of the
strength of the repulsive centers and the left center becomes
attractive whereas the right center is still repulsive. (b)
$\bar{\hbar}=4$ and $\bar{\hbar}=10$, both centers are attractive.}
\end{figure}
The dif\/ferences between the Type IIa and Type IIb potentials,
increasing with $\bar\hbar$, are shown in the next f\/igure.
\begin{figure}[h]\centering
\includegraphics[height=4.2cm]{Guilarte-fig25}
\caption{3D graphics of the quantum
potential $\hat{V}^{(0)}$ for ${\bf a}={\bf 1}={\bf b}$ and ${\bf a}={\bf 1}$, ${\bf b}={\bf 0}$
in the cases $\bar{\hbar}=1$, $\bar{\hbar}= 10$. In this range we
note the dif\/ferences between Type IIa and IIb quantum potentials.}
\end{figure}
Below we shall discuss the coincidences and dif\/ferences of the three
distinct points of view.
\subsection{The spectral problem}\label{sec6.2}
Both for the Type IIa and Type IIb systems the spectral problem in
the scalar sectors is separable in elliptic coordinates:
\begin{gather*} \hat{h}^{(0)\choose
(2)}\psi^{(0)\choose(2)}_E(u,v)=E\psi^{(0)\choose(2)}_E(u,v) , \qquad
\psi^{(0)\choose(2)}_E(u,v)=\eta^{(0)\choose(2)}_E(u)\zeta^{(0)\choose(2)}_E(v),\\
\left[-\bar\hbar^2(u^2-1)\frac{d^2}{du^2}-\bar\hbar^2u\frac{d}{du}+
2(1+\delta)u-2Eu^2\pm\bar\hbar(-1)^a(1+\delta)\sqrt{\frac{u^2-1}{2(1+\delta)u-\kappa}}\,
\right]\!\eta^{(0)\choose(2)}_E(u)\!\!\\
\qquad{} =I\eta^{(0)\choose(2)}_E(u),
\\
\left[-\bar\hbar^2(1-v^2)\frac{d^2}{dv^2}+\bar\hbar^2v\frac{d}{dv}+
2(1-\delta)v+2Ev^2\pm\bar\hbar(-1)^b(1-\delta)\sqrt{\frac{1-v^2}{2(1-\delta)v-\kappa}} \,\right]\!
\zeta^{(0)\choose(2)}_E(v)\!\!\\
\qquad{} =-I\zeta^{(0)\choose(2)}_E(v) .
\end{gather*}
The separated ODE's are, however, much more dif\/f\/icult (non-linear)
than in the Type I system and there is no hope of f\/inding explicit
eigenvalues and eigenfunctions.
\subsection{Bosonic ground states}\label{sec6.3}
We shall therefore concentrate on searching for the ground states.
First, the bosonic zero modes:
\[
\hat{C}_+\Psi_0^{(0)}(u,v)=0, \qquad
\hat{C}_-\Psi_0^{(2)}(u,v)=0 .
\]
The separation ansatz
\[
\psi^{(0)\choose
(2)}_0(u,v)=\eta^{(0)\choose(2)}_0(u)\zeta^{(0)\choose(2)}_0(v)
\]
makes these equations equivalent to:
\begin{gather*}
e^u_1\nabla_u^- \eta_0^{(0)}(u)=0 , \qquad e^u_1\nabla_u^+
\eta_0^{(2)}(u)=0, \qquad e^v_2\nabla_v^- \zeta_0^{(0)}(v)=0 ,
\qquad -e^v_2\nabla_v^+ \zeta_0^{(2)}(v)=0 ,
\end{gather*}
or,
\begin{gather*}
\left(\bar\hbar\frac{d}{du}\mp(-1)^a\sqrt{\frac{2(1+\delta)u-\kappa}{u^2-1}}\right)\eta^{(0)\choose(2)}_0(u)=0
, \\
\left(\bar\hbar\frac{d}{dv}\mp(-1)^b\sqrt{\frac{2(1-\delta)v+\kappa}{1-v^2}}\right)\zeta^{(0)\choose(2)}_0(v)=0
.
\end{gather*}
The solutions, i.e.\ the bosonic zero modes, are:
\[
\eta_0^{(0)\choose(2)}(v)=\exp\left[\pm
\frac{F_a(u;\kappa)}{\bar\hbar}\right], \qquad
\zeta_0^{(0)\choose(2)}(v)=\exp \left[\pm
\frac{G_b(v;\kappa)}{\bar\hbar}\right].
\]
It is not possible to calculate analytically the norm in these
cases, but we of\/fer a numerical integration of the $F=0$ bosonic
ground states:
\[ N(\bar{\hbar};\kappa,a,b)=2\int_{-1}^1\, dv\int_1^{\infty}\,
du\, \frac{u^2-v^2}{\sqrt{u^2-1}\, \sqrt{1-v^2}} \exp \left[\frac{2 \, F_a(u;\kappa)}{\bar{\hbar}}\right]\exp \left[\frac{2 \, G_b(v;\kappa)}{\bar{\hbar}}\right] \] in the
next f\/igures. We observe that there is one normalizable bosonic
ground state of zero energy of Type IIa and one of Type IIb once the
value of $a$ is set to be one. Remarkably, the Type IIb zero mode
disappears (the norm becomes inf\/inity) at the classical limit.
\begin{figure}[h]\centering
\begin{minipage}[b]{75mm}\centering
\includegraphics[height=4cm]{Guilarte-fig26}
\caption{Numerical plot of
$N(\bar{\hbar};3,1,1)$ as function of $\bar{\hbar}$ for
$\delta=1/2$.}
\end{minipage}\qquad
\begin{minipage}[b]{75mm}\centering
\includegraphics[height=4cm]{Guilarte-fig27}
\caption{Numerical
plot of $N(\bar{\hbar};3,1,0)$ as function of $\bar{\hbar}$ for
$\delta=1/2$.}
\end{minipage}
\end{figure}
Some 3D plots of the Type IIa and Type IIa bosonic zero modes for
several values of $\bar\hbar$ are shown in the next f\/igures:
\begin{figure}[h]\centering
\includegraphics[height=2.7cm]{Guilarte-fig28} \ \includegraphics[height=2.7cm]{Guilarte-fig29}
\caption{3D graphics of the ground state
probability density $|\Psi^{(0)}(x_1,x_2)|^2$, for $\delta=1/2$,
$\kappa=3$, and ${\bf a}={\bf b}={\bf 1}$ (or $|\Psi^{(2)}(x_1,x_2)|^2$ for
${\bf a}={\bf b}={\bf 0}$). Cases: $\bar{\hbar}=0.2$, $\bar{\hbar}= 2$,
$\bar{\hbar}=4$ and $\bar{\hbar}=10$. The norms are
$N(0.2)=0.004914$, $N(2)=2.83912$, $N(4)=22.8914$ and
$N(10)=511.092$.}
\end{figure}
\begin{figure}[h]\centering
\includegraphics[height=2.7cm]{Guilarte-fig30} \
\includegraphics[height=2.7cm]{Guilarte-fig31}
\caption{3D graphics of the ground state
probability density $|\Psi^{(0)}(x_1,x_2)|^2$, for $\delta=1/2$,
$\kappa=3$, and ${\bf a}={\bf 1}$, ${\bf b}={\bf 0}$ (or $|\Psi^{(2)}(x_1,x_2)|^2$
for ${\bf a}={\bf 0}$, ${\bf b}={\bf 1}$). Cases: $\bar{\hbar}=0.2$, $\bar{\hbar}=
2$, $\bar{\hbar}=4$ and $\bar{\hbar}=10$. And the norms are $N(0.2)
= 9.61622 \cdot 10^{20} $, $N(2)=473.903$, $N(4)=287.687$ and
$N(10)=1399.71$.}
\end{figure}
\subsection{Fermionic ground states}\label{sec6.4}
Second, the fermionic ground states of zero energy:
\[
\hat{C}_+\Psi_0^{(1)}(u,v)=0 , \qquad
\hat{C}_-\Psi_0^{(1)}(u,v)=0 .
\]
Unlike the bosonic case where the logic is {\it or} instead
of {\it and}, note that both equations must be satisf\/ied by
the fermionic zero modes. The separation ansatz
\[
\psi_0^{(0)1}(u,v)=\eta_0^{(0)1}(u)\zeta_0^{(0)1}(v) , \qquad
\psi_0^{(0)2}(u,v)=\eta_0^{(0)2}(u)\zeta_0^{(0)2}(v)
\]
makes these equations tantamount to
\begin{gather*}
e^u_1\left(\nabla_u^++\frac{\bar\hbar u}{u^2-v^2}\right)
\psi_0^{(1)1}(u,v)+e^v_2\left(\nabla_v^+-\frac{\bar\hbar v}{u^2-v^2}\right) \psi_0^{(1)2}(u,v)=0, \\
-e^v_2\left(\nabla_v^--\frac{\bar\hbar v}{u^2-v^2}\right)
\psi_0^{(1)1}(u,v)+e^u_1\left(\nabla_u^-+\frac{\bar\hbar u}{u^2-v^2}\right)
\psi_0^{(1)2}(u)=0 ,
\end{gather*}
or,
\begin{gather*}
\bar\hbar\frac{d\eta_0^{(1)1}}{du}+\left(\frac{d
F_a}{du}+\frac{\bar\hbar u}{u^2-v^2}\right)\eta_0^{(1)1}(u)=0
, \qquad \bar\hbar\frac{d\eta_0^{(1)2}}{du}-\left(\frac{d
F_a}{du}-\frac{\bar\hbar u}{u^2-v^2}\right)\eta_0^{(1)2}(u)=0,
\\ \bar\hbar\frac{d\zeta_0^{(1)1}}{dv}-\left(\frac{d
G_a}{dv}+\frac{\bar\hbar v}{u^2-v^2}\right)\zeta_0^{(1)1}(v)=0
, \qquad \bar\hbar\frac{d\zeta_0^{(1)2}}{dv}+\left(\frac{d
G_a}{dv}-\frac{\bar\hbar v}{u^2-v^2}\right)\zeta_0^{(1)2}(v)=0 .
\end{gather*}
The fermionic zero modes have the form of the linear combination:
\begin{gather*}
\Psi_0^{(1)}(u,v)=\frac{1}{\sqrt{u^2-v^2}}\left\{A_1\left(
\begin{array}{c} 0\\
\exp [-\frac{(F_a(u;\kappa)-G_b(v;\kappa)}{\bar{\hbar}}]\\ 0 \\
0
\end{array} \right) + A_2\left(
\begin{array}{c} 0\\0\\
\exp [\frac{F_a(u;\kappa)-G_b(v;\kappa)}{\bar{\hbar}}]\\
0
\end{array} \right)\right\}.
\end{gather*}
Because the norm is
\[
N(\bar{\hbar};\kappa,a,b)=2\int_{1}^{\infty}\int_{-1}^1 \frac{du
dv}{\sqrt{(u^2-1)(1-v^2)}} \left( A_1^2
e^{-2\frac{F_a(u;\kappa)-G_b(v;\kappa)}{\bar{\hbar}}}+A_2^2
e^{2\frac{F_a(u;\kappa)-G_b(v;\kappa)}{\bar{\hbar}}}\right) .
\]
only $A_1=0$ or $A_2=0$ are normalizable, depending on the choice of
$a$ and $b$.
Again, these integrals cannot be computed analytically but the
outcome of numerical calculations is shown in the next f\/igures for
several values of $\bar\hbar$.
\begin{figure}[h]\centering
\includegraphics[height=3.5cm]{Guilarte-fig32} \qquad\quad \includegraphics[height=3.5cm]{Guilarte-fig33}
\caption{Numerical plots of
$N(\bar{\hbar};3,1,1)$ and $N(\bar{\hbar;3,1,0})$ for $A_1=1$,
$A_2=0$ and $\delta=1/2$ ($N(\bar{\hbar};3,0,0)$ and
$N(\bar{\hbar;3,0,1})$ for $A_1=0$, $A_2=1$).}
\end{figure}
In this case we see that the fermionic zero mode of Type IIa does
not have a classical limit whereas the Type IIb fermionic ground
state behaves smoothly near $\bar\hbar=0$.
3D plots of fermionic zero modes, of both Type IIa and IIb, are
shown in the last f\/igures.
\begin{figure}[h]\centering
\includegraphics[height=2.65cm]{Guilarte-fig34}\
\includegraphics[height=2.65cm]{Guilarte-fig35}
\caption{Graphics of
$|\Psi_0^{(1)(1)}(x_1,x_2)|^2$ for $A_1=1$, $A_2=0$, $\delta=1/2$,
$\kappa=3$, and the sign combination: ${\bf a}={\bf b}={\bf 1}$. Cases:
$\bar{\hbar}=0.2, 2, 4$ and $10$. These graphics also represent
$|\Psi_0^{(1)(2)}(x_1,x_2)|$ with $A_1=0$, $A_2=1$, $\delta=1/2$,
$\kappa=3$, and ${\bf a=b=0}$. The norms are $N(0.2)= 1.03792 \cdot
10^{22} $, $N(2)=251.908$, $N(4)=45.7495$ and $N(10)=25.207$.}
\includegraphics[height=2.7cm]{Guilarte-fig36}\
\includegraphics[height=2.7cm]{Guilarte-fig37}
\caption{Graphics of the function
$|\Psi^{(1)(1)}(x_1,x_2)|^2$ for $A_1=1$, $A_2=0$, $\delta=1/2$,
$\kappa=3$, and the sign combination: ${\bf a}={\bf 1}$, ${\bf b}={\bf 0}$. Cases:
$\bar{\hbar}=0.2, 2, 4$ and $10$. These graphics also represent
$|\Psi_0^{(1)(2)}(x_1,x_2)|^2$ for $A_1=0$, $A_2=1$ and ${\bf a}={\bf 0}$,
${\bf b}={\bf 1}$. The norms are $N(0.2)=0.05046$, $N(2)=1.46657$,
$N(4)=3.62192$ and $N(10)=9.20207$.}
\end{figure}
\section{Summary}\label{sec7}
In this paper we have built and studied two types of supersymmetric
quantum mechanical systems starting from two Coulombian centers of
force. Our theoretical analysis could be of interest in molecular
physics seeking supersymmetric spectra closely related to the
spectra of homonuclear or heteronuclear diatomic molecular ions,
e.g., the hydrogen molecular ion or the same system with a proton
replaced by a deuteron. In $H_2^+$ for instance, the f\/irst case, the
value of the non-dimensional quantization parameter is checked to be
$\bar\hbar=0.7$:
\[
\hbar = 1.05 \cdot 10^{-34} \ {\rm kg} \cdot {\rm m}^2 \cdot {\rm
s}^{-1} , \qquad \sqrt{m d \alpha} = 1.493 \cdot
10^{-34} \ {\rm kg }\cdot{\rm m}^2 \cdot {\rm s}^{-1} ,
\]
using the international system of units (SI).
The f\/irst type is def\/ined from a superpotential that solves the
Poisson equation with the potential of the two centers as the
source. In this case, the spectral problem is shown to be equivalent
to entangled families of Razavy and Whittaker--Hill equations. Using
the property of the quasi-exact solvability of these systems, many
eigenvalues of the SUSY system corresponding to bound states have
been identif\/ied when the strengths of the two centers are dif\/ferent.
If the strengths are equal things become easier and some bound
states are also found. In summary, for our simplest choice of Type~I
superpotential the main features of the spectrum are the following:
\begin{itemize}\itemsep=0pt
\item There is an inf\/inite set of discrete energy eigenvalues in the
$F=0$ Bose sub-space of the Hilbert space:
\[
E_n=2\frac{m\alpha^2}{\hbar^2}
(1+\delta)^2\left(1-\frac{1}{(n+1)^2}\right) .
\]
The ionization energy is: $E_\infty =2\frac{m\alpha^2}{\hbar^2}
(1+\delta)^2$, the threshold of the continuous spectrum.
\item There is a sub-space of dimension $n+1$ of degenerate
eigenfunctions with energy $E_n$. The eigenvalues of the symmetry
operator $\hat{I}$
\[
I_{nm}=\frac{\hbar^2}{4}\left((\lambda_{nm}-(n+1)^2\right) ,
\qquad m=1,2, \dots , n+1 ,
\]
where $\lambda_{nm}$ are the $n+1$ roots of the polynomial that
solves the Razavy equation, label a~basis of eigefunctions in each
energy sub-space.
\item In the case $\delta=1$ the eigenfunctions can be
explicitly found. Moreover, the system enjoys a discrete symmetry
under center exchange: $v\leftrightarrow -v$ $(r_1\leftrightarrow
r_2)$. The energy eigenfunctions come in even/odd pairs of functions
of $v$ with respect to this ref\/lection. This fact suggests that the
purely bosonic two f\/ixed centers Hamiltonian
\[
H=-{1\over 2m}\nabla^2+{\alpha\over r_1}+{\alpha\over r_2}+C
\]
enjoys a hidden supersymmetry (if the constant $C$ is greater than
$8\frac{m\alpha^2}{\hbar^2}$) of the kind recently unveiled in
\cite{Ply} for classy one-dimensional systems. This hidden
supersymmetry is spontaneously broken because the even/odd ground
states have positive energy:
$E_\pm=C-8\frac{m\alpha^2}{\hbar^2}>0$.
\item There are eigenfunctions in the $F=1$ Fermi sub-space for
the same eigenvalues $E_n$, $n>0$. Analytically, the Fermi
eigenfunctions are obtained from the Bose eigenfunctions through the
action of $\hat{Q}_+$.
\end{itemize}
The second type starts from superpotentials solving the
Hamilton--Jacobi equation. There are two non-equivalent sign
combinations giving two sub-classes. Both Type IIa and Type IIb
superpotentials are def\/ined in terms of incomplete and complete
elliptic integrals of the f\/irst and second kind. The superpotential
in this approach is no more than the Hamilton's characteristic
function for zero energy and f\/lipped potential. The separability of
the HJ equation in elliptic coordinates means that we can f\/ind a
``complete'' solution of this equation. The spectral problem is,
however, hopeless for this Type.
All the zero-energy ground states, bosonic and fermionic, Type I and
Type IIa/IIb, dif\/ferent strengths and equal strengths, have been
obtained. The cross section ($x_2=0$) of the probability density of
some of the ground states are shown, for the sake of comparison, in
these f\/inal Tables. It is remarkable that, despite of being
analytically very dif\/ferent, Type I and Type II zero modes show
similar patterns.
\begin{table}[htdp]\centering \caption{$\bar{\hbar} = 1$, $\delta=1/2$.}
\vspace{1mm}
\begin{tabular}{|@{}c@{}|@{}c@{\,\,}c@{\,\,}c@{}|} \hline \tsep{2ex}
$|\Psi_{0}^{(0/1)}(x_1,x_2)|^2$ & Type I & $\begin{array}{@{}c@{}} {\rm Type \ IIa}\\ \kappa=3\end{array}$
& $\begin{array}{@{}c@{}} {\rm Type \ IIb} \\ \kappa=3 \end{array}$ \bsep{1ex} \\
\hline \tsep{2ex} $\begin{array}{@{}c@{}} {\rm Bosonic}\\ {\rm zero \ mode}\end{array}$ &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig38}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig39}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig40}} \bsep{2ex}
\\
\hline \tsep{2ex} $\begin{array}{@{}c@{}} {\rm Fermionic}\\ {\rm zero\ mode}\end{array}$ &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig41}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig42}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig43}} \bsep{2ex} \\
\hline
\end{tabular}
\end{table}
\begin{table}[htdp]\centering
\caption{$\bar{\hbar} = 2$, $\delta=1/2$.}
\vspace{1mm}
\begin{tabular}{|@{}c@{}|@{}c@{\,\,}c@{\,\,}c@{}|} \hline \tsep{2ex}
$|\Psi_{0}^{(0/1)}(x_1,x_2)|^2$ & Type I & $\begin{array}{@{}c@{}} {\rm Type \ IIa}\\ \kappa=3\end{array}$
& $\begin{array}{@{}c@{}} {\rm Type \ IIb} \\ \kappa=3 \end{array}$ \bsep{1ex} \\
\hline \tsep{2ex} $\begin{array}{@{}c@{}} {\rm Bosonic}\\ {\rm zero \ mode}\end{array}$ &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig44}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig45}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig46}} \bsep{2ex}
\\
\hline \tsep{2ex} $\begin{array}{@{}c@{}} {\rm Fermionic}\\ {\rm zero \ mode}\end{array}$ &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig47}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig48}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig49}} \bsep{2ex} \\
\hline
\end{tabular}
\end{table}
\begin{table}[htdp]\centering
\caption{$\bar{\hbar} = 4$, $\delta=1/2$.}
\vspace{1mm}
\begin{tabular}{|@{}c@{}|@{}c@{\,\,}c@{\,\,}c@{}|} \hline \tsep{2ex}
$|\Psi_{0}^{(0/1)}(x_1,x_2)|^2$ & Type I & $\begin{array}{@{}c@{}} {\rm Type \ IIa}\\ \kappa=3\end{array}$
& $\begin{array}{@{}c@{}} {\rm Type \ IIb} \\ \kappa=3 \end{array}$ \bsep{1ex} \\
\hline \tsep{2ex} $\begin{array}{@{}c@{}} {\rm Bosonic}\\ {\rm zero \ mode}\end{array}$ &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig50}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig51}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig52}} \bsep{2ex}
\\
\hline \tsep{2ex} $\begin{array}{@{}c@{}} {\rm Fermionic}\\ {\rm zero \ mode}\end{array}$ &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig53}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig54}} &
\parbox{4cm}{\includegraphics[width=4cm]{Guilarte-fig55}} \bsep{2ex} \\
\hline
\end{tabular}
\end{table}
\subsection*{Acknowledgements}
We are grateful to Mikhail Iof\/fe for many enlightening lessons and
conversations on SUSY quantum mechanics as well as letting us know
about reference \cite{Bondar}. JMG thanks Nigel Hitchin for sending
him his unpublished lecture notes on the Dirac operator.
Finally, we recognize f\/inancial support from the Spanish DGICYT and
the Junta de Castilla y Le\'on under contracts: FIS2006-09417,
VAO13C05.
\pdfbookmark[1]{References}{ref}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,973
|
Honza Čopák introduced us to virtual reality. We could not possibly hope for better introduction! Virtual reality impressed us in many aspects and areas as well as a place for potential utilization of cryptocurrencies and blockchain technology.
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Our first digital footprint in virtual reality is 3D model of Blockchain Legal created by Honza. Thank you, Honza!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,963
|
{"url":"https:\/\/nbviewer.jupyter.org\/github\/christopherbronner\/MTA_Turnstile_Data\/blob\/master\/MTA_ridership.ipynb","text":"MTA Turnstile Data\u00b6\n\nThe Metropolitan Transportation Authority (MTA) operates the New York City Subway. On their website, the MTA publishes data from the turnstiles in its subway stations. For each turnstile, passenger entries into and exits out of the subway station are logged accumulatively for four-hour periods: each turnstile has an entry and an exit counter and the data essentially provide the counter values every four hours.\n\nIn this notebook, we will first explore and prepare the turnstile data. Then we will determine the busiest stations and characterize stations as commuter origins or commuter destinations. Thereafter, we will explore the evolution of ridership over the course of the day and the year. Finally, we will build a linear regression model that reproduces the daily ridership.\n\nIn\u00a0[1]:\nfrom IPython.core.display import display, HTML\ndisplay(HTML(\"<style>.container { width:60% !important; }<\/style>\"))\n\n\nFirst, some imports:\n\nIn\u00a0[40]:\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib import gridspec\nfrom mpl_toolkits.basemap import Basemap\n\n\nData Exploration and Preparation \u00b6\n\nThe data comes in indiviual files containing the turnstile data of one week. For now, we will explore one exemplary file from the week preceding 3\/24\/18.\n\nIn\u00a0[3]:\ndata = pd.read_csv('turnstile_180324.txt')\n# Fixes read-in problem of last column name:\ndata = data.rename(columns = {data.columns[-1]:'EXITS'})\n# Remove irrelevant columns\ndata = data.drop(['DIVISION','DESC','LINENAME'], axis=1)\n\nIn\u00a0[4]:\ndata.shape[0]\n\nOut[4]:\n197149\n\nThere are 197,149 samples in this data set.\n\nIn\u00a0[5]:\ndata.head()\n\nOut[5]:\nC\/A UNIT SCP STATION DATE TIME ENTRIES EXITS\n0 A002 R051 02-00-00 59 ST 03\/17\/2018 00:00:00 6552626 2219139\n1 A002 R051 02-00-00 59 ST 03\/17\/2018 04:00:00 6552626 2219140\n2 A002 R051 02-00-00 59 ST 03\/17\/2018 08:00:00 6552626 2219140\n3 A002 R051 02-00-00 59 ST 03\/17\/2018 12:00:00 6552626 2219140\n4 A002 R051 02-00-00 59 ST 03\/17\/2018 16:00:00 6552626 2219140\n\nThe columns C\/A, UNIT, and SCP denote identifiers for a specific turnstile. STATION is the name of the subway station. ENTRIES and EXITS are the accumulated counter values of the turnstile (not the number of entries in the given time interval).\n\nWe will create additional columns to make data evaluation easier:\n\nIn\u00a0[6]:\n# Create timestamp column\ndata['timestamp'] = pd.to_datetime(data['DATE'] + ' ' + data['TIME'])\n\n# Create column with length of interval since last data entry\ndata['interval'] = data['timestamp'] - data['timestamp'].shift(1)\n\n# Calculate number of entries\/exits in the preceding interval\ndata['ENTRY_DIFF'] = data['ENTRIES'] - data['ENTRIES'].shift(1)\ndata['EXIT_DIFF'] = data['EXITS'] - data['EXITS'].shift(1)\n\n\nOut[6]:\nC\/A UNIT SCP STATION DATE TIME ENTRIES EXITS timestamp interval ENTRY_DIFF EXIT_DIFF\n0 A002 R051 02-00-00 59 ST 03\/17\/2018 00:00:00 6552626 2219139 2018-03-17 00:00:00 NaT NaN NaN\n1 A002 R051 02-00-00 59 ST 03\/17\/2018 04:00:00 6552626 2219140 2018-03-17 04:00:00 04:00:00 0.0 1.0\n2 A002 R051 02-00-00 59 ST 03\/17\/2018 08:00:00 6552626 2219140 2018-03-17 08:00:00 04:00:00 0.0 0.0\n3 A002 R051 02-00-00 59 ST 03\/17\/2018 12:00:00 6552626 2219140 2018-03-17 12:00:00 04:00:00 0.0 0.0\n4 A002 R051 02-00-00 59 ST 03\/17\/2018 16:00:00 6552626 2219140 2018-03-17 16:00:00 04:00:00 0.0 0.0\n\nIn order to better understand the structure of the data set, we can look at histograms for the time at which turnstile counter data are reported and the length of the reporting intervals.\n\nIn\u00a0[7]:\nfig, ax = plt.subplots(1,2, figsize=(12,4), gridspec_kw={'wspace': 0.3})\n\n#Plot histogram of times\nax[0].hist(data['TIME'].apply(pd.Timedelta) \/ pd.Timedelta('1 hour'),\nbins=48)\nax[0].set_title('Reporting Times')\nax[0].set_xlabel('Hour of Day')\nax[0].set_xticks(range(24)[::2])\nax[0].set_xlim(0,24)\nax[0].set_ylabel('Samples in Data Set')\n\n# Plot histogram of interval lengths\nax[1].hist(data['interval'].dropna() \/ pd.Timedelta('1 hour'), bins=100)\nax[1].set_title('Reporting Intervals')\nax[1].set_xlabel('Interval Length (Hours)')\nax[1].set_ylabel('Samples in Data Set')\nplt.show()\n\n\nMost reporting times come in four-hour intervals, but at two separate time sequences: a little more than half the reporting times are at 0, 4, 8, 12, 16, and 20 hours (using the 24-hour format) and a large amount of the rest are reported at 1, 5, 9, 13, 17, and 21 hours.\n\nThere are a few samples with other reporting times but we will not take those into account. The small amount of reporting intervals of -168 hours are an artefact stemming from the time difference between the last sample of one turnstile and the first of the next turnstile. We will also eliminate those samples.\n\nIn the following, we remove:\n\n\u2022 The first sample for each turnstile (we can't calculate the difference to the previous one)\n\u2022 Samples with a negative number of entries or exits per interval\n\u2022 Samples with more than 4000 entries\/exits per interval (corresponds to 1 entry\/exit in every 4 seconds, which is the highest frequency (aside from artifacts) found in the data)\n\u2022 Samples with interval lengths that are not around 4 hours\nIn\u00a0[8]:\n# Remove first entry of each turnstile\ndata['NEW_SCP'] = ~ (data['SCP'] == data['SCP'].shift(1))\ndata = data[ (data['NEW_SCP']==False)\n# Remove negative no. of entries\/exits:\n& (data['ENTRY_DIFF']>=0) & (data['EXIT_DIFF']>=0) # only positive entries\/exits\n# Remove entry\/exit rates that are too high\n& (abs(data['ENTRY_DIFF'])<4000) & (abs(data['EXIT_DIFF'])<4000)\n# Remove intervals that are not around 4 hours long\n& (data['interval']\/pd.Timedelta('1 hour')>3.9) & (data['interval']\/pd.Timedelta('1 hour')<4.5)\n].drop('NEW_SCP', axis=1)\n\nIn\u00a0[9]:\ndata.shape[0]\n\nOut[9]:\n189301\n\nAfter eliminating these anomalous samples, we are still left with 189,301 which is about 96% of the original samples.\n\nOf the remaining, realistic entry\/exit data, we can plot a histogram of the number of samples with a certain entry\/exit frequency:\n\nIn\u00a0[10]:\nplt.hist(data['ENTRY_DIFF'], bins=100, alpha=0.5, label='Entries')\nplt.hist(data['EXIT_DIFF'], bins=100, alpha=0.5, label='Exits')\nplt.yscale('log', nonposy='clip')\nplt.xlabel('Entries in a Time Interval')\nplt.ylabel('Samples in Data Set')\nplt.legend()\nplt.show()\n\n\nBusiest Stations\u00b6\n\nWe now want to find out what the busiest stations are. For this purpose we will look at both entries and exits at any given station.\n\nIn\u00a0[11]:\n# Group data by station\ntotal_riders = data.groupby(['STATION'], as_index=False) \\\n[['ENTRY_DIFF','EXIT_DIFF']].sum()\n# Columns for sum and difference of entries and exits\ntotal_riders['TOTAL_DIFF'] = \\\ntotal_riders['ENTRY_DIFF'] + total_riders['EXIT_DIFF']\ntotal_riders['ENTRY_EXIT_DEFICIT'] = \\\ntotal_riders['ENTRY_DIFF'] - total_riders['EXIT_DIFF']\n\n\nThe total_riders DataFrame contains the summed-up entries and exits for each station as well as their sum and difference. By sorting the DataFrame we can immediately tell the busiest stations. 34 St.\/Penn Station is the busiest with 1.7 million turnstile crossings in this week in March 2018.\n\nIn\u00a0[12]:\ntotal_riders.sort_values(by=['TOTAL_DIFF'], ascending=False).head()\n\nOut[12]:\nSTATION ENTRY_DIFF EXIT_DIFF TOTAL_DIFF ENTRY_EXIT_DEFICIT\n59 34 ST-PENN STA 893112.0 768412.0 1661524.0 124700.0\n229 GRD CNTRL-42 ST 781770.0 698025.0 1479795.0 83745.0\n57 34 ST-HERALD SQ 608904.0 572023.0 1180927.0 36881.0\n45 23 ST 624999.0 459962.0 1084961.0 165037.0\n14 14 ST-UNION SQ 580247.0 503932.0 1084179.0 76315.0\n\nWe now want to visualize the weekly ridership for each station on a map. The file stations_conversion.csv contains the geographic coordinates for each station. We will load these data and merge it with the total_riders DataFrame.\n\nIn\u00a0[13]:\nstations = pd.read_csv('stations_conversion.csv')\n\nIn\u00a0[14]:\ntotal_riders = pd.merge(total_riders, stations, on='STATION', how='inner')\n\nOut[14]:\nSTATION ENTRY_DIFF EXIT_DIFF TOTAL_DIFF ENTRY_EXIT_DEFICIT GTFS Latitude GTFS Longitude\n0 1 AV 129461.0 143324.0 272785.0 -13863.0 40.730953 -73.981628\n1 103 ST 173796.0 116370.0 290166.0 57426.0 40.795379 -73.959104\n2 103 ST-CORONA 117369.0 84928.0 202297.0 32441.0 40.749865 -73.862700\n3 104 ST 13785.0 6985.0 20770.0 6800.0 40.688445 -73.841006\n4 110 ST 58883.0 48748.0 107631.0 10135.0 40.795020 -73.944250\n\nNow we can plot the stations as markers on a Basemap map with the size of the markers corresponding to the total ridership of the station. The data are displayed on a satellite image retrieved using arcgisimage.\n\nIn\u00a0[34]:\nplt.figure(figsize=(8, 8))\n\n# Create map with basemap\nm = Basemap(projection='cyl', resolution='i',\nllcrnrlat = total_riders['GTFS Latitude'].min(),\nllcrnrlon = total_riders['GTFS Longitude'].min(),\nurcrnrlat = total_riders['GTFS Latitude'].max(),\nurcrnrlon = total_riders['GTFS Longitude'].max())\nm.arcgisimage(service='ESRI_Imagery_World_2D',\nxpixels=1500, verbose=True)\n\n# Draw stations with marker size according to ridership\nfor line in total_riders.iterrows():\nx,y = m(line[1]['GTFS Longitude'],line[1]['GTFS Latitude'])\nsize = line[1]['TOTAL_DIFF'] \/ 100000\nplt.plot(x, y, 'o', markersize=size, color='red', alpha=0.5,\nmarkeredgewidth=1, markeredgecolor='white')\n\nplt.show()\n\nhttp:\/\/server.arcgisonline.com\/ArcGIS\/rest\/services\/ESRI_Imagery_World_2D\/MapServer\/export?bbox=-74.19055,40.576127,-73.75540500000001,40.903125&bboxSR=4326&imageSR=4326&size=1500,1127&dpi=96&format=png32&f=image\n\n\nFrom this map visualization we can tell that the busiest stations are located in Manhattan, specifically in Midtown.\n\nCommute\u00b6\n\nUsing the MTA ridership data, we can learn about commute in New York City by identifying stations that people commute to (commuter destinations) and stations that people commute from (commuter origins).\n\nWe create two subsets of the data, one for the morning (data_am) and one for the evening (data_pm), and group them by station. We then merge them and create a new column am_pm_difference that reflects the difference in ridership in the morning and in the evening at a given station. More preciesly, a large am_pm_difference arises from more entries in the morning or exits in the evening, as well as less entries in the evening and less exits in the morning.\n\nIn\u00a0[22]:\n# Morning data grouped by station\ndata_am = data[(data['TIME']=='08:00:00') | \\\n(data['TIME']=='09:00:00')]\\\n.groupby('STATION', as_index=False).sum()\n# Evening data grouped by station\ndata_pm = data[(data['TIME']=='20:00:00') | \\\n(data['TIME']=='21:00:00')]\\\n.groupby('STATION', as_index=False).sum()\n\n# Merge morning and evening data\ncommute = pd.merge(data_am, data_pm, on='STATION', suffixes=['am','pm'])\n\n# Calculate difference\ncommute['am_pm_difference'] \\\n= commute['ENTRY_DIFFam'] + commute['EXIT_DIFFpm'] \\\n- commute['ENTRY_DIFFpm'] - commute['EXIT_DIFFam']\n\n\nThis difference (which is a measure for how many riders commute from this station) is plotted in the following histogram. It is immediately clear that more stations are commuter origins than commuter destinations.\n\nIn\u00a0[24]:\nplt.hist(commute[abs(commute['am_pm_difference'])<100000]['am_pm_difference'], bins=100)\nplt.xlabel('AM-PM Difference')\nplt.ylabel('Number of Stations')\nplt.show()\n\n\nWe now want to plot these data for each station. First, we exclude anomalous values. Then, we merge the commute DataFrame with the station locations and plot them using Basemap.\n\nIn\u00a0[25]:\ncommute = commute[abs(commute['am_pm_difference'])<50000]\n\nIn\u00a0[27]:\ncommute = pd.merge(commute, stations, on='STATION', how='inner')\n\nIn\u00a0[33]:\nfig = plt.figure(figsize=(8, 8))\n\n# Create map with basemap\nm = Basemap(projection='cyl', resolution='i',\nllcrnrlat = commute['GTFS Latitude'].min(),\nllcrnrlon = commute['GTFS Longitude'].min(),\nurcrnrlat = commute['GTFS Latitude'].max(),\nurcrnrlon = commute['GTFS Longitude'].max())\nm.arcgisimage(service='ESRI_Imagery_World_2D',\nxpixels = 1500, verbose= True)\n\n# Draw stations with marker sizes according to AM-PM difference\nfor line in commute.iterrows():\nx,y = m(line[1]['GTFS Longitude'],line[1]['GTFS Latitude'])\ndifference = line[1]['am_pm_difference']\nmarker_size = 2000\nif difference > 0: # Commuter origins\nsize = difference \/ marker_size\ncolor = 'blue'\nelse: # Commuter destinations\nsize = - difference\/ marker_size\ncolor = 'red'\n\nplt.plot(x, y, 'o', markersize=size, color=color, alpha=0.5,\nmarkeredgewidth=1, markeredgecolor='white')\n\nplt.show()\n\nhttp:\/\/server.arcgisonline.com\/ArcGIS\/rest\/services\/ESRI_Imagery_World_2D\/MapServer\/export?bbox=-74.073643,40.576127,-73.75540500000001,40.903125&bboxSR=4326&imageSR=4326&size=1500,1541&dpi=96&format=png32&f=image","date":"2021-01-28 06:01:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18348988890647888, \"perplexity\": 10754.468433091723}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610704835901.90\/warc\/CC-MAIN-20210128040619-20210128070619-00123.warc.gz\"}"}
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Newcomers' Info
What is NRDC-ITA
Affiliated Units
Contributing Nations
Former Commanders
NRDC-ITA Mission in Afghanistan
NATO Response Force
Steadfast Jackal 21
Eagle Meteor 19
Eagle Light III
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Liutenant General Giuseppe Emilio Gay
Former Commander
Lieutenant General Giuseppe Emilio Gay attended the Italian Military Academy in Modena, and was commissioned as a 2nd Lieutenant in 1971.
Following two years of specialisation training in Turin, he was assigned as a First Lieutenant to the 182nd "Garibaldi" Armoured Regiment and later, as a Captain, he commanded a tank company in the 13th "M.O. Pascucci" Tank Battalion.
Following the Basic and Advanced Courses of the Italian Army General Staff College, he served as a Staff Officer at the North Eastern District's Estate & Facilities Office, the Army General Staff's Personnel Division, the Army Corps' HQ (Chief G3), the Army General Staff's Logistic Division (Chief G4) and the 1st Command Force of Defence (COS).
LTG Gay's command experience includes Commander of the 7th Tank Battalion "M.O. Di Dio" in Vivaro, Commander of the 1st Armoured Regiment in Teulada, Deputy Commander and Commander of the 132nd "Ariete" Armoured Brigade in Pordenone – during this assignment he commanded the Multinational Brigade West in Pec (Kosovo 1999 – 2000) – Deputy Commander of the Kosovo Force, in Pristina (2003 – 2004), Deputy Commander of the Allied Rapid Reaction Corps (2004 – 2007) – during this appointment he spent a tour as Deputy Commander (Stability) of ISAF IX in Kabul (2006) - and Commander of Land Forces Support HQ.
He has received numerous military decorations, badges and ribbons. Besides this he has been awarded the Italian Army Bronze Medal for Gallantry, the First Class Medal "Don Alfonso HENRIQUES" of the Portuguese Army, the German Army Gold Cross for Honour and Meritorious Officer Cross with Swords of Malta's Sovereign Military Order. He also was conferred with the title of Commander of the Italian Republic Order of Merit and the title of Knight of the Italian Military Order.
LTG Gay holds a Bachelor's degree and a Master's degree in Strategic Sciences from the University of Turin, a Bachelor's degree in International and Diplomatic Sciences from the University of Trieste and a post-graduate degree in Classical Sciences from the "Accademia Agostiniana" – Lateran University of Rome.
Lieutenant General Gay assumed the appointment of Commander NRDC-ITA on 4th September 2007.
On 16 April 2010 He was appointed Commander of the Italian Army Application School, situated in Turin.
He is married to Anna and they have two adult children.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,492
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The Mi Vida Loca Remix palette features 24 rectangular 1.1g eyeshadows set in a square cardboard palette. Interestingly, the palette actually doesn't have a flip cover. Instead, it's housed in a black and silver embossed cardboard slipcover. The packaging is not as convenient as taking the palette out and putting it back into the "box" takes a lot longer than just flipping a lid. With that said, with the size of the palette, it's not really designed to be convenient or travel-friendly.
In terms of the color range, this palette pretty much as it all. From rainbow shimmers to matte neutrals, Mi Vida Loca Remix offers all the staples, including hard-to-find reds and yellows. In terms of the formulation, the majority are pigmented, soft, and smooth. There are a couple that are slightly patchy but given the high matte to shimmer ratio of the palette, I'm still very impressed.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 248
|
require 'coveralls'
Coveralls.wear! do
add_filter 'spec'
end
require 'bigdecimal'
require 'spec_helper/operator_testing'
require 'spec_helper/schema_utils'
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,465
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{"url":"http:\/\/physics.stackexchange.com\/questions\/38561\/magnetic-field-lines?answertab=active","text":"# Magnetic field lines\n\nWhen I searched the net about magnetic field lines, Wikipedia told something about contour lines and that magnetic materials placed along a magnetic field has some specific loci, which i did not understand. Can someone explain it to me in details. Also how do we know that tangent to a magnetic field line gives the direction of magnetic field? I am very confused.\n\n-\nMagnetic field lines are defined so that the the tangents to them give the direction of the magnetic field at each point. It's possible: one may extend any field line by moving infinitesimally in the direction of the field, and repeat that many times. What exactly are you confused by? Which page you were reading? What was unclear about that page to you? Are you serious that you want the users here to explain \"in details\" something that is probably describe with lots of details on Wikipedia, a source contributed by millions of people? \u2013\u00a0 Lubo\u0161 Motl Sep 28 '12 at 12:09\n\nBoth electric & magnetic field lines are introduced as an aid for visualizing electric both fields. These lines are imaginary straight or curved paths along which a free (isolated) pole would travel when placed in magnetic field. In case of electric field lines, the term unit charge is used. The other properties are almost similar for both. These concepts of field lines were introduced by Faraday and both of these fields are related by Maxwell equations.\n\nThe direction of these lines is from North pole to South pole outside the magnet whereas in the inside, it's the other way around. They're thought to flow but, no actual movement occurs. And, they're always continuous closed loops extending throughout the magnetic source and never intersect each other. The crowding or sparsity of these field lines depend upon the intensity of the magnetic source.\n\nThese field lines are best understood by a home-experiment which has been followed for years. Take a wire and cardboard. Throw some metal fillings over the board. Pass some current through the wire and tap the board gently. The fillings rearrange themselves in the form of curves. And the direction of this field is given by Maxwell's right hand cork screw rule. Or, just use a bar magnet instead of catching a live-wire.\n\nWhy are tangents then? Place a magnetic compass somewhere in a magnetic field (say, a bar magnet like the one below). The needle would align tangentially to the magnetic lines of force. Still, if you aren't able to follow, have a look at \"how tangents are drawn\"...\n\n-\n\nI think Crazy Buddy has covered most of it. But I'm just adding a couple of points that may (or may not) help your understanding of this -\n\nThe magnetic field lines (or electric field, or any field lines for that matter) are defined such that the tangent to the field line at a point gives the direction of the field at that point. That's all there is to it, really.\n\nBut since it is after all just a curve in space, it has to have an equation. A plot of these equations will help you visualize what the field lines look like. As this guy at Physics Forum has explained, the equation for a magnetic field line (in polar co-ordinates) in 2 dimensions is given by $$\\frac{B_r}{B_\\theta} = \\frac{dr}{rd\\theta}$$\n\nwhere $B_r$ is the radial component of the magnetic field and $B_\\theta$ is the angular component.\n\nSo a plot of these equations will give you lines that look like the ones you see in textbooks and Wikipedia.\n\nNote: This is the field line equations only for a magnetic dipole (a regular magnet). If you have a more complicated magnetic system, these field equations will no longer hold.\n\n-","date":"2014-03-14 05:16:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6825152635574341, \"perplexity\": 319.8703729833833}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1394678686979\/warc\/CC-MAIN-20140313024446-00017-ip-10-183-142-35.ec2.internal.warc.gz\"}"}
| null | null |
Atlantans
Naaz Malek: Physician's Assistant and 2020 Wonder Woman
Melanie McGriff : Speech Language Pathologist and 2020 Wonder Woman
Naaz Malek | Lead Physician's Assistant, Emory Decatur and Emory Hillandale Hospital
One of the hardest hit parts of the country during COVID-19 has been New York City, which reached its peak case count in April. Medical professionals across the country put their lives on pause to go and assist, including Naaz Malek, the Lead Physician's Assistant at Emory Decatur and Emory Hillandale Hospital.
"My experience working in the Bronx was very humbling," she says. "I witnessed an entire city go comatose because of something invisible. Most of my staff members were volunteers from different parts of the country who put their lives on hold to care for COVID-19 patients. … I am grateful for the opportunity and honored to have been of assistance."
Originally a freelance journalist, Naaz's journey into healthcare was shaped by another crisis—the recession of 2008. "My once upon a time dream of becoming a doctor, which I was too timid to pursue, started to surface as I covered story after story of the devastation brought on to families because of the economy. I felt helpless behind a camera holding a microphone. I wanted to do something." So she applied to PA school and shortly after graduated with a Master's in Health Sciences from the University of South Alabama. "I enjoy working in a fast-paced environment where there is something new and different to do every day. I knew going into clinicals that emergency medicine would be the best field for me," says Naaz.
Like so many in the medical field, Naaz has witnessed the pain and anxiety the lack of social and physical interaction has caused so many. "It hurts me to see my patients alone in their rooms, especially the ones who are elderly, … don't speak English as their first language or are adults barely above the age of 18. These patients feel lost, alone and terrified without a loved one at their side." This change has also affected Naaz and her coworkers. "The pandemic has taken away our most valuable non-verbal means of communication: our personal touch. … I haven't truly seen my coworkers' faces in almost six months. I can hear them laugh, but I can't see them smile."
Recently, Naaz has set up a drive-thru COVID testing site in partnership with Emergent Testing. The site is located at 3110 Lawrenceville-Suwanee Road in Suwanee and offers results within 24-48 hours.
EmoryHealthcare.org
Wyndi Kappes October 13, 2020
Food Expert Alton Brown Talks About Weight Loss and his Four-List Method
40 Tips for Finding Love After 40
Michelle Middlebrooks: Battalion Chief and 2020 Wonder Woman
Wonder Women Host Partner Artisan Beauté
Lori Ellwood: COVID Response Team Manager and 2020 Wonder Woman
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,035
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Stéphane Traineau, född den 16 september 1966 i Cholet, Frankrike, är en fransk judoutövare.
Han tog OS-brons i herrarnas halv tungvikt i samband med de olympiska judotävlingarna 1996 i Atlanta.
Han tog OS-brons igen i samma viktklass i samband med de olympiska judotävlingarna 2000 i Sydney.
Referenser
Källor
Externa länkar
Sports-reference.com
Franska judoutövare
Franska olympiska bronsmedaljörer
Olympiska bronsmedaljörer 1996
Olympiska bronsmedaljörer 2000
Tävlande vid olympiska sommarspelen 1988 från Frankrike
Tävlande i judo vid olympiska sommarspelen 1988
Tävlande vid olympiska sommarspelen 1992 från Frankrike
Tävlande i judo vid olympiska sommarspelen 1992
Tävlande vid olympiska sommarspelen 1996 från Frankrike
Tävlande i judo vid olympiska sommarspelen 1996
Tävlande vid olympiska sommarspelen 2000 från Frankrike
Tävlande i judo vid olympiska sommarspelen 2000
Män
Födda 1966
Levande personer
Personer från Cholet
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,335
|
\section{Introduction}
Sequential recommendations become a crucial task in our daily recommendation scenarios. It is able to learn from a series of correlated browse sequences (e.g., a person may buy shirts, pants, shoes, etc. together), and recommend some new items given the sequence.
Recent years have witnessed a significant advance and success in sequential recommendation.
The prosperity of neural networks (NN) motivates the community to design NN-based sequential recommendation models \cite{hidasi2015session,kang2018self}.
Recently, user privacy has drawn more concerns, and regulations such as General Data Protection Regulation (GDPR)\footnote{https://gdpr-info.eu/} and California Consumer Privacy Act (CCPA)\footnote{https://oag.ca.gov/privacy/ccpa} are set to protect user privacy.
To further preserve users' privacy, Federated Learning (FL) techniques \cite{mcmahan2017communication}, as a new privacy-preserving paradigm, are incorporated with the recommender system.
Prior works of federated recommendation tackle the privacy problem by incorporating FL \cite{ammad2019federated,chai2020secure,han2021deeprec,luopaper1}.
Among them, \cite{ammad2019federated} combines the FL with collaborative filtering as a basic paradigm for the federated recommendation.
Reference \cite{chai2020secure} propose federated matrix factorization for making privacy-preserving recommendations.
Reference \cite{han2021deeprec} resort to tacking the real situation problem by incorporating the sequential recommendation model GRU4REC \cite{hidasi2015session} with FL.
However, current federated recommender systems are facing some challenges.
Firstly, the majority of federated recommendation models only consider the model performance and the privacy-preserving ability, while ignoring the optimization of the communication process. Thus the communication overhead could be a heavy burden for the system.
Secondly, most of the federated recommenders are designed for heterogeneous systems, causing unfairness problems during the federation process.
For example, the clients with a large data size would normally share a higher weight in the model aggregation process, then the aggregated model would turn to prefer this client. The model performance in these clients may be obviously better than those of small clients.
Besides, a large proportion of current federated sequential recommender systems do not sufficiently consider personalized recommendations. In other words, all clients share the model with the same parameters. This could degrade the model performance since different clients may have different interests.
To tackle the aforementioned challenges, we hereby propose a \underline{C}ommunication efficient and \underline{F}air \underline{Fed}erated personalized \underline{S}equential \underline{R}ecommendation algorithm (CF-FedSR), which is a model agnostic algorithm and thus could be fit to other tasks with appropriate adjustments.
Specifically, we first perform client selection and sampling based on client clustering. The client is clustered according to the latent representation thus similar clients could benefit from each other for faster training.
Secondly, we propose a fairness-aware algorithm for model aggregation.
Here we consider the heterogeneity among different clients, and fairness is achieved by designing an adaptive aggregation algorithm to eliminate the impact. Detailedly, we add the fairness coefficient when aggregating the models.
Moreover, the personalized is achieved by local fine-tuning and model adaption.
Our extensive experiments over real-world datasets show the effectiveness of our proposed algorithms, efficiency in time and computing resources, and a tradeoff between privacy and recommendation performance.
Overall, the main contributions of this work are summarized as follows.
{$\bullet$} We propose a Communication efficient and Fair Federated personalized Sequential Recommendation algorithm. To reduce the communication overhead,
we utilize adaptive client selection and sampling based on client clustering to accelerate the training process.
{$\bullet$} We investigate the fairness and personalization problem in the federated sequential recommendation, and propose an adaptive
model aggregation and personalization algorithm as the solution.
The fairness is ensured by adaptively aggregating the uploaded model parameters according to the client feature and performance, and the personalization is achieved by local fine tunes and model adaption.
{$\bullet$} Evaluation on real-world datasets demonstrates the superiority of the proposed model over representative methods. The proposed model brought 9.12\% improvement over competing methods on average while requiring less communication round.
\section{Problem Formulation and Definition}
\label{sec:pf}
In this section, we formally formulate the federated sequential recommendations task and state the current approaches' shortcomings. Also, we define fairness in the federated recommender system.
The federated sequential recommendations task is to train a global recommendation model for a server and many distributed and privacy concerning clients while not disclosing the raw data from the clients.
However, current approaches care less about the communication efficiency, fairness, and personalization of the federated recommenders. We hope to bridge this gap in this paper.
Here we also define the \textit{fairness} of the federated recommender system as the variance between different clients.
For client $\{c_1,...,c_n\}$, with corresponding model performance $\{p_1,...,p_n\}$, then $fairness = variance(\{p_1,...,p_n\})$.
\section{Proposed Method}
In this section, we first describe the overall system architecture, then we detailed introduce our proposed method, including client selection, sampling based on client clustering, fairness-aware model aggregation, and personalization module.
\subsection{Overall System Architecture}
\label{subsec:syss}
A typical example of federated recommender system is illustrated in Fig.~\ref{fig:pathdemo}.
Here each client has a local recommender engine to recommend items to local users. The local recommender engine performs local model training, exchanges models with the recommendation server, and makes local recommendations. The recommendation server gathers information from all the clients and distributes the aggregated learning model to the clients. In this way, the server and the clients exchange models instead of raw data, which preserves the users' privacy.
Based on the standard FL algorithm as described beforehand. We try to improve it by designing a communication efficient and fair federated learning algorithm for the sequential recommendation, termed CF-FedSR.
\begin{figure}[!t]
\centering
\includegraphics[width=0.47\textwidth,height=0.19\textwidth]{m1.png}
\caption{Illustration of the overall system architecture}
\label{fig:pathdemo}
\end{figure}
\subsection{Communication efficient and Fair Federated Sequential Recommendation Algorithm (CF-FedSR)}
\label{subsec:pfl}
This section introduces our proposed Communication efficient and Fair Federated Sequential Recommendation algorithm (CF-FedSR).
To enhance the model performance, improve the fairness and reduce the communication overhead simultaneously, we specifically design client selection, sampling based on client clustering, fairness-aware model aggregation, and a personalization module.
The client selection module introduces the adaptive selector to select appropriate clients to participate in training.
Then sampling relies on client clustering to adaptively select representative clients.
Also, the parameter aggregation module performs fairness-aware parameter aggregation.
Moreover, the personalization module uses fine-tune techniques for personalized recommendations.
\subsubsection{Client selection}
Federated learning requires lots of transmissions of model parameters between servers and clients. However, we find not all communication is necessary. In the beginning stage, the global model performance is limited due to random initialization.
Allowing all the clients to participate in all training rounds may be suboptimal since small clients may negatively affect the model performance in the beginning stage.
Since for small clients, due to their limited data sizes or low learning rates, they may not contribute enough to improve the model.
Instead, the updates by these small clients may even negatively affect the model aggregation.
To cope with this problem, an adaptive selector is adopted on the server-side to save training time and computing power, as well as to improve the performance.
We introduce two criteria for selecting appropriate clients: (i) the number of data samples $|D_i|$ in client $c_i$ exceeds a threshold $\lambda_1$, or (ii) the training epoch $t_i$ of client $c_i$ exceeds a threshold $\lambda_2$.
If a client $c_i$ satisfy either of the two criteria, we select it to participate in the training.
In this way, the system can get rid of the interference from the updates from small clients in the beginning stage. Through adaptive client selection, the server can train the global model faster and achieve better performance.
\subsubsection{Sampling based on client clustering}
We also propose client clustering to reduce the communication and computation complexity further.
Each client is represented by short-term and long-term interest, as illustrated in Figure~\ref{fig:client_rep}, which is represented by the concatenate of the recent interacted item embeddings.
Specifically, we choose the recent $v_1$ items to represent the user's short-term interest and the recent $v_2$ items to represent the user's long-term interest. Obviously we have $v_2 > v_1$.
Then we calculate the average embeddings of the selected items.
After that, we concatenate the long-term and short-term interests to form the client representation.
Several clustering techniques can be adapted to cluster similar users, such as K-means \cite{hartigan1979algorithm} and Mean shift \cite{comaniciu2002mean}.
We sample clients proportionally from each cluster; thus, the sampled clients should be more representative than the randomly sampled.
\begin{figure}[t]
\centering
\includegraphics[width=0.44\textwidth,height=0.24\textwidth]{p5.png}
\caption{Client representation}
\label{fig:client_rep}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.46\textwidth,height=0.2\textwidth]{p6.png}
\caption{Client clustering}
\label{fig:client_clu}
\end{figure}
\subsubsection{\textcolor{black}{Fairness aware parameter aggregation}}
Firstly, we review the FedAvg \cite{konevcny2016federated}, which is a commonly used federation aggregation strategy. The formula is denoted as:
\begin{equation}
w_{t+1} = \sum_{k \in S_t} \frac{n_k}{n} w_k^{t+1}
\end{equation}
where $w_{t+1}$ is the model parameter in time $t+1$, and $n_k$ is the sample size for client $k$.
According to the formula, a larger sample size would result in a more significant contribution to the global model for clients.
However, the FedAvg algorithm suffers from several problems.
Though the overall model performance may be satisfactory, the local model performance for each client may not be promising.
The main reason is that the data for clients are Non-IID, then the local datasets are usually heterogeneous between clients in terms of size and distribution.
Therefore, it is unfair for some clients since they may have a small data sample size and are largely ignored in the parameter aggregation process.
To tackle the heterogeneous problem, we design a weighting strategy to encourage a more fair distribution in this work.
Firstly, we propose a fairness-aware parameter aggregation method to deal with the varying model performance among clients.
To start with, we represent the performance for client $k$ as $p_k$.
For federated recommendation scenarios, commonly adopted evaluation metrics are Hit Rate (HR) and Normalized Discounted Cumulative Gain (NDCG).
Thus we use the sum of evaluation metrics to represent the performance, denoted as:
\begin{equation}
p_{k} = \text{HR}_k + \text{NDCG}_k,
\end{equation}
Then we normalize it:
\begin{equation}
p_{k}^{\prime} = \frac{p_k}{\sum_{k \in S_t} p_k}.
\end{equation}
After that, we apply an activation function. The activation function should satisfy decreasing in range $[0,1]$. In this work, the activation function is $f(x) = (\frac{1}{2})^x $.
\begin{equation}
\hat{p}_k = \text{Activation}(p_{k}^{\prime}).
\end{equation}
Then we normalize it again,
\begin{equation}
\hat{p}_k^{\prime} = \frac{p_k}{\sum_{k \in S_t} p_k},
\end{equation}
where the ${p_k^{\prime}}$ is the fairness-aware factor.
Secondly, we alleviate the impact of the data sample size.
For the unfairness caused by the imbalance sample size, one way to eliminate this effect is to reduce the impact of the parameter of data sampling size. In this way, the aggregation formula is denoted as:
We set the weight parameter as $q_k$. Intuitively, it is denoted as:
\begin{equation}
q_k = \frac{n_k}{n}.
\end{equation}
Then we use an activation function. The activation function should satisfies increasing in range $[0,1]$. In this work, the activation function is $f(x) = \sqrt{x} $.
\begin{equation}
\hat{q}_k = \text{Activation}(q_{k}^{\prime}).
\end{equation}
Then we have:
\begin{equation}
\hat{q}_k^{\prime} = \frac{q_k}{\sum_{k \in S_t} q_k}.
\end{equation}
Lastly, we combine these two factors and form the final parameter aggregation function:
\begin{align}
& o_k = \alpha \hat{p}_k^{\prime} + \beta \hat{q}_k^{\prime},\\
& w_{t+1} = \sum_{k \in S_k} \frac{o_k}{\sum_{i \in S_t} o_i} w_k^{t+1},
\end{align}
where $w_{t+1}$ is the global model at time $t+1$, $\alpha$ and $\beta$ are two hyper-parameters.
\subsubsection{Personalization}
Prior researches show that FL could be used to train a high-performing global model for federated recommendation \cite{wu2021fedgnn,qi2020fedrec}. The trained model, however, is more of generality but lacks personalization. Therefore, we fine-tune the global model with the client's local data to make personalized recommendations. We aggregate the global and fine-tuned model to balance generality and personalization. Thus the client could benefit from the personalization process. We denote the process as:
\begin{align}
& w_{local} = w_{global} - \eta \bigtriangledown L(w, d),\\
& w = \gamma\cdot w_{local} + (1-\gamma)\cdot w_{global},
\end{align}
where $\gamma$ is a hyper-parameter.
The model could adaptively balance the generalizable and personalization by adjusting the hyper-parameters.
Additionally, note we only transmit the embedding layer in the training process, as the skeleton network, while letting the remaining part of the network be trained localized.
\section{Experiments and Analysis}
\label{sec:exp}
This section presents our experimental results over three real datasets by comparing our proposed algorithms' performance against other reasonable baselines.
Besides, we demonstrate the effectiveness for each component through an ablation study.
We also analyze the impact of different experimental settings, such as embedding size and different numbers of clusters.
Our experiments are designed to answer the following Research Questions (RQs):
\emph{$\bullet$ RQ1:} Does CF-FedSR show superiority of existing methods in federated sequential recommendation in terms of model performance, communication cost, and fairness?
\emph{$\bullet$ RQ2:} What is the influence of various components in the CF-FedSR framework? Are those components necessary?
\emph{$\bullet$ RQ3:} What is the impact of the hyper-parameters settings in CF-FedSR?
\subsection{Experiment Setup}
\label{exp_set}
\subsubsection{Dataset Description}
We have conducted experiments using real-world datasets to evaluate our proposed method.
A detailed description of the total number of user clicks, the number of items and clicks, and the average number of clicks is listed in Table~\ref{tab:description}.
These datasets vary in domains, platforms, and sparsity.
Here we give the detailed description of the datasets.
\emph{$\bullet$ Amazon:} A series of datasets introduced in \cite{mcauley2015image}, comprising large corpora of product reviews crawled from Amazon.com. Top-level product categories on Amazon are treated as separate datasets. We consider two categories, \emph{Beauty} and \emph{Video}. This dataset is notable for its high sparsity and variability.
\emph{$\bullet$ Wikipedia:} This dataset contains one month of edits on Wikipedia \cite{wiki2021}. The editors who made at least 5 edits and the 1,000 most edited pages are filtered out as users and items for recommendation. This dataset contains 157,474 interactions in total.
\begin{table}[t]
\centering
\caption{Description of datasets}
\label{tab:description}
\scalebox{1}
{
\begin{tabular}{lllll}
\hline
Dataset & \# of user & \# of item & \# of click & Avg. length\\
\hline
Beauty & 52204 & 57289 & 394908 & 7.56\\
Video & 31013 & 23715 & 287107 & 9.26\\
Wikipedia & 8227 & 1000 & 157474 & 19.14\\
\hline
\end{tabular}
}
\end{table}
\subsubsection{Baselines}
We aim to explore the performance of federated learning in a deep learning model for sequential recommendations.
We use the \emph{GRU4REC} \cite{hidasi2015session} as the backbone sequential recommendation model. The model combines RNNs to model user sequences for the sequential recommendation.
We compare our proposed CF-FedSR algorithm with the popular \emph{FedAvg} \cite{konevcny2016federated} method.
FedAvg is a classical federated learning algorithm that allows many clients to train a global model collaboratively.
Recall that CF-FedSR is designed to improve FedAvg in communication efficiency, fairness, and personalization.
Also, we compare them with centralized training settings.
\subsubsection{Evaluation Metrics}
We use the following two metrics to evaluate the performance of our proposed algorithms and existing algorithms.\\
1) \emph{Hit Rate (HR)}. To calculate the proportion of predicted items that are accurate. In this work, we evaluate the top-5 and top-10 items for comparison. \\
2) \emph{Normalized Discounted Cumulative Gain (NDCG)}. To calculate the ranking position of correctly predicted items. When the rank list is predicted correctly, the NDCG is high. If predicted item ranks exceed the recommended limit (5 or 10 in our settings), the NDCG value is zero.
\subsubsection{Experiment Settings}
For performance evaluation, we use the leave-one-out strategy, following \cite{kang2018self,luopaper2}. For each user, we held out their latest interaction for the test set, the second last for validation and the remaining data for training.
Also, we adopt a commonly used negative sampling method to reduce heavy computation, in accordance to \cite{kang2018self,luopaper2}.
Specifically, we randomly sample 100 negative items, and rank these items together with the ground-truth item.
\begin{table*}[]
\caption{Experiment results compared with baseline methods. }
\label{overall_exp}
\centering
\scalebox{0.93}[0.93]{
\begin{tabular}{lllllllllllll}
\hline
& \multicolumn{4}{c}{Beauty} & \multicolumn{4}{c}{Video} & \multicolumn{4}{c}{Wikipedia} \\
Model & HR@5 & NDCG@5 & HR@10 & NDCG@10 & HR@5 & NDCG@5 & HR@10 & NDCG@10 & HR@5 & NDCG@5 & HR@10 & NDCG@10 \\\hline
Central & 0.3223 & 0.2435 & 0.4482 & 0.2835 & 0.3283 & 0.2648 & 0.4922 & 0.2848 & 0.8982 & 0.8767 & 0.9181 & 0.8828 \\
FedAvg & 0.2651 & 0.1828 & 0.3740 & 0.2173 & 0.2720 & 0.1858 & 0.3987 & 0.2259 & 0.7842 & 0.7606 & 0.8077 & 0.7681 \\
CF-FedSR & 0.2869 & 0.2036 & 0.4073 & 0.2509 & 0.2951 & 0.2008 & 0.4223 & 0.2517 & 0.8469 & 0.8199 & 0.8722 & 0.8279 \\
Impro. & 8.22\% & 11.38\% & 8.90\% & 15.46\% & 8.49\% & 8.07\% & 5.92\% & 11.42\% & 8.00\% & 7.80\% & 7.99\% & 7.79\%\\\hline
\end{tabular}}
\end{table*}
\subsubsection{Implementation Details}
The number of users used in each round of model training is 128, and the total number of the epoch is 200.
The ratio of dropout is 0.3. Adam \cite{kingma2014adam} is selected as the optimizer,
and the default learning rate is 0.001.
We tune hyper-parameters using the validation set, and
terminate training if validation performance doesn't improve
for five successive epochs.
Unless stated otherwise, we use the same hyper-parameters of FedAvg and CF-FedSR.
We report the average performance scores over the five repetitions.
Our evaluation target is to compare the recommendation performance in a federated context,
thus we don't compare it to other centralized algorithms since they're not well suited to federated settings.
\subsection{ Performance Comparison (RQ1) }
\subsubsection{Evaluation on model performance}
We have compared different methods' performances in Table~\ref{overall_exp}.
We have the following observations:
$\bullet$
Our proposed CF-FedSR significantly outperforms the state-of-the-art federated recommender systems in all datasets.
Compared with FedAvg, CF-FedSR achieves on average 7.92\% and 10.32\% relative improvements on HR and NDCG, respectively.
Several advantages of CF-FedSR support its superiority:
(1) client selection and sampling helps to find more representative and useful clients to participate in training;
(2) fairness aware aggregation algorithm help to bridge the gap between clients with various data distribution and facilitate the training.
$\bullet$ The centralized models perform better than those decentralized models.
Compared with CF-FedSR, centralized models achieves in average 10.25\% and 15.20\% relative improvements in HR and NDCG, respectively.
There are two reasons:
(1) centralized models models are better as they can directly train the model without noise or pseudo-labeled items.
In the meantime, the federated learning framework achieves better privacy-preserving ability than centralized training because clients do not reveal raw data to the server.
It could be viewed as the tradeoff of the recommendation performance to privacy.
(2)
though FL theoretically would have similar performance with the centralized settings. However, due to the data heterogeneous, distributed systems are harder to train since they lack the capacity to model the global data structures.
\begin{table}[]
\caption{Comparison of overall performance and converge speed among CF-FedSR and competing methods.}
\label{converge}
\centering
\begin{tabular}{lllll}
\hline
Model & HR@10 & NDCG@10 & Converge round \\\hline
FedAvg & 0.3740 & 0.2173 & \~75 \\
CF-FedSR & 0.4073 & 0.2509 & \~67 \\
Impro. & 8.90\% & 15.46\% & -10.67\% \\\hline
\end{tabular}
\end{table}
\subsubsection{Evaluation on communication efficiency}
We evaluate the converge speed together with the model performance in Table~\ref{converge}.
We observe that the CF-FedSR converges much more quickly than FedAvg.
Thus the communication cost is saved by transmitting less round.
The reason is that we adopt several techniques to reduce communication loss.
We strictly limit the participation of small clients at the beginning stage of training, which prevents the inference from these small clients.
Additionally, we carefully select representative clients to participate through clustering, making the system converge faster.
In a nutshell, CF-FedSR can enhance the model performance while boosting communication efficiency.
\subsubsection{Evaluation on fairness}
Recall that we define the \textit{fairness} as the variance among clients in Section~\ref{sec:pf}. A lower value of variance means the system is more fair.
The experimental results is shown in Figure~\ref{fairness}.
The result demonstrate that our proposed method could decrease the variance and boost the model performance in the meanwhile, showing the effectiveness of our proposed method.
The superiority of our proposed method come from our specially designed adaptive aggregation algorithm.
\begin{figure}[t]
\centering
\caption{Comparison of overall performance and fairness among CF-FedSR and competing methods.}
\label{fairness}
\centering
\includegraphics[width=0.46\textwidth,height=0.17\textwidth]{p7.png}
\end{figure}
\subsection{Ablation Study (RQ2)}
Specifically, we repeat the experiment by removing one module from the proposed CF-FedSR model and test the performance.
The detailed information of variants is listed as follows.
\textbf{$\bullet$ CF-FedSR-Variation 1 (w/o client selection \& sampling)}: The proposed CF-FedSR without the client selection \& sampling component.
\textbf{$\bullet$ CF-FedSR-Variation 2 (w/o fair aggregation)}: The proposed CF-FedSR without fair aggregation component.
\textbf{$\bullet$ CF-FedSR-Variation 3 (w/o personalization)}: The proposed CF-FedSR without the personalization module.
From Table~\ref{ablation}, we observe that the CF-FedSR achieves the best results on all datasets, which verifies the importance of each critical module.
Among them, the experimental results of CF-FedSR-Variation 3 drop sharply, which proves that the lack of personalization could significantly decrease the learning ability of the framework.
\begin{table}[!h]
\caption{Ablation study.}
\label{ablation}
\centering
\scalebox{0.98}[0.98]{
\begin{tabular}{lllll}
\hline
& \multicolumn{4}{c}{Wikipedia} \\
Model & HR@5 & NDCG@5 & HR@10 & NDCG@10 \\\hline
CF-FedSR & 0.8469 & 0.8199 & 0.8722 & 0.8279 \\
CF-FedSR-Variation 1 & 0.8039 & 0.7631 & 0.8396 & 0.7751 \\
CF-FedSR-Variation 2 & 0.8259 & 0.7968 & 0.8610 & 0.8083 \\
CF-FedSR-Variation 3 & 0.7984 & 0.7703 & 0.8358 & 0.7801 \\\hline
\end{tabular}
}
\end{table}
\begin{table}[!h]
\caption{CF-FedSR performance with respect to different embedding sizes $d$.}
\label{para_d}
\centering
\begin{tabular}{lllll}
\hline
Dataset & \multicolumn{4}{c}{Beauty} \\\hline
Dimension & HR@5 & NDCG@5 & HR@10 & NDCG@10 \\\hline
d=10 & 0.2631 & 0.1986 & 0.3689 & 0.2296 \\
d=20 & 0.2726 & 0.2003 & 0.3821 & 0.2389 \\
d=50 & 0.2869 & 0.2036 & 0.4073 & 0.2509 \\
d=100 & 0.2965 & 0.2073 & 0.4124 & 0.2516 \\\hline
\end{tabular}
\end{table}
\begin{table}[!h]
\caption{CF-FedSR performance with respect to different cluster number $k$.}
\label{para_k}
\centering
\begin{tabular}{lllll}
\hline
Dataset & \multicolumn{4}{c}{Beauty} \\\hline
\# clusters & HR@5 & NDCG@5 & HR@10 & NDCG@10 \\\hline
k=3 & 0.2846 & 0.2013 & 0.4029 & 0.2453 \\
k=5 & 0.2869 & 0.2036 & 0.4073 & 0.2509 \\
k=10 & 0.2912 & 0.2065 & 0.4068 & 0.2527 \\\hline
\end{tabular}
\end{table}
Without the client selection \& sampling module, the performance of the framework still decreases obviously.
Throughout these three ablation experiments, it turns out that each module improves the model performance from different aspects and is meaningful.
\subsection{Hyper-parameter study (RQ3)}
\subsubsection{Sensitivity of CF-FedSR to embedding size}
Table~\ref{para_d} illustrates the analysis of the effect of embedding on the \emph{Beauty} dataset.
We repeat the experiment with default parameters and with different values for the number of embedding size, i.e. with $d \in \{10, 20, 50, 100\}$.
We find that performance is continually improved with the increase of embedding size.
Presumably, this is because higher dimension of item embedding could bring more information from different aspect.
However, a larger embedding size would have a higher communication cost. Thus it is a tradeoff between the communication cost and the model performance.
\subsubsection{Sensitivity of CF-FedSR to number of clusters}
This experiment demonstrates the sensitivity of CF-FedSR to the number of clusters hyper-parameter. We ran CF-FedSR using the default parameters in subsection 5.1 but with different values for the number of clusters, i.e. with $k \in \{3, 5, 10\}$. The result of this experiment is presented in Table~\ref{para_k}.
We observe that its performance improves and then drops when the number of cluster increases from
3 to 10.
\section{Conclusion}
\label{sec:con}
In this paper, we study the problem of the federated sequential recommendations and propose the CF-FedSR framework.
The key component for CF-FedSR includes client selection, sampling based on client clustering, fairness-aware model aggregation, and a personalization module.
These techniques are adopted to ensure fairness and personalization in the federated recommendation, boost the recommendation and ease the communication cost in the meanwhile.
We conducted extensive experiments over real datasets and observed that our proposed framework effectively makes federated sequential recommender systems more accurate and reduces the communication overhead.
\section{Acknowledgements}
This work was supported in part by the Changsha Science and Technology Program International and Regional Science and Technology Cooperation Project under Grants kh2201026, the Hong Kong RGC grant ECS 21212419, the Technological Breakthrough Project of Science, Technology and Innovation Commission of Shenzhen Municipality under Grants JSGG20201102162000001, InnoHK initiative, the Government of the HKSAR, Laboratory for AI-Powered Financial Technologies, the Hong Kong UGC Special Virtual Teaching and Learning (VTL) Grant 6430300, and the Tencent AI Lab Rhino-Bird Gift Fund.
|
{
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| 4,692
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{"url":"https:\/\/zbmath.org\/authors\/?q=ai%3Acampo.antonio","text":"## Campo, Antonio\n\nCompute Distance To:\n Author ID: campo.antonio Published as: Campo, Antonio; Campo, A.\n Documents Indexed: 40 Publications since 1978 Co-Authors: 40 Co-Authors with 33 Joint Publications 831 Co-Co-Authors\nall top 5\n\n### Co-Authors\n\n 3 single-authored 5 Morrone, Biagio 3 Lacoa, Ulises 3 Ridouane, El Hassan 3 Salazar, Abraham J. 2 Chow, Louis C. 2 Manca, Oronzio 2 Raydan, Marcos 2 Yoshimura, Toshio 1 Alhama, Francisco 1 Amon, Cristina H. 1 Auguste, Jean-Claude 1 Ben Beya, Brahim 1 Ben Cheikh, Nader 1 Cabezas-G\u00f3mez, Luben 1 Campo, Leon J. 1 Celentano, Diego J. 1 Chang, Jane Y. 1 Chikh, Salah 1 Dehghan Takht Fooladi, Mehdi 1 Fern\u00e1ndez-Seara, Jos\u00e9 1 Ghafoori-Fard, H. 1 Guimar\u00e3es, Luiz Gustavo Monteiro 1 Guzella, Matheus dos Santos 1 Hammami, Fay\u00e7al 1 Hanin, Leonid G. 1 Hern\u00e1ndez-Morales, Bernardo 1 Hitt, Darren L. 1 Kacimov, Anvar R. 1 Landon, Mark D. 1 Liao, Shijun 1 Lili, Taieb 1 Marucho, Marcelo D. 1 Morales, Juan Carlos 1 Morales, Juanc C. 1 Mujumdar, Arun Sadashiv 1 Nazari-Golshan, A. 1 Nourazar, S. Salman 1 Papari, Mohammad Mehdi 1 Perret, J. S. 1 Ribeiro dos Santos, Gustavo 1 Saadatmandi, Abbas 1 Sablani, S. S. 1 Schuler, Carlos 1 Sekuli\u0107, Du\u0161an P. 1 Sieres, Jaime 1 Stuffle, R. Eugene 1 Yang, Yi 1 Zueco, Joaqu\u00edn\nall top 5\n\n### Serials\n\n 11 International Journal of Heat and Mass Transfer 7 International Journal of Numerical Methods for Heat & Fluid Flow 5 Numerical Methods for Partial Differential Equations 2 Applied Mathematics and Computation 2 Mathematical Problems in Engineering 1 Computers & Mathematics with Applications 1 International Journal for Numerical Methods in Fluids 1 International Journal of Systems Science 1 Journal of Applied Mechanics 1 Journal of Computational Physics 1 Journal of Fluid Mechanics 1 Mechanics Research Communications 1 Heat and Technology 1 Applied Mathematics Letters 1 International Journal of Differential Equations and Applications 1 International Journal of Modern Physics C 1 International Journal of Applied and Computational Mathematics\nall top 5\n\n### Fields\n\n 24 Classical thermodynamics, heat transfer\u00a0(80-XX) 23 Fluid mechanics\u00a0(76-XX) 12 Numerical analysis\u00a0(65-XX) 9 Partial differential equations\u00a0(35-XX) 4 Mechanics of deformable solids\u00a0(74-XX) 2 Computer science\u00a0(68-XX) 1 Ordinary differential equations\u00a0(34-XX) 1 Calculus of variations and optimal control; optimization\u00a0(49-XX) 1 Statistical mechanics, structure of matter\u00a0(82-XX) 1 Systems theory; control\u00a0(93-XX)\n\n### Citations contained in zbMATH Open\n\n20 Publications have been cited 169 times in 152 Documents Cited by Year\nAnalytic solutions of the temperature distribution in Blasius viscous flow problems.\u00a0Zbl\u00a01007.76014\nLiao, Shijun; Campo, Antonio\n2002\nA modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations.\u00a0Zbl\u00a01308.65134\nNazari-Golshan, A.; Nourazar, S. S.; Ghafoori-Fard, H.; Yildirim, A.; Campo, A.\n2013\nTheoretical analysis of the exponential transversal method of lines for the diffusion equation.\u00a0Zbl\u00a00953.65063\nSalazar, A. J.; Raydan, M.; Campo, A.\n2000\nNon-iterative estimation of heat transfer coefficients using artificial neural network models.\u00a0Zbl\u00a01121.80312\nSablani, S. S.; Kacimov, A.; Perret, J.; Mujumdar, A. S.; Campo, A.\n2005\nRandom heat transfer in flat channels with timewise variation of ambient temperature.\u00a0Zbl\u00a00391.76067\nCampo, Antonio; Yoshimura, Toshio\n1979\nNumerical solution of the heat conduction equation with the electro-thermal analogy and the code PSPICE.\u00a0Zbl\u00a01061.65076\nAlhama, Francisco; Campo, Antonio; Zueco, Joaqu\u00edn\n2005\nA new minimum volume straight cooling fin taking into account the \u201clength of arc\u201d.\u00a0Zbl\u00a01137.49314\nHanin, Leonid; Campo, Antonio\n2003\nApproximate solution of the nonlinear heat conduction equation in a semi-infinite domain.\u00a0Zbl\u00a01203.80026\nYu, Jun; Yang, Yi; Campo, Antonio\n2010\nAxial conduction in laminar pipe flows with nonlinear wall heat fluxes.\u00a0Zbl\u00a00391.76068\nCampo, Antonio; Auguste, Jean-Claude\n1978\nEffect of surface radiation on buoyant convection in vertical triangular cavities with variable aperture angles.\u00a0Zbl\u00a01140.80370\nSieres, Jaime; Campo, Antonio; Ridouane, El Hassan; Fern\u00e1ndez-Seara, Jos\u00e9\n2007\nThe Legendre-tau technique for the determination of a source parameter in a semilinear parabolic equation.\u00a0Zbl\u00a01200.65077\nSaadatmandi, Abbas; Dehghan, Mehdi; Campo, Antonio\n2006\nRemarkable improvement of the L\u00e9v\u00eaque solution for isoflux heating with a combination of the transversal method of lines (TMOL) and a computer-extended Fr\u00f6benius power series.\u00a0Zbl\u00a01189.76384\nCampo, Antonio; Amon, Cristina H.\n2005\nOptimum plate separation in vertical parallel-plate channels for natural convective flows: Incorporation of large spaces at the channel extremes.\u00a0Zbl\u00a00925.76646\nMorrone, Biagio; Campo, Antonio; Manca, Oronzio\n1997\nHeat removal of in-tube viscous flows to air with the assistance of arrays of plate fins. I: Theoretical aspects involving 3-D, 2-D and 1-D models.\u00a0Zbl\u00a00999.76118\nCampo, Antonio\n2000\nOn the solution of a 2-D, parabolic, partial differential energy equation subjected to a nonlinear convective boundary condition via a simple solution for a uniform, Dirichlet boundary condition.\u00a0Zbl\u00a00823.76078\nCampo, Antonio; Lacoa, Ulises\n1995\nHeat transfer and pressure drop characteristics of laminar air flows moving in a parallel-plate channel with transverse hemi-cylindrical cavities.\u00a0Zbl\u00a01125.80316\nRidouane, El Hassan; Campo, Antonio\n2007\nPrediction of unsteady states in lid-driven cavities filled with an incompressible viscous fluid.\u00a0Zbl\u00a01263.76022\nHammami, Fay\u00e7al; Ben-Cheikh, Nader; Campo, Antonio; Ben-Beya, Brahim; Lili, Taieb\n2012\nOn the asymptotic solution of the Graetz-Nusselt problem for short $$x \\to 0$$ and large $$x \\to \\infty$$ with partial usage of finite differences.\u00a0Zbl\u00a01116.76426\nCampo, Antonio\n2004\nOn approximate solutions for unsteady conduction in slabs with uniform heat flux.\u00a0Zbl\u00a00924.65086\nSalazar, Abraham; Campo, Antonio; Morrone, Biagio\n1998\nMeshless approach for computing the heat liberation from annular fins of tapered cross section.\u00a0Zbl\u00a01134.80006\nCampo, Antonio; Morrone, Biagio\n2004\nA modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations.\u00a0Zbl\u00a01308.65134\nNazari-Golshan, A.; Nourazar, S. S.; Ghafoori-Fard, H.; Yildirim, A.; Campo, A.\n2013\nPrediction of unsteady states in lid-driven cavities filled with an incompressible viscous fluid.\u00a0Zbl\u00a01263.76022\nHammami, Fay\u00e7al; Ben-Cheikh, Nader; Campo, Antonio; Ben-Beya, Brahim; Lili, Taieb\n2012\nApproximate solution of the nonlinear heat conduction equation in a semi-infinite domain.\u00a0Zbl\u00a01203.80026\nYu, Jun; Yang, Yi; Campo, Antonio\n2010\nEffect of surface radiation on buoyant convection in vertical triangular cavities with variable aperture angles.\u00a0Zbl\u00a01140.80370\nSieres, Jaime; Campo, Antonio; Ridouane, El Hassan; Fern\u00e1ndez-Seara, Jos\u00e9\n2007\nHeat transfer and pressure drop characteristics of laminar air flows moving in a parallel-plate channel with transverse hemi-cylindrical cavities.\u00a0Zbl\u00a01125.80316\nRidouane, El Hassan; Campo, Antonio\n2007\nThe Legendre-tau technique for the determination of a source parameter in a semilinear parabolic equation.\u00a0Zbl\u00a01200.65077\nSaadatmandi, Abbas; Dehghan, Mehdi; Campo, Antonio\n2006\nNon-iterative estimation of heat transfer coefficients using artificial neural network models.\u00a0Zbl\u00a01121.80312\nSablani, S. S.; Kacimov, A.; Perret, J.; Mujumdar, A. S.; Campo, A.\n2005\nNumerical solution of the heat conduction equation with the electro-thermal analogy and the code PSPICE.\u00a0Zbl\u00a01061.65076\nAlhama, Francisco; Campo, Antonio; Zueco, Joaqu\u00edn\n2005\nRemarkable improvement of the L\u00e9v\u00eaque solution for isoflux heating with a combination of the transversal method of lines (TMOL) and a computer-extended Fr\u00f6benius power series.\u00a0Zbl\u00a01189.76384\nCampo, Antonio; Amon, Cristina H.\n2005\nOn the asymptotic solution of the Graetz-Nusselt problem for short $$x \\to 0$$ and large $$x \\to \\infty$$ with partial usage of finite differences.\u00a0Zbl\u00a01116.76426\nCampo, Antonio\n2004\nMeshless approach for computing the heat liberation from annular fins of tapered cross section.\u00a0Zbl\u00a01134.80006\nCampo, Antonio; Morrone, Biagio\n2004\nA new minimum volume straight cooling fin taking into account the \u201clength of arc\u201d.\u00a0Zbl\u00a01137.49314\nHanin, Leonid; Campo, Antonio\n2003\nAnalytic solutions of the temperature distribution in Blasius viscous flow problems.\u00a0Zbl\u00a01007.76014\nLiao, Shijun; Campo, Antonio\n2002\nTheoretical analysis of the exponential transversal method of lines for the diffusion equation.\u00a0Zbl\u00a00953.65063\nSalazar, A. J.; Raydan, M.; Campo, A.\n2000\nHeat removal of in-tube viscous flows to air with the assistance of arrays of plate fins. I: Theoretical aspects involving 3-D, 2-D and 1-D models.\u00a0Zbl\u00a00999.76118\nCampo, Antonio\n2000\nOn approximate solutions for unsteady conduction in slabs with uniform heat flux.\u00a0Zbl\u00a00924.65086\nSalazar, Abraham; Campo, Antonio; Morrone, Biagio\n1998\nOptimum plate separation in vertical parallel-plate channels for natural convective flows: Incorporation of large spaces at the channel extremes.\u00a0Zbl\u00a00925.76646\nMorrone, Biagio; Campo, Antonio; Manca, Oronzio\n1997\nOn the solution of a 2-D, parabolic, partial differential energy equation subjected to a nonlinear convective boundary condition via a simple solution for a uniform, Dirichlet boundary condition.\u00a0Zbl\u00a00823.76078\nCampo, Antonio; Lacoa, Ulises\n1995\nRandom heat transfer in flat channels with timewise variation of ambient temperature.\u00a0Zbl\u00a00391.76067\nCampo, Antonio; Yoshimura, Toshio\n1979\nAxial conduction in laminar pipe flows with nonlinear wall heat fluxes.\u00a0Zbl\u00a00391.76068\nCampo, Antonio; Auguste, Jean-Claude\n1978\nall top 5","date":"2022-11-27 15:03:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5923671722412109, \"perplexity\": 12160.22746234901}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710409.16\/warc\/CC-MAIN-20221127141808-20221127171808-00476.warc.gz\"}"}
| null | null |
Q: Proof formula for triangle: $b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$ given is a triangle with corners A,B,C and corresponding sides a, b, c.
D is a point somewhere between A and B.
I have to proof:
$b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$
Unfortunately I have no idea - I just tried with some pythagoras but only got nonsense.
So I'll be glad if you can give me a hint where to start, then I might get it :)
Kind regards!
A: You need at the fondation of your work the cosinus rule (Al Kashi), which is the extension of Phytagoras.
That is (1) $$a^2=b^2+c^2-2bc \cos \angle A$$
The same way you have (2) $$|CD|^2=b^2+|AD|^2-2b|AD|\cos \angle A$$
Then
$b^2|BD| +a^2|AD| - c|CD|^2 =a^2|AD|+b^2(c-|AD|)-c(b^2+|AD|^2-2b|AD|\cos \angle A)$
$b^2|BD| +a^2|AD| - c|CD|^2 =(a^2-b^2)|AD|-c|AD|^2+2bc|AD|\cos \angle A$
$b^2|BD| +a^2|AD| - c|CD|^2 =(c^2-2bc\cos \angle A)|AD|-c|AD|^2+2bc|AD|\cos \angle A$
$b^2|BD| +a^2|AD| - c|CD|^2 =c|AD|(c-|AD|)=c|AD||BD|$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,703
|
package org.apache.accumulo.test.randomwalk.security;
import java.util.Properties;
import java.util.Random;
import org.apache.accumulo.core.client.AccumuloException;
import org.apache.accumulo.core.client.AccumuloSecurityException;
import org.apache.accumulo.core.client.Connector;
import org.apache.accumulo.core.security.SystemPermission;
import org.apache.accumulo.test.randomwalk.State;
import org.apache.accumulo.test.randomwalk.Test;
public class AlterSystemPerm extends Test {
@Override
public void visit(State state, Properties props) throws Exception {
Connector conn = state.getConnector();
WalkingSecurity ws = new WalkingSecurity(state);
String action = props.getProperty("task", "toggle");
String perm = props.getProperty("perm", "random");
String targetUser = WalkingSecurity.get(state).getSysUserName();
SystemPermission sysPerm;
if (perm.equals("random")) {
Random r = new Random();
int i = r.nextInt(SystemPermission.values().length);
sysPerm = SystemPermission.values()[i];
} else
sysPerm = SystemPermission.valueOf(perm);
boolean hasPerm = ws.hasSystemPermission(targetUser, sysPerm);
// toggle
if (!"take".equals(action) && !"give".equals(action)) {
if (hasPerm != conn.securityOperations().hasSystemPermission(targetUser, sysPerm))
throw new AccumuloException("Test framework and accumulo are out of sync!");
if (hasPerm)
action = "take";
else
action = "give";
}
if ("take".equals(action)) {
try {
conn.securityOperations().revokeSystemPermission(targetUser, sysPerm);
} catch (AccumuloSecurityException ae) {
switch (ae.getSecurityErrorCode()) {
case GRANT_INVALID:
if (sysPerm.equals(SystemPermission.GRANT))
return;
throw new AccumuloException("Got GRANT_INVALID when not dealing with GRANT", ae);
case PERMISSION_DENIED:
throw new AccumuloException("Test user doesn't have root", ae);
case USER_DOESNT_EXIST:
throw new AccumuloException("System user doesn't exist and they SHOULD.", ae);
default:
throw new AccumuloException("Got unexpected exception", ae);
}
}
ws.revokeSystemPermission(targetUser, sysPerm);
} else if ("give".equals(action)) {
try {
conn.securityOperations().grantSystemPermission(targetUser, sysPerm);
} catch (AccumuloSecurityException ae) {
switch (ae.getSecurityErrorCode()) {
case GRANT_INVALID:
if (sysPerm.equals(SystemPermission.GRANT))
return;
case PERMISSION_DENIED:
throw new AccumuloException("Test user doesn't have root", ae);
case USER_DOESNT_EXIST:
throw new AccumuloException("System user doesn't exist and they SHOULD.", ae);
default:
throw new AccumuloException("Got unexpected exception", ae);
}
}
ws.grantSystemPermission(targetUser, sysPerm);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 761
|
Christ dismantled the top-down hierarchy of the Law, with its static God pinned down in the Temple and its priests and power abuses, and put in its place an evolving network of the Spirit.
2000 years later it appears that this change needs to happen again.
The Complex Christ is a book about how such a transformation might take place. It argues that renewal of the Church will not be a revolution - precipitated quickly from the top down, but an evolution - a slow, viral change that transforms the very genes of the institution.
The book takes Christ life and passion as an archetype for change. Seeing the city space - the place where we are forced to interact with the 'other' - as the true home of the Church, The Complex Christ sets out how the urban church might find inspiration for renewal from the science of emergence.
Kester Brewin is a teacher and founder member of Vaux, a community of artists and city-lovers based in South London.
Since 1998 they have been seeking the divine in the urban, and the book reflects something of their journey through art, urban theory, dirt, gift, poetry and liturgy. This is his first book.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 3,338
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\section{Basic definitions}\label{sec_def}
Let $G$ be a finite group and $F$ a field. If $C$
is a subset of $G$, then we define $\q C$ to be the element $\sum_{g
\in C}g$ in the group algebra $FG$, and we call $\q C$ a \emph{simple quantity} of $FG$. Given a subset $C \subseteq G$ and an integer $m$, we define
$C^{(m)}=\{g^m : g \in C\}$. For any $x \in FG$, where $x =
\sum_{g\in G} a_gg$, we define $x^{(m)} = \sum_{g\in G}a_gg^m$.
Given a subset $X \subseteq FG$, we denote the subspace spanned by $X$ by
$FX$. If $\m L$ is a collection of subsets of $G$, we let $\q{\m L}$ denote the set $\{\q C : C \in \m L\}$.
A subalgebra $S$ of the group algebra $FG$ is called a \emph{Schur ring} (or \emph{S-ring}) over $G$ if there are disjoint nonempty subsets $T_1,\dots,T_n$ of $G$ such that $\q T_1,\dots, \q T_n$ form a basis for $S$, with the following properties,
\begin{enumerate}
\item[(i)] For every $i$ there is some $j$ such that $T_i^{(-1)}=T_j$.
\item[(ii)] $T_1=\{1\}$, and $G=T_1\cup T_2\cup \cdots \cup T_n$.
\end{enumerate}
The sets $T_1,\dots,T_n$ are called \emph{basic sets} of $S$ and are said to
form a \emph{Schur partition} of $G$. The corresponding $\q
T_1,\dots,\q T_n$ are called the \emph{basic quantities} of $S$. If $S$
satisfies condition (i), but perhaps not (ii), then $S$ is
called a \emph{pseudo S-ring} (or \emph{PS-ring}). The two sets $\{1\}$, $G\setminus\{1\}$ always form a Schur partition; the corresponding S-ring is called the \emph{trivial} S-ring over $G$.
Given $x, y \in FG$, where $x=\sum_{g \in G}a_gg,y=\sum_{g\in G}b_gg$, their \emph{Hadamard product} is defined by
$$x \circ y = \sum_{g\in G}a_gb_gg.$$
There is a well-known purely algebraic description of S-rings and PS-rings in terms of closure under the Hadamard product, avoiding reference to the combinatorial notion of basic sets (for proofs, see \cite[Proposition 3.1]{muzychuk} and \cite[Lemma 1.3]{muzychuk2}, or \cite[Theorem 1.7, Corollary 1.8]{kerby_masters}):
\begin{thm}\label{folklore}
Let $A$ be a subalgebra of $FG$. Then $A$ is a $PS$-ring if and
only if $A$ is closed under $\circ$ and $^{(-1)}$.
\end{thm}
\begin{cor}\label{folklore_cor}
Let $A$ be a subalgebra of $FG$. Then $A$ is an $S$-ring if and only if $A$ is closed under $\circ$ and $^{(-1)}$ and contains $1$ and $\q G$.
\end{cor}
\begin{defn}
Let $S_1$ and $S_2$ be S-rings over groups $G_1$ and $G_2$
respectively. An algebra isomorphism $\phi : S_1 \to S_2$ is called
an \emph{isomorphism} of S-rings if $\phi$ maps basic quantities to
basic quantities, i.e., if for every basic set $C$ of $S_1$ there is
some basic set $D$ of $S_2$ such that $\phi(\q C)=\q D$.
\end{defn}
The following well-known result gives a purely algebraic characterization of S-ring isomorphisms (an elementary proof may be found in \cite[Theorem 3.4]{kerby_masters}):
\begin{thm}\label{isohad}
Let $S_1$ and $S_2$ be S-rings over groups $G_1$ and $G_2$
respectively. Then an $F$-algebra isomorphism $\phi: S_1 \to S_2$ is
an S-ring isomorphism if and only if $\phi$ respects the Hadamard
product, i.e., if and only if $\phi(x \circ y) = \phi(x) \circ
\phi(y)$ for all $x,y \in S_1$.
\end{thm}
Elsewhere in the literature, for clarity, an isomorphism of S-rings is sometimes called an \emph{algebraic isomorphism}, in contrast to the notion of a \emph{combinatorial isomorphism} of S-rings, which, properly speaking, is an isomorphism of the association schemes corresponding to the S-rings.
If $\q C$ is any simple quantity contained in an S-ring $S$, then we say that $C$ is an \emph{$S$-set}. The following is also well-known (For a proof, see \cite[Theorem 3.7, Corollary 6.8]{kerby_masters}):
\begin{thm}\label{isosimp}
Let $\phi: S_1 \to S_2$ be an isomorphism of S-rings over groups $G_1$ and
$G_2$ respectively.
Let $C \subseteq G_1$ be an $S_1$-set. Then $\phi(\q C)=\q D$ for some $S_2$-set $D \subseteq G_2$, and we write $\phi(C)=D$. Moreover,
\begin{enumerate}
\item[(i)] $|C|=|D|$, and
\item[(ii)] If $C$ is a subgroup then $D$ is also a subgroup.
\end{enumerate}
\end{thm}
An isomorphism from an S-ring onto itself is called an \emph{automorphism}. The set of automorphisms of an S-ring $S$ is denoted $\Aut(S)$. Every group automorphism in
$\Aut(G)$ naturally induces an S-ring isomorphism of $S$ (onto a possibly distinct S-ring); such an isomorphism is called a \emph{Cayley isomorphism}, and the two S-rings are called \emph{Cayley-isomorphic}.
It is possible to consider S-rings where the coefficient field $F$ is replaced by an arbitrary ring $R$ (with unity). The collection of Schur partitions is affected only by the characteristic of $R$; so whether $R$ is $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{C}$, or any other ring of characteristic 0 makes no difference, as far as the collection of Schur partitions is concerned. Nevertheless, Theorem \ref{folklore} and Corollary \ref{folklore_cor} both become false over any commutative ring $R$ which is not a field; for this reason, we will always take our coefficient ring to be a field $F$. Over rings of nonzero characteristic, the collection of Schur partitions includes all those associated with zero characteristic in addition possibly to others. (For proof of these statements, see \cite[Example 1.10, Theorem 1.12]{kerby_masters}.) It is possible for the isomorphism class of the automorphism group $\Aut(S)$ to change depending on the characteristic of the coefficient field $F$ (see Example \ref{ex_nzchar} below). Throughout the literature on S-rings, most authors assume a coefficient ring of characteristic zero. We will also need this assumption in \S\ref{sec_cyc} when we show that the automorphism group of an S-ring over a cyclic group is abelian; indeed, this theorem becomes false over any ring of nonzero characteristic, as Example \ref{ex_nzchar} will show.
\section{Central and rational S-rings}\label{sec_rat}
\begin{defn}
A PS-ring $S$ over a group $G$ is \emph{central} if $S \subseteq
Z(FG)$, i.e., $S$ is contained in the center of the group algebra.
\end{defn}
\begin{remark}
This is equivalent to requiring that every basic set $T_i$ be a union of conjugacy classes of $G$.
\end{remark}
Of course, over an abelian group every S-ring is central. We are
primarily interested in a special class of central S-rings known
as rational S-rings:
\begin{defn}
A PS-ring $S$ over a group $G$ is \emph{rational} if for every $x\in
S$ and $\phi \in\Aut(G)$, we have $\phi(x)=x$.
\end{defn}
\begin{remark}
This is equivalent to requiring that every basic set $T_i$ be a union of automorphism
classes of $G$, where the \emph{automorphism classes} of $G$ are the orbits of $\Aut(G)$ acting on $G$ in the natural way.
\end{remark}
The following provides a method for constructing many interesting central and rational S-rings:
\begin{thm}\label{latsring}
Let $G$ be any finite group, and let $\m{L}$ be any sublattice of
the lattice of normal subgroups of $G$. Then the vector space $F\q{\m
L}$ is a central PS-ring over $G$ with the
following properties, for all $H, K \in \m L$:
\begin{enumerate}
\item[(i)]$\q H^{(-1)} = \q H$
\item[(ii)] $\q H \circ \q K = \q{H\cap K}$
\item[(iii)] $\q H\ \q K = |H\cap K|\q{HK}$
\item[(iv)] $F\q{\m L}$ is an S-ring if and only if $1, G \in \m L$.
\item[(v)] $F\q{\m L}$ is rational if and only if $\m L$ consists entirely of characteristic subgroups.
\end{enumerate}
\end{thm}
\begin{proof}
(i) is clear since $H$, as a subgroup, is closed under inverses.
(ii) is immediate from the definition of the Hadamard product. (iii)
is clear since, by elementary group theory, every element of $HK$
can be written in $|H\cap K|$ ways as a product of an element in $H$
with an element in $K$. Now, since the subgroups $H$ and $K$ are
normal, $HK$ is also a subgroup of $G$; since $\m L$ is a lattice,
we have $H\cap K, HK \in\m L$. Thus, (i)--(iii) show that $F\q{\m L}$
is closed under $^{(-1)}$, multiplication, and the Hadamard product,
so by Theorem \ref{folklore}, $F\q{\m L}$ is a PS-ring. Since each
subgroup $H$ is normal, we have $g\q Hg^{-1}=\q{gHg^{-1}}=\q H$ for
all $g\in G$ and $H \in\m L$, so that $\q H \in Z(FG)$. It follows
that $F\q{\m L}$ is a central PS-ring.
Now, if $1, G \in \m L$ then $1, \q G \in F\q{\m L}$, so by Corollary
\ref{folklore_cor}, $F\q{\m L}$ is an S-ring. Suppose conversely
that $F\q{\m L}$ is an S-ring. Let $L=\bigcap_{H\in\m L} H$, so that
$L \in \m L$. If $|L|>1$, then since $L \subseteq H$ for every spanning
element $\q H$ of $F\q{\m L}$, it follows that every element of $F\q{\m L}$ with a non-zero coefficient of $1\in G$ also has other non-zero
coefficients (namely, the other elements of $L$ have non-zero
coefficients), so that $1 \notin F\q{\m L}$, contrary to the assumption that $F\q{\m L}$ is an S-ring. So we must have $L=1$, hence $1 \in \m L$, as desired. Now define
$M=\prod_{H\in\m L} H$. If $M\neq G$, then since $H \subseteq M$ for
every spanning element $\q H$ of $F\q{\m L}$, it follows that the nonzero
coefficients of every element $x \in F\q{\m L}$ are contained in $M$,
so that $\q G \notin F\q{\m L}$, again contradicting that $F\q{\m L}$ is an S-ring. So $G \in \m L$ as desired.
Finally, $F\q{\m L}$ is rational if and only if $\phi(\q H)=\q H$ for
every $\phi\in\Aut(G)$ and each spanning element $\q H$ of $F\q{\m L}$;
this holds if and only if $\phi(H)=H$ for each $H\in\m L$, i.e. if
and only if each $H\in\m L$ is characteristic.
\end{proof}
If $G$ is a cyclic group, then every subgroup of $G$ is
characteristic; consequently, the construction of Theorem
\ref{latsring} produces only rational S-rings. In this context, we
may state the main theorem of \cite{muzychuk} as follows:
\begin{thm}\emph{(Muzychuk)}\label{ratcyc}Every rational S-ring over a finite cyclic group may be constructed as in Theorem \ref{latsring}.
\end{thm}
There are rational S-rings over abelian $p$-groups
which cannot be constructed as in Theorem \ref{latsring} (see
Example \ref{nonlatsring}). However, there are other types of groups
for which Theorem \ref{latsring} produces the complete set of
rational S-rings. We mention an example, whose proof can be found in \cite[Theorem 2.6]{kerby_masters}:
\begin{thm}\label{ratdih}Every rational S-ring over a finite dihedral group may be constructed as in Theorem \ref{latsring}.
\end{thm}
\section{Characteristic subgroups of abelian $p$-groups}\label{sec_auto}
In this section, we review the description of the automorphism classes and characteristic subgroups of finite abelian $p$-groups. This topic was considered in 1905 and 1920 by G. A. Miller \cite{miller,miller2} and again, independently, in 1934 by Baer, who considered the more general case of periodic abelian groups \cite{baer}, and finally in 1935 by Birkhoff \cite{birkhoff_subabel}. The historical nature of these early works is such that they are not easy reading; the present author confesses that in many places the statements made in them, and particularly the proofs, do not always seem clear. A more recent treatment may be found in \cite{kerby_turner} (or \cite{kerby_masters}), which includes proofs of all the results below as well as a detailed description of the exceptional case $p=2$.
Throughout this section, $G$ will denote a finite abelian $p$-group: $$G=Z_{p^{\lambda_1}}\times Z_{p^{\lambda_2}}\times \cdots \times Z_{p^{\lambda_n}},$$ where $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. We define $\h \lambda(G)$ to be the tuple $(\lambda_1,\dots,\lambda_n)$. For convenience, we will often use the convention $\lambda_0=0$. As we will be working extensively with such tuples of integers, it will be useful to introduce some notation for dealing with them:
\begin{defn}
Given tuples $\h a=(a_1,\dots,a_n)$ and $\h b=(b_1,\dots,b_n)$ with integer entries, define
\begin{align*}
\h a \leq \h b &\text{ if $a_i \leq b_i$ for all $i\in\{1,\dots,n\}$}\\
\h a \wedge \h b &= (\min\{a_1,b_1\}, \min\{a_2,b_2\}, \dots, \min\{a_n,b_n\})\\
\h a \vee \h b &= (\max\{a_1,b_1\}, \max\{a_2,b_2\}, \dots, \max\{a_n,b_n\})
\end{align*}
\end{defn}
Define $\Lambda(G)$ to be the set of tuples $$\Lambda(G)=\{\h a : \h 0 \leq \h a \leq \h \lambda(G)\}.$$ It is evident that $\Lambda(G)$, under the partial order $\leq$, forms a finite lattice in which $\wedge$ and $\vee$ are the greatest lower bound and least upper bound respectively.
Given a tuple $\h a \in \Lambda(G)$, we define $T(\h a)$ to be the set of elements $g\in G$ for which the $i$th component of $g$ has order $p^{a_i}$:
$$T(\h a)=\{(g_1,g_2,\dots,g_n) \in G : |g_i|=p^{a_i}\text{ for all $i=1,\dots,n$}\}.$$
Note that the sets $T(\h a)$ partition the group $G$. If $g \in T(\h a)$, we say that the \emph{type} of $g$ is $T(\h a)$.
The following is straightforward to prove:
\begin{lemma}\label{lemtype}
If two elements have the same type, then they are in the same automorphism class.
\end{lemma}
\begin{defn}
Given a type $T(\h a)$, the automorphism class of $G$ containing $T(\h a)$ is denoted $O(\h a)$.
\end{defn}
\begin{defn}\label{defcan}
A tuple $\h a=(a_1,\dots,a_n)$ and its corresponding type $T(\h a)$ are called \emph{canonical} if for all $i\in\{1,\dots,n-1\}$,
\begin{enumerate}
\item[(i)] $a_i \leq a_{i+1}$ and
\item[(ii)] $a_{i+1}-a_i \leq \lambda_{i+1}-\lambda_i$.
\end{enumerate}
\end{defn}
The definition of ``canonical" is justified by the following theorem.
\begin{thm}\label{unican}
Every automorphism class contains a unique canonical type.
\end{thm}
\begin{example}
Let $G=Z_2\times Z_8=Z_2\times Z_{2^3}=\langle s\rangle\times\langle
t\rangle$. Then there are 6 automorphism classes of $G$, namely:
\begin{align*}
O(0,0)&=T(0,0)=\{1\},\\
O(0,1)&=T(0,1)=\{t^4\},\\
O(0,2)&=T(0,2)=\{t^2,t^6\},\\
O(1,1)&=T(1,1)\cup T(1,0)=\{s,st^4\},\\
O(1,2)&=T(1,2)=\{st^2,st^6\},\\
O(1,3)&=T(1,3)\cup T(0,3)=\{t,st,t^3,st^3,t^5,st^5,t^7,st^7\}.\\
\end{align*}
\end{example}
Having established a sufficiently detailed description of the automorphism classes for our purposes, we now turn to the characteristic subgroups. We let $\Char(G)$ denote the lattice of characteristic subgroups of $G$.
\begin{defn}
Given a tuple $\h a \in \Lambda(G)$, we define the subgroup $R(\h a)=\underset{\h b\leq \h a}\cup T(\h b)$ and call $R(\h a)$ the \emph{regular subgroup below} $\h a$.
\end{defn}
\begin{remark}
We use the term ``regular", following Baer (\cite{baer}). But this concept of regular should not be confused with the notion of a regular permutation group, nor of a regular $p$-group.
\end{remark}
\begin{thm}\label{canchar}
$R(\h a)$ is a characteristic subgroup if and only if $\h a$ is canonical.
\end{thm}
The following is easily verified by direct calculation:
\begin{thm}\label{Rlat}For any $\h a, \h b \in \Lambda(G)$,
\begin{enumerate}
\item[(i)] $R(\h a) \cap R(\h b) = R(\h a \wedge \h b)$
\item[(ii)] $\langle R(\h a), R(\h b) \rangle = R(\h a \vee \h b)$
\item[(iii)] $|R(\h a)|=p^{\sum_{i=1}^n a_i}$
\end{enumerate}
\end{thm}
From (i) and (ii), it follows that the regular characteristic subgroups form a sublattice of $\Char(G)$. Using Theorem \ref{canchar}, this then implies that if $\h a$ and $\h b$ are canonical tuples then so are $\h a \wedge \h b$ and $\h a \vee \h b$. (This is also not difficult to verify directly.) Thus the canonical tuples form a sublattice of $\Lambda(G)$; this sublattice will be denoted by $\m C(G)$.
The following theorem is of great importance to us:
\begin{thm}[Miller-Baer]\label{charreg}
If $p \neq 2$, then every characteristic subgroup of $G$ is regular.
\end{thm}
We then have the following important corollaries:
\begin{cor}\label{corcong}
If $p \neq 2$, then $\Char(G) \cong \m C(G)$.
\end{cor}
\begin{cor}[Miller-Baer-Birkhoff]\label{num_autos}
For any prime $p$, the number of automorphism classes of $G=Z_{p^{\lambda_1}}\times Z_{p^{\lambda_2}}\times \cdots \times Z_{p^{\lambda_n}}$ is $$\prod_{i=1}^n(\lambda_i-\lambda_{i-1}+1).$$
If $p\neq 2$, then this is also the number of characteristic subgroups of $G$.
\end{cor}
\begin{proof}
We apply Corollary \ref{corcong}, observing that the canonical tuples $\h a$ are precisely those which satisfy $a_{i-1} \leq a_i \leq a_{i-1}+\lambda_i-\lambda_{i-1}$ for each $i \in \{1,\dots,n\}$. Thus there are $\lambda_i-\lambda_{i-1}+1$ choices for each coordinate $a_i$, and the first statement follows. The second follows from the first and Theorems \ref{canchar} and \ref{charreg}.
\end{proof}
We define $\m W(G)$ to be the subalgebra of elements of $FG$ which are invariant under $\Aut(G)$. Thus, $\m W(G)$ is the maximum rational S-ring over $FG$; its basic sets are the automorphism classes of $G$. We then obtain one last corollary:
\begin{cor}\label{charind}
If $p \neq 2$, then $\q{\Char(G)}$ is a basis for $\m W(G)$.
\end{cor}
\begin{proof}
Clearly $\q{\Char(G)} \subseteq \m W(G)$. By Theorem \ref{unican}, the S-ring $\m W(G)$ is spanned by $\{\q{O(\h a)} : \h a \in \m C(G)\}$. For any canonical tuple $\h a\in\m C(G)$, we can write $$\q{O(\h a)} = \q{R(\h a)}-\sum_{\underset{\h b < \h a}{\h b \in \m C(G)}} \q{O(\h b)},$$
and, since $\q{R(\h a)} \in \q{\Char(G)}$, it follows by induction that each $\q{O(\h a)}$ is in the span of $\q{\Char(G)}$. Since $\dim(\m W(G))=|\Char(G)|$ by Corollary \ref{num_autos}, the result follows.
\end{proof}
\begin{example}\label{nonlatsring}
Let $G=Z_p\times Z_{p^3}$ for an odd prime $p$. Then the characteristic subgroups of $G$ are shown in Table \ref{char13}.
\end{example}
\begin{table}
\centering
\caption{Characteristic subgroups of $G=Z_p\times Z_{p^3}$ for odd prime $p$}\label{char13}
\begin{tabular}{lr}
\begin{tabular}{|l|l|} \hline
$H_1$&R(0,0)\\ \hline
$H_2$&R(0,1)\\ \hline
$H_3$&R(0,2)\\ \hline
$H_4$&R(1,1)\\ \hline
$H_5$&R(1,2)\\ \hline
$H_6$&R(1,3)\\ \hline
\end{tabular}
&
\begin{tabular}{r}
\psset{xunit=.2cm,yunit=.2cm,labelsep=2.5mm}
\begin{pspicture*}(-10,-10)(10,22)
\Rput[t](0,0){$H_1$}
\Rput[t](0,5){$H_2$}
\psline(0,-.5)(0,1.5)
\Rput[t](-5,10){$H_3$}
\Rput[t](5,10){$H_4$}
\psline(-1.5,4.5)(-3.5,6.5)
\psline(1.5,4.5)(3.5,6.5)
\Rput[t](0,15){$H_5$}
\psline(-3.5,9.5)(-1.5,11.5)
\psline(3.5,9.5)(1.5,11.5)
\Rput[t](0,20){$H_6$}
\psline(0,14.5)(0,16.5)
\end{pspicture*}
\end{tabular}
\end{tabular}
\end{table}
\begin{example}
Let $G=Z_p\times Z_{p^3}\times Z_{p^5}$ for an odd prime $p$. Then the characteristic subgroups of $G$ are shown in Table \ref{char135}. Note that the sublattice of $\Char(G)$ between $H_5=R(0,1,2)$ and $H_{14}=R(1,2,3)$ forms a cube; i.e., this sublattice is isomorphic to the boolean lattice $\m P(X)$ of subsets of a set $X$ of cardinality 3.
\end{example}
\begin{table}
\centering
\caption{Characteristic subgroups of $G=Z_p\times Z_{p^3}\times Z_{p^5}$ for odd prime $p$}\label{char135}
\begin{tabular}{lr}
\begin{tabular}{|l|l|} \hline
$H_1$&R(0,0,0)\\ \hline
$H_2$&R(0,0,1)\\ \hline
$H_3$&R(0,0,2)\\ \hline
$H_4$&R(0,1,1)\\ \hline
$H_5$&R(0,1,2)\\ \hline
$H_6$&R(0,1,3)\\ \hline
$H_7$&R(0,2,2)\\ \hline
$H_8$&R(0,2,3)\\ \hline
$H_9$&R(0,2,4)\\ \hline
$H_{10}$&R(1,1,1)\\ \hline
$H_{11}$&R(1,1,2)\\ \hline
$H_{12}$&R(1,1,3)\\ \hline
$H_{13}$&R(1,2,2)\\ \hline
$H_{14}$&R(1,2,3)\\ \hline
$H_{15}$&R(1,2,4)\\ \hline
$H_{16}$&R(1,3,3)\\ \hline
$H_{17}$&R(1,3,4)\\ \hline
$H_{18}$&R(1,3,5)\\ \hline
\end{tabular}
&
\begin{tabular}{r}
\psset{xunit=.2cm,yunit=.2cm,labelsep=2.5mm}
\begin{pspicture*}(-15,-5)(15,47)
\Rput[t](0,0){$H_1$}
\Rput[t](0,5){$H_2$}
\psline(0,-.5)(0,1.5)
\Rput[t](-5,10){$H_3$}
\Rput[t](5,10){$H_4$}
\psline(-1.5,4.5)(-3.5,6.5)
\psline(1.5,4.5)(3.5,6.5)
\Rput[t](-5,15){$H_5$}
\Rput[t](5,15){$H_{10}$}
\psline(-5,9.5)(-5,11.5)
\psline(5,9.5)(5,11.5)
\psline(3.5,9.5)(-3.5,11.5)
\Rput[t](-5,20){$H_6$}
\Rput[t](0,20){$H_7$}
\Rput[t](5,20){$H_{11}$}
\psline(-5,14.5)(-5,16.5)
\psline(-4.25,14.5)(-1.5,16.5)
\psline(-3.5,14.5)(3.5,16.5)
\psline(5,14.5)(5,16.5)
\Rput[t](-5,25){$H_8$}
\Rput[t](0,25){$H_{12}$}
\Rput[t](5,25){$H_{13}$}
\psline(-3.5,19.5)(-1.5,21.5)
\psline(-1.5,19.5)(-3.5,21.5)
\psline(3.5,19.5)(1.5,21.5)
\psline(1.5,19.5)(3.5,21.5)
\psline(-5,19.5)(-5,21.5)
\psline(5,19.5)(5,21.5)
\Rput[t](-5,30){$H_9$}
\Rput[t](5,30){$H_{14}$}
\psline(5,24.5)(5,26.5)
\psline(1.5,24.5)(4.25,26.5)
\psline(-3.5,24.5)(3.5,26.5)
\psline(-5,24.5)(-5,26.5)
\Rput[t](-5,35){$H_{15}$}
\Rput[t](5,35){$H_{16}$}
\psline(-5,29.5)(-5,31.5)
\psline(5,29.5)(5,31.5)
\psline(3.5,29.5)(-3.5,31.5)
\Rput[t](0,40){$H_{17}$}
\psline(-3.5,34.5)(-1.5,36.5)
\psline(3.5,34.5)(1.5,36.5)
\Rput[t](0,45){$H_{18}$}
\psline(0,39.5)(0,41.5)
\end{pspicture*}
\end{tabular}
\end{tabular}
\end{table}
The following theorem gives a generalization of preceding two examples which will be important to us in \S\ref{sec_main}:
\begin{thm}\label{latbool}
Let $X$ be a (finite) set containing $n$ elements, and let
$$G=Z_p \times Z_{p^3}\times\cdots\times\mathbb Z_{p^{2n-1}}.$$
Then there is an embedding $\psi$ of the boolean lattice $\m P(X)$ into $\Char(G)$. Further, $\psi$ has the property that for all subsets $Y_1, Y_2$ of $X$, $|Y_1|=|Y_2| \iff |\psi(Y_1)|=|\psi(Y_2)|$.
\end{thm}
\begin{proof}
We will map $\m P(X)$ onto the sublattice of $\Char(G)$ between $R(0,1,2,\dots,n-1)$ and $R(1,2,3,\dots,n)$ in the following way: Write $X=\{x_1,\dots,x_n\}$; then, for $Y \subseteq X$ define $\psi(Y)=R(\h a)$, where $\h a=(a_1,\dots,a_n)$ is given by
$$a_i=\begin{cases}i-1, & \text{if $x_i\notin Y$}\\ i, & \text{if $x_i\in Y$}\end{cases}$$
Note that $\psi$ is well-defined since each the tuple $\h a$, as thus defined, is always canonical, since the difference between two consecutive components is either 0, 1, or 2. We have
\begin{align*}
\psi(Y_1 \cap Y_2)&=R(\h a(Y_1 \cap Y_2))=R(\h a(Y_1) \wedge \h a(Y_2))\\
&= R(\h a(Y_1)) \cap R(\h a(Y_2)) = \psi(Y_1) \cap \psi(Y_2)
\end{align*} and
\begin{align*}
\psi(Y_1 \cup Y_2)&=R(\h a(Y_1 \cup Y_2))=R(\h a(Y_1) \vee \h a(Y_2)) \\
&= \langle R(\h a(Y_1)), R(\h a(Y_2)) \rangle = \langle \psi(Y_1), \psi(Y_2)\rangle,
\end{align*} so that $\psi$ is a lattice homomorphism. By construction $\psi$ is injective, so $\psi$ is an embedding. Theorem \ref{Rlat}(iii) shows that $|\psi(Y)|=p^{\frac{n(n-1)}2+|Y|}$, and the last claim follows.
\end{proof}
\section{Main Theorem}\label{sec_main}
We now turn to our main result. In this section, we take all S-rings over a coefficient field $F$ of characteristic $0$.
\begin{thm}\emph{(Main Theorem)}\label{mainthm}
Let $p$ be an odd prime. Every finite group can be represented as the automorphism group of a rational S-ring over an abelian $p$-group.
\end{thm}
The proof relies on three key ideas:
\begin{enumerate}
\item[(1)] Every finite group can be represented as the automorphism group of a finite distributive lattice,
\item[(2)] Every distributive lattice can be embedded in a boolean lattice, and
\item[(3)] The lattice of characteristic subgroups of the group $Z_p \times Z_{p^3}\times\cdots\times Z_{p^{2n-1}}$ contains a boolean sublattice of order $2^n$.
\end{enumerate}
The proof of the Main Theorem, in outline, goes as follows: Given a group $G$, by (1) there is a distributive lattice with automorphism group isomorphic to $G$. By (2), this lattice may be embedded in a boolean lattice, which by (3) may in turn be embedded as a sublattice of the lattice of characteristic subgroups of an appropriate abelian $p$-group $P$. Finally, using Theorem \ref{latsring}, we construct the rational S-ring over $P$ associated with this lattice and show that the automorphism group of this S-ring is isomorphic to G.
(1) was shown by Birkhoff in \cite{birkhoff}; a proof may also be found in
\cite[\S7]{kerby_masters}, \cite{gratzer_birkhoff1}, or \cite{gratzer_birkhoff2} (see also \cite{foldes}). (2) is a standard result in lattice theory (also first proven by Birkhoff); a proof may be found in \cite[Theorem 5.12]{davey}, \cite[20.1]{hermes}, \cite[11.3]{crawley}, or \cite[Theorem 6.6]{kerby_masters}. We have already shown (3) in Theorem \ref{latbool} above. So all that remains is to verify the last step of our argument. Before doing this, however, we will need to give a more explicit description of the embedding of (2):
Recall that an element $j$ of a lattice $\m L$ is \emph{join-reducible} if there exist $x<j$ and $y<j$ with $j=x\vee y$. Otherwise $j$ is said to be \emph{join-irreducible}.
\begin{thm}[Birkhoff]\label{latdist}Let $\m L$ be a finite distributive lattice and let $J \subseteq \m L$ be the set of join-irreducible elements. Then the map $\phi(x)=\{j \in J : j \leq x\}$ is a embedding of $\m L$ into $\m P(J)$.
\end{thm}
In what follows, we will need the following fact:
\begin{thm}\label{latsize}
In Theorem \ref{latdist}, $|\phi(\alpha(x))|=|\phi(x)|$ for all $\alpha \in \Aut(\m L)$ and $x\in \m L$.
\end{thm}
\begin{proof}
Since lattice isomorphisms permute the join-irreducible elements among themselves, we have
\begin{align*}
\phi(\alpha(x)) &= \{j \in J : j \leq \alpha(x)\}
= \{\alpha(j) \in J : \alpha(j) \leq \alpha(x)\}\\
&= \{\alpha(j) \in J : j \leq x \} = \alpha(\phi(x)),
\end{align*}
hence $|\phi(\alpha(x))|=|\alpha(\phi(x))|=|\phi(x)|$.
\end{proof}
Now we are able to prove the Main Theorem. Given a finite group $G$,
by (1) there is a
finite distributive lattice $\m D$ with $\Aut(D) \cong G$. By
Theorem \ref{latdist}, there is an embedding $\phi$ of $\m D$ into
the boolean lattice $\m P(J)$, where $J$ is the set of
join-irreducible elements of $\m D$. In turn, by Theorem
\ref{latbool}, there is an embedding $\psi$ of $\m P(J)$ into the
lattice of characteristic subgroups $\Char(P)$ of an appropriate
abelian $p$-group $P$. Let $\m L=\psi(\phi(D))\cup\{1\}\cup\{P\}$. Then by Theorem \ref{latsring}, $F\q{\m L}$ is an S-ring over $P$. To prove the theorem, we will show that $\Aut(F\q{\m L}) \cong G$.
Since every automorphism of $\m L$ fixes $1$ and $P$ and hence restricts to an automorphism of $\psi(\phi(D))$, it follows that $\Aut(\psi(\phi(D))) \cong \Aut(\m L)$. So since $D \cong \psi(\phi(D))$, we then have $G \cong \Aut(D) \cong \Aut(\psi(\phi(D))) \cong \Aut(\m L)$. So
it suffices to show $\Aut(\m L) \cong \Aut(F\q{\m L})$.
We define a map $\chi : \Aut(\m L) \to \Aut(F\q{\m L})$ by $\chi(\beta) : \q H \mapsto \q{\beta(H)}$ for $H\in\m L$. We need to justify that $\chi$ is well-defined, i.e., that $\chi(\beta)$ actually is an S-ring automorphism of $\m L$. After that, it will be clear that $\chi$ is an injective homomorphism. Proving the surjectivity of $\chi$ will then complete the proof.
Let $\beta\in \Aut(\m L)$. Since Corollary \ref{charind} implies that $\q{\m L}$ is linearly independent and hence forms a basis for $F\q{\m L}$, it is clear that $f=\chi(\beta)$ is a well-defined bijective linear map from $F\q{\m L}$ to itself. To show that $f$ is an S-ring automorphism, we just need to show that it preserves the Hadamard and ordinary products. Applying Theorem \ref{latsring}, we have
\begin{align*}
f(\q H \circ \q K) &= f(\q{H\cap K}) = \q{\beta(H\cap K)} = \q{\beta(H)\cap\beta(K)} \\
&= \q{\beta(H)} \circ \q{\beta(K)} = f(\q H) \circ f(\q K),
\end{align*}
so $f$ preserves the Hadamard product. Before we can show $f$ preserves the ordinary product, we first need the result that for all $H \in \m L$, $|\beta(H)|=|H|$. For $H=1$ or $H=P$ this is trivially true since in these cases $\beta(H)=H$. For any other $H$, we may write $H=\psi(\phi(x))$ for some $x\in D$. Now, letting $\alpha\in\Aut(D)$ be the composite function $\alpha=\phi^{-1}\psi^{-1}\beta\psi\phi$, we see that showing $|\beta(H)|=|H|$ is equivalent to showing $|\psi(\phi(\alpha(x)))|=|\psi(\phi(x))|$, which by Theorem \ref{latbool} is equivalent to showing $|\phi(\alpha(x))|=|\phi(x)|$, which in turn holds by Theorem \ref{latsize}. So we have proven $|\beta(H)|=|H|$. In particular, if $H,K\in\m L$, we have
$$|H\cap K| = |\beta(H\cap K)|=|\beta(H)\cap\beta(K)|,$$
and from this, by applying Theorem \ref{latsring}, we obtain
\begin{align*}
f(\q H\ \q K) &= f(|H \cap K|\q{HK})
= |H\cap K| f(\q{HK})\\
&= |H\cap K| \q{\beta(HK)}
= |H\cap K| \q{\beta(H)\beta(K)}\\
&= |\beta(H)\cap\beta(K)|\q{\beta(H)\beta(K)}
= \q{\beta(H)}\ \q{\beta(K)}
= f(\q H)f(\q K),
\end{align*}
so $f$ preserves the ordinary product, which proves $f\in\Aut(F\q{\m L})$.
Now we only need to show that $\chi$ is surjective.
Suppose $\gamma$ is any S-ring automorphism of $F\q{\m L}$. For any $H \in \m L$, we have $\gamma(\q H)=\q K$ for some
subgroup $K\leq G$ by Theorem \ref{isosimp}(ii). Since $F\q{\m L}$ is a rational S-ring, $K$ is
necessarily characteristic. By the linear independence of
characteristic subgroups (Corollary \ref{charind}),
we must have $K \in \m L$. Thus $\gamma$ permutes the basis elements
$\{\q H : H \in \m L\}$ of $F\q{\m L}$. We can then define a bijection
$\beta : \m L \to \m L$ by setting $\beta(H)=K$ where $K$ is the
subgroup of $G$ such that $\gamma(\q H)=\q K$, so that
$\q{\beta(H)}=\gamma(\q H)$ for all $H\in \m L$. Once we show that $\beta$ is a lattice automorphism of $\m L$, we will have $\gamma=\chi(\beta)$, which will complete the proof.
Theorem
\ref{latsring} implies
\begin{align*}
\q{\beta(H \cap K)}
&= \gamma(\q{H \cap K})
=\gamma(\q H \circ \q K)\\
&=\gamma(\q H) \circ \gamma(\q K)
=\q{\beta(H)} \circ \q{\beta(K)}
=\q{\beta(H) \cap \beta(K)}
\end{align*}
so that $\beta(H\cap K)=\beta(H)\cap\beta(K)$. Observe that $|\beta(H)|=|H|$ by Theorem \ref{isosimp}(i), hence
$$|H\cap K|=|\beta(H\cap K)|=|\beta(H)\cap\beta(K)|,$$
so a further application of Theorem \ref{latsring} implies
\begin{align*}
\q{\beta(HK)}
&= \gamma(\q{HK})
= \gamma\left(\frac1{|H\cap K|}\q H\ \q K\right)
= \frac1{|H\cap K|}\gamma(\q H)\gamma(\q K)\\
&= \frac1{|H\cap K|}\q{\beta(H)}\ \q{\beta(K)}
= \frac1{|\beta(H)\cap\beta(K)|}\q{\beta(H)}\ \q{\beta(K)}
= \q{\beta(H)\beta(K)}
\end{align*}
so that $\beta(HK)=\beta(H)\beta(K)$. This proves that $\beta$ is a lattice automorphism of $\m L$, as desired.
\section{Automorphisms of S-rings over cyclic groups}\label{sec_cyc}
In this section, we again take all S-rings over a coefficient field $F$ of characteristic zero.
In \cite{leung_man2}, Leung and Man give a recursive classification
of all S-rings over cyclic groups. We give a brief description of
this classification. They give three basic methods of
constructing S-rings over a group $G$:
\begin{enumerate}
\item[(I)] Given a subgroup $\Omega \leq \Aut(G)$, let $T_1,\dots,T_n$
be the orbits of $\Omega$ acting on $G$. Then $T_1,\dots,T_n$ form a
Schur partition of $G$.
\item[(II)] Suppose $G=H\times K$ for nontrivial subgroups $H, K \leq G$, and suppose $S_H$ is an S-ring over $H$ with basic sets
$C_1,\dots,C_h$ and $S_K$ is an S-ring over $K$ with basic sets
$D_1,\dots,D_k$. Then the product sets $C_iD_j$, $1 \leq i \leq h, 1
\leq j\leq k$, form a Schur partition of $G$.
\item[(III)] Suppose $H$ and $K$ are nontrivial, proper subgroups of $G$ with $H \leq
K$ and $H \unlhd G$, and let $S_K$ be an S-ring over $K$ with basic
sets $C_1,\dots,C_k$ and $S_{G/H}$ be an S-ring over $G/H$ with
basic sets $D_1,\dots,D_k$, and suppose that $\pi(S_K)=F(K/H)\cap
S_{G/H}$, where $\pi: G \to G/H$ is the natural projection map,
extended to a natural projection map of the group algebra $FG$ onto
$F(G/H)$. Then
$$G=C_1\cup\cdots\cup C_k\cup\{ \pi^{-1}(D_i) : i\in\{1,\dots,k\}, D_i \nsubseteq K/H
\}$$ forms a Schur partition of $G$.
\end{enumerate}
In the case of cyclic groups, the S-ring constructed in (I) is called a \emph{cyclotomic} S-ring (a more general definition of cyclotomic S-rings, together with some of their applications, is found in \cite{muzychuk_ponomarenko}).
The S-ring constructed in (II) is denoted $S_H \cdot S_K$ and is
called the \emph{dot product} (also denoted $S_H \otimes S_K$ and called the \emph{tensor product}) of $S_H$ and $S_K$. The S-ring
constructed in (III) is denoted $S_K \wedge S_{G/H}$ and is called
the \emph{wedge product} (or \emph{generalized wreath product}) of $S_K$ and $S_{G/H}$.
The main theorem of Leung and Man (\cite[Theorem 3.7]{leung_man2}) may then be stated as follows:
\begin{thm}
Every nontrivial S-ring over a cyclic group $G$ is either cyclotomic, a dot product, or a wedge product.
\end{thm}
Note that the constructions (I), (II), and (III) can be used to
produce S-rings over an arbitrary group $G$, but if $G$ is not
cyclic then it is not necessarily true that all S-rings can be
constructed using these methods. For instance, over elementary abelian $p$-groups we have the following (for proofs, see \cite[Theorem 8.4, Example 8.5]{kerby_masters}):
\begin{thm}Let $p$ be a prime with $p\geq 5$. Then there are S-rings
over $Z_p\times Z_p$ which cannot be constructed as in (I), (II), or (III).
\end{thm}
\begin{thm}There are S-rings over $Z_3\times Z_3 \times Z_3 \times Z_3$ which cannot be constructed as in (I), (II), or (III).
\end{thm}
In contrast, all S-rings over $Z_3 \times Z_3$ and $Z_3 \times Z_3 \times Z_3$ can be constructed as in (I). In \cite[Question 8.6]{kerby_masters}, the question was asked of whether all S-rings over an elementary abelian 2-group can be constructed as in (I). We can now give a negative answer to this:
\begin{thm}There are S-rings over $Z_2^6$ which cannot be constructed as in (I), (II), or (III).
\end{thm}
\begin{proof}
Consider $G=Z_2^6$ as the additive group of the finite field $F_{64}$. Since $x^6+x^4+x^3+x+1$ is primitive over $\mathbb{F}_2$, $F_{64}^\times$ has a generator $\omega$ with $\omega^6+\omega^4+\omega^3+\omega+1=0$. The action of multiplication by $\langle \omega^9 \rangle$ partitions $F_{64}^\times$ into 9 orbits (each of size 7),
$$C_i=\{\omega^{i+9j} : j \in \{0..6\}\},\quad i=0,\dots,8.$$
We have found using MAGMA \cite{magma} that
$$\{0\},C_0\cup C_1\cup C_2\cup C_3\cup C_4, C_5\cup C_6\cup C_7\cup C_8$$
forms a Schur partition of $G$ whose corresponding S-ring $S$ cannot be constructed as in (I), (II), or (III) (For (II) and (III), this is fairly obvious: since basic sets have size 35, 28, and 1, it is clear that $S$ has no nontrivial proper $S$-subgroup, i.e., $S$ is a \emph{primitive} S-ring.)
\end{proof}
\begin{remark}
An exhaustive enumeration using MAGMA shows that all S-rings over $Z_2^n$ for $n \leq 4$ can be constructed as in (I). The question in the case $n=5$ remains open; this remaining case would probably still be feasible to answer by exhaustive enumeration, if one reduced the computation by taking advantage of the symmetry of the group.
\end{remark}
We now turn to the main result of this section:
\begin{thm}\label{autcyc}
If $S$ is an S-ring over a cyclic group $Z_n$, then $\Aut(S)$ is abelian.
\end{thm}
Before proving this, we need a few elementary lemmas, proofs of which may be found in \cite[Lemmas 8.8, 8.9, 8.10]{kerby_masters}. A version of
Lemma \ref{lemwie} can be found in \cite[Theorem 23.9]{wielandt},
while more general versions of Lemmas \ref{autexp} and
\ref{localstr} can be found in \cite[Propositions 3.1, 3.4]{muzychuk2}.
\begin{lemma}[Wielandt]\label{lemwie}
Suppose $S$ is an S-ring over an abelian group $G$ and $m$ is an
integer relatively prime to $|G|$. If $\q C \in S$, then also $\q
C^{(m)}\in S$. Moreover, if $\q C$ is a basic element of $S$ then
$\q C^{(m)}$ is also a basic element of $S$.
\end{lemma}
\begin{lemma}\label{autexp}
Suppose $S$ is an S-ring over an abelian group $G$ and
$\phi\in\Aut(S)$. Then for any simple quantity $\q C\in S$ and any
integer $m$ relatively prime to $|G|$, we have
$\phi(C^{(m)})=\phi(C)^{(m)}$.
\end{lemma}
\begin{lemma}\label{localstr}
Suppose $S$ is a cyclotomic S-ring over $Z_n$. Let $\phi$ be an
automorphism of $S$. Then for any basic set $T$ of $S$, there is an
integer $m$ relatively prime to $n$ such that $\phi(T)=T^{(m)}$.
\end{lemma}
\begin{remark}
Lemma \ref{lemwie} says that over an abelian group, every Cayley isomorphism is actually an automorphism of the S-ring. Lemma \ref{autexp} says that the Cayley automorphisms are in the center of the automorphism group of the S-ring. Finally, Lemma \ref{localstr} says that for a cyclotomic S-ring over a cyclic group, every S-ring automorphism
``locally" behaves like a Cayley automorphism. (In Lemma \ref{localstr}, the assumption that the S-ring be cyclotomic is actually unnecessary. However, the generalization omitting this hypothesis is a much deeper result which we will not need to use directly; it may be found in \cite[Theorem 1.1']{muzychuk2}, where it plays a key role in the proof of Muzychuk's result cited in Theorem \ref{cyciso}.)
\end{remark}
\begin{proof}[Proof of Theorem \ref{autcyc}]
If $S$ is a trivial S-ring, then $\Aut(S)$ is also trivial, hence
abelian, and we are done. So first consider the case that $S$ is
cyclotomic. Each basic set of $S$ then consists of elements of the
same order. Let $\m T_d$ be the collection of basic sets of $S$
containing elements of order $d$. We can consider $\Aut(S)$ as a
permutation group acting on the basic sets of $S$, and by Lemma
\ref{localstr}, for each $d$, $\Aut(S)$ permutes the basic sets of
$\m T_d$ among themselves. If we let $A_d$ denote the restriction of
$\Aut(S)$ to $\m T_d$, then $\Aut(S)$ is a subdirect product of all
the $A_d$'s, so it suffices to show that each $A_d$ is abelian. Fix
a divisor $d$ of $n$, and let $T$ be a basic set in $\m T_d$. For
any $k$ relatively prime to $n$, $T^{(k)}$ is another basic set of
$\m T_d$ by Lemma \ref{lemwie}, and every basic
set of $\m T_d$ has this form, for if $T'$ is any basic set in $\m
T_d$, there is an integer $l$ relatively prime to $n$ such that
$T^{(l)}\cap T' \neq \emptyset$, hence $T^{(l)}=T'$. Now, by Lemma
\ref{localstr}, $\phi(T)=T^{(m)}$ for some $m$ relatively prime to
$n$. It follows by Lemma \ref{autexp} that for any basic set
$T^{(k)} \in \m T_d$,
$$\phi(T^{(k)})=\phi(T)^{(k)}=(T^{(m)})^{(k)}=(T^{(k)})^{(m)}.$$
Thus $\phi(T')=(T')^{(m)}$ for any basic set $T' \in \m T_d$. From
this it is evident that $A_d$ is abelian.
Now suppose $S$ is a dot product $S_H\cdot S_K$. By induction we may
assume $\Aut(S_H)$ and $\Aut(S_K)$ are abelian. Let $\phi$ be an
element of $\Aut(S)$. Since $H$ is the unique subgroup of order $|H|$ in $G$, Theorem \ref{isosimp}(i,ii) implies $\phi(\q H)=\q H$, and similarly $\phi(\q K)=\q K$. So $\phi(S_H)$ is an S-ring over $H$ which
is isomorphic to $S_H$. By the result of Muzychuk cited in Theorem
\ref{cyciso}, the only such S-ring is $S_H$ itself, so we must have
$\phi(S_H)=S_H$ and likewise $\phi(S_K)=S_K$. So $\phi|_{S_H} \in
\Aut(S_H)$ and $\phi|_{ S_K}\in\Aut(S_K)$. Given another
automorphism $\psi \in\Aut(S)$, and any basic set $CD$ of $S$, where
$C$ is a basic set of $S_H$ and $D$ is a basic set of $S_K$, we have
$\phi(\psi(\q C))=\psi(\phi(\q C))$ and $\phi(\psi(\q
D))=\psi(\phi(\q D))$ since $\Aut(S_H)$ and $\Aut(S_K)$ are abelian,
hence \begin{align*} \phi(\psi(\q{CD}))&=\phi(\psi(\q C\ \q
D))=\phi(\psi(\q C)\psi(\q D))=\phi(\psi(\q C))\phi(\psi(\q D))\\
&=\psi(\phi(\q C))\psi(\phi(\q D))=\psi(\phi(\q C)\phi(\q
D))=\psi(\phi(\q C\ \q D))=\psi(\phi(\q{CD}))
\end{align*}
which proves that $\phi$ and $\psi$ commute, so $\Aut(S)$ is
abelian.
Finally suppose $S$ is a wedge product $S_K \wedge S_{G/H}$. As above,
given an automorphism $\phi \in \Aut(S)$, we have $\phi|_{S_K}
\in\Aut(S_K)$. Now define $F$-linear maps $\pi : FG \to F(G/H)$
and $\pi': F(G/H) \to FG$ by
\begin{align*}
&\pi: g \mapsto gH \\
&\pi': gH \mapsto g\q H.
\end{align*}
The following relations are easily checked:
\begin{align*}\label{pipi}
\pi(\pi'(x)) &= |H|x \\
\pi(xy) &= \pi(x)\pi(y)\\
\pi'(xy) &= \frac1{|H|}\pi'(x)\pi'(y)
\end{align*}
Consider the linear map $\phi^* : S_{G/H} \to S_{G/H}$ given by the composite function $\phi^* =\frac1{|H|}\pi\phi\pi'$. We need to justify that this is well-defined.
Note that if $B$ is a basic set of $S_{G/H}$ then $\pi'(\q B)=\q C$ where $C$ is a basic set of $S_G$ and $C$ is a union of cosets of $H$, so that $\q C\ \q H=|H|\q C$. Applying $\phi$ to boths sides gives $\phi(\q C)\q H=|H|\phi(\q C)$, so that the basic set $D=\phi(C)$ of $S_G$ is also union of cosets of $H$. Then $\pi(\q D)=|H|\q E$ for a basic set $E$ of $S_{G/H}$. Putting this together, we have
$$\phi^*(\q B)=\frac1{|H|}\pi(\phi(\pi'(\q B)))=\frac1{|H|}\pi(\phi(\q C))=\frac1{|H|}\pi(\q D)=\q E,$$
so that $\phi^*$ is well-defined and maps basic quantities to basic quantities. Now, we have
\begin{align*}
\left(\frac1{|H|}\pi'\phi^{-1}\pi\right)\phi^* &= \frac1{|H|^2}\pi'\phi^{-1}\pi\pi'\phi\pi
= \frac1{|H|}\pi'\phi^{-1}\phi\pi=\frac1{|H|}\pi'\pi=1_{FG},
\end{align*}
so $\phi^*$ is a bijection. Finally,
\begin{align*}
\phi^*(xy)&=\frac1{|H|}(\pi\phi\pi')(xy)=\frac1{|H|^2}\pi(\phi(\pi'(x)\pi'(y)))\\
&=\frac1{|H|^2}\pi(\phi(\pi'(x)))\pi(\phi(\pi'(y)))=\phi^*(x)\phi^*(y)
\end{align*}
so $\phi^*$ is an S-ring automorphism of $S_{G/H}$.
By induction we may assume
$\Aut(S_K)$ and $\Aut(S_{G/H})$ are abelian. Then, given another
automorphism $\psi\in\Aut(S)$ and a basic set $C$ of $S$, to complete the proof, it suffices to show $\phi(\psi(\q C))=\psi(\phi(\q C))$. We have
two cases: If $C\subseteq K$, then
$$\phi(\psi(\q C))=\phi|_{S_K}(\psi|_{S_K}(\q C))=\psi|_{S_K}(\phi|_{S_K}(\q C))=\psi(\phi(\q C))$$
and we are done. Suppose instead that $C\nsubseteq K$. Then $C$ is a
union of cosets of $H$, hence $\pi'(\pi(\q C))=|H|\q C$. As above,
$\phi(C)$ is also a union of cosets of $H$. Similarly, so are $\psi(C)$, $\phi(\psi(C))$, and $\psi(\phi(C))$. It follows that
\begin{align*}
(\pi'\pi\phi)(\q C) &= \pi'(\pi(\q{\phi(C)})) =
|H|\q{\phi(C)}=|H|\phi(\q C)
\end{align*}
and likewise
\begin{align*}
(\pi'\pi\psi)(\q C) &= |H|\psi(\q C) \\
(\pi'\pi\phi\psi)(\q C) &= |H|(\phi\psi)(\q C) \\
(\pi'\pi\psi\phi)(\q C) &= |H|(\psi\phi)(\q C)
\end{align*}
From all of this it follows that
\begin{align*}(\phi\psi)(\q C)&=\frac1{|H|}(\pi' \pi\phi\psi)(\q C)
=\frac1{|H|^2}(\pi' \pi\phi\pi'\pi\psi)(\q
C)\\
&=\frac1{|H|^3}(\pi'
\pi\phi\pi'\pi\psi\pi'\pi)(\q C)\\
&=\frac1{|H|}(\pi' \phi^*\psi^*\pi)(\q C) =
\frac1{|H|}(\pi'
\psi^*\phi^*\pi)(\q C)\\
&=\frac1{|H|^3}(\pi'
\pi\psi\pi'\pi\phi\pi'\pi)(\q C)\\
&=\frac1{|H|^2}(\pi'
\pi\psi\pi'\pi\phi)(\q C)=\frac1{|H|}(\pi'
\pi\psi\phi)(\q C)=(\psi\phi)(\q C),
\end{align*}
so that $\phi$ and $\psi$ commute, as desired.
\end{proof}
We observe that Theorem \ref{autcyc} is false if the coefficient field $F$, or more generally the coefficient ring $R$, has nonzero characteristic:
\begin{example}\label{ex_nzchar}
Let $R$ have characteristic $n>0$. Set $G=Z_{4n}=\langle t\rangle$ and
$H=Z_n\leq G$. Define $S_H$ to be the S-ring $S=S_H \wedge S_{G/H}$
where $S_H$ is the trivial S-ring over $H$ and $S_{G/H}$ is the full
group algebra $R(G/H)$. Then $S$ has five basic sets
$$\{1\},Z_n-\{1\},T_1,T_2,T_3$$ where $T_i=t^iZ_n$. Then in $RG$ we have $\q T_i\q T_j=nt^{i+j}\q Z_n=0$ for all
$i,j\in\{1,2,3\}$ while $(\q Z_n-1)\q T_i = -\q T_i$. Thus $\Aut(S)
\cong \Sym_3$ is non-abelian. Over a ring with characteristic zero,
this same Schur partition gives an S-ring with automorphism group
isomorphic to $Z_2$.
\end{example}
We observe that the Leung-Man classification of S-rings over cyclic groups does not hold if the coefficient field has nonzero characteristic (see, e.g., \cite[Example 1.11]{kerby_masters}), but Example \ref{ex_nzchar} shows that this is not the only reason that Theorem \ref{autcyc} fails in this case. This suggests two problems, both of which we can expect to be difficult:
\begin{problem}
Classify the S-rings over cyclic groups for coefficient rings of nonzero characteristic.
\end{problem}
\begin{problem}
Describe the automorphism groups of such S-rings.
\end{problem}
\section{Converse to Muzychuk's Theorem}\label{sec_conv}
A remarkable theorem of Muzychuk states:
\begin{thm}[\cite{muzychuk2}]\label{cyciso}Two S-rings over a cyclic group $Z_n$ are isomorphic if and only if they are identical. \end{thm}
We prove a converse to this result. But first we need the following lemma:
\begin{lemma}\label{lemnonchar}
Let $G$ be a finite group which is not cyclic. Then $G$ has a subgroup which is not characteristic.
\end{lemma}
\begin{proof}
By way of contradiction, suppose every subgroup of $G$ is
characteristic. Then in particular every subgroup of $G$ is normal.
If $G$ is non-abelian, then $G$ is a Hamiltonian group (i.e., a nonabelian group in which every subgroup is normal) and we may write $G = Q \times A$ where $Q$ is an
8-element quaternion group $\langle i,j\rangle$ and $A$ is abelian
\cite[9.7.4]{scott}. But in this case $\langle i \rangle$ is a
subgroup of $G$ which is not characteristic, since there is an
automorphism of $Q$ mapping $\langle i \rangle$ to $\langle j
\rangle$ and this automorphism extends to an automorphism of $G$.
Therefore $G$ must be abelian.
Since $G$ is not cyclic, some Sylow $p$-subgroup of $G$ is not
cyclic and, by the Fundamental Theorem of finitely-generated abelian
groups, we may write $G = \langle t \rangle \times \langle s \rangle
\times A$ where $|t|=p^a$ and $|s|=p^b$ for some $a$ and $b$ where
$1 \leq a \leq b$. Then $\langle s \rangle$ is not characteristic,
since an automorphism $\phi$ is determined by setting $\phi(s)=ts$,
$\phi(t)=t$, and $\phi(a)=a$ for all $a \in A$.
\end{proof}
We remark that, by a similar method of proof, Lemma \ref{lemnonchar}
may be extended to infinite non-abelian groups and to finitely
generated abelian groups. However, there are non-cyclic infinitely
generated abelian groups in which every subgroup is characteristic,
an example being the direct sum $\underset{\text{$p$ prime}}\sum
Z_p$.
\begin{thm}\label{thm_conv}
Let $G$ be a finite group which is not cyclic. Then there exist distinct Cayley-isomorphic S-rings $S_1$ and $S_2$ over $G$.
\end{thm}
\begin{proof}
By Lemma \ref{lemnonchar}, let $H$ be a subgroup of $G$ which is not
characteristic. Choose some $\phi \in \Aut(G)$ such that $\phi(H)
\neq H$. Then $S_1=F\{1,\q H,\q G\}$ and $S_2=F\{1,\q{\phi(H)},\q G\}$
are S-rings over $G$ which are Cayley-isomorphic. We only need to
verify that they are distinct. The basic quantities of $S_1$ are
$\{1,\q H-1,\q G-\q H\}$ while the basic quantities of $S_2$ are
$\{1,\q{\phi(H)}-1,\q G-\q{\phi(H)}\}$. If $S_1=S_2$ then the basic
quantities of the two S-rings must be the same (in some order), so
either $\q H-1=\q{\phi(H)}-1$ or $\q H-1=\q G-\q{\phi(H)}$. The former
is impossible since $H \neq \phi(H)$. The latter would imply $G=H
\cup \phi(H)$, which is impossible, since no group is the union of
two proper subgroups.
\end{proof}
\section{Some examples}\label{sec_ex}
\begin{example}\label{nonlatsring}
Let $G=Z_p\times Z_{p^3}$ for an odd prime $p$. Let $H_1,\dots,H_6$ be the characteristic subgroups of $G$, in the order shown in Table \ref{char13}. Using Theorem \ref{folklore}, it is easy to check that $S=F\{1,\q H_2,\q H_3+\q H_4,\q H_5,\q G\}$ is a rational S-ring. We show that $S$ cannot be constructed as in Theorem \ref{latsring}. Suppose $S=F\q{\m L}$ for some lattice $\m L$. By Corollary \ref{charind}, the elements $\{\q H : H \in \m L\}$ are linearly independent, hence form a basis for $S$. So $\dim S=|\m L|$. Now $\dim S=5$, yet, by applying Corollary \ref{charind} again, it is easy to see that $1, H_2, H_5$, and $G$ are the only four subgroups of $G$ which are $S$-sets, hence $|\m L|\leq 4$, a contradiction.
\end{example}
Over other abelian $p$-groups, it is easy to construct many rational S-rings similar to Example \ref{nonlatsring}, where the basic quantities are sums of characteristic subgroups, chosen in such a way as to ensure closure under the Hadamard and ordinary product; but, thus far, it has not been possible to extend this construction to give a complete classification of rational S-rings over abelian $p$-groups. Some of the difficulty is indicated by the following example:
\begin{example}
Let $G=Z_p\times Z_{p^3}\times Z_{p^5}$ where $p$ is any prime. Let $H_1,\dots,H_{18}$ be the regular characteristic subgroups of $G$, as in Table~\ref{char135}. By direct computation, one can show using Corollary~\ref{folklore_cor} that $$S=F\{1,\q H_5,\q H_6+\q H_7-\q H_{8}-\q H_{11},\q H_8+3\q H_{11}-\q H_{12}-\q H_{13},\q H_{14},\q G\}$$ is an S-ring over $G$ if and only if $p=3$. It is of course also possible to present $S$ in terms of its basic elements:
$$S=F\{1,\q O_2+\q O_3+\q O_4+\q O_5,\q O_6+\q O_7+\q O_{14},
\q O_{12}+\q O_{13},\q O_8+\q O_{10}+\q O_{11},\q O_9+\q O_{15}+\q O_{16}+\q O_{17}+\q O_{18}\},$$
where $O_i=O(\h a)$ for $H_i=R(\h a)$. This S-ring does not have a basis consisting of sums of characteristic subgroups. This example also shows that choice of prime $p$ can make a difference in determining whether a partition of automorphism classes of $G$ is a Schur partition or not, and that it is not merely a question of whether $p=2$. For a further discussion of this and related examples, see \cite[Example 5.28]{kerby_masters} (But note that the initial basis given there for $S$ is erroneous; we have given a correct basis above).
\end{example}
Such examples lead us to a conjecture:
\begin{conj}
Let $(\lambda_1,\dots,\lambda_n)$ be a tuple of integers, $1 \leq \lambda_1 \leq \cdots \leq \lambda_n$. Then, as $p \to \infty$, the set of Schur partitions of the abelian $p$-group
$$G=Z_{p^{\lambda_1}} \times \cdots \times Z_{p^{\lambda_n}}$$
is eventually constant, and for sufficiently large $p$, every rational S-ring over $G$ has a basis consisting of sums of characteristic subgroups (where, here we are abusing notation slightly by thinking of the Schur partitions as partitions of $\m C(G)$, rather than of $G$ itself).
\end{conj}
One natural approach to classifying rational S-rings over abelian $p$-groups would involve (1) answering this conjecture, (2) giving an explicit description of which sums of characteristic subroups are allowed, and (3) attempting to understand the apparently ``exceptional" rational S-rings which occur when $p$ is small relative to $n$.
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Tazria 2019
04/04/2019 04/04/2019 by The Hebron Fund
Parshat Tazria
By: Rabbi Moshe Goodman, Kollel Ohr Shlomo, Hebron
לשכנו תדרשו
Inviting the Holy Presence in Our Holy Land
On the Eighth Day You Shall Circumcise
There are only two positive commandments in the Torah that are so severe that abstaining from them renders the transgressor liable of "karet," spiritual "incision from the People": circumcision and Korban Pesach. Interestingly, these commandments involve "incision". In circumcision, it is the foreskin; in Korban Pesach, the slaughtering of the lamb. Also, there is a connection between these mitzvoth as the Korban Pesach cannot be eaten by one who is not circumcised himself or even when one of his sons (under Bar-mitzva age) or servants are not circumcised. Simply, we may explain that the fact that these two commandments are so important and also interconnected is due to the fact that these commandments represent the initiation of our People into the covenant of Hashem. Circumcision was the first commandment given to Avraham our Father in Hebron as part of the initiation of him and his offspring into the covenant with Hashem. Korban Pesach was the initiation of our People into the covenant of Hashem in the Exodus.
Examining this matter more deeply, we see that the term for circumcision in Hebrew, "brit mila," can literally mean "covenant of the word." The Arizal explains that Moshe Rabeinu's state (at the time of the Exodus) as "uncircumcised of tongue," was actually indicative of the entire People's state in Exile, i.e "spiritual muteness". Therefore, on the day of the Exodus, every year, we celebrate "Pe" – "Sach," i.e the "talking mouth," in telling the wonders of Hashem during the Exodus. Indeed, tractate Psachim, the tractate that deals with the laws of Pesach, begins with a discourse about the power of words, and how one should be careful to use only speech that is pure and good.
The Torah (Dvarim 10, 16) says: "and you shall circumcise your hearts, and stiffen your necks never more." This means that, along with the physical and verbal "circumcision" we just mentioned, there must also be a "circumcision of the heart." How can we truly purify ourselves of slander, impure speech, etc., if our hearts harbor thoughts and feelings of anger, hatred, lust, etc.?
Yet deeper is a united vision coming from the inspiration of Hebron, the City of Unity, the place of Avraham's covenant with Hashem. Through this inspiration we realize the common meaning of covenant both in circumcision and Korban Pesach. While circumcision highlights the covenant with Hashem inherent in the People Israel, the Korban Pesach highlights this covenant with Hashem in regard to the Holy Presence in the Temple – the "soul" of our Holy Land. The fact that we are missing the Korban Pesach today shows us that we must take this matter to our hearts, speech, and actions. The Holy Presence is the Origin and Light of our souls, and therefore Its redemption, through our efforts to glorify It in our Holy Land and Temple, is ultimately also the redemption of our People's thoughts, speech, and actions, let this be, Amen.
Real Stories from the Holy Land #309
"Wanted: Families or singles to resettle ancient city of Hebron. For details contact Rabbi M. Levinger."
This unassuming newspaper advertisement captured the attention of many Israelis in 5728 (1968). Rabbi Moshe Levinger and a group of like-minded individuals determined that the time had come to return home to the newly liberated heartland of the Land of Israel. In Hebron, the Park Hotel's Arab owners were delighted to accept the cash-filled envelope which Rabbi Levinger placed on the front desk. In exchange, they agreed to rent the hotel to an unlimited amount of people for an unspecified period of time. The morning of Passover eve saw the Levinger family along with families from Israel's north, south and center packed their belongings for Hebron. They quickly cleaned and koshered the half of the hotel's kitchen allotted to them and began to settle in. Eighty-eight people celebrated Passover Seder that night in the heart of Hebron. Two days later, Rabbi Levinger announced to the media that the group intended to remain in Hebron. Dignitaries, Knesset members and Israelis from near and far streamed to the Park Hotel to encourage the pioneers. Defense Minister Moshe Dayan was anxious to remove the pioneers from the hotel. He suggested that they move to the military compound overlooking Hebron. Six weeks later, the pioneers moved to the military compound. Rabbi Levinger insisted on accommodations for 120 people even though they numbered less than half at that time. Rabbi Levinger was accused of being an unrealistic dreamer. Within a few short weeks however, he was proven correct. The 120 places in the military compound could not accommodate the hundreds of people who wanted to be part of the renewed of Jewish life in Hebron, city of the Patriarchs.
Source: hebronfund.org
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Categories Torah Portion Tags Torah Portion
Shemini 2019
Passover in Hebron 2019
Cave of the Patriarchs (Machpelah)
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L'Australian Indoors (Sydney, Australie) est un tournoi de tennis féminin du circuit professionnel WTA.
Une seule édition de l'épreuve a été organisée en 1985, remportée par Pam Shriver en simple et en double (avec Elizabeth Smylie).
Palmarès
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Lien externe
Site de la WTA
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{"url":"https:\/\/mathoverflow.net\/questions\/213055\/number-of-representations-as-sums-of-squares-in-rings-of-integers-of-number-fiel","text":"# Number of representations as sums of squares in rings of integers of number fields\n\nLet $K$ be some number field, $\\mathcal O_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\\alpha \\in \\mathcal O_K$, and consider the quantity $r_{n,K}(\\alpha)$, which denotes the number of solutions to the equation $$\\alpha = x_1^2 + x_2^2 + \\ldots + x_n^2$$ with $x_1, x_2, \\ldots, x_n \\in \\mathcal O_K$.\n\nWhen $K=\\mathbb Q$ and $\\mathcal O_K=\\mathbb Z$, the precise formulas for the computation of $r_{n,K}(\\alpha)$ were developed for all $n \\leq 4$ (perhaps for other $n$'s as well, but I am not familiar with them). Are there any formulas for $r_{n,K}(\\alpha)$ when $[K:\\mathbb Q] > 1$? I am especially interested in the cases when $n = 2, 3, 4$ and $K$ is a real quadratic extension of $\\mathbb Q$.\n\n\u2022 On page 11 of \"A first course on modular forms\" by Diamond and Shurman it is shown that a generating function counting solutions can be made into a modular firm for a congruence subgroup. Using this, the weight of the form, its level, some analysis to find the corresponding vector space of modular forms you may actually be able to recover the generating function. This works over the rationals, and I guess that you can use Hilbert modular forms over totally real fields (such as real quadratic fields). \u2013\u00a0Pablo Aug 5 '15 at 1:09\n\u2022 Have a look at the following survey for the subtleties of the ternary case over totally real number fields (the quaternary case is simpler): arxiv.org\/abs\/1402.1332 \u2013\u00a0GH from MO Aug 5 '15 at 12:20\n\nThere are some results known in the $n = 4$ case. In particular, in 1928 Gotzky (see Mathematische Annalen volume 100 pages 411-437) proved a formula for the number of representations of a totally positive integer $\\alpha \\in \\mathcal{O}_{K}$ as a sum of four squares of elements in $\\mathcal{O}_{K}$ for $K = \\mathbb{Q}(\\sqrt{5})$. In 1960, Harvey Cohn wrote a paper (in the American Journal of Mathematics, pages 301-322) determining formulas when $K = \\mathbb{Q}(\\sqrt{2})$ and $K = \\mathbb{Q}(\\sqrt{3})$. A 1961 paper of Cohn addresses the question of sums of three squares over $\\mathbb{Q}(\\sqrt{2})$ (where there is a formula) and $\\mathbb{Q}(\\sqrt{3})$ (where there might not be as clean a formula).\n\nKate Thompson, a recent Ph.D. student of Jonathan Hanke and Daniel Krashen, has been working on extending results about sums of four squares to other quadratic number fields.\n\nFor $n=4$, in addition to what Jeremy mentioned, there is some addition information in the answers to sum of squares in ring of integers. John Goes has also done some work but it seems his preprint is not yet available.\n\nThere are also some results for $n=2$ and $n=3$.\n\nFor $n=3$, see Donkar's 1977 American Journal paper. He uses quaternion algebras over number fields like he did earlier over $\\mathbb Q$. He has fairly general results with just an assumption on the even primes of the number field, and he worked out some very explicit statements in several examples such as $\\mathbb Q(\\sqrt 5)$, $\\mathbb Q(\\sqrt 2)$ and $\\mathbb Q(\\sqrt 17)$.\n\nFor $n=2$, there has at least been work for $\\mathbb Q(\\sqrt 2)$. See Michele Elia & Chirs Monico's paper On the Representation of Primes in $\\mathbb Q(\\sqrt 2)$ as Sums of Squares (JP Journal of Algebra, Number Theory, 2007).\n\n(There has been other work over imaginary quadratic fields when $n=2$, but I'm not sure about other results over totally real fields right now. There are also results for $n>4$.)","date":"2019-10-19 16:12:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.915676474571228, \"perplexity\": 185.2504425249768}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986696339.42\/warc\/CC-MAIN-20191019141654-20191019165154-00541.warc.gz\"}"}
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Sheetz Will Increase Salaries Despite Overtime Rule Injunction
ALTOONA, Pa. — Sheetz Inc. will continue with its plans to give all salaried employees a pay increase that provides a minimum base salary of $47,500 per year despite a federal judge's injunction preventing a new overtime regulation from going into effect.
The convenience store chain's decision to institute a base salary was made in conjunction with the proposed Fair Labor Standards Act rule from the U.S. Department of Labor, which called for an increase to the minimum salary for salaried employees and was set to take effect Dec. 1.
"Since our founding in 1952, the success and satisfaction of our employees at Sheetz has been vital to the accomplishments of the company itself," said Joe Sheetz, president and CEO of Sheetz, Inc. "This announcement represents our constant efforts toward attracting and retaining the best talent and being a great place to work. It is a commitment that reaches beyond compensation, to the offering of excellent benefits and a great balance between work and family."
The pay increase affects approximately 270 employees and is expected to cost approximately $1 million annually.
Sheetz operates more than 535 c-stores and employs more than 17,500 people in Pennsylvania, West Virginia, Maryland, Virginia, Ohio and North Carolina.
Corporate & Store Operations
Sheetz Expanding E15 Availability Thanks to $7M Federal Grant
Sheetz Begins Free GED Program for Employees
Sheetz Invests $16.8M in Wages for Its Store-Level Employees
Legislative, Regulatory & Legal
DOL Focuses on Salary Threshold in Overtime Rule Appeal
Joe Sheetz Named Convenience Store News' 2018 Retailer Executive of the Year
Sheetz Signs Multi-Year Sponsorship Pact With Richmond Raceway
Sheetz to Open Technology & Innovation Hub
USA Today Readers Choose Sheetz as Best Regional Fast Food
NACS Gives Labor Department Its Opinion on Overtime Rule
Sheetz Rolls Out Customized Loyalty Program
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Home › JR Cigar Blog & Videos - Reviews, Lists & More Cigar News - The Latest News in the World of Cigars › Drew Estate: Liga Privada H99 Connecticut Corojo
Drew Estate: Liga Privada H99 Connecticut Corojo
10 years ago, Drew Estate bestowed upon the cigar world one of the highest rated and most popular cigars in history. The Liga Privada is no longer just a cigar line, but a brand name on its own. Its symbolic lion logo can be seen on ashtrays, hats, cutters and even tattoos. It has become a symbol of power, flavor and above all quality. It represents the transition of Drew Estate from infused cigars to high-end premium cigars. While we have seen additions such as the T52 and the little brother line, Undercrown, fans of the brand have been waiting in earnest for a new release under the Liga Privada flag. Last year, cigar shops rejoiced with the announcement of the newest Liga Privada release.
Drew Estate has announced the release of the Liga Privada H99 Connecticut Corojo. It is handcrafted with a blend of aged Honduran and Nicaraguan filler tobaccos. It is held together with a bold and rich Mexican San Andres binder. The finishing touch is a stunning Corojo wrapper grown in the heart of the Connecticut River Valley.
According to a press release from Drew Estate, the H99 will be featured in a single toro size for now, although more sizes are planned for later. This rich blend of tobaccos promises to deliver a rich and complex blend that is spicy, rich and slightly sweet.
I don't know about the rest of you, but I can hardly wait for this cigar. I have yet to be disappointed with any Liga Privada or Unico release. Each one of this cigars has its spot in my regular rotation, and I'm willing to be the H99 will find its place right there as well. Although a shipping date has not yet been announced, keep checking back here for more updates and sales info on the brand new Drew Estate Liga Privada H99 Connecticut Corojo. If you are looking to buy any Drew Estate cigars online, shop right here at JR Cigars.
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Gift bags are a great tool to welcome guests, honor parents on mother and father's days, or use as prizes at VBS or youth retreats. All of our Church Gift Bags are customized for your specific ministry, event or need. Our gift bags come in a variety of materials, colors and sizes. Gift bags can be used to pull your welcome packet together, or hold gifts for guest such as mugs, t-shirts and pens. If you don't see what you're looking for, call us at 866-654-6127. We have more options and ideas for customizing gift bags that we would love to talk to you about.
A great way to bring your welcome package together!
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{"url":"http:\/\/torus.math.uiuc.edu\/cal\/math\/cal?year=2017&month=01&day=01&interval=next+12+months®exp=Harmonic+Analysis+and+Differential+Equations","text":"Seminar Calendar\nfor Harmonic Analysis and Differential Equations events the next 12 months of Sunday, January 1, 2017.\n\n.\nevents for the\nevents containing\n\nQuestions regarding events or the calendar should be directed to Tori Corkery.\n December 2016 January 2017 February 2017\nSu Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa\n1 2 3 1 2 3 4 5 6 7 1 2 3 4\n4 5 6 7 8 9 10 8 9 10 11 12 13 14 5 6 7 8 9 10 11\n11 12 13 14 15 16 17 15 16 17 18 19 20 21 12 13 14 15 16 17 18\n18 19 20 21 22 23 24 22 23 24 25 26 27 28 19 20 21 22 23 24 25\n25 26 27 28 29 30 31 29 30 31 26 27 28\n\n\n\nTuesday, February 14, 2017\n\nHarmonic Analysis and Differential Equations (HADES)\n1:00 pm \u00a0 in 347 Altgeld Hall,\u00a0\u00a0Tuesday, February 14, 2017\n Del Edit Copy\nSubmitted by vzh.\n Tobias Ried (Karlsruhe Institute of Technology)Gevrey smoothing of weak solutions of the homogeneous Boltzmann equation for Maxwellian moleculesAbstract: We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \\in L^1_2({\\mathbb R}^d) \\cap L \\log L({\\mathbb R}^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity. This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers. (Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)\n\nTuesday, April 4, 2017\n\nHarmonic Analysis and Differential Equations\n1:00 pm \u00a0 in 347 Altgeld Hall,\u00a0\u00a0Tuesday, April 4, 2017\n Del Edit Copy\nSubmitted by berdogan.\n Michael Goldberg (U. Cincinnati)To Be Announced","date":"2017-03-27 08:30:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4524725377559662, \"perplexity\": 573.9957168491774}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218189466.30\/warc\/CC-MAIN-20170322212949-00535-ip-10-233-31-227.ec2.internal.warc.gz\"}"}
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{"url":"http:\/\/www.sciforums.com\/threads\/new-motherboard.2635\/","text":"# new motherboard\n\nDiscussion in 'Computer Science & Culture' started by dexter, Mar 2, 2001.\n\nNot open for further replies.\n1. ### dexterROOTRegistered Senior Member\n\nMessages:\n689\nhello, i was just courious if anyone had anything on the new gigabyte mother board, the 7dx. from what i hear it seems pretty nice, they fixed the bugs from a earlier version. the reason i am asking is becaseu i am in the market for a new motherboard, AMD or course, able to hold 1 ghz athalon.\n\nalso, if you know of any other good motherboards, that are below 250 (mabe evan 200) please tell me.\n\nthanx\n\nMessages:\n4,127\nIf you're willing to wait a few months, I think the dual-processor AMD boards will be available. I'm holding off upgrading untill then (dual-GHz thunderbirds! sweet!).\n\nIf not, the Abit KT7 is pretty cheap (only $215 Canadian which is ... ~$140 US)\n\n5. ### dexterROOTRegistered Senior Member\n\nMessages:\n689\ndual 1ghz thunderbird, arrrgghhh...... that would be sweet, but i am on a budget, since i do not have the money, and am fixing computers for my dad to pay for it, i think i can only get 1 procesor for now. mabe in a few years or months i will replace the m\/b and get another procesor?","date":"2018-10-16 08:37:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.23005691170692444, \"perplexity\": 4596.41899457813}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583510415.29\/warc\/CC-MAIN-20181016072114-20181016093614-00290.warc.gz\"}"}
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{"url":"https:\/\/tex.stackexchange.com\/questions\/132422\/self-defined-width-of-the-pseudocode-box-with-algorithmicx","text":"# Self-defined Width of the Pseudocode Box with algorithmicx?\n\nThis post has beautifully shown how to make the algorithm pseudocode box as wide as one column or two columns.\n\nI am now writing a pseudocode with many if layers. So a one-column box cannot suffice. However, the layers are not that many to span over two columns. So my problem is that if I use the two-column box, the RHS is too empty and the comments are too far away from the code.\n\nOne-column, too narrow. Two-column, too wide.\n\nCan I define the width of the algorithm box myself, say 1.5 columns wide?\n\nYou can place the algorithmic environment inside a minipage of desired width (for example, 1.5\\columnwidth):\n\n\\documentclass{IEEEtran}\n\\usepackage{algpseudocode}% http:\/\/ctan.org\/pkg\/algorithmicx\n\\usepackage{lipsum}% http:\/\/ctan.org\/pkg\/lipsum\n\\usepackage{float}% http:\/\/ctan.org\/pkg\/float\n\\floatstyle{boxed} % Box...\n\\restylefloat{figure}% ...figure environment contents.\n\\begin{document}\n\\section{A section}\n\\begin{figure*}\n\\centering\n\\caption{Euclid\u2019s algorithm}\\label{euclid}\n\\begin{minipage}{1.5\\columnwidth}\n\\begin{algorithmic}[1]\n\\Procedure{Euclid}{$a,b$}\\Comment{The g.c.d. of a and b}\n\\State $r\\gets a\\bmod b$\n\\While{$r\\not=0$}\\Comment{We have the answer if r is 0}\n\\State $a\\gets b$\n\\State $b\\gets r$\n\\State $r\\gets a\\bmod b$\n\\EndWhile\\label{euclidendwhile}\n\\State \\textbf{return} $b$\\Comment{The gcd is b}\n\\EndProcedure\n\\end{algorithmic}\n\\end{minipage}\n\\end{figure*}\n\n\\lipsum[1-15]% dummy text\n\\end{document}\n\n\nTo reduce the box size itself, it is far more convenient to remove the boxed float style, and mere wrap the minipaged algorithmic environment inside an \\fbox:\n\n\\documentclass{IEEEtran}\n\\usepackage{algpseudocode}% http:\/\/ctan.org\/pkg\/algorithmicx\n\\usepackage{lipsum}% http:\/\/ctan.org\/pkg\/lipsum\n\\usepackage{float}% http:\/\/ctan.org\/pkg\/float\n%\\floatstyle{boxed} % Box...\n\\restylefloat{figure}% ...figure environment contents.\n\\begin{document}\n\\section{A section}\n\\begin{figure*}\n\\centering\n\\caption{Euclid\u2019s algorithm}\\label{euclid}\n\\fbox{\\begin{minipage}{1.5\\columnwidth}\n\\begin{algorithmic}[1]\n\\Procedure{Euclid}{$a,b$}\\Comment{The g.c.d. of a and b}\n\\State $r\\gets a\\bmod b$\n\\While{$r\\not=0$}\\Comment{We have the answer if r is 0}\n\\State $a\\gets b$\n\\State $b\\gets r$\n\\State $r\\gets a\\bmod b$\n\\EndWhile\\label{euclidendwhile}\n\\State \\textbf{return} $b$\\Comment{The gcd is b}\n\\EndProcedure\n\\end{algorithmic}\n\\end{minipage}}\n\\end{figure*}\n\n\\lipsum[1-15]% dummy text\n\\end{document}\n\n\u2022 cool it works! only one minor adjsutment to be done. The upper and lower boundaries of the algorithm box remain unchanged, still 2 columns. How to make it also on the minipage? THanks:) Sep 9, 2013 at 15:55","date":"2022-09-27 17:43:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8657186031341553, \"perplexity\": 6047.58597715714}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030335054.79\/warc\/CC-MAIN-20220927162620-20220927192620-00377.warc.gz\"}"}
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\section{Introduction}
Applied researchers often claim that the risk difference (RD) is a more heterogeneous effect measure than the relative risk (RR) and odds ratio (OR); see \cite{poole2015risk} and references therein. Based on surveys of meta-analyses, they found \emph{empirical evidence} that the null hypotheses of homogeneity are rejected more often for the RD than for the RR and OR. However, some epidemiologists have pointed out that the current empirical evidence for the claim that the RD is more heterogeneous is not satisfactory. Specifically, \cite{poole2015risk} point out that the current empirical results on RD being more heterogeneous may be due to difference in statistical power of tests used. From this perspective, the high power for testing the null of RD or RR may in fact be used as an argument to support the use of RD or RR over OR.
On the other hand, some argue that there are \emph{theoretical grounds} for this claim. For example, \cite{schmidt2016comments} point out that if $p_{00}=0.27$, $p_{10}=0.46, p_{01}>0.81,$ then the RD is not possible to be homogeneous; here $p_{va} = P(Y(V=v, A=a))$ denotes the risk among treatment group $a$ and covariate group $v$. See \cite{omalley2022discussion} for a similar example. \cite{ding2015differential} theoretically quantify the homogeneity of different effect scales. Under the no-interaction condition for any effect measure, the four outcome probabilities $p_{va}, v=0,1,a=0,1$ lie in a three dimensional space, called the the homogeneity space. \cite{ding2015differential} compare the three-dimensional volume of the homogeneity space in $\mathbb{R}^4$ and found that the homogeneity space of the RD has the smallest volume, and that of the OR has the largest volume.
In this note, we argue that the current theoretical arguments for the heterogeneity of the RD is not satisfactory either. These arguments reflect that certain effect measures are variation independent of a nuisance parameter that is easier to interpret (i.e. the baseline risk), rather than the homogeneity of these measures. Note that former is decided by humans, but the latter is decided by nature. We also point out that the volume of the homogeneity space depends on the coordinate system one chooses. Although \cite{ding2015differential}'s calculation holds under the probability scales, one can construct alternative scales under which the volume of the homogeneity space is larger for the RR than the OR.
Unless we know which coordinate system the nature prefers, it seems very difficult to argue
which null hypothesis of homogeneity is more likely to hold. Alternatively, one can say under the prior information of a specific coordinate system, one effect measure is more heterogeneous than another.
\section{Background and notation}
Let $A = 0,1$ be a binary exposure, $Y=0,1$ be a binary response, and $V=0,1$ be a binary covariate. Define $p_{av} = P(Y\mid A=a, V=v)$ and $p_a(V) = P(Y\mid A=a, V).$ We shall focus on the following effect measures in this article: $RR(v) = p_{1v}/p_{0v}, RD(v) = p_{1v}-p_{0v}, OR(v) = p_{1v}(1-p_{0v})/\{p_{0v}(1-p_{1v})\}.$
Unlike the OR, both RD and RR are \emph{variation dependent} on the baseline risk $p_0(V)$. For example, if $p_0(v) = 0.5$, then the range of possible values for the $RD(v)$ is restricted to be $[0.5,-0.5],$ while that for the $RR(v)$ is restricted to be $[0,2].$ In contrast, under the same condition, the range for the $OR(v)$ coincides with its range without this condition, i.e. $[0,\infty)$.
Take the RR as an example. First, consider the saturated Poisson model:
\begin{flalign*}
\log RR(V) &= \alpha_0^P + \alpha_1^P V; \\
\log p_0(V) &= \beta_0^P + \beta_1^P V,
\end{flalign*}
where $
\beta_0^P = \log p_{00},
\beta_1^P = \log p_{10} - \log p_{00},
\alpha_0^P = \log p_{01} - \log p_{00},
\alpha_1^P = \log p_{11} - \log p_{10} - (\log p_{01} - \log p_{00}).
$ The model on $p_0(V)$ is a nuisance model as it is not of primary interest. However, as pointed out above, it is variation dependent on $RR(V).$
\cite{richardson2017modeling} discover that the so-called odds ratio $OP(v) = p_{1v}p_{0v}/\{(1-p_{1v})(1-p_{0v})\}$ is \emph{variation independent} of both the RR and RD. In other words, the range of possible values for $RD(v)$ or $RR(v)$ does not depend on the value of $OP(v)$. Based on this, an alternative model for estimating the RR is
\begin{flalign*}
\log RR(V) &= \alpha_0^P + \alpha_1^P V; \\
\log OP(V) &= \beta_0^R + \beta_1^R V,
\end{flalign*}
where $
\beta_0^P = \log OP_{0\cdot},
\beta_1^P = \log OP_{1\cdot} - \log OP_{0\cdot},
\alpha_0^P = \log p_{01} - \log p_{00},
\alpha_1^P = \log p_{11} - \log p_{10} - (\log p_{01} - \log p_{00}).
$
\section{Volumes of the Homogeneity Spaces}
\cite{ding2015differential} impose a uniform distribution on $(p_{00},p_{10},p_{01})$, which induces a distribution on $(\beta_0^P, \beta_1^P, \alpha_0^P)$. They then calculated the probability that $(\beta_0^P, \beta_1^P, \alpha_0^P)$ is compatible with the condition $\alpha_1^P=0$ under this induced distribution.
Because $\alpha_1^P$ is variation independent of $(\beta_0^P, \beta_1^P, \alpha_0^P)$, and hence $(p_{00},p_{10},p_{01})$, it is not surprising that the probability is smaller than 1. In fact, according to \cite{ding2015differential}, this probability is 0.75.
\cite{ding2015differential} conducted a similar calculation for the OR. Following a similar argument, since $\alpha_1^L \defeq \log odds_{11} - \log odds_{10} - (\log odds_{01} - \log odds_{00})$ is variation independent of $(p_{00},p_{10},p_{01})$, the corresponding probability is 1.
The key point in \cite{ding2015differential}'s arguments is that there are certain values of $(p_{00},p_{10},p_{01})$ that are not compatible with $\alpha_1^P=0$, but are compatible with $\alpha_1^L=0$. Hence under any $(p_{00},p_{10},p_{01})$, it is more likely that $\alpha_1^P=0$ is rejected.
However, there is no reason to believe that nature would impose a prior distribution on $(p_{00},p_{10},p_{01})$, which are variation independent of $\alpha_1^L$ but variation dependent of $\alpha_1^P$. For example, if the nature imposes a prior distribution on $(\alpha_0^P, \beta_0^R, \beta_1^R)$, then the corresponding probability for $RR$ would be 1 since $(\alpha_0^P, \beta_0^R, \beta_1^R)$ is variation independent of $\alpha_1^P$. Similarly, since $(\alpha_0^P, \beta_0^R, \beta_1^R)$ is variation independent of $\alpha_1^L,$ the corresponding probability for $OR$ would also be 1.
To further illustrate this point, we consider a nuisance parameter, $$\eta(Y|A,V) \defeq \left|\log \dfrac{(1-p_0(V))(p_1(V)+0.5)}{(1-p_1(V))p_0(V)}\right|.$$
The nuisance parameter is made up so that is variation independent of $RR$, but variation dependent of $OR$.
We note that although this measure seems unnatural for human beings, there is no reason to think this is unnatural for nature.
Following the notation above, define the following saturated model
\begin{flalign*}
\log RR(Y|A,V) &= \alpha_0^P + \alpha_1^P V; \\
\log \eta(Y|A,V) &= \beta_0^\eta + \beta_1^\eta V.
\end{flalign*}
Following similar arguments as before, if the nature imposes a prior distribution on $(\alpha_0^P, \beta_0^\eta, \beta_1^\eta)$, then the probability that $(\alpha_0^P, \beta_0^\eta, \beta_1^\eta)$ is compatible with the condition $\alpha_1^P = 0$ (or equivalently, $RR=1$) would still be 1, but the corresponding probability for $\alpha_1^L = 0$ (or equivalently, $OR=1$) would be smaller than 1!
\section{Concluding remarks}
In this note, we present counterarguments for the claim that nature would think RD and RR to be more heterogeneous than OR. We argue that this common perception is due to some parameterizations being more natural to human beings.
In the literature, it is often claimed that if a measure is more homogeneous under one scale, then it would be preferable to use this scale as a measure of the treatment effect for the sake of generalizability \citep[e.g.][]{ding2015differential}. We argue that in some contexts that rely on generalizability such as meta-analysis, collapsibility is also important as it reflects the generalizability of findings within
the same data set. Furthermore, as advocated in \cite{richardson2017modeling}, one should not choose the effect measure based on statistical properties anyway.
\bibliographystyle{apalike}
\section{Summary}
\label{sec:summary}
\begin{itemize}
\item Motivation: many meta-analyses report that the RD is more heterogeneous than the RR and OR
\begin{itemize}
\item Specifically, this means the null hypotheses of homogeneity are rejected more often for RD than for the RR and OR.
\end{itemize}
\item Question 1: Is it more ``likely'' that the null hypothesis of homogeneity holds (exactly) for RD than for the RR and OR
\begin{itemize}
\item The null hypothesis always holds with probability zero, so we need to define what we mean by ``likely''
\item \cite{schmidt2016comments} thinks there is ``theoretical grounds for why the odds ratio is ... less heterogeneous''. They points outs that if $p_{0A}=0.27$, $p_{1A}=0.46, p_{0B}>0.81$ then the RD is not possible to be homogeneous.
\item \cite{ding2015differential} mathematically quantifies this idea. They essentially two definitions to define the ``likelihood'' here.
\begin{itemize}
\item Under the no-interaction condition for any effect measure, the four outcome probabilities lies in a three dimensional space, called the \emph{the homogeneous space}.
\item Definition 1: Volumes of the homogeneous space
\item Definition 2: Three-dimensional volume of the homogenous space in $\mathbb{R}^4$.
\end{itemize}
\item Critique: the volume is coordinate system-dependent!
\item Conclusion: as volume of the homogeneous space can be larger for RD under one coordinate system but larger for OR under another coordinate system, unless we know which coordinate system the nature prefers, it is hopeless to argue which null hypothesis of homogeneity is more ``likely'' to hold.
\begin{itemize}
\item Alternatively we can say under the prior information of a specific coordinate system, one measure is more heterogenous than another one.
\end{itemize}
\end{itemize}
\item Question 2: for a given alternative, are tests of heterogeneity more powerful for one measure than the other?
\begin{itemize}
\item \cite{poole2015risk} pointed out that the current empirical results on RD being more heterogeneous may be due to difference in statistical power of tests used.
\item If this \emph{were} always true for one measure, then it is a reason \emph{in favor of} using that methods, rather than the other way around (but we don't think this would hold in general).
\item We agree with \cite{poole2015risk}, but bring this one step further by arguing that for a given alternative, it may be inherently more difficult to test heterogeneity for one measure than another.
\begin{itemize}
\item If we use empirical averages for point estimates of risks, then we could compare the ``true'' p-value by repeat sampling from the data generating distribution (which we know in simulation)
\item So under these alternatives, it is not because we don't have more powerful tests for one measure, but because of the sampling mechanism, it is harder to test heterogeneity for one measure than another.
\item (To be done) by theoretical calculation we might be able to give guidance on what are the values of $(p_{0A},p_{0B},p_{1A},p_{1B})$ such that it is easier to test heterogeneity for one measure than another.
\end{itemize}
\item Although it may be tempting to compare the ``likelihood'' of the truth falling into a region where it is inherently more difficulty to test the heterogeneity for one measure, we caution against this as the volume is coordinate-system dependent.
\item A not-so-naive reader may think this is due to variational dependence, a linear or Poisson regression is better at detecting interaction than the logistic regression. At least for the $2$ by $2$ case, simulation results disagree.
\begin{itemize}
\item Linear regression and the variational independent RD regression \citep{richardson2016modeling} give virtually the same results (and they should!)
\item In more complicated settings such as linear covariate case, we warn that the linear/Poisson regression that practitioners are likely to be wrong \emph{a priori} and hence give biased point estimates. In this case, a comparison of power would be meaningless.
\end{itemize}
\end{itemize}
\item Seems hopeless: what is next?
\begin{itemize}
\item We should not choose measure based on statistical property anyway!
\item Even if we prefer one measure for the sake of generalizability, collapsibility/transportability should be a more important issue: that deals with generalizability of findings within the same data set!
\item This a problem that is theoretically hopeless and practically not important.
\end{itemize}
\end{itemize}
\section{Set-up}
Let $A = 0,1$ be the binary exposure, $Y=0,1$ be the binary response, and $V=0,1$ for the binary covariate. Define $p_{va} = P(Y(V=v,A=a)).$ We assume $A$ is randomized.
The saturated Poisson model is
\begin{flalign*}
\log RR(Y|A,V) &= \alpha_0^P + \alpha_1^P V; \\
\log p_0(V) &= \beta_0^P + \beta_1^P V.
\end{flalign*}
Under randomization, it is easy to verify that
\begin{flalign*}
\beta_0^P &= \log p_{00} \\
\beta_1^P &= \log p_{10} - \log p_{00} \\
\alpha_0^P &= \log p_{01} - \log p_{00} \\
\alpha_1^P &= \log p_{11} - \log p_{10} - (\log p_{01} - \log p_{00}).
\end{flalign*}
The saturated RR model as described in \cite{richardson2016modeling} is
\begin{flalign*}
\log RR(Y|A,V) &= \alpha_0^P + \alpha_1^P V; \\
\log OP(Y|A,V) &= \beta_0^R + \beta_1^R V.
\end{flalign*}
Under randomization, it is easy to verify that
\begin{flalign*}
\beta_0^P &= \log OP_{0\cdot} \\
\beta_1^P &= \log OP_{1\cdot} - \log OP_{0\cdot} \\
\alpha_0^P &= \log p_{01} - \log p_{00} \\
\alpha_1^P &= \log p_{11} - \log p_{10} - (\log p_{01} - \log p_{00}).
\end{flalign*}
\subsection{Domain of the Homogeneity Spaces}
\cite{ding2015differential} impose a uniform distribution on $(p_{00},p_{10},p_{01})$, which induces a distribution on $(\beta_0^P, \beta_1^P, \alpha_0^P)$. They then calculated the probability that $(\beta_0^P, \beta_1^P, \alpha_0^P)$ is compatible with the condition $\alpha_1^P=0$ under this induced distribution.
Because $\alpha_1^P$ is variation independent of $(\beta_0^P, \beta_1^P, \alpha_0^P)$, and hence $(p_{00},p_{10},p_{01})$, it is not surprising that the probability is smaller than 1 (according to \cite{ding2015differential}, it is 0.75).
\cite{ding2015differential} conducted a similar calculation for the OR. Following a similar argument, since $\alpha_1^L \defeq \log odds_{11} - \log odds_{10} - (\log odds_{01} - \log odds_{00}).$ is variation independent of $(p_{00},p_{10},p_{01})$, the corresponding probability is 1.
The key point in \cite{ding2015differential}'s arguments is that there are certain values of $(p_{00},p_{10},p_{01})$ that are not compatible with $\alpha_1^P=0$, but are compatible with $\alpha_1^L=0$. Hence under any $(p_{00},p_{10},p_{01})$, it is more likely that $\alpha_1^P=0$ is rejected.
\bigskip
However, there is no reason to believe that nature would impose a prior distribution on $(p_{00},p_{10},p_{01})$, which are var. ind. of $\alpha_1^L$ but var. dep. of $\alpha_1^P$. For example, if the nature imposes a prior distribution on $(\alpha_0^P, \beta_0^R, \beta_1^R)$, then the corresponding probability for $RR$ would be 1 since $(\alpha_0^P, \beta_0^R, \beta_1^R)$ is var. ind. of $\alpha_1^P$. Similarly, since $(\alpha_0^P, \beta_0^R, \beta_1^R)$ is var. ind. of $\alpha_1^L$ (?) the corresponding probability for $OR$ would also be 1.
\bigskip
To further illustrate this point, we consider a nuisance parameter, $$\eta(Y|A,V) \defeq \left|\log \dfrac{(1-p_0(V))(p_1(V)+0.5)}{(1-p_1(V))p_0(V)}\right|.$$
The nuisance parameter is made up so that is variation independent of $RR$, but variation dependent of $OR$.
We note that although this measure seems ``unnatural'' for human beings, there is no reason to think this is ``unnatural'' for the nature.
Following the notation above, define the following saturated model
\begin{flalign*}
\log RR(Y|A,V) &= \alpha_0^P + \alpha_1^P V; \\
\log \eta(Y|A,V) &= \beta_0^\eta + \beta_1^\eta V.
\end{flalign*}
Following a similar arguments as before, if the nature imposes a prior distribution on $(\alpha_0^P, \beta_0^\eta, \beta_1^\eta)$, then the corresponding prob. for $RR$ would still be 1, but the corresponding prob. for $OR$ would be smaller than 1!
\subsection{Volume of the Homogeneous Spaces}
Using the great circle puzzle as an example. \cite{ding2015differential} shows that although the measure for $S_m(p)$ and $S_o(p)$ are both zero, they have different ``three-dimensional volumes!'' As a metaphor, this corresponds to the intuition that although both a great circle and a small circle on a unit sphere have measure 0, the great circle should have large ``density'' than the small circle. This is supported by the fact that the length of a great circle is larger than a small circle.
However, as we argued before, there is no reason that the nature adopts the coordinate system $(p_{00},p_{01},p_{10},p_{11}).$ For example, if the nature adopts an alternative coordinate system $(\alpha_0^P,\alpha_1^P,\beta_0^R,\beta_1^R)$ (note that these four quantities are var. ind. so that they can be used to define an coordinate system), then the ``three dimension volume'' for the corresponding $S_m(p)$ would be 1, which is the maximal possible volumes under this coordinate system. One could also tell a similar story using $(\alpha_0^P,\alpha_1^P,\beta_0^\eta,\beta_1^\eta)$.
\subsection{Heterogeneity Comparison Based on Statistical Inference}
I think statistical uncertainty and the common parameterization we use is responsible for the observed ``heterogeneity''. For example, from a Bayesian perspective, although there is no reason to believe the parameterization $(p_{00},p_{01},p_{10},p_{11})$ is more ``natural'' for the nature, it is indeed more natural for human beings. So if we impose a prior distribution using this coordinate system, then surely we find the posterior probability of violating $\alpha_1^P=0$ is higher than $\alpha_1^L=0$. But that is due to the parameterization we choose: not due to the truth.
The things may be more tricky from a Frequentist perspective. It is possible that due to the variation dependence, the Poisson model is better at detecting interaction. For example, if $p_{00} = 0.3, p_{01} = 0.7, p_{10}=0.7, p_{11}=0.8$. I yet need to do simulation on this, and with other examples, such as the one in \cite{poole2015risk}. But my intuition is that one may tell some interaction with Poisson model because of the range problem.
Update: simulation results disagree. (see Section \ref{sec:summary}).
\section{Concluding remarks}
\begin{enumerate}
\item The consensus in this literature, as stated by \cite{ding2015differential}, seems to be that if the treatment effect is more homogeneous under one scale, then it would be preferable to use this scale as a measure of the treatment effect (for the sake of generalizability!). However, we think collapsibility is a more important issue than homogeneity, especially in the context of meta-analysis.
\item We agree with \cite{poole2015risk} that if the empirical evidence for the heterogeneity is due to the improved power for detecting difference in RD or RR, then this is not a bad thing. In fact, this may be used as an argument to support the use of RD or RR.
\item We disagree with the implication of \cite{ding2015differential} that the nature would think RD and RR to be more heterogeneous than OR. We argue that it is due to the parameterization that is more natural to human beings.
\item We also comment that in general, unless in the unusual saturated case, the Poisson/linear model for RR/RD is likely to be wrong. Hence the RR/RD model as described in \cite{richardson2016modeling} is still preferable.
\end{enumerate}
\bibliographystyle{apalike}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Static quantities}
From the grid of simulated state points we build a corresponding grid of
translational ($D_{trans}$) and diffusional ($D_{rot}$) coefficients, defined as:
\begin{equation}
D_{trans} = \lim_{t\rightarrow+\infty} \frac{1}{N} \sum_i \frac{\langle \|{\bf x}_i(t) - {\bf x}_i(0)\|^2 \rangle}{6 t}
\end{equation}
\begin{equation}
D_{rot} = \lim_{t\rightarrow+\infty} \frac{1}{N} \sum_i \frac{\langle \| \Delta\Phi_i \|^2 \rangle}{4 t}
\end{equation}
where $\Delta\Phi_i = \int_0^t {\bf \omega}_i dt$,
${\bf x}_i$ is position of the center of mass and $ {\bf \omega}_i$ is the angular velocity of ellipsoid $i$.
By proper interpolation, we evaluate the
isodiffusivity lines, shown in Fig. \ref{Fig:isod}.
Results show a striking decoupling of the translational and rotational
dynamics. While the translational isodiffusivity lines
mimic the swallow-like shape of the coexistence between the isotropic liquid and the crystalline phases (as well as the MMCT prediction for the glass transition\cite{LetzSchilLatz}), rotational isodiffusivity lines reproduce qualitatively the shape of the I-N coexistence.
As a consequence of the the swallow-like shape, at large fixed $\phi$, $D_{trans}$ increases by increasing the particle's anisotropy, reaching its maximum at $X_0\approx 0.5$ and $X_0\approx 2$.
Further increase of the anisotropy results in a decrease of $D_{trans}$. For all $X_0$, an increase of $\phi$ at constant $X_0$ leads to a significant suppression of $D_{trans}$, demonstrating that $D_{trans}$ is controlled
by packing.
The iso-rotational lines are instead mostly controlled by $X_0$, showing a progressive slowing down of the rotational dynamics independently from the translational behavior. This suggests that on moving along a path of
constant $D_{trans}$, it is possible to progressively decrease the rotational dynamics, up to the point where rotational diffusion arrest and all rotational motions become hindered.
Unfortunately, in the case of monodisperse HE, a
nematic transition intervenes well before this point is reached. It is thus stimulating to think about the possibility of designing a system of hard particles in which the nematic transition is inhibited by a proper
choice of the disorder in the particle's shape/elongations.
We note that the slowing down of the rotational dynamics is consistent with MMCT predictions of a nematic glass for large $X_0$ HE\cite{LetzSchilLatz}, in which orientational degrees of freedom start to freeze approaching the isotropic-nematic transition line, while translational degrees of freedom mostly remain ergodic.
\begin{figure}[tbh]
\vskip 1cm
\includegraphics[width=.47\textwidth]{Fig2}
\caption{Isodiffusivity lines. Solid lines are isodiffusivity lines from translational diffusion coefficients
$D_{trans}$ and dashed lines are isodiffusivities lines from rotational diffusion coefficients $D_{rot}$.
Arrows indicate decreasing diffusivities. Left and right arrows refer to rotational diffusion
coefficients. Diffusivities along left arrow are: $1.5$, $0.75$, $0.45$, $0.3$, $0.15$.
Diffusivities along right arrow are: $1.5$, $0.75$, $0.45$, $0.3$, $0.15$, $0.075$, $0.045$.
Central arrow refers to
translational diffusion coefficients, whose values are: $0.5$, $0.3$, $0.2$, $0.1$, $0.04$, $0.02$.
Thick long-dashed lines are FM and TME coexistence lines from Fig.\ref{Fig:grid}}.
\label{Fig:isod}
\end{figure}
To support the possibility that the slowing down of the dynamics
on approaching the nematic phase originates from a close-by glass transition,
we evaluate the self part of the intermediate scattering function $F_{self}$
\begin{equation}
F_{self}(q,t) = \frac{1}{N} \langle \sum_j e^{i{\bf q}\cdot({\bf x}_j(t) - {\bf x}_j(0))} \rangle
\end{equation}
and the second order orientational correlation function
$C_2(t)$ defined as \cite{AllenFrenkelDyn}
$C_2(t) = \langle P_2(\cos\theta(t))\rangle $,
where $P_2(x) = (3 x^2 - 1) / 2$ and $\theta(t)$ is the angle between the symmetry axis at time $t$ and at time $0$.
The $C_2(t)$ rotational isochrones
are found to be very similar to rotational isodiffusivity lines.
These two correlation functions never show a clear two-step relaxation decay in the entire
studied region, even where the isotropic phase is metastable, since the system can not be
significantly over-compressed.
As for the well known hard-sphere case, the amount of over-compressing
achievable in a monodisperse system is rather limited. This notwithstanding, a comparison of
the rotational and translational correlation functions reveals that the onset of dynamic slowing down and glassy
dynamics can be detected by the appearance of stretching.
Fig. \ref{Fig:corrshape} contrasts the shape of $F_{self}$, evaluated at $q=q_{max}$, where $q_{max}$ is
the $q$ corresponding to the first maximum of the center-of-mass static structure factor, and $C_2(t)$
at $\phi=0.50$ for different $X_0$ values with best-fit based on
an exponential ($ \sim \exp[-t/\tau]$) and a stretched exponential ($ \sim \exp[-(t/\tau)^\beta]$) decay. As a criteria to avoid including in the fit the short-time ballistic contribution, we limit the time-window to
times larger than $t^*$, defined for $F_{self}$ and $C_2$
as the time at which the autocorrelation
function of center-of-mass velocity ${\bf v}$ (${\phi}_{vv}(t) \equiv \frac{1}{N}\sum_i \langle {\bf v}_i(t){\bf v}_i(0) \rangle$ ) and of angular velocity respectively
(${\phi}_{\omega\omega}(t) \equiv \frac{1}{N}\sum_i \langle {\bf\omega}_i(t) {\bf\omega}_i(0) \rangle$) reaches $1/e$ of its initial value.
\begin{figure}[tbh]
\includegraphics[width=.5\textwidth]{Fig3}
\caption{Shape of $F_{self}$ and $C_2$ at $\phi=0.50$ for different
$X_0$. Symbols are data from MD simulations.
Solid lines are fits to exponential functions, while long-dashed lines are fits to
stretched exponentials ($\beta$ is the stretching parameter).
$t^*$ is the time at which correlation functions $\phi_{vv}$ and $\phi_{\omega\omega}$, for $F_{self}$ and $C_2$
respectively, reach $1/e$ of their initial values.
Top: Prolate ellipsoids with $X_0=3.2$, $C_2$ shows a significant stretching while $F_{self}$ decays exponentially.
Center: $X_0=1.0$ for $F_{self}$ and $X_0=1.1$ for $C_2$, the dashed line is
the theoretical decay of a free rotator $C_2^f$
($C_2^f(t) = 1-\frac{3}{2}\frac{t}{\tau_f}\exp[-t^2/\tau_{f}^2]\tilde{\Phi}(t/\tau_f)$, where
$\tau_{f}^2=1/\phi_{\omega\omega}(0)$ and $\tilde{\Phi}(t)=\int_0^t \exp[x^2] dx$).
Bottom: Oblate ellipsoids with $X_0=0.348$. }
\label{Fig:corrshape}
\end{figure}
We note that $F_{self}$ shows an exponential behaviour close to the I-N transition ($X_0=3.2$,$\ 0.3448$)
on the prolate and oblate side, in agreement
with the fact that translational isodiffusivities lines do not exhibit any peculiar behaviour close to the I-N line. Only when $X_0\approx 1$,
$F_{self}$ develops a small stretching, consistent with the minimum of the swallow-like curve observed in the
fluid-crystal line \cite{HardSpheresExp,HardSpheresSim}, in the jamming locus as well as in the predicted behavior of the glass line for HE\cite{LetzSchilLatz} and for small elongation dumbbells\cite{DumbellChongGoetze,DumbellChongFra}. Opposite behavior is seen for the case of the
orientational correlators. $C_2$ shows stretching at large anisotropy, i.e. at small and large $X_0$ values,
but decays within the microscopic time for almost spherical particles.
In this quasi-spherical limit, the decay is well represented by the decay of a free rotator\cite{FreeRotator}. Previous studies of the rotational dynamics of
HE\cite{AllenFrenkelDyn} did not report stretching in $C_2$, probably due to the smaller values of $X_0$ previously investigated and to the present increased statistic which allows us to follow the full decay of the correlation functions.
Fig. \ref{Fig:corrshape} clearly shows that $C_2$ becomes stretched approaching the I-N transition while $F_{self}$ remains exponential on approaching the transition.
To quantify the amount of stretching in $C_2$ we show in Fig. \ref{Fig:betavsX0}
the $X_0$ dependence of $\tau$ and $\beta$ for three different values of $\phi$.
In all cases, slowing down of the characteristic time and stretching increases progressively on approaching the I-N transition. It is interesting to observe that the amount of stretching appears to be more pronounced in the case of oblate HE compared to prolate ones.
A similar (slight) asymmetry between oblate and prolate HE can be observed in the lines reported in Figure $2$.
In summary, we have shown that clear precursors of dynamic slowing down and stretching can be observed in the region of the phase diagram where a (meta)stable isotropic phase can be studied. Despite the monodisperse character of the present system prevents the possibility of observing a clear glassy dynamics,
our data suggest that a slowing down in the orientation degree of freedom --- driven by the elongation of the particles --- is in action. The main effect of this shape-dependent slowing down is
a decoupling of the translational and rotational dynamics which generates an almost perpendicular crossing of the $D_{trans}$ and $D_{rot}$ isodiffusivity lines.
This behavior is in accordance with MMCT predictions, suggesting two glass transition mechanisms, related respectively to cage effect (active for $0.5 \lessapprox X_0 \lessapprox 2$) and to pre-nematic order ($X_0 \lessapprox 0.5$, $X_0 \gtrapprox 2$) \cite{LetzSchilLatz}.
It remains to be answered if it is possible to find a suitable model, for example
polydisperse in size and elongation, for which nematization can be sufficiently destabilized, in analogy to the destabilization of crystallization induced by polydispersity in hard-spheres.
\begin{figure}[tbh]
\vskip 1cm
\includegraphics[width=.45\textwidth]{Fig4}
\caption{$\beta_{C_2}$ and $\tau_{C_2}$ are obtained from fits of $C_2$ to a stretched exponential for $\phi=0.40,0.45$ and $0.50$. Top: $\tau_{C_2}$ as a function
of $X_0$. Bottom: $\beta_{C_2}$ as a function of $X_0$. The time window used
for the fits is chosen in such a way to exclude the microscopic short times ballistic
relaxation (see text for details). For $0.588 < X_0 < 1.7$ the orientational relaxation is
exponential.}
\label{Fig:betavsX0}
\end{figure}
We acknowledge support from MIUR-PRIN.
We also thank A. Scala for suggesting code optimization and
taking part to the very early stage of this project.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,189
|
\section{Introduction}
Since 2002, the spectrometer SPI \citep{Vedrenne2003_SPI} on board the INTErnational Gamma-Ray Astrophysics Laboratory \citep[INTEGRAL satellite][]{Winkler2003_INTEGRAL} has been observing astrophysical high-energy phenomena in the hard X-ray and soft $\gamma$-ray range between 20~keV and 8~MeV. In this energy range, measured $\gamma$-ray spectra are dominated by instrumental background (BG) photons. These originate mainly from nuclear de-excitation reactions and continuum processes, such as bremsstrahlung, of the instrument and satellite material, being exposed to cosmic-rays (CRs). There is no stand-alone BG model available to deal with SPI observations in an extensive, consistent, and elaborate way. In this paper, we show how to construct a self-consistent BG model for SPI data analysis from a data base of spectral parameters over the INTEGRAL mission years.
The spectra in each of the 19 high-purity Ge detectors of SPI can be characterised by a large number of instrumental $\gamma$-ray lines on top of a broken power-law shaped continuum. The main features of the spectra among detectors, energy, and time stay constant and change gradually according to solar activity and detector degradation. \citet{Diehl2018_BGRDB} illustrated how the SPI spectral BackGround Response Data Base (BGRDB) is created, maintained, and checked for consistency over the entire mission time. In general, the BGRDB contains fitted spectral parameters per detector and time. The time integration is chosen as either one INTEGRAL orbit of three\footnote{In early 2015, the INTEGRAL spacecraft performed several orbit adjustment manoeuvre, to safely bring the satellite back to Earth in 2029. For this reason, the INTEGRAL orbit is now merely 2.7~days long, and will decrease with ongoing mission time.} days, or the time between two detector restoration\footnote{About twice a year, the lattice structure of the SPI Germanium detectors is repaired from cosmic-ray radiation by heating up the camera for several days to $\approx 100~\mrm{^{\circ}C}$. This is called an annealing.} periods, which is typically half a year. These different time scales are a key issue for the understanding and the construction of a $\gamma$-ray telescope BG model - to record sufficient BG data for reliable fits to the spectral, and to be able to trace gradual changes in the spectral response.
This paper is structured as follows: In Sec.~\ref{sec:data_analysis}, we describe, how SPI data analysis is generally treated (Sec.~\ref{sec:general_method}), and describe the parts of the SPI BGRDB which are required for BG modelling in more detail. Based on this, we explain how we construct a self-consistent BG model in Sec.~\ref{sec:bg_model_construction}, with full algorithm detail (Sec.~\ref{sec:general_approach}) and a discussion about the underlying foundation (Sec.~\ref{sec:why_does_this_work}). In Sec.~\ref{sec:fit_adequacy}, we show how to evaluate the BG model fit adequacy for different test cases for point-like and diffuse emission (Sec.~\ref{sec:data_sets}), which are then further discussed considering their temporal BG behaviour (Sec.~\ref{sec:time_scale}) in Sec.~\ref{sec:test_cases}.
\begin{figure*}[!ht]
\centering
\subfloat[Typical $5\times5$ rectangular dithering strategy with SPI, marked with dots and sequenced by number above the insets. Each dot represents one pointing with a field of view of $16^{\circ} \times 16^{\circ}$ and is $2.1^{\circ}$ away from the next/neighbouring pointing. The celestial source is marked with a blue star symbol at the position $(l/b) = (35.8^{\circ} / -3.9^{\circ})$. In each inset panel, ranging from 0 to 2.75, the relative detector pattern of how the source would be seen by SPI is shown. The dashed lines in each sub-panel indicate how different the expected sky pattern appears and changes from pointing to pointing with respect to a flat $1:1:...:1$-ratio pattern. \label{fig:sky_pattern}]{\includegraphics[width=1.4\columnwidth,trim=1.0cm 1.1cm 2.4cm 2.1cm,clip=true]{det_pat_str.pdf}}\\
\subfloat[Zoom of panel 13 in Fig.~\ref{fig:sky_pattern}. The relative detector pattern is shown; the dashed line at 1.0 marks a flat detector pattern. \label{fig:pat_pan13}]{\includegraphics[width=0.813\columnwidth,trim=2.5cm 1.9cm 2.4cm 2.4cm,clip=true]{diss_plot_panel13_zoom.pdf}}~~~~
\subfloat[Shadowgram equivalent to detector pattern of panel 13. \label{fig:dets_pan13}]{\includegraphics[width=0.687\columnwidth,trim=2.8cm 0.8cm 2.8cm 0.8cm,clip=true]{det_pat_sky_panel13.pdf}}
\caption{Detector pattern and shadowgram of a celestial source near the optical axis of SPI. From \citet{Siegert2017_PhD}.}
\end{figure*}
\section{SPI data analysis}\label{sec:data_analysis}
\subsection{General method - fitting time series}\label{sec:general_method}
SPI data analysis and source flux extraction is performed for individual energy bins. Each energy bin\footnote{The smallest public available bin size in SPI data analysis is 0.5~keV. The chosen bin width for the actual data set, however, is arbitrary. To increase the statistics and to reduce the time for the analysis at the same time (number of bins), for example, the bin width can be increased. Note, that spectral information is lost when large energy bins are used.} is treated separately, even though the bins are connected, e.g. by instrumental resolution. This dependence is taken into account when the BG model is built.
The counting of photons into one energy bin over time obeys the Poisson statistics. From this, the likelihood, $\mathcal{L}(\boldsymbol{\theta}|D)$, can be calculated, given the data set, $D$, and the model parameters $\boldsymbol{\theta}$. The data in SPI per energy bin is structured as a vector of magnitude $\mrm{"number~of~detectors"}\times\mrm{"number~of~pointings"} = \left\|d\right\| \times \left\|p\right\|$, where $\left\|d\right\|$ would be equal to 19 if all individual detectors\footnote{SPI also records events which occur on multiple detectors within a short time, classified as double, triple, and high-order detector hits, which are individually saved in SPI data sets.} of SPI are used, and $\left\|p\right\|$ equals the number of observations, $N_{obs}$, in a specific data set.
A pointing, $p$, is an observation unit for a specific amount of time, $T_p$, typically 0.5 to 1.0~h, and a specific region in the sky, marked by the galactic longitude and latitude $(l_p/b_p)$. The satellite is staring during this time at this position, and is recording data. Depending on the type of observation (diffuse/point-like; persistent/transient), $N_{obs}$ pointings are observed over larger regions in the sky, or only around the target source. Sub-pointing time intervals, e.g. for gamma-ray bursts, can also be analysed. The likelihood is given by
\begin{equation}
\mathcal{L}(\boldsymbol{\theta}|D) = \prod_{p=1}^{N_{obs}} \frac{m_p^{d_p} \exp(-m_p)}{d_p!}\mrm{,}
\label{eq:likelihood}
\end{equation}
where $m_p$ is the model to describe (to be fitted to) the data $d_p$ per pointing $p$. The log-likelihood (C-stat) is then given by
\begin{equation}
\mathcal{C}(\boldsymbol{\theta}|D) = -2 \ln\mathcal{L}(\boldsymbol{\theta}|D) = -2 \sum_{p=1}^{N_{obs}} \left( d_p \ln m_p - m_p - \ln d_p! \right) \mrm{.}
\label{eq:loglikelihood}
\end{equation}
We will use the likelihood and related values (such as C-stat or $\chi^2$-values) and the degrees of freedom (dof: number of data points minus the number of fitted parameters) in a maximum likelihood fit as a goodness-of-fit criterion for our BG modelling method in Sec.~\ref{sec:fit_adequacy}.
SPI data are typically dominated by instrumental BG\footnote{Only in a few cases during the INTEGRAL mission, transient sources appeared which out-shined the BG. One example would be the microquasar V404 Cygni \citep[e.g.][]{Rodriguez2015_v404,Roques2015_V404,Siegert2016_V404,Jourdain2017_V404} during revolutions 1554--1557.}, so that a subtraction in neither spectral, nor in the $\left\|d\right\| \times \left\|p\right\|$-dimension would provide reasonable results. Furthermore, there is no absolute BG model for SPI as there are variations in all dimensions which change according to the unpredictable solar activity. The goal is hence to obtain a description of the BG, from the data itself, but independent from the celestial source of interest. The key to such a BG model is found in the way, the data is modelled, and how the different model terms (can) influence each other:
\begin{equation}
m_p = \sum_{t} \sum_{j} R_{jp} \sum_{k=1}^{N_S} \theta_{k,t} M_{kj} + \sum_{t'} \sum_{k=N_S+1}^{N_S+N_B} \theta_{k,t'} B_{kp}\mrm{.}
\label{eq:spimodfit_model}
\end{equation}
Here, $M_{kj}$ is the k-th of $N_S$ sky images (celestial emission models) to which the instrumental response function (IRF, coded-mask shadowing), $R_{jp}$, is applied for each pointing $p$ and image element (e.g. pixel), $j$. The $N_B$ BG models $B_{kp}$ are independent of the IRF in each observation, which means the background is assumed completely independent from the shadowing of the mask and therefore independent of spacecraft repointings. Both model parts, sky and BG, can depend on time, and on different scales, $t$ for celestial sources, and $t'$ for the BG. While the BG timing depends on the instrument materials, being activated and decaying on various time-scales as a consequence of solar activity, the variability of the sky emission is only subject to the physics of the sources themselves. This means that the change of the BG amplitude, $\theta_{k,t'}$, depends on the process which resulted in the BG $\gamma$-ray photon. This may change on very short time scales, e.g. prompt emission after an intense dose of CR particles from a solar flare, or very long scales, e.g. from a radioactive build-up, when the creation of radioactive material happens on shorter time-scales than the decay time.
\subsection{Spectral background and response data base}\label{sec:SPI-BGRDB}
The SPI BGRDB by \citet{Diehl2018_BGRDB} spans the energy range between 20 and 2000~keV per INTEGRAL orbit, and consists of parameters for 383 $\gamma$-ray lines, each modelled with four parameters, and two continuum parameters, depending on the energy\footnote{This includes both data formats, the "single events" data (SE) from 20 to 1392~keV and from 1745 to 2000~keV, as well as the "pulse-shape-discriminated" data (PSD) from 490 to 2000~keV.}. In the energy range between 1 and 8~MeV\footnote{Above 2~MeV, the public available data are termed "high-energy events" (HE), for which the smallest energy binning is set to 1~keV, instead of 0.5~keV for the lower energies.}, the count statistics drops rapidly towards larger energies, so that longer integration times are required with increasing energy to obtain robust fitting results. For the BGRDB above 2~MeV, the time between two annealing periods is used to fit 614 $\gamma$-ray lines on top of a broken power-law-shaped continuum (Weinberger et al. 2019, in prep.). The low- and high-energy bands were subdivided into smaller energy regions, $e$, to limit the number of fitted parameters per fit.
As a function of energy, $E$, the SPI spectra, $F(E)$, per detector $d$, orbit $r$, and energy range $e$, are described by
\begin{eqnarray}
F_{der}(E) & = & C(E; \boldsymbol{c}_{der}) + \sum_{i} L_i(E;\boldsymbol{l}_{der}^i)\mrm{,~with} \\
\boldsymbol{c}_{der} & = & \left(C_{0,der},\alpha_{der}\right)\mrm{,~and}\nonumber\\
\boldsymbol{l}_{der}^i & = & \left(A_{0,der}^i,E_{0,der}^i,\sigma_{der}^i,\tau_{der}^i\right)\mrm{.}\nonumber
\label{eq:fit_fun_spectrum}
\end{eqnarray}
Here, $C(E;\boldsymbol{c}_{der})$ and $L(E;\boldsymbol{l}_{der}^i)$ describe the shape of the $\gamma$-ray continuum and lines with the parameter tuples $\boldsymbol{c}_{der} = (C_0,\alpha)_{der}$ and $\boldsymbol{l}_{der}^i = (A_0,E_0,\sigma,\tau)_{der}^i$, respectively, where $i$ is the index of the i-th $\gamma$-ray line in the band $e$. The parameters $C_0$ and $\alpha$ are the amplitude and the spectral index of the power-law function $C(E)$, normalised at the pivot energy $E_C$. This shape is motivated from the superposition of many power-law-like continuum processes in the satellite:
\begin{align}
C(E;C_0,\alpha) & = C_0 \left(\frac{E}{E_C}\right)^{\alpha} \label{eq:conti_full}
\end{align}
The parameters $A_0$, $E_0$, and $\sigma$ represent the amplitude, centroid, and width of a symmetric Gaussian function, with $\tau$ accounting for the detector degradation. The line shape that results from cosmic-ray bombardment is physically motivated \citep{Kretschmer2011_PhD}: As the regular lattice structure is deteriorated, the electronic structure changes, and charge carriers may be temporarily trapped. This leads to a time delay in the charge collection process, and thus in the charge pulse of the read-out electronics. The non-trapped portion of a charge cloud is proportional to $\exp(-\kappa x)$ \citep{Debertin1988_gamma}, where $x$ is the distance to Ge detector electrode, and $\kappa$ is the absorption coefficient. Since these parameters cannot be determined independently, a combined degradation parameter $\tau$ is used. The convolution of a symmetric Gaussian, $G(E;A_0,E_0,\sigma)$, with an exponential tail function, $T(E;\tau)$, leads to an asymmetric line shape, $L(E;A_0,E_0,\sigma,\tau)$ \citep[see also Eqs.~(3)--(6) in][]{Diehl2018_BGRDB}:
\begin{align}
G(E;A_0,E_0,\sigma) & = A_{0} \exp \left(- \frac{(E-E_{0})^2}{2\sigma} \right) \label{eq:gaussian} \\
T(E;\tau) & = \frac{1}{\tau} \exp \left( - \frac{E}{\tau} \right) \quad \forall E > 0 \label{eq:tail} \\
L(E;A_0,E_0,\sigma,\tau) & = (G \otimes T)(E) = \nonumber \\
& = \sqrt{\frac{\pi}{2}} \frac{A_{0} \sigma}{\tau} \exp \left( \frac{2 \tau (E-E_{0}) + \sigma^2}{2 \tau^2} \right) \nonumber \\
& \erfc \left( \frac{\tau (E-E_{0}) + \sigma^2}{\sqrt{2} \sigma \tau} \right) \label{eq:cls_function_full}
\end{align}
\citet{Diehl2018_BGRDB} finds that there are families of $\gamma$-ray lines in the instrumental background of SPI, which are characterised by their detector patterns. The authors define several groups of lines, of which we recap the most prominent: The "Ge-like" lines show higher count rates for inner detectors ($00$--$06$) with respect to outer detectors ($07$--$18$). These include excitation of nuclei in the SPI camera, such as Ge and Al, but also related and produced isotopes, such as Ga, Zn, and Mg. "Bi-like" lines show the opposite detector pattern, as their main source are in the materials of the SPI anticoincidence shield, made of bismuth-germanate (BGO). Related isotopes in the vicinity of Bi (e.g. Pb, Ra, Tl) and also isotopes from the actinide alpha-decay chains (e.g. Ac, Th, U) show this behaviour. To a lesser extent, also material from mountings (Ti, V), wires (Cu, Co, Fe), and other instruments aboard INTEGRAL (Cd, Te, Cs) show this pattern.
The most important finding from monitoring the long term trends in this spectral response data base is, that the patterns of individual isotopes stay constant on all time scales. This also means that isotopes which produces different line energies show the same pattern. Only detector failures change this behaviour - but then, the patterns are again constant. This is the prime assumption of how a self-consistent background model is constructed, and is further explained in Sec.~\ref{sec:general_approach}. Since the spectral shape changes with time (see above), the detectors may respond differently to this degradation, and the prompt and delayed background emission is different for different energies, the SPI BGRDB includes the information to reconstruct the expected spectral response of the SPI camera and predict the instrumental background as a whole at any point in time in high spectral resolution.
\section{Construction of the background model}\label{sec:bg_model_construction}
\subsection{General approach}\label{sec:general_approach}
At each specific energy, $E$, we model the BG by two components: photons from continuum processes, and from one (or more) $\gamma$-ray line-producing processes. In narrow energy bins, i.e. less than a few times the instrumental resolution\footnote{Between 20 and 8000~keV, the instrumental resolution is determined from narrow background lines, and ranges from 1.7 to 8.5~keV.}, we construct the BG by a combination of \textit{the continuum} and \textit{the lines}, i.e. combining all continuum-like processes and line-like processes to two individual BG models. Depending on the energy, either of the BG components may dominate, Sec.~\ref{sec:point_source_emission} (Fig.~\ref{fig:flux_npar_aic_crab}). Per each pointing, $p$, the factor $\sum_{j} R_{jp} M_{kp} \equiv S_{kp}$ changes for each sky component $k$, i.e. the detector pattern - which detectors are illuminated and how strong - changes with time / pointing. This is illustrated in Fig.~\ref{fig:sky_pattern} for the standard INTEGRAL $5 \times 5$ observation scheme of a point-like source.
On the other hand, for a specific \textit{physical process} inside the satellite, the detector patterns from the BG, $B_{kp}$, do not change. For our two BG model components, $k=c$ for \textit{the continuum}, and $k=l$ for \textit{the lines}, hence follows: $B_{cp} = \mrm{const.} = B^c$ and $B_{lp} = \mrm{const.} = B^l$. What might change as a function pointing (=time), $p$, are the amplitudes $\theta_c(p)$ and $\theta_l(p)$. Therefore only these are determined in the maximum likelihood fit, following Eqs.~(\ref{eq:loglikelihood}) and (\ref{eq:spimodfit_model}), see also Secs.~\ref{sec:BG_changes} and \ref{sec:time_scale}.
Following these considerations \citep[see also][Sec.~2.4]{Diehl2018_BGRDB}, the SPI BG model for a particular detector, $d$, at a specific pointing, $p$, and energy bin, $e$, is written as
\begin{eqnarray}
B_{d,e,p} & = & \theta^c_{e,p} \times B^c_{d,e,p} + \theta^l_{e,p} \times \sum_{i=i(e)}^{N_{lines}(e)} B^l_{d,e,p} = \label{eq:line_one_bg_model} \\
& = & \theta^c_{e,p} \times C(\boldsymbol{c}_{d,e,p(r)}) \cdot t'^c_{e,p} + \nonumber\label{eq:line_two_bg_model}\\
& + & \theta^l_{e,p} \times \sum_{i=i(e)}^{N_{lines}(e)} L_i(\boldsymbol{l}_{d,e,p(r)}^i) \cdot t'^l_{i;e,p}\mrm{.}\nonumber
\label{eq:total_bg_model}
\end{eqnarray}
In Eq.~(\ref{eq:line_one_bg_model}), the BG is factorised into the fitted amplitudes for \textit{continuum} and \textit{lines}, $\theta^c_{e,p}$ and $\theta^l_{e,p}$, respectively, the spectral parts, and a temporal part. Here, individual pointings may be scaled by the same amplitude parameter (see Sec.~\ref{sec:time_scale}). For the \textit{lines} component, we sum over the individual lines $i$, which contribute to one energy bin, $e$. In principle, each line amplitude could be fitted individually, but this is only robust for strong BG lines. The spectral parameters per pointing, $\boldsymbol{c}_{d,e,p(r)}$ and $\boldsymbol{l}_{d,e,p(r)}^i$, depend on in which revolution the pointing occurred, hence $p(r)$\footnote{This is true because the SPI BGRDB was created on an orbit time scale. This can be generalised to arbitrary time bins in spectral parameter data bases, e.g. above 1~MeV where the parameters are being determined on "annealing" time scale, i.e. the average over half a year.}, and are fixed to the values from the SPI BGRDB in the maximum likelihood fit.
The detector pattern per energy and process is included as each detector is assigned a specific value according to the spectral response at that time and energy. Thus, $C_{d,e,p(r)}$ and $\sum_{i=i(e)}^{N_{lines}(e)} L_{i;d,e,p(r)}$ completely determine the spectral as well as detector ratio information required for BG modelling. The tracer functions $t'^c_{e,p}$ and $t'^l_{i;e,p}$ then allow for a relative weighting between pointings for \textit{continuum} and \textit{line} processes, respectively. These are parts of the specific BG modelling per science case, and thus are independent from the SPI BGRDB, hence do not follow $p(r)$. In general, each process can have its own variability, being prompt from cosmic-ray excitation and instantaneous de-excitation, or delayed when there is a longer lifetime of the produced isotopes included. The latter can lead to long-lasting radioactive decays, e.g. after a strong solar flare which created a lot of radioactive material (e.g. $\mrm{^{48}V}$ with a half-life time of 16 days), or to radioactive build-up when the decay-time is considerably larger than the production rate \citep[e.g. $\mrm{^{60}Co}$ with a half-life time of 5.27 years, cf.][]{Diehl2018_BGRDB}. The long-term trends are traced already by the SPI BGRDB as the amplitudes are determined on three-day time scales (Sec.~\ref{sec:BG_changes}), so that the remaining prompt BG emission processes at each energy can be traced by one temporal function, i.e. $t'^c_{e,p} = t'^l_{i;e,p} := t'_{e,p}$.
\subsection{Temporal behaviour}\label{sec:BG_changes}
At individual energies, i.e. for small energy bins - not integrating over the energy range of the physical process, the instrumental $\gamma$-ray line BG patterns, $B^l$, become a function of energy because of detector degradation. This does not mean that the physical processes change - the pattern per process is constant - but the pattern at a particular energy can change. This is caused by the degradation of individual detectors on different time scales due to their individual constitutions, behaviours, and reactions to particle irradiation. These changes in detector patterns versus energy and time are taken into account by using the SPI BGRDB. This data base traces the small but important changes, as it has been created on a three-day time scale for energies below 2~MeV \citep{Diehl2018_BGRDB}. For higher energies, i.e. lower BG rate, the BG count statistics is rarely sufficient to determine the gradual change in degradation, so that a mean value over half a year is determined.
The SPI BG amplitude of a certain instrumental process is in general unpredictable. In SPI data analysis, these BG variations are approximated at first order by "tracers", i.e. rates of onboard radiation monitors, such as the SPI anti-coincidence shield count rate, or the rate of saturating Ge detector events ($\gtrsim 15$~MeV; GeDSat). These rates trace the cosmic-ray particle flux that leads to instrumental $\gamma$-ray BG. The variations of these BG time series during one orbit are of the order of 1\%, i.e. $<1\%$ from pointing to pointing.
These approaches work well for all energy bins, i.e. as small as 0.5~keV or even 1~MeV continuum bands. But it is important to note that the described tracing relies on the prompt effect of cosmic-ray irradiation, which is true for many, but not all BG-generating processes in the satellite. Any BG process on longer scales is not represented by such a tracer, but instead is already implemented in the SPI BGRDB. In the special case of mid-term radioactivities, i.e. longer than several pointings, solar-flare-induced background lines or radioactive build-ups may be traced directly by the respective exponential decay law with the isotope's characteristic decay time \citep[cf. Fig. 15 of][showing the examples $\mrm{^{48}V}$ and $\mrm{^{60}Co}$]{Diehl2018_BGRDB}. Thus instead of relying on independent rates, the natural time scale of the process can also be used.
This tracing provides a good first-order description of the BG model variation on shorter time scales below one orbit and down to pointing-by-pointing. When the BG is fitted to the data per energy bin, however, the appropriate BG re-scaling has to be determined. This depends on energy, because the average BG count rate changes with energy due to different contributions, and strengths of \textit{continuum} and \textit{lines}, and the different intrinsic time-scales of the processes, be they prompt or delayed. This is discussed in Sec.~\ref{sec:time_scale}.
With the experience of hundreds of analysed X- and $\gamma$-ray sources during 16 years of the INTEGRAL mission, in different sky regions and $\gamma$-ray energies, it has become evident that for a large energy range between $\approx 200$ and $8000$~keV, the saturating Ge detector events (GeDSat) are sufficient to trace the inter-pointing variations. The 511~keV BG line, for example, follows strongly the rate of the side-shield assembly of the SPI anticoincidence shield \citep[SSATOTRATE][]{Skinner2014_511,Siegert2016_511}. Below 200~keV, the BG rate is very high and can often be determined on a pointing time-scale, so that no tracer is required at all if the source is also strong. Note that GeDSat does not necessarily explain the full variation from one pointing to the next, and requires re-scaling in our method. Likewise, at the steps of the SPI BGRDB, the BG must be carefully investigated and re-scaled (Sec.~\ref{sec:time_scale}).
\subsection{Discussion - why does this work?}\label{sec:why_does_this_work}
In a single pointing, the mask patterns for sources in the field of view can be very strong, so that a determination of detector patterns from the BG ("BG response") is not possible on this time scale. Off-observations, for example at high latitudes, can be used if broad energy bins are analysed (continuum sources). For detailed spectroscopy in small energy bins, however, the BG patterns change smoothly with time due to detector degradation so that the actual spectra should be used to determine the BG. If many pointings of the same observation are combined, the source imprints due to the mask smear out, so that the resulting spectra per detector for a longer time scale appear as due to BG only. In Fig.~\ref{fig:pattern_smear_out}, we show how $S_{kp}$ smears out for a point source, observed with a $5 \times 5$-dithering strategy of INTEGRAL. For a source contribution of 10\% (90\%) to the total recorded spectrum, the cumulative pattern of BG plus source is varying by less than 1\% (10\%) after only 20 pointings. During one INTEGRAL orbit, 50--90 pointings are performed, so that the contribution of a residual mask pattern from even strong sources would be less than 1\%. This means that the SPI BGRDB \citep{Diehl2018_BGRDB} is mostly free of source contributions of any type, and that in this way, a BG model for single pointings can be constructed.
The detector patterns from instrumental BG continuum, $B^c$, are nearly flat around a value of 1.0 \citep{Diehl2018_BGRDB}. The limit, $\lim\limits_{N_p \rightarrow +\infty}{ \frac{1}{N_p} \sum_{p=1}^{N_p} S_{kp} }$, also evolves to a constant value of 1.0 across all detectors, so that any additional source contribution appears in the time-integrated spectra as an increased BG continuum. For individual pointings however, as analysed in a data set, thus, the amplitude of any source contribution can be determined in the fit because the expected patterns from BG and sky are now clearly different. The maximum likelihood fit including BG continuum, BG lines, and sources, will, as a consequence of this procedure, scale down \textit{the continuum} to "give way" to possible source counts.
\begin{figure}[!ht
\includegraphics[width=\columnwidth]{pattern_smear_out_stddev.pdf
\caption{Variance of the detector patterns as shown for a typical $5 \times 5$-dithering pattern from Fig.~\ref{fig:sky_pattern}. For each added pointing, we calculate the standard deviation across the 19 Ge detectors, and also consider different source (right axis) to BG (left axis) ratios. For a specific ratio, the variance changes with the number of added pointings to the power of $-0.8$. Thus, even for strong source contributions, the mask pattern will smear out over the course of one INTEGRAL orbit.
\label{fig:pattern_smear_out
\end{figure}
As a result, the SPI BGRDB, fitted to the data, provides already a good first-order BG model for any INTEGRAL/SPI observation. For detailed and high-resolution spectroscopy of both continuum as well as $\gamma$-ray line sources - especially at fine energy binning - the characteristics of the parameter values in the data base must be investigated further to ensure consistency of the spectral description, in order to build a stable and robust BG model (see also Secs.~\ref{sec:BG_changes} and \ref{sec:test_cases}):
In the case of partial coding of sources during one INTEGRAL orbit, the relative counts of detectors at the rim of the SPI camera (parts of the outer detectors) may be increased on a per cent level (see above), so that the derived background patterns may be slightly skewed. This can also happen with strong sources in the fully coded field of view, leading to residual "average patterns" from these sources. The additional pattern, which is mixed with the true background pattern in the SPI BGRDB, is the sum of all detector responses from sources seen by SPI in a particular orbit (e.g. the sum of all patterns in Fig.~\ref{fig:sky_pattern}). This \textit{can} introduce weak coding in the partially coded field of view, but has marginal effects on the known sources. It affects mainly the energy range below $\approx 100~\mrm{keV}$, and can be accounted for by adding the expected "average pattern" as a third background model component.
\begin{table*}[!ht]%
\begin{tabular}{lrrrrccl}
Energy & Bin size & $N_{ebins}$ & $N_{obs}$ & $T_{exp}$ & Type & BGRDB & Comments \\
\hline
508--514 & 6.0 & 1 & 12587 & 24.3 & diffuse/line & o & inner Galaxy / also weak point sources\\
1795--1820 & 0.5 & 50 & 92867 & 201.0 & diffuse/line & a & full sky \\
2500--3500 & 1000.0 & 1 & 11029 & 21.6 & diffuse/continuum & a & inner Galaxy \\
\hline
50--100 & 2.0 & 25 & 3771 & 7.2 & point-like/continuum & o & \\
508--514 & 6.0 & 1 & 3771 & 7.2 & point-like/continuum & o & annihilation emission possible \\
1790--1840 & 0.5 & 100 & 3771 & 7.2 & point-like/continuum & o & no $\mrm{^{26}Al}$ expected\\
2500--3500 & 1000.0 & 1 & 3185 & 6.0 & point-like/continuum & a & \\
\hline
\end{tabular}
\caption{Summary of data sets used to show the performance of the BG modelling method. The first three columns define the analysed energy range and the chosen energy bin size in units of keV, and the resulting number of bins, $N_{ebins}$. The fourth and fifth column hold the number of pointings, $N_{obs}$, and the resulting exposure time, $T_{exp}$ in units of Ms for a working detector, from the chosen observations. In the "Type" column, the expected emission types, diffuse vs. point-like, and line vs. continuum emission, are listed. The "BGRDB" column shows on which data base was used to construct the background model, either per one INTEGRAL orbit (o), or per half a year (annealing, a). In the comments, additional characteristics of the data set or the expected emission are mentioned.}
\label{tab:data_sets}
\end{table*}
\section{Evaluation of the background model performance}\label{sec:fit_adequacy}
If the fluxes of the analysed sources are known, for example from previous analysis, or can be expected from theoretical considerations, it is possible to estimate the contribution of source counts in the data set. Consequently, the fit adequacy can be investigated, if the sources would be ignored. In general, the covariance between BG and source amplitude parameters is nearly zero, however will turn to negative values if the source contribution is weak or absent. This is shown and explained in Appendix~\ref{sec:covar_matrix}, Figs.~\ref{fig:correl_matrix_1795}--\ref{fig:corr_source_lines}, for the case of a line signal at 1809~keV (see also Sec.~\ref{sec:1809_bg_analysis}). As a consequence, the BG (time) scaling and the source contributions (also their time scaling if needed) should be optimised simultaneously.
As an absolute figure-of-merit to determine a fit's quality, we use the "modified $\chi^2_{\gamma}$-statistics" \citep{Mighell1999_chi2} as defined by
\begin{equation}
\chi^2_{\gamma} := \sum_{i=1}^{N_{obs}} \frac{\left[ d_i + min(d_i,1) - m_i \right]^2}{d_i + 1}\mrm{.}
\label{eq:mod_chi2}
\end{equation}
As an alternative to $\chi^2_{\gamma}$ and to distinguish between models, we use the Akaike Information Criterion \citep[AIC,][]{Akaike1974_AIC,Cavanaugh1997_AIC}. The AIC is defined as
\begin{equation}
AIC = 2 N_{par} - 2 \ln\mathcal{L}(D|\theta_i) + \frac{2N_{par}^2 + 2N_{par}}{N_{obs} - N_{par} -1}\mrm{.}
\label{eq:definition_AIC}
\end{equation}
This figure-of-merit is similar to the reduced $\chi^2$-value by penalising models with more parameters. Using relative likelihoods (absolute AIC differences, $\Delta\mrm{AIC}$), we identify on which time basis the BG has to be re-adjusted in the different cases of Sec.~\ref{sec:test_cases}. Note, that its absolute value has no proper meaning \citep{Burnham2004_AICBIC}. In general, the smaller the AIC, the more preferred a model is. It accounts for the appropriate likelihood of the data generating process, so that testing different assumptions on the BG (and source) scaling provides the most adequate number of BG (and source) parameters (Sec.~\ref{sec:time_scale}). The $\chi^2_{\gamma}$ and AIC values are \textit{calculated} from the best-fit $\ln\mathcal{L}$-value for reference. In general, one should avoid to rely on (reduced) $\chi^2$ estimations when dealing with Poisson-distributed data, however the $\chi^2_{\gamma}$-statistics is specifically adapted for this case \citep{Mighell1999_chi2}, and can provide a general measure.
\subsection{Data sets}\label{sec:data_sets}
In order to illustrate different applications of the BG modelling method, we choose four energy regions to perform our analysis towards point-like and diffuse emission. We demonstrate the performance of our method on line-like emission at 511~keV, binned into one energy bin between 508 and 514~keV, as would be used for imaging. The second-strongest $\gamma$-ray line at 1809~keV of decaying $\mrm{^{26}Al}$ is analysed between 1795 and 1820~keV in 0.5~keV bins to also show a fine-binned but low-statistics case. Analysis of diffuse emission in a continuum band is presented between 2.5 and 3.5~MeV, summed into one bin of 1~MeV. For point-like emission, we also use a low-energy band, from 50 to 100~keV in 2~keV bins. We make use of Crab observations, performing tests in the same energy bands as for diffuse emission. A summary of the data sets and on which time basis the BGRDB is used, is found in Tab.~\ref{tab:data_sets}.
\subsection{Background changes with time}\label{sec:time_scale}
As described above, the relative weighting of background amplitudes between individual pointings can be determined by a tracer, but which may not fully cover the actual variations. In the following, we use pre-defined time steps to re-scale the BG model, as shown in Tab.~\ref{tab:time_scales}, and judge which scaling, i.e. how many fit parameters, are preferred.
\begin{table}[!ht
\begin{tabular}{lrrrr}
Scaling & Orbits & Pointings & Days & Hours \\
\hline
\verb|Const| & $\infty$ & $\infty$ & $\infty$ & $\infty$ \\
\multirow{4}{*}{\texttt{DetFail}} & at 140 & at 5521 & at 1435.42 & - \\
& at 214 & at 10251 & at 1659.46 & - \\
& at 775 & at 42443 & at 3337.50 & - \\
& at 930 & at 52239 & at 3799.67 & - \\
\verb|Anneal| & 60--70 & 1600--4500 & 170--210 & 4000--5000 \\
\verb|30n| & 30 & 1450--2150 & 90 & 2160 \\
\verb|20n| & 20 & 950--1450 & 60 & 1440 \\
\verb|10n| & 10 & 400--800 & 30 & 720 \\
\verb|5n| & 5 & 190--410 & 15 & 360 \\
\verb|3n| & 3 & 110--250 & 9 & 216 \\
\verb|2n| & 2 & 70--170 & 6 & 144 \\
\verb|1n| & 1 & 50--90 & 3 & 52--72 \\
\verb|24p| & 1/3 & 24 & 3/4--1 & 16--24 \\
\verb|12p| & 1/6 & 12 & 3/8--1/2 & 8--12 \\
\verb|6p| & 1/12 & 6 & 3/16--1/4 & 4--6 \\
\verb|3p| & 1/24 & 3 & 3/32--1/8 & 2--3 \\
\verb|2p| & 1/36 & 2 & 3/48--1/12 & 1--2 \\
\verb|1p| & 1/72 & 1 & 3/96--1/16 & 0.5--1 \\
\hline
\end{tabular}
\caption{Background re-scaling times used in the performance check. The number of orbits, pointings, days, and hours are presented for the largest used data set, the $\mrm{^{26}Al}$-case with 92867 pointings, comprising the full sky over 13.5 mission years. A \texttt{Const} background means, that only one background parameter per component is used for the entire data set, i.e. the background variability is fixed. The \texttt{1p} case allows for large variability as each observation pointing obtains its own background amplitude. The time nodes at which the four SPI detectors, 02, 17, 05, and 01 failed, are given in INTEGRAL Julian Days (IJD) and marked as \texttt{DetFail}. Intermediate background time scalings are chosen either inherent from the spectral background and response data base (\texttt{1n} and \texttt{Anneal}) or from selected times inbetween.}
\label{tab:time_scales}
\end{table}
In the case of diffuse emission, for instance, the expected detector pattern changes more smoothly than that for a point source. In addition, INTEGRAL/SPI data sets as they are typically analysed for diffuse emission cover very large areas in the sky, of which many are expected to include little or no source flux but only instrumental BG. For example, $\mrm{^{26}Al}$ is distributed dominantly in the plane of the Galaxy, but high-latitude observations are included in the data sets to get a better leverage on the absolute BG model. The data sets for large-scale diffuse emission thus include as many data/pointings as possible. In return this means, that the BG has to be re-scaled whenever emission from the sky - in certain regions, given by the emission models - is expected to change its contribution to the total by a significant amount. This could also be, when observing the same sky region at different mission times with different background levels. As a consequence, the sensitivity for diffuse emission with respect to analyses of point sources is reduced.
If a particular point source is investigated, the data sets are typically constrained to the region in which the source is located. When this same source is monitored from time to time, the BG level should be adjusted accordingly - not because the source flux might have changed\footnote{Of course, the source flux can also change, e.g. in the case of X-ray binaries. This just means that the fit is not stabilised by a constant source and varying BG any more, but rather independent for individual times.}, but because the BG level varies on longer time scales (e.g. solar cycle).
\section{Test cases}\label{sec:test_cases}
In the following Sec.~\ref{sec:diffuse_emission}, we will present the different diffuse emission cases and explain the choice of parameters. In Sec.~\ref{sec:point_source_emission}, we illustrate the findings for one point source, and highlight the differences as mentioned above.
\subsection{Diffuse emission - large data sets}\label{sec:diffuse_emission}
\subsubsection{Positron annihilation emission in the bulge}\label{sec:511_bg_analysis}
The 511~keV emission in the Milky Way from the annihilation of electrons with positrons is concentrated in the bulge region \citep[see][for a review]{Prantzos2011_511}. Here we choose only INTEGRAL observations in a central area around $(l/b)=(0^{\circ}/0^{\circ})$ with an extent of $\Delta l \times \Delta b = 15^{\circ} \times 15^{\circ}$. Due to the large field of view of SPI, also emission regions out to $|b|$ and $|l| \approx 20^{\circ}$ are included, and have to be modelled. We use the best-fitting emission model from \citet{Siegert2016_511} to characterise the 511~keV morphology in the bulge. This model consists of four 2D-Gaussian-shaped templates to represent the bulge and the superimposed disk. Here, we combine the individual components into one map.
\begin{figure}[!ht]
\centering
\subfloat[AIC vs. number of fitted BG parameters. \label{fig:npar_aic_511}]{\includegraphics[width=0.49\textwidth,trim=0.38in 0.65in 0.0in 0.49in,clip=true]{511_aic_npar_analysis_skip.pdf}}\\
\subfloat[AIC vs. 511~keV flux. \label{fig:flux_aic_511}]{\includegraphics[width=0.49\textwidth,trim=0.38in 0.65in 0.0in 0.49in,clip=true]{flux_aic_analysis_511_skip_new.pdf}}
\cprotect\caption{Variation of the AIC in fits of the 511~keV emission band against the number of fitted background parameters per component (top). The bottom panel shows variations with the measured flux values for each fit, indicating the time scales in blue, and the corresponding number of fitted parameters in red. The corresponding reduced $\chi^2$-values are indicated on the right axis.}
\end{figure}
Clearly, a minimum is found at a BG time-scale of three days, i.e. one INTEGRAL orbit (Fig.~\ref{fig:npar_aic_511}). Between a time scale of 8~h and the annealing time scale (\verb|12p|--\verb|Anneal|), the measured flux shows only slight variations, indicating adequate and stable fits (Fig.~\ref{fig:flux_aic_511}). Very long (\verb|DetFail| / \verb|Const|) and very short ($\lesssim 8$~h; \verb|1p|--\verb|6p|) time scales, on the other hand, provide bad fits or over-fit the data, respectively, which leads to false flux values. For the optimum AIC at 3~d ($476 \times 2 = 952$ fitted BG parameters; \verb|1n|), the 511~keV flux is determined to $(1.6 \pm 0.1) \times 10^{-3}~\mrm{ph~cm^{-2}~s^{-1}}$, which is consistent with previous measurements considering a narrow and a broad bulge component, together with a disk in this region \citep[e.g.][]{Skinner2014_511,Siegert2016_511}.
The reduced $\chi^2$-value at the optimal AIC is 0.9927 for 203184 dof. This is $2.3\sigma$ from the canonically desired value of 1.0, and thus not "over-fitting" the data. In general in the 511~keV case, the fit, considering the $\chi^2$-statistics, is always acceptable: Between the \verb|Const| BG model with 2 parameters and a fit per each individual pointing with 25172 parameters, the reduced $\chi^2$ changes between 1.0111 and 0.9881, i.e. $+3.6\sigma$ and $-3.5\sigma$ from an optimal $\chi^2$-fit value of 1.0. The BG scaling closest to $\chi^2=1.0$ is found at \verb|2n|, i.e. one BG parameter every second INTEGRAL orbit (approximately six days). The minimum reduced-$\chi^2$ value is found for a fit per pointing, \verb|1p| (0.5--1.0~h), but which is largely penalised by the AIC, and thus represents an over-parametrised fit.
\subsubsection{Full-sky emission of radioactive $\mrm{^{26}Al}$}\label{sec:1809_bg_analysis}
The radioactive isotope $\mrm{^{26}Al}$ is produced in massive stars and ejected by winds and core-collapse supernovae. With a half-life time of 717~kyr, $\mrm{^{26}Al}$ traces the ongoing nucleosynthesis in the Milky Way. These nuclei decay via $\beta^+$-decay to an excited state of $\mrm{^{26}Mg}$, which is short-lived (476~fs) and de-excites to the stable ground state of $\mrm{^{26}Mg}$ by the emission of a $E_{lab} = 1808.63$~keV $\gamma$-ray photon \citep[cf.][for example]{Oberlack1996_26Al,Plueschke2001_26Al,Diehl2006_26Al,Diehl2010_ScoCen,Kretschmer2013_26Al,Bouchet2015_26Al,Siegert2017_PhD}. In this work, we use the Maximum Entropy Map from COMPTEL \citep{Plueschke2001_26Al} for our sky model to determine the BG scaling.
\begin{figure}[!ht
\includegraphics[width=\columnwidth,trim=0.32in 0.42in 0.59in 1.00in,clip=true]{aic_analysis_26Al.pdf
\caption{Study of background scaling in the $\mrm{^{26}Al}$ region. The bottom panel shows the AIC, similar to Fig.~\ref{fig:npar_aic_511}, but as a function of energy. From the optimal AIC at each energy bin, the flux is determined (black data points, upper panel, left axis). In addition, we show the fitted background (gray), divided into continuum (blue) and line (green) contributions (upper panel, right axis). To the black data points, a degraded Gaussian line on top of a power-law-shaped continuum is fitted, and shown as $1$, $2$, and $3\sigma$ uncertainty bands. The dotted line marks the zero flux level.
\label{fig:flux_npar_aic_1809
\end{figure}
Here, we use a 0.5~keV binning to resolve the 1809~keV $\gamma$-ray line between 1795 and 1820~keV. This allows us to show that the optimal BG scaling depends on energy, as the BG intensity rises when strong instrumental BG lines have to be taken into account. In Fig.~\ref{fig:flux_npar_aic_1809}, we show the result of our BG analysis in the $\mrm{^{26}Al}$ case for diffuse emission. While the energy bins containing the $\mrm{^{26}Al}$-line and a complex of strong BG lines (green), require a BG scaling between 30~d and half a year (\verb|30n|--\verb|Anneal|), the neighbouring continuum is sufficiently well fitted by only 1 to 29 parameters per BG component (\verb|Const|--\verb|Anneal|). Note, how also the weak instrumental BG line around 1797~keV is captured by our BG modelling method. In Appendix~\ref{sec:appendix_chi2}, we show the same analysis, based on the minimal reduced $\chi^2$-value in each energy bin. The major difference between the AIC and the $\chi^2$ profiles is in the erratic behaviour among bins, and the large spread in the choice of the optimal number of parameters. The line contribution also favours larger number of parameters - similar to the AIC analysis. The number of parameters in the $\mrm{^{26}Al}$ line is very large (5396), for which reason the errors bars on the derived fluxes are much larger in the optimal $\chi^2$-case than in the optimal AIC case. We compare the spectral parameters of the $\mrm{^{26}Al}$ region in Tab.~\ref{tab:26Al_spectral_fits}. All parameter are consistent, except for the line width, being as small as the instrumental resolution in the optimal $\chi^2$-case, and kinematically broadened for the optimal AIC case. Especially the 1.8~MeV line flux is consistent with previous studies of the full $\mrm{^{26}Al}$ sky \citep[e.g.][]{Bouchet2015_26Al,Siegert2017_PhD}.
\begin{table}[!ht
\begin{tabular}{lrrrrr}
Case & $C_0$ & $\alpha$ & $I_L$ & $\Delta E_{peak}$ & $\Gamma_L$ \\
\hline
$\chi^2_{opt}$ & $3.9^{+3.6}_{-2.7}$ & $93^{+309}_{-119}$ & $1.57^{+0.14}_{-0.13}$ & $0.41^{+0.05}_{-0.05}$ & $3.17^{+0.14}_{-0.13}$ \\
$\mrm{AIC}_{opt}$ & $7.1^{+3.5}_{-3.6}$ & $35^{+92}_{-135}$ & $1.79^{+0.11}_{-0.10}$ & $0.27^{+0.05}_{-0.05}$ & $3.81^{+0.14}_{-0.14}$ \\
\hline
\end{tabular}
\caption{Spectral parameters of the $\mrm{^{26}Al}$-line region, selected for the optimal $\chi^2$-case and the optimal AIC case, $\chi^2_{opt}$ and $\mrm{AIC}_{opt}$, respectively. The units are $10^{-6}~\mrm{ph~cm^{-2}~s^{-1}~keV^{-1}}$, $1$, $10^{-3}~\mrm{ph~cm^{-2}~s^{-1}}$, $\mrm{keV}$, and $\mrm{keV}$ for the continuum amplitude $C_0$, the power-law index $\alpha$, the line flux $I_L$, the line centroid shift $\Delta E_{peak} = E_{peak} - E_{lab}$, and line width (FWHM) $\Gamma_L$, respectively.}
\label{tab:26Al_spectral_fits}
\end{table}
\subsubsection{Diffuse soft $\gamma$-ray continuum}\label{sec:3000_bg_analysis}
Above $\approx 100$~keV and up to several tens of MeV, the diffuse continuum emission in the Milky Way is dominated by inverse Compton scattering \citep{Strong2005_gammaconti}, with contributions from bremsstrahlung and positron annihilation \citep{Beacom2006_511,Knoedlseder07,Strong2010_CR_luminosity,Lacki2014_SF_MeV_GeV}. Here, we use a continuum band between 2500 and 3500~keV to illustrate how our BG modelling method performs for diffuse continuum emission. We combine all photon events into one large energy bin of 1~MeV binwidth to increase the photon number statistics, and to illustrate the capabilities of our method in a broad band analysis.
\begin{figure}[!ht
\includegraphics[width=\columnwidth,trim=1.04in 0.69in 0.95in 0.97in,clip=true]{flux_aic_analysis_MeV.pdf
\caption{Variations of the measured flux between 2.5 and 3.5~MeV for each fit, similar to Fig.~\ref{fig:flux_aic_511}. The time scales are indicated in blue, and the corresponding number of fitted parameters in red. The corresponding reduced $\chi^2$-values are shown on the right axis of the inset.
\label{fig:flux_aic_3000
\end{figure}
For our analysis, we use the central part of the Milky Way, similar to Sec.~\ref{sec:511_bg_analysis}. Because the "high-energy" data sets of SPI, between 2 and 8~MeV, are very sensitive to solar activity, we removed periods in which solar flares strongly imprint and enhance BG intensities of nuclear de-excitation lines, such as $\mrm{^{2}H}$ at 2.2~MeV, $\mrm{^{12}C}$ at 4.4~MeV, or $\mrm{^{16}O}$ at 6.1~MeV. Consequently, the high-energy data sets are smaller than in the 511~keV case. We use an inverse Compton template from GALPROP \citep{Strong2011_GALPROP} to fit the morphology in the 2.5--3.5~MeV band.
The optimal AIC solution is found at a re-scaling time of $\approx 8$~h (\verb|12p|), corresponding to 2630 BG fit parameters for 176537 dof (Fig.~\ref{fig:flux_aic_3000}). The reduced $\chi^2$ value at this point is 1.0107, $3.2\sigma$ away from the desired $\chi^2/\nu=1.0$-solution. Based on the reduced $\chi^2$-value only, the optimal solution would be found at a 1~h BG scaling (\verb|1p|), but which is clearly disfavoured by the AIC, penalising too many fit parameters. The fitted flux values at $3.0\pm0.5$~MeV for acceptable fits range between $1.7$ and $5.3 \times 10^{-6}~\mrm{ph~cm^{-2}~s^{-1}~keV^{-1}}$. From these fits, a detection significance between $3.4$ and $7.0\sigma$ is found. At the optimum AIC, the diffuse continuum flux, based on the used inverse Compton template, is determined to $(2.7\pm0.7) \times 10^{-6}~\mrm{ph~cm^{-2}~s^{-1}~keV^{-1}}$. This is consistent with measurements from COMPTEL \citep{Strong1999_COMPTEL_MeV}, considering the different solid angles covered in the analyses.
\subsection{Point source emission - small and partitioned data sets}\label{sec:point_source_emission}
We use the Crab observations of INTEGRAL/SPI from 14 mission years. As a comprehensive example to be compared to the above cases, we show the energy regions from 50--100~keV in 2~keV bins, the positron annihilation line region between 508 and 514~keV in one 6~keV bin, the $\mrm{^{26}Al}$ region between 1790 and 1840~keV in 0.5~keV binning, and one 1~MeV bin between 2.5 and 3.5~MeV. Unlike in the diffuse emission examples, the Crab should only show continuum emission, and no $\gamma$-ray lines\footnote{Emission lines from decaying $\mrm{^{44}Ti}$ could be expected but are below the sensitivity limit for SPI.}. Similar to the high-energy case for diffuse emission, we apply special selection criteria to the Crab observations to avoid strong solar activity, for which reason also the Crab data set above 2~MeV is smaller than below. The choice of energy regions allows for direct comparison of the diffuse and line emission cases.
\begin{figure*}[!ht
\includegraphics[width=\textwidth,trim=0.60in 0.00in 0.10in 0.44in,clip=true]{crab_analysis_aic_all.pdf
\caption{Study of background scaling in the Crab spectrum at different energies with varying energy binning, similar to Fig.~\ref{fig:flux_npar_aic_1809}. The bottom panel shows the AIC as a function of energy. From the optimal AIC at each energy bin, the flux is determined (black data points, upper panel, left axis). The fitted background is shown as gray histogram, divided into continuum (blue) and line (green) contributions (upper panel, right axis). The black data points are fitted by a power-law-shaped continuum, either only using the 50--100~keV range (pink band), only the 1790--1840~keV range (yellow band), or all four ranges together (red band).
\label{fig:flux_npar_aic_crab
\end{figure*}
In Fig.~\ref{fig:flux_npar_aic_crab},we show the results of our BG modelling method for the Crab and the time scaling analysis in four different bands with varying energy bin size. In the energy range, between 50 and 100~keV, it is evident that there is a large difference in BG re-scaling with time, as the BG flux changes rapidly with energy. Especially below $\approx 80$ and above $\approx 90$~keV, the instrumental BG \textit{line} contributions dominate over the instrumental BG \textit{continuum}, so that the BG scaling time is optimal between 4 and 8~h (\verb|6p|--\verb|12p|). In the \textit{continuum}-dominated region, 3~d (\verb|1n|) scaling and thus fewer parameters are enough for an optimal BG scaling. The resulting spectrum in this energy band is smooth and power-law like. Similar to the diffuse emission in the centre of the Galaxy, the BG amplitude in the 511~keV line band should be fitted every three days. There is no enhancement of a possible narrow 511~keV line in the spectrum, as determined from the spectral fit over all analysed energy bands (see below). Likewise, the energy band between 1790 and 1840, in which emission from $\mrm{^{26}Al}$ at 1809~keV could be expected, is sufficiently scaled once between each detector failure or annealing period (\verb|DetFail|--\verb|Anneal|) in the \textit{continuum}, and every 60--90~d (\verb|20n|--\verb|30n|) in the comparably strong instrumental \textit{line} complex between 1805 and 1813~keV. No $\mrm{^{26}Al}$ line is detected, which reinforces our method, being able to systematically suppress instrumental BG lines, and only reveals the source spectrum. At higher energies, between 2.5 and 3.5~MeV, the Crab flux is about the same order of magnitude as the diffuse emission in the galactic centre. However, the emission is concentrated into one point, so that the re-scaling must be less often than in the galactic diffuse case - here only once per three days (\verb|1n|). Also here, the difference in the uncertainties is evident between diffuse emission and point-like emission. While the galactic centre shows a flux uncertainty of $7.0 \times 10^{-7}~\mrm{ph~cm^{-2}~s^{-1}~keV^{-1}}$, the Crab emission is uncertain by $0.9 \times 10^{-7}~\mrm{ph~cm^{-2}~s^{-1}~keV^{-1}}$, even though the effective exposure times are about three times larger in the galactic centre (25~Ms) with respect to the Crab (7.5~Ms).
\begin{table}[!ht
\begin{tabular}{lrrr}
Energy range & $A_0$ & $\alpha$ & $E_m$~[keV]\\
\hline
50--100~keV & $1.367(1) \times 10^{-3}$ & $-2.073(3)$ & $75$ \\
1790--1840~keV & $3.2(5) \times 10^{-6}$ & $-5^{+15}_{-22}$ & $1815$ \\
Full range & $10.53^{+0.12}_{-0.17}$ & $-2.073(3)$ & $1$ \\
\hline
\end{tabular}
\caption{Spectral parameters of the Crab spectrum, derived from different energy bands. We normalise the spectra with $A_0$ in units of $\mrm{ph~cm^{-2}~s^{-1}~keV^{-1}}$ at different energies $E_m$ to avoid numerical problems.}
\label{tab:spec_params_crab}
\end{table}
We check for consistency in our analysis by fitting the Crab spectrum, using three different energy bands: (a) between 50 and 100~keV, (b) between 1790 and 1840~keV, or (c) all data points between 50 and 3500~keV. The fitting results are shown in Tab.~\ref{tab:spec_params_crab}, demonstrating consistency between the analyses, and further substantiating our BG modelling method. The Crab spectrum is determined to be $(10.53^{+0.12}_{-0.17}) \times E^{-2.073 \pm 0.003}~\mrm{ph~cm^{-2}~s^{-1}~keV^{-1}}$ between 50 and 3500~keV with a reduced $\chi^2$ value of 1.32 for 165 dof. This is consistent with previous SPI results reported for the Crab \citep[][with 1.2~Ms of observation time]{Jourdain2009_Crab}, and reduces the uncertainties by the increased amount of exposure.
\section{Conclusion}
In this work, we showed, how to construct BG models for INTEGRAL/SPI data from a data base of spectral parameters which characterise the response as well as the BG properties of individual components of the instrument. In particular we illustrate our method on a few characteristic examples, including point-like, diffuse, continuum, and $\gamma$-ray line emission.
Our method is based on long-term monitoring and understanding of instrumental BG, which is separated into \textit{continuum} BGs and $\gamma$-ray \textit{line} BGs. These physical components show individual, but characteristic detector patterns on the SPI detector array, and are constant over time. For individual energies (energy bins, not integrating over a $\gamma$-ray line, for example), the detector patterns change smoothly with time due to the change in their responses, mainly caused by detector degradation and the influence of solar activity. This is taken into account by our detailed spectral fits of INTEGRAL/SPI data per unit time \citep{Diehl2018_BGRDB}.
The time basis for these spectral fits must be chosen:
\begin{enumerate}
\item long enough to smear out residual celestial patterns which are imprinted by the coded mask,
\item short enough to trace gradual variations of detector responses, and
\item long enough to accumulate enough statistics for reliable fits.
\end{enumerate}
Point 1 is fulfilled after 15--25 INTEGRAL pointings, depending on the strength of the source (Fig.~\ref{fig:pattern_smear_out}). We note that a correlation of our BG model with celestial emission is to be minimised by the way the BG is re-normalised in time, and want to point out that the more pointings are added for determining the BG response, the smaller the correlation will be. For example, a source contribution of 10\% (50\%) is smeared out to less than 1\% (5\%) after a standard $5 \times 5$-dithering pattern. This means one INTEGRAL orbit (50--90 pointings) would smear out the expected detector pattern of a single point source almost completely ($<3\%$ for any source strength; $<0.5\%$ for typical source contributions of less than 10\%), so that the correlation is minimal. At the same time scale, we find significant changes in the detector response ($\gamma$-ray line shape parameters) for the strongest $\gamma$-ray lines, so that point 2 is also fulfilled using one INTEGRAL orbit, or three days. Depending on the energy, i.e. BG count rate, the spectral fits are adequate (point 3) on a time scale of one orbit below energies of about 1--2~MeV. Above this energy, a spectral fit per detector requires more statistics to provide robust results. We choose the time span between SPI detector annealings to determine spectral parameters above 1--2~MeV. This reduces the accuracy for the line broadenings within half a year, but this can be accounted for by correcting for a linear trend using the orbit-by-orbit fits if required. We find, however, that fits per half-year period provide accurate results, as shown for the $\mrm{^{26}Al}$-case in (diffuse) line emission. If continuum sources are investigated and broader spectral bands are analysed, a spectral data base per orbit is also useful.
We show that our BG modelling method is reproducible and also results in values for celestial sources which are in concordance with literature values. The optimal choice of BG re-scaling as a function of time, i.e. the number of parameters at each energy, depends on the emission type (diffuse or point-like), the spectral regime (low-energy or high-energy; dominated by instrumental \textit{continuum} or \textit{lines}), and the width of the energy bins analysed. We provide examples of how to determine adequate fits for INTEGRAL/SPI analysis, based on the Akaike Information Criterion. This penalises too many and disfavours too few parameters, and we compare the results to reduced $\chi^2$-values, which are commonly used in such analyses, but should be taken with care when using Poisson-distributed data with a low mean count rate. This BG method can be applied to a multitude of analysis cases, from short-term (e.g. gamma-ray bursts, state-variable X-ray binaries) to long-term (e.g. persistent sources, diffuse emission), continuum emission (e.g. synchrotron, bremsstrahlung, inverse Compton) and $\gamma$-ray line searches (e.g. decay, excitation, annihilation), as well as a combination of all of these at the same time. We show that our method eliminates most of the systematic uncertainties considering BG modelling with INTEGRAL/SPI.
\begin{acknowledgements}
This research was supported by the German DFG cluster of excellence 'Origin and Structure of the Universe'. The INTEGRAL/SPI project has been completed under the responsibility and leadership of CNES; we are grateful to ASI, CEA, CNES, DLR, ESA, INTA, NASA and OSTC for support of this ESA space science mission.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Pseudomphrale crenata är en tvåvingeart som först beskrevs av Becker 1913. Pseudomphrale crenata ingår i släktet Pseudomphrale och familjen fönsterflugor.
Artens utbredningsområde är Iran. Inga underarter finns listade i Catalogue of Life.
Källor
Fönsterflugor
crenata
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Keep up with emerging tech and showcase your brand with the Calypso Qi Wireless Charging Pad. Featuring a one of a kind classic design, with polish metal trim and customizable PU leather charging surface. Simply place your Qi-enabled phone on top to charge, and built-in LED light will illuminate to indicate charge in progress. The Calypso is Qi Standard listed and CE & FCC certified. Each comes with retail packaging.
Click here to see a list of Qi-enabled phones.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,125
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Q: libfaac: Queue input is backward in time I am using libav along with libfaac to encode audio into aac.
following is the logic:
frames[n]
i = 0 ;
while (there are frames)
{
cur_frame = frames[i];
av_encode_audio(frame, ...., &frame_finished);
if( frame_finished )
{
i++;
}
}
but I am getting this annoying warning for few frames "queue input is backward in time !"
A: The answer is very simple, you are not supposed to pass the same frame again to the libfaac,
so even if the frame_finished is not 1 you should still go to the next frame.
it should be as follows:
frames[n]
i = 0 ;
while (there are frames)
{
cur_frame = frames[i];
av_encode_audio(frame, ...., &frame_finished);
i++;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,137
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\section{Introduction}
Hadrons carrying heavy quarks, {\it i.e.} charm or bottom, are important probes
in high energy hadronic collisions.
Heavy quark-antiquark pairs are mainly produced in initial hard scattering
processes of partons.
While some of the produced pairs form bound quarkonia, the vast majority
hadronizes into particles carrying open heavy flavor.
The latter do not only provide a crucial baseline for quarkonia measurements
but are also of prime interest on their own.
Heavy-flavor measurements in p+p collisions provide an important proving ground
for quantum chromodynamics (QCD).
Because of the large quark masses, charm and bottom production can be treated
perturbatively (pQCD) even at small momenta \cite{mangano93}.
This is in distinct contrast to the production of particles carrying light
quarks only which can be evaluated within the pQCD framework only for
sufficiently large momenta.
Systematic studies in p+p and d+Au collisions should be sensitive to the
nucleon parton distribution functions as well as nuclear modifications of
these such as shadowing \cite{lin96}.
In Au+Au collisions, heavy quarks present a unique probe for the created hot
and dense medium.
Important observables in addition to heavy flavor yields are energy loss
\cite{dokshitzer01,armesto05} and azimuthal anisotropy \cite{lin03,greco04}
as well as quarkonia suppression \cite{matsui86} or enhancement
\cite{pbm00,thews01,andronic03}.
At RHIC, the PHENIX and STAR experiments study heavy-quark production
indirectly via the measurement of electrons from semileptonic decays of
hadrons carrying charm or bottom.
In addition, STAR directly reconstructs hadronic $D$ meson decays in p+p
and d+Au collisions.
The mid rapidity electron spectrum from heavy-flavor decays measured by PHENIX
in p+p collisions at $\sqrt{s} = 200$~GeV \cite{phenix_e_pp} is shown in
Fig.~\ref{fig1} (left panel).
A leading order PYTHIA calculation and a recent next-to-leading order (FONLL)
pQCD calculation \cite{FONLL} are compared to the data.
The corresponding STAR measurement \cite{star_dau} agrees with these data
within the substantial uncertainties.
While the spectrum is significantly harder than predicted by PYTHIA, the FONLL
calculation describes the shape better but still leaves room for further
heavy-flavor production beyond the included NLO processes, {\it e.g.} via
jet fragmentation.
Bottom decays are expected to be essentially irrelevant for the electron cross
section at $p_T < 3$~GeV/c and become significant only for $p_T > 4$~GeV/c.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig_1a.eps}
\includegraphics[width=0.49\textwidth]{fig_1b.eps}
\caption{Invariant differential cross section of electrons from heavy-flavor
decays in p+p collisions at 200 GeV in comparision with PYTHIA and FONLL pQCD
calculations (left panel). Transverse momentum spectra of $D^0$ mesons and
electrons from heavy-flavor decays in p+p and d+Au collisions at 200 GeV
(right panel).}
\label{fig1}
\end{center}
\end{figure}
$D^0$ meson spectra, measured by STAR in p+p and d+Au collisions at
$\sqrt{s_{NN}} = 200$~GeV are shown in Fig.~\ref{fig1} (right panel) together
with electron spectra from heavy-flavor decays \cite{star_dau}.
It is important to note that the electron and $D$ meson data are compatible
with each other, {\it i.e.} the measured electron spectra agree within errors
with spectra calculated for semileptonic $D$ meson decays.
The nuclear modification factor $R_{dA}$ is calculated as the ratio of the
d+Au electron spectrum to the spectrum from p+p collisions scaled with the
number of underlying nucleon-nucleon binary collisions.
Averaged over the range $1 < p_T < 4$~GeV/c, STAR
measures\nolinebreak \cite{star_dau} $R_{dA} = 1.3 \pm 0.3 \pm 0.3$,
which is consistent within errors with binary scaling as expected for a
point-like hard pQCD process.
A modest Cronin enhancement is not excluded by the data.
Electron spectra measured by PHENIX in d+Au collisions as function of
centrality confirm the observed binary scaling \cite{phenix_e_pp}.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig_2a.eps}
\includegraphics[width=0.49\textwidth]{fig_2b.eps}
\caption{Heavy-flavor electron $p_T$ spectra for different Au+Au centrality
selections at 200 GeV (scaled for clarity) compared to fits to the binary
scaled p+p measurement (left panel). Nuclear modification factor $R_{AA}$ in
the range $2.5 < p_T < 5$~GeV/c as function of the number of participant
nucleons for $\pi^0$ and electrons from heavy-flavor decays (right panel).}
\label{fig2}
\end{center}
\end{figure}
For Au+Au collisions at $\sqrt{s_{NN}} = 200$~GeV, PHENIX has demonstrated
\cite{phenix_e_auau} that the yield of electrons from heavy-flavor decays
is consistent with binary scaling in the range $0.8 < p_T < 4$~GeV/c, which
is entirely dominated by charm decays.
Preliminary electron data, however, indicate a strong modification of the
spectral shape in central collisions.
Relative to binary scaled p+p data, electrons from heavy-flavor decays are
significantly suppressed at high $p_T$ in central Au+Au collisions as shown in
Fig.~\ref{fig2} (left panel) \cite{phenix_e_raa}.
This observation is consistent with a scenario where quarks suffer energy
loss while propagating through the hot and dense medium created at RHIC.
The nuclear modification factor $R_{AA}^{2.5-5.0}$, defined as the ratio
of the yield of electrons from heavy-flavor decays in Au+Au collisions in
the range $2.5 < p_T < 5$~GeV/c to the binary scaled yield in p+p collisions
in the same $p_T$ range, is shown as function of the number of participant
nucleons $N_{part}$ in Fig.~\ref{fig2} (right panel) together with the same
quantity for neutral pions as measured by PHENIX \cite{phenix_pion_raa}.
The high $p_T$ electron suppression is comparable to the pion suppression,
but the experimental uncertainties are still substantial and do not allow
to establish the centrality dependence of heavy quark energy loss.
A complementary observable related to the interaction of heavy quarks with
the medium created in Au+Au collisions is elliptic flow.
The elliptic flow strength $v_2$ for electrons from heavy-flavor decays
is shown as function of $p_T$ in Fig.~\ref{fig3}.
The PHENIX data \cite{phenix_e_v2} and preliminary results from STAR
\cite{star_e_v2} are compared with two recombination model calculations
\cite{greco04}.
Charm quark flow is consistent with the experimental data but the
uncertainties are currently too large to exclude a scenario where charmed
hadrons acquire a non-zero $v_2$ through being produced via the coalescence
of a non-flowing charm quark and a flowing light quark.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth,viewport=0 2 540 398,clip]{fig_3.eps}
\caption{Elliptic flow strength $v_2$ of electrons from heavy-flavor decays
in Au+Au collisions at 200 GeV as function of $p_T$ in comparision with
recombination model calculations with and without charm quark flow.}
\label{fig3}
\end{center}
\end{figure}
PHENIX has measured J/$\psi$ in the dielectron channel at mid rapidity and in
the dimuon channel at forward and backward rapidities in p+p and d+Au
collisions at 200 GeV \cite{phenix_jpsi_pp,phenix_jpsi_dau}.
The shape of the rapidity distribution shown in Fig.~\ref{fig4} (left panel)
for p+p collisions agrees well with Color Octet Model (COM) and PYTHIA
calculations using different parton distribution functions.
The measured total cross section is consistent with predictions from COM
and Color Evaporation Model (CEM) calculations.
In d+Au collisions weak cold nuclear matter effect are observed.
The ratio of the rapidity distribution from d+Au collisions to the properly
scaled p+p measurement show in Fig.~\ref{fig4} (right panel) indicates both
weak absorption as well as weak shadowing of the gluon distribution function
in nuclear matter \cite{phenix_jpsi_dau}.
Given the current uncertainties, it is difficult to disentangle these small
effects.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{fig_4a.eps}
\includegraphics[width=0.49\textwidth]{fig_4b.eps}
\caption{J/$\psi$ differential cross section, multiplied with the dilepton
branching ratio, as function of rapidity in p+p collisions at 200 GeV compared
with a PYTHIA calculation (left panel). Ratio of d+Au and appropriately scaled
p+p J/$\psi$ rapidity distributions compared with model calculations
$^{21,22}$ (right panel).}
\label{fig4}
\end{center}
\end{figure}
In a deconfined medium the yield of J/$\psi$ might either be reduced due
to the expected screening of the attractive QCD potential \cite{matsui86}
or possibly even enhanced via coalescence \cite{thews01} or statistical
recombination \cite{pbm00,andronic03}.
While scenarios leading to a strong J/$\psi$ enhancement are disfavored by
first data from Au+Au collisions at 200 GeV \cite{phenix_jpsi_auau}, a
definite answer will emerge only from the currently ongoing analysis of the
high statistics Au+Au data sample recorded in RHIC Run-4.
\vspace*{-0.4cm}
\section*{References}
\vspace*{-0.3cm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,341
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import {Component, OnInit, OnDestroy} from '@angular/core';
import {Hotel, OfferRequest} from '../../model/backend-typings';
import {Store} from '@ngrx/store';
import {AppState} from '../../reducers/index';
import {RequestStatusEnum, getRequestStatusValues} from '../../model/RequestStatusEnum';
import {OfferRequestService} from '../../shared/offer-request.service';
import {Observable} from 'rxjs/Observable';
import {Subscription} from 'rxjs/Subscription';
import 'rxjs/add/operator/pluck';
import {Router, ActivatedRoute} from "@angular/router";
@Component({
selector: 'offer-request-edit-page',
template: `
<offer-request-edit
[offerRequest]="offerRequest | async"
[hotels]="hotels | async"
[requestStatusList]="requestStatusList"
(saveOfferRequest)="saveOfferRequest($event)">
</offer-request-edit>
`
})
export class OfferRequestEditPageComponent implements OnInit, OnDestroy {
hotels: Observable<Hotel[]>;
offerRequest: Observable<OfferRequest>;
requestStatusList: RequestStatusEnum[];
private campId: number;
private campIdSubscription: Subscription;
private offerRequestSubscription: Subscription;
constructor(private route: ActivatedRoute,
private store: Store<AppState>,
private offerRequestService: OfferRequestService,
private router: Router) {
this.campIdSubscription = this.route.params.pluck<string>('campId')
.map(id => parseInt(id))
.subscribe(id => this.campId = id);
this.offerRequestSubscription = this.route.params.pluck<string>('offerRequestId')
.map(offerRequestId => parseInt(offerRequestId))
.subscribe(offerRequestId => {
if (!isNaN(offerRequestId)) {
this.offerRequest = this.offerRequestService.getOfferRequest(offerRequestId);
} else {
this.offerRequest = Observable.of({lastStatusChange: new Date()} as OfferRequest);
}
});
this.hotels = this.store.select<Hotel[]>('hotels');
this.requestStatusList = getRequestStatusValues();
}
ngOnInit() {
}
ngOnDestroy() {
this.campIdSubscription.unsubscribe();
this.offerRequestSubscription.unsubscribe();
}
saveOfferRequest(offerRequest: OfferRequest) {
this.offerRequestService.saveOfferRequest(this.campId, offerRequest).subscribe(
() => this.goToCamp()
);
}
private goToCamp() {
this.router.navigate(['/camps/', this.campId]);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,616
|
Each program and project that is brought to life through the Foundation is created with the underlying goal of building a stronger and more dynamic infrastructure for the arts.
Improve the quality of life for Western Australians by providing access to culture activities and the performing arts.
• Provide programs and services that encourage WA artists and communities to work together, in ways that foster artistic excellence, accessibility, civic engagement, and cultural understanding.
• Increase public understanding of WA's creative economy; its artists, cultural organisations, and creative industries.
• Enhance the Foundation's ability to sustain an effective, innovative, and resilient organisation with both focus and agility.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,189
|
Gentlewoman style: At home with Phyllis Wang
I can't remember if I told you, but I'm working on my next book!
It's going to be beauty-focused and as part of my research, I've been re-reading my last book, The New Garconne: How to be a Modern Gentlewoman. Weirdly – or perhaps not – I haven't had much need to revisit it since it was published in 2016. But like most books, when I do re-read it, it's interesting to notice the parts that have a new or different relevance.
I loved interviewing comedienne Phyllis Wang. She was one of the last people I approached and was recommended by my then commissioning editor, Camilla. I was looking for women who embodied the 'gentlewoman' aesthetic and values of shopping consciously yet with their own 'garconne style' twist. Phyllis certainly veered to the eccentric end of the scale, which was totally fine!
I did the interview by phone and we didn't get to meet IRL until a year later when, on a whim, film director Emma Miranda Moore and I Eurostar-ed to Paris to make a short film about her. Reader, she was a dream interviewee; funny, smart and insightful and I still revisit her wisdom on writing discipline years later.
Here's the 'director's cut' of the interview featured in the book, plus the short film we made…
I was born in New York but raised mainly in Los Angeles, California. I had a normal, upper middle class Taiwanese-American upbringing where my father was a businessman and my mother was a homemaker. I went to very good private schools and when I graduated from Berkeley I focused on art history and also theatre. I went to drama school in London and during that time I was also doing quite a few things in fashion.
My fashion relationship has always been instinctual. I met a designer named Huishan; he liked the way I looked and he asked me to kind of help out. When a young designer starts everybody does a bit of everything and so I would wear his clothes but at the same time we travelled to China together and I helped with the design process and the production.
When you meet people in different situations there's an energy and there's an interest. I don't know how to separate them. The energy is also reflected in how you dress and that attracts like-minded people. It's really from those experiences that I ended up doing different things.
When I started comedy, it was actually by talking to some guy at a party. We were having a tit for tat about something and a friend of mine was like, you know Phyllis, you have a very wry, dry sense of humour, you should think about stand-up. At that moment I was going to a lot of castings and auditions for a feature film. I missed doing stage work so that's when I started writing my own material and performing it, developing it that way.
My taste came from my family; my father was quite stylish and so is my mum. She used to buy a lot of Donna Karan. She never worked in an office, it was just an aesthetic she really liked and appreciated. It was through both of them that I found a sense of my own taste.
My father was always really interested in Chinese art. He collected a lot of bronzes and painting s and ceramics and I think I was influenced by his love of the Asian culture and also art. And then I decided I wanted to study what he loved, because it became what I loved. When you're studying art history in the States, they have a very European focus. For them it's Western art history that is considered "art". But with my background, it's only one of the arts. Different cultures have their own evolution of art and aesthetics and so I did both.
I do love my clothes! When I shop, whether it's a vintage shop or a grandma shop or a high street label or a luxury brand, it's really the details and the workmanship and the form that I'm totally into.
I love Thomas Tait leather jackets. I used to think Junya Watanabe did the best leather jackets but then when Thomas came about I was like, wow this guy does the best leathers. His take on the Perfecto is super-unique.
I remember a friend of mine saying, "you really need to stop wearing this intellectual, architectural bullshit. No guy's going to want to fuck you if you dress like that". And I said, but I'm not dressing for a guy to fuck me. In fact, I'm kind of trying to avoid that. Maybe it was my prudish Asian side but it never dawned on me to dress for a man. I always just dressed how I felt.
When I'm living with something, I have to like it. Most of the things I have are based on both function and aesthetics. I wouldn't say it's only aesthetics, I think I'm more on the path of whatever works, works. I have a lot of furniture that my mother gave me or things like that, that's not my taste but it's become a part of my life. It's emotional and practical.
My stand up comedy started on the stage. At a lot of the open mikes you discovered material on stage in front of an audience and from those seeds that were planted or spoken when I was on stage, there might be something very interesting that could be developed and pushed further. Then I would come home and try to remember it. Or if I'd recorded the five minutes that I did, I'd listen to it again and develop it more from there. A lot of the writing for me was almost speaking it.
A big part of what stand up comedians do, we'll constantly have a notebook and we're jotting down different ideas. It's a constant process. Always observing the world around you and also the world inside of you in relation to the world around you. There's a certain sense of understanding yourself.
I do make an effort to make some time to write every day. And I'm not even saying OK you have to be funny. I'm just saying, OK Phyllis, write what is inside of you and if you have to, make it funny after. Or if it's just letting it out, then that's OK as well.
I'm a bit of a mess cat! So I'll buy a beautiful notebook and I'll start something in it but then I'll still have the old notebook because I haven't finished it. So all of a sudden I've accumulated several notebooks, all over the house and then I'll have that one point bringing all those notebooks together with those audio recordings and syphon through all that information. It's like a circle for me.
When I cut my hair, I had a picture of Louise Brooks. I'd been watching a lot of her silent films and I'd been reading her book and I like her as a woman. There's something about her that's very strong, very feminine but also quite boyish. She's daring and it was that quality that I was feeling when I cut my hair.
If I like something I'll probably buy it. I don't know if it's because I'm older but I find that I like things less and less. I'm being a bit choosier and in a way it's a great thing. I like to think that because I choose everything, it is a conscious form of consumption.
My mother taught me it's really important to take care of your skin. I veer towards organic skincare products. I always go back to Dr Hauschka. I have a lot of essential oils; I have oils for the chakras. I'm not very knowledgeable about Ayurvedic medicine but I do find that the smell of the essential oils do different things to me. If I have to put make up on I use Shu Uemura pre makeup skincare. When I'm performing and I know we're filming I'll make sure my skin is as clean as possible. I'll use foundation and a bit of powder. It's mainly about the skin; it reflects how you feel. I find when my skin is horrible it's because I'm not feeling so great, it's not living, it's not full of life.
I really like going across the street and having a coffee at the bar. In the morning it's part of my routine, sometimes I do need to get out of the house because otherwise if I'm writing I won't leave the house the whole day sometimes. If I'm working on editing material, then just being around other people is quite nice. Paris is great for that. I never got that feeling in London where you could just go, walk into a café, and stand at the counter and have a coffee. Paris has a café culture, which I really like.
The place where I produce stand up comedy shows is a hundred metres from my house. It's called Chez Georges. It's a wine bar that's been around since the 1950s. People have been going there for sixty years and it used to be a place where young musicians would go and test out their new sounds and young comedians would test out their material. Sometimes even if I don't perform there, I go there to have a glass of wine and chat with the locals and it's a really good sense of community that I really like. It's more and more rare to have that in a major city.
I'm the type of person who needs to move around a lot. There are moments I really enjoy being around people and I find the exchange and the conversation fantastic, and the engagement, even on social media very nourishing. But I cannot live just on that. I need moments when I can just be within myself, by myself. And so after travelling a lot, there's a moment when I need to be in one place at one time and focus.
Sometimes we can spread ourselves too thin and it's the nature of the modern world. I'm still trying to learn how to manage it. And how much of it is important in terms of my professional career and how much of it is just the way we communicate with people.
I still prefer holding a book and reading it, as opposed to reading on a tablet. It's become sensual in a way. I like writing on a piece of paper, I like the way things come in contact with me or I come in contact with things and objects.
I actually have quite a large collection of DVDs and yeah I could watch them on Netflix but I also like that I have a library of DVDs like I have a library of books. I have some silly films I like and I have some great directors that I like as well, and they're all there where I can see them, pull them out and put them back and rediscover. I haven't found a way to catalogue or create a library online.
I have a lot of novels, philosophical books and psychology books. And I have a lot of Freud books; he wrote a whole volume on comedy. Every time I'm really into something I'm like, I need a book about that. Maybe that's my way of consuming it. I need to know about it by reading about it before I feel like I know anything really.
I don't know if it's because I'm Asian but I just feel like as Asians we work really hard in whatever we do. But as an Asian woman I always found myself effacing myself, never really saying what I wanted. So in a way my career organically developed on its own. In my 20s I didn't think about anything, I didn't think about what my future was going to look like, I just followed my gut. Sometimes it worked and sometimes it made me think, OK I really need to be more ambitious and more focused.
Not everything you do in life will be successful, but that's how it is. And actually that's part of why we continue. Sometimes I feel like for me, comedy is about changing the way people think, through laughter. It's not just cracking one joke after the other. If I want to get to that place I'm going have to go through moments where I'm not going to crack the laugh. Not everything we do will be successful and it's quite a good journey in order to do what we want.
I respect those people who are ambitious and see clearly the structure of how they need to go and where they need to go. I think good for you, kudos to you. It's just I don't believe I work that way. Sometimes I wish I did. But a lot of what I do and perhaps it's my job, it's like I have to let it be, I have to let it brew. And sometimes I feel my function is to create room for myself to do that. It can't be driven my ambition, it just can't.
For more interviews, buy my book, The New Garconne: How to be a Modern Gentlewoman here.
WORDS: Disneyrollergirl/Navaz Batliwalla
IMAGES: Elise Toide
NOTE: Most images are digitally enhanced. Some posts use affiliate links and PR samples. Please read my privacy and cookies policy here
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Disneyrollergirl Beauty, Books, News books, films, gentlewoman style, Phyllis Wang, The New Garconne
We will be with our friends again Into this: Emily Adams Bode's abode
SuWu
22 April, 2020 @ 2:04 pm
Hello Navaz, this is brilliant! I have your first book and as a matter of fact when I was looking through my stacks the other day, there it was. Still a wonderful read. I CAN. NOT. WAIT. for your new book. Stay healthy!
Thank you so much SuWu, your encouragement means a lot! x
Fantastic news, Navaz. I used to buy a lot of style books but New Garconne was the first one to really resonate and I still refer back to it. Aspirational stuff as I look like a spud, but one can dream!! Take care and keep up the excellent work.
24 April, 2020 @ 12:20 am
I'm sure you do NOT look like a spud! Thank you so much for the lovely words! x
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,360
|
Q: Building a carousel in React I am trying to build a carousel from scratch in React. I want the active card to be in the middle and the others to show a portion of them on the side. When I put overflow hidden, it hides the portions.
How do I make it so that the side portions appear but I still maintain the background image blur?
here is my jsx code:
import React from 'react'
import './Header.css'
import {images} from '../../constants'
import Blur from 'react-blur'
const Header = () => {
return (
<Blur className="app__header" img={images.bgcake} blurRadius={35} style={{height: "100vh"}} >
<div className="app__header-carousel">
<div className="app__header-carousel-1">
<img className="app__header-carousel-img" alt="pink cake" src={images.carousel01} />
<div className="app__header-carousel-content">
<h3 className="app__header-h3">Fresh Baked | Fathers Day</h3>
<h1 className="app__header-h1">MORNING BERRY</h1>
<p className="app__header-p">Lorem ipsum dolor sit amet consectetur adipisicing elit. Itaque, omnis odit similique dolor ab maiores quae aliquam beatae ad deleniti porro ducimus eum provident fuga quo doloremque animi impedit unde.</p>
<button type="button" className="app__explore-button">Order</button>
<button type="button" className="app__order-button">Explore</button>
</div>
</div>
<div className="app__header-carousel-1">
<img className="app__header-carousel-img" alt="red velvet" src={images.redVelvet} />
<div className="app__header-carousel-content">
<h3 className="app__header-h3">Fresh Baked | Fathers Day</h3>
<h1 className="app__header-h1">RED VELVET</h1>
<p className="app__header-p">Lorem ipsum dolor sit amet consectetur adipisicing elit. Itaque, omnis odit similique dolor ab maiores quae aliquam beatae ad deleniti porro ducimus eum provident fuga quo doloremque animi impedit unde.</p>
<button type="button" className="app__explore-button">Order</button>
<button type="button" className="app__order-button">Explore</button>
</div>
</div>
<div className="app__header-carousel-1">
<img className="app__header-carousel-img" alt="pink cupcake" src={images.carousel03} />
<div className="app__header-carousel-content">
<h3 className="app__header-h3">Fresh Baked | Fathers Day</h3>
<h1 className="app__header-h1">CUP CAKES</h1>
<p className="app__header-p">Lorem ipsum dolor sit amet consectetur adipisicing elit. Itaque, omnis odit similique dolor ab maiores quae aliquam beatae ad deleniti porro ducimus eum provident fuga quo doloremque animi impedit unde.</p>
<button type="button" className="app__explore-button">Order</button>
<button type="button" className="app__order-button">Explore</button>
</div>
</div>
</div>
</Blur>
)
}
export default Header
here is my css code:
.app__header-carousel{
display: flex;
position: absolute;
top: 10%;
right: 10%;
left: 10%;
overflow: hidden;
}
.app__header-carousel-1{
display: flex;
justify-content: space-between;
height: 80vh;
width: 100%;
margin-right: 3em;
}
.app__header-carousel-img-1{
width: 80%;
}
.app__header-carousel-content{
background: #FDFFD5;
padding: 4rem 10rem 5rem 5rem;
}
.app__header-h3{
color: #76D9D2;
margin-bottom: 0.65rem;
}
.app__header-h1{
color: #F397B0;
font-size: 4rem;
margin-bottom: 0.65rem;
}
.app__header-p{
margin-bottom: 5rem;
}
.app__explore-button{
background: #2D2D2D;
color: beige;
width: 8rem;
margin-right: 0.5rem;
padding: 5px;
border: none;
border-radius: 5px;
cursor: pointer;
}
.app__order-button{
background: #F397B0;
color: #2D2D2D;
width: 8rem;
margin-right: 0.5rem;
padding: 5px;
border: none;
border-radius: 5px;
cursor: pointer;
}
How do I make it appear how I want it to?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,970
|
Here Are All The Male Matchups For The Finals At The 2019 World Pro
April 26, 2019 admin 0 Comments finals, male, Matchups, Pro, World
Image Source: Averi Clements for Jiu-Jitsu Times
After a very exciting few days, the climax of the 2019 UAEJJF World Pro is nearly upon us. Friday will be dedicated to all of the black belt finals matches, which will take place one by one in the center of Abu Dhabi's Mubadala Arena. While anything is possible (and we've already seen a few upsets), there are some clear favorites — and some not-so-clear favorites — in the running for gold this year.
Here are all the men vying for the championship this year:
56kg – Hiago George vs. Carlos Alberto Da Silva
Carrying the momentum of wins at this year's Pans and Euros, George is hoping to move a couple places further up the podium at the World Pro this year after winning bronze last year in the 62kg category. To get there, though, he'll have to defeat the reigning 56kg champion. George submitted his way to his place in the finals today, but da Silva isn't going to give up a second World Pro title so easily.
62kg – Joao Batista [Gabriel] De Sousa vs. Joao Miyao
De Sousa and Miyao both cruised their way to the finals this year, but their smooth sailing stops here. Miyao is well established as one of the best and most famous jiu-jitsu competitors of his generation, but de Sousa is a force to be reckoned with. He won the World Pro as a brown belt last year not long before earning his black belt, and even an athlete as dominant as Miyao will have to work hard and be extremely careful to pull off a win over this rising star.
69kg – Isaac Doederlein vs. Paulo Miyao
This isn't the first time Doederlein has faced Miyao. The two have matched up several times, and so far, Miyao has bested Doederlein in every one of their meetings. Miyao also won the World Pro last year, and this (in combination with his previous wins over Doederlein) makes him the clear favorite to win this time around. However, if Doederlein can learn from his past mistakes with Miyao, he might just be able to pull off a serious upset in Abu Dhabi.
77kg – Oliver Lovell vs. Tommy Langaker
Image Source: Kitt Canaria for Jiu-Jitsu Times
Lovell claimed one of the day's upsets when he took Levi Jones-Leary out of the running early with a kneebar. Lovell went on to remove Ffion Davies' coach Darragh O'Conaill from the running as well, and we're expecting an exciting match between him and Langaker. The latter competitor has been consistently medaling at some of the world's top competitions, including Worlds and Pans, and while many fans were anticipating seeing him and Jones-Leary battling it out for gold, Lovell has proven he's more than worthy of the opportunity.
85kg – Rudson Mateus vs. Faisal Al Ketbi
Faisal Al Ketbi is the UAE's shining star in jiu-jitsu. He won silver at last year's World Pro, and this year, he beat Devhonte Johnson and DJ Jackson on his way to the finals. He'll be facing 2019 Euros champ Rudson Mateus, who had to defeat William Dias and 2019 Masters World Pro runner-up David Willis to make it to this point. This match will be huge for Al Ketbi as he competes for gold to the sound of deafening cheers from his fellow countrymen and women, and given his dominant performances throughout the day, he might just be able to gain the edge over the formidable Mateus.
94kg – Kaynan Duarte vs. Adam Wardzinski
In what is certainly one of the most exciting finales in the male division, Atos standout Duarte (who beat Erberth Santos earlier today) will face Wardzinski (who famously choked out Leandro Lo at Pans this year). Duarte has bested Wardzinski in each of the three matches they've had against each other so far, and this combined with he's only lost one match since October make him a narrow favorite to win gold. Given how close their last match was, though, Duarte can't afford to make even the tiniest mistake with his opponent. Wardzinski has become somewhat of a regular in the World Pro finals, and this could finally be the year he takes home the gold.
Watch what Wardzinski had to say about his matches for the day:
110kg – Joao Gabriel Rocha vs. Yahia Al Hammadi
Emotions will be running high during the last final of the day. A win from Al Hammadi would bring immeasurable pride and joy to the UAE, and he'll be carrying the weight of the hopes of his entire nation on his shoulders as he rolls for gold. A win for Rocha, though, would be a fairy tale ending to his incredibly inspirational story. Rocha very recently won his five-year-long battle with cancer, getting the news just days after defeating Marcus "Buchecha" Almeida at BJJ Stars. He was all smiles after winning the 110kg Brazilian qualifier, and winning World Pro gold would be incredibly emotionally significant for both him and his fans.
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The Best Setup for a Darce Choke – BJJ Fanatics
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,873
|
2014 Lok Sabha elections to clash with exams
Published Jan 6, 2014, 1:24 pm IST
Updated Mar 19, 2019, 4:55 am IST
The electoral rolls are expected to be ready by the end of January.
The 2014 Lok Sabha polls will be held in five or six phases starting mid-April and continuing into early May.
New Delhi: The Election Commission of India is scheduled to hold the 2014 Lok Sabha polls in five or six phases starting mid-April and continuing into early May.
Andhra Pradesh, Orissa and Sikkim will be electing their new Assemblies alongside. An official announcement to this effect is to be made in late February or early March.
The dates are under consideration for a meeting with the Union home secretary to finalise deployment of paramilitary forces before the announcement of the poll schedule.
The term of the current Lok Sabha expires on June 1 and the new House has to be constituted by May 31. Nearly 80 crore people are expected to exercise their franchise.
The electoral rolls are likely to be released by the end of this month. A top poll panel source said, "The announcement of the poll schedule would be done in the last days of February or at best the first two-three days of March."
He added, "We already have a list of electoral rolls. We need to update it. We expect that the lists will be ready before January-end."
The last Lok Sabha polls were held in five phases from April 16 to May 13 and the counting of votes was done on May 16, 2009. The announcement for the last general election was made March 2, 2009. There were then 714 million voters against 671 million voters in the 2004 Lok Sabha polls. A total of 1.1 crore poll personnel, including security forces, are to be deployed for the elections.
Read more: Lok Sabha elections to be held from mid-April in 5-6 phases
Among those, 5.5 million civilian staff would be put in place to ensure free and fair elections. An estimated eight lakh polling stations are to be set up for polling across the country. In addition, plans are afoot to deploy around 12 lakh EVMs. The poll panel is likely to secure another 2.5 lakh EVMs in mid-February.
In the states of Andhra Pradesh, Orissa and Sikkim, the poll panel is all set to place two EVMs alongside so that voters can vote in both the elections.
The term of the 294-member Andhra Pradesh Assembly is till June 2, 2014; that of the 147-member Orissa House till June 7, 2014; and that of the 32-member Sikkim Assembly till May 21, 2014. The poll panel has to grant around two weeks for government formation before the term of the Lok Sabha or a state Assembly expires.
"There will be a final meeting of the commission with the Union home secretary for securing the availability of security forces for poll duty," sources said. Currently, chief electoral officers of various states are in touch with the state DGPs over availability of state police forces for the polls. The list of Central government employees to be deployed as micro-observers in sensitive polling stations is also being finalised.
Poll officials feel a multi-phased general election is best suited for a country of India's size and electorate. Also, states like Jammu and Kashmir and Chhattisgarh have to be accorded special treatment on account of insurgency-related problems.
The Election Commission has not received any request from any state for early polls though there is speculation that Haryana may opt for an early election.
Tags: andhra, lok sabha elections, electoral rolls, announcement, schedule, new assemblies election
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,262
|
{"url":"https:\/\/zenodo.org\/record\/5571808\/export\/json","text":"Presentation Open Access\n\n# Evolution of magnetic activity on the main sequence as a function of spectral type using Kepler data\n\nMathur, Savita; et al.\n\n### JSON Export\n\n{\n\"files\": [\n{\n\"self\": \"https:\/\/zenodo.org\/api\/files\/a2fcbe12-3e36-4f1a-80b7-82357fbc93c9\/PLATO_conf2021_SMathur.pdf\"\n},\n\"checksum\": \"md5:c262864707cede7e267abc3f7f8b50c7\",\n\"bucket\": \"a2fcbe12-3e36-4f1a-80b7-82357fbc93c9\",\n\"key\": \"PLATO_conf2021_SMathur.pdf\",\n\"type\": \"pdf\",\n\"size\": 7299797\n}\n],\n\"owners\": [\n55692\n],\n\"doi\": \"10.5281\/zenodo.5571808\",\n\"stats\": {\n\"unique_views\": 82.0,\n\"views\": 85.0,\n\"version_views\": 85.0,\n\"version_unique_views\": 82.0,\n\"volume\": 364989850.0,\n\"version_volume\": 364989850.0\n},\n\"doi\": \"https:\/\/doi.org\/10.5281\/zenodo.5571808\",\n\"conceptdoi\": \"https:\/\/doi.org\/10.5281\/zenodo.5571807\",\n\"bucket\": \"https:\/\/zenodo.org\/api\/files\/a2fcbe12-3e36-4f1a-80b7-82357fbc93c9\",\n\"html\": \"https:\/\/zenodo.org\/record\/5571808\",\n\"latest_html\": \"https:\/\/zenodo.org\/record\/5571808\",\n\"latest\": \"https:\/\/zenodo.org\/api\/records\/5571808\"\n},\n\"conceptdoi\": \"10.5281\/zenodo.5571807\",\n\"created\": \"2021-10-17T15:24:29.260327+00:00\",\n\"updated\": \"2021-10-18T08:53:50.029392+00:00\",\n\"conceptrecid\": \"5571807\",\n\"revision\": 3,\n\"id\": 5571808,\n\"access_right_category\": \"success\",\n\"doi\": \"10.5281\/zenodo.5571808\",\n\"description\": \"<p>Stellar magnetic activity studies are very important for different fields of astrophysics. Several spectroscopic surveys have been aimed at characterizing the magnetic activity of solar-like stars, especially to look for cycles. These surveys were mostly led to put the Sun into context and in time compared to other stars. By investigating the magnetic activity of other stars with different conditions (rotation periods, metallicity…), we can provide additional constraints to dynamo models. Stellar magnetic activity has a direct impact on the habitability of exoplanets hosted by those stars. Consequently, it is important to understand how magnetic activity evolves in time and as a function of spectral type.<\/p>\\n\\n<p>The recent catalog of rotation periods and photometric magnetic activity proxies for more than 55,000 stars observed by the Kepler mission opens the possibility to study the surface magnetic activity of a large number of stars. In this talk, we will present a subsample of main-sequence stars in order to compare the Sun to Sun-like stars and show the effect of metallicity using high-resolution spectroscopic data. While we see an interesting behavior as a function of metallicity, we also find that the magnetic activity of the Sun is comparable to the one of stars selected to be very similar to the Sun based on effective temperature, metallicity, and Rossby number, which is the ratio of rotation period and the convective turnover time and is a key parameter in dynamo theory. For all the stars of our sample, we also compute ages based on models taking into account the most recent theory of angular momentum transport that reproduce rotation rates for the Kepler asteroseismic sample. This allows us to study the evolution of magnetic activity as a function of Rossby number and age, providing a more complete picture to understand the changes in the dynamo behaviors during the life of the star until the terminal age main sequence.<\/p>\",\n\"id\": \"CC-BY-4.0\"\n},\n\"title\": \"Evolution of magnetic activity on the main sequence as a function of spectral type using Kepler data\",\n\"relations\": {\n\"version\": [\n{\n\"count\": 1,\n\"index\": 0,\n\"parent\": {\n\"pid_type\": \"recid\",\n\"pid_value\": \"5571807\"\n},\n\"is_last\": true,\n\"last_child\": {\n\"pid_type\": \"recid\",\n\"pid_value\": \"5571808\"\n}\n}\n]\n},\n\"communities\": [\n{\n\"id\": \"plato2021\"\n}\n],\n\"publication_date\": \"2021-10-15\",\n\"creators\": [\n{\n\"orcid\": \"0000-0002-0129-0316\",\n\"affiliation\": \"Instituto de Astrofisica de Canarias\",\n\"name\": \"Mathur, Savita\"\n},\n{\n\"name\": \"et al.\"\n}\n],\n\"access_right\": \"open\",\n\"resource_type\": {\n\"type\": \"presentation\",\n\"title\": \"Presentation\"\n},\n\"related_identifiers\": [\n{\n\"scheme\": \"doi\",\n\"identifier\": \"10.5281\/zenodo.5571807\",\n\"relation\": \"isVersionOf\"\n}\n]\n}\n}\n85\n50\nviews","date":"2023-01-30 11:14:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.30848607420921326, \"perplexity\": 3768.672053199868}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499816.79\/warc\/CC-MAIN-20230130101912-20230130131912-00382.warc.gz\"}"}
| null | null |
\section{Introduction}
In the eighteenth century, Leonhard Euler's studied factorial series (see. e.g. \cite{V2007}). His interest of these series has also lead us to study \textit{Euler's factorial series} which is defined as follows:
\begin{equation*}
F(t) \coloneqq {}_2F_0(1,1|t)=\sum_{n=0}^\infty n!t^n.
\end{equation*}
Further, we write $F_p(t)=\sum_{n=0}^\infty n!t^n$ when we consider the Euler's factorial series in $p$-adic domain. It is in a class of series which can be written by $\sum_{n\geq 0} a_n n! t^n$, where $a_n$ satisfies certain properties. These series are called \textit{F-series} and they were introduced by V. G. Chirski\u{i} \cite{C1989,C1990}. Euler's factorial series has also connections to quantum physics (see e.g. \cite{BW2015,F1997}).
It is clear that using standard Archimedean metric, Euler's factorial series converges only when $t=0$. However, using $p$-adic metric, the series converges for all integers $t$. Hence, an interesting question is, how close to zero the term $F_p(t)$ can be using $p$-adic metric. Or even more, if we look at a linear form
\begin{equation}
\label{eq:DefLambda}
\Lambda_p=\lambda_0+\lambda_1F_p(\alpha_1)+\lambda_2F_p(\alpha_2)+\ldots+\lambda_kF_p(\alpha_k),
\end{equation}
where $k \geq 1$, numbers $\lambda_j$ are integers, $\lambda_j \ne 0$ for at least one index $j$ and numbers $\alpha_j$ are distinct non-zero integers, can this be zero? If not, how close is it to zero in $p$-adic metric? Especially, could we prove any explicit results for this?
In 2004, D. Bertrand, V. G. Chirski\u{i} and J. Yebbou \cite{BCY2004} showed the following result concerning to $F$-series: Let $K$ be an algebraic number field of degree $\kappa$ over $\mathbb{Q}$. Consider linear forms of linearly independent $F$-series over $\mathbb{K}(x)$ whose coefficients are in $\mathbb{Z}_\mathbb{K}$ and which satisfy certain extra conditions relating to a certain differential system. Assume also that one of the $F$-series is $\equiv 1$. Then, there is an infinite collection of intervals containing a prime number $p$ such that for some valuation $v \mid p$ the linear form is not zero. A $v$-adic lower bound for the linear form is also given in \cite[Theorem 1.1]{BCY2004}. These results in article \cite{BCY2004} are effective.
Could we prove something similar but explicit considering linear forms in Euler's factorial series? May some other conditions help us to maintain these results or be useful to obtain the results? Or do we know any more precis results about the terms $F_p(t)$ which do not equal to zero?
In 2015, V. G. Chirski\u{i} \cite{C2015} proved that there exists infinitely many primes such that $F_p(1) \neq 0$. Three years later, T. Matala-aho and W. Zudilin \cite{MZ2018} showed the following condition:
\begin{theorem}
\label{thm:Zudilin}
\cite[Theorem 1]{MZ2018}
Given $t \in \mathbb{Z}\setminus \{0\}$, let $R \subseteq \mathbb{P}$, where $\mathbb{P}$ denotes the set of primes, be such that
\begin{equation}
\label{eq:OldCOndC}
\limsup_{n \to \infty} c^n n! \prod_{p \in R} \left|n!\right|_p^2 =0, \quad \text{where } c=c(t;R):=4 |t|\prod_{p \in R} |t|_p^2.
\end{equation}
Then either there exists a prime $p \in R$ for which $F_p(t)$ is irrational, or there are two distinct primes $p, q \in R$ such that $F_p(t) \neq F_q(t)$.
\end{theorem}
Hence, for some integers $t$ and for some prime $p$ the term $F_p(t)$ cannot be zero.
In 2019, A.-M. Ernvall-Hyt\"onen, T. Matala-aho and L. Sepp\"al\"a \cite{EHMS2019} studied more about these type of conditions when they considered the case where $k =1$ and $\lambda_1 \ne 0$ in formula \eqref{eq:DefLambda}. They showed that if condition \eqref{eq:OldCOndC} is satisfied for all sets $R \setminus S$, where $S$ is any finite subset of $R$, then there exists infinitely many primes $p \in R$ such that $\Lambda_p \neq 0$. This raised a question, what kind of sets could satisfy condition \eqref{eq:OldCOndC}. Hence, in the same article, they showed that the set $R$, which consists of primes in more than $\varphi(m)/2$ residue classes in the reduced residue system modulo $m \geq 3$, satisfies the condition \eqref{eq:OldCOndC} and there are infinitely primes in $R$ such that $\Lambda_p \neq 0$. Even more, they also proved the following, conditional result for primes in $\varphi(m)/2$ residue classes:
\begin{theorem}
\label{thm:EHMaS}
\cite[Theorem 8]{EHMS2019}
Assume the Generalized Riemann Hypothesis. Let $m \geq 3$ be a given integer. Assume that $R$ is any union of primes in $\psi(m)/2$ residue classes in reduced residue system modulo $m$. Then there is a value $d_m$ such that if $t$ is any non-zero integer satisfying the bound
\begin{equation*}
4\left|t\right|\prod_{p \in R} \left|t \right|_p^2<d_m,
\end{equation*}
then there exists a prime number $p \in R$ for which $a-bF_p(\xi) \neq 0$. Here $a$ and $b \neq 0$ are fixed integers.
\end{theorem}
Further, very recently A.-M. Ernvall-Hyt\"onen, T. Matala-aho and L. Sepp\"al\"a \cite{EHMS2022} also shoved an explicit, $p$-adic lower bound for $\Lambda_p$ with $k=1$, $\lambda_1 \ne 0$ and $\alpha_1=\pm p^a$ when $p^a$ large enough.
In 2020, L. Sepp\"al\"a \cite{S2020} generalized the results for the Euler's factorial series at algebraic integer points and for linear forms $\Lambda_p$ with $k \geq 1$. She showed if a type \eqref{eq:OldCOndC} condition holds, then for some valuation $v$ we must have $\Lambda_v \neq 0$. She also proved that if we consider a set $R$ which consists of primes in more than $k\varphi(m)/(k+1)$ residue classes in reduced residue system modulo $m$, then for a valuation $v$ which belongs to a certain set and for which we have $v \mid p$ for some $p \in R$, we have $\Lambda_v \ne 0$. Even more, she also proved explicit lower bounds for $\left|\Lambda_v\right|_v$.
In this article, we first prove that if a type \eqref{eq:OldCOndC} condition holds for a set of primes $R$ and for $\Lambda_p$ with $k \geq 1$, then $\Lambda_p \neq 0$ for some $p \in R$. This is a slight improvement to Corollary 9.1 in \cite{S2020}. Then, we generalize Theorem \ref{thm:EHMaS} for linear forms $k \ge 1$ and prove an explicit version of it. Finally, we prove explicit $p$-adic lower bounds for the terms $\Lambda_p$, where numbers $p$ are taken from a set containing some primes in arithmetic progressions from at least $k\varphi(m)/(k+1)$ residue classes in reduced residue system modulo $m$. The results are described in more detailed way in Section \ref{sec:results}.
Throughout the article $p$ denotes a prime number and $\mathbb{P}$ is a set of prime numbers. We also use abbreviation GRH for the Generalized Riemann Hypothesis.
\section{Results}
\label{sec:results}
In this Section, we describe the main results.
Before the actual results, we describe some definitions and notations. First we define term $c$ for integers $k, \alpha_1,\ldots, \alpha_k$ and set $R$:
\begin{equation}
\label{def:c1}
c\coloneqq c(\overline{\alpha}; R)=2^{k}\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)^{k}\prod_{p \in R}\left(\max_{1 \leq i \leq k}\{\left|\alpha_i\right|_p\}\right)^{k+1}.
\end{equation}
Further, in Table \ref{table:a1a2} we define numbers $a_1(m,k)$ and $a_2(m)$ for different integres $m$.
\begin{table}[!h]
\centering
\begin{tabular}{|c c c|}
\hline
$m$ & $a_1(m,k)$ & $a_2(m)$ \\
\hline
$3\leq m \leq 432$ & $e^{-4.017\varphi(m)k}$ & $\log\left(\max\{m,\sqrt{1865}\}\right)+4.017$ \\
$433\leq m\leq 10^5$ & $e^{-k\left(12.647\varphi(m)+0.292\right)}$ & $\log m+\frac{0.292}{\varphi(m)}+12.647$ \\
$10^5< m<4\cdot 10^5$ & $e^{-k\left(8.184\varphi(m)+0.052\right)}$ & $\log m+\frac{0.052}{\varphi(m)}+8.184$ \\
$4\cdot 10^5\leq m<10^{29}$ & $e^{-k\left(5.563\varphi(m)+0.032\right)}$ & $\log m+\frac{0.032}{\varphi(m)}+5.563$ \\
$m \geq 10^{29}$ & $e^{-k\left(2.897\varphi(m)+0.001\right)}$ & $\log m+\frac{0.001}{\varphi(m)}+2.897$ \\
\hline
\end{tabular}
\caption{Definitions for $a_1(m,k)$ and $a_2(m)$ for different integers $m$.}
\label{table:a1a2}
\end{table}
The first result tells that if a certain condition holds, then a linear form in functions $F_p$ must be non-zero.
\begin{theorem}
\label{thm:limSupCond}
Let $k \geq 1$ be an integer and let $\lambda_0, \lambda_1, \ldots, \lambda_k$ be integers such that $\lambda_i \neq 0$ for at least one index $i$. Further, assume that numbers $\alpha_1,\alpha_2,\ldots, \alpha_k$ are non-zero integers. Let $R$ be a non-empty set of prime numbers such that condition
\begin{equation}
\label{eq:limSupCond}
\limsup_{n \to \infty} c^n(kn+k)!\cdot(kn+k)\prod_{p \in R}\left|(kn)!n!\right|_p=0
\end{equation}
holds. Then for at least one prime number $p \in R$ we have
\begin{equation*}
\lambda_0+\lambda_1F_p(\alpha_1)+\ldots+\lambda_kF_p(\alpha_k) \neq 0.
\end{equation*}
\end{theorem}
The previous result raises a question for what kind of non-empty set the condition \eqref{eq:limSupCond}. Next we prove that primes in enough many arithmetic progressions satisfy the condition if one extra assumption for number $c(\overline{\alpha}; R)$ is assumed.
\begin{theorem}
\label{thm:dmGene}
Assume that $k \geq 1$, $m \geq 3$ and $\lambda_0, \lambda_1, \ldots,\lambda_k$ be integers such that $\lambda_j \neq 0$ for at least one index $j$. Even more, assume also that GRH holds. Further, suppose that $R$
is any union of primes in $k\psi(m)/(k+1)$ residue classes in the reduced residue system modulo $m$.
If $\alpha_1, \alpha_2,\ldots, \alpha_k$ are any non-zero integers for which $c(\overline{\alpha}; R)$ satisfies the bound
\begin{equation}
\label{eq:cUpper}
c(\overline{\alpha}; R)
<\frac{a_1(m,k)}{(ke)^k}e^{-k\varphi(m)\log\left(\max\{m,\sqrt{1865}\} \right)},
\end{equation}
where we denote $\overline{\alpha}=(\alpha_1,\alpha_2,\ldots, \alpha_k)$, then there exists a prime number $p \in R$ for which
\begin{equation*}
\lambda_0+\lambda_1F_p(\alpha_1)+\lambda_2F_p(\alpha_2)+\ldots+\lambda_kF_p(\alpha_k)\neq 0.
\end{equation*}
\end{theorem}
\begin{remark}
\label{remark:ExistsC}
We notice that there actually is sets $R$ such that condition \eqref{eq:cUpper} holds: For example, let $k=1$, $m=3$, $R$ containing all primes which are congruent to $1$ modulo $3$ and $\alpha_1=\alpha_k=p \in R$ such that $p> 3760e^{9.034}$. Then inequality \eqref{eq:cUpper} is satisfied.
\end{remark}
Since the previous theorem tells us that under certain conditions there is a prime such that a linear form does not equal to zero, a natural question is, how close it is to zero and how large the possible prime number is. To answer this question, we first denote
\begin{equation}
\label{def:A1mk}
A_1(m,k)
:=
\begin{cases}
27.640k\varphi(m)+40.160k+0.054 &\text{if } 3 \leq m \leq 36 \\
\left(0.001\varphi(m)+1.502\right)k &\text{if } m \geq 37
\end{cases},
\end{equation}
\begin{equation*}
A_2(m,k):=
\begin{cases}
0.017 k^2+ 0.033 k+0.011 &\text{if } 3 \leq m \leq 36 \\
0.385 k^2 + 0.761 k+0.202 &\text{if } m \geq 37
\end{cases}
\end{equation*}
and
\begin{equation*}
D(m,k,\overline{\alpha}; R):=-\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right).
\end{equation*}
Now we are ready to describe the next result:
\begin{theorem}
\label{thm:lowerBound}
Assume that numbers $k,\lambda_j$ and $m$ and set $R$ are defined as in Theorem \ref{thm:dmGene}. Assume also GRH and that $\alpha_1,\alpha_2, \ldots, \alpha_k$ are pairwise distinct non-zero integers. Let us also denote the set
\begin{equation}
\label{eq:interval}
R \cap \left[2,\frac{2k\log H}{D(m,k,\overline{\alpha}; R)}+2k\right)
\end{equation}
by $R'$, and assume that $c(\overline{\alpha}; R')$ is smaller than the right-hand side of inequality \eqref{eq:cUpper}. Suppose also that
$\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\} \leq H$, where
\begin{equation}
\label{eq:BoundH}
\frac{\log H}{D(m,k,\overline{\alpha}; R')}
\geq \max\left\{1866, m^{\varphi(m)}+1,\max_{1\leq j \leq k}\{|\alpha_j|\}+1, e^{\frac{2A_1(m,k)}{D(m,k,\overline{\alpha}; R')}}+1\right\}.
\end{equation}
Then
\begin{multline*}
\left|\lambda_0+\lambda_1F_p(\alpha_1)+\lambda_2F_p(\alpha_2)+\ldots+\lambda_kF_p(\alpha_k)\right|\\
>H^{-\left(\frac{k^2}{D(m,k,\overline{\alpha}; R')}\log\log H-\frac{k^2\log D(m,k,\overline{\alpha}; R')}{D(m,k,\overline{\alpha}; R')}+\frac{2k^2+3k\log 2-2k}{D(m,k,\overline{\alpha}; R')}+\frac{2k^2A_1(m,k)}{D(m,k,\overline{\alpha}; R')^2}+A_2(m,k)\right)}
\end{multline*}
for some $p \in R'$.
\end{theorem}
\begin{remark}
According to Remark \ref{remark:ExistsC}, taking a large enough $H$ it is possible to find number $c(\overline{\alpha}; R')$ such that inequality \eqref{eq:cUpper} holds. Further, this also means that for large enough $H$ bound \eqref{eq:BoundH} is satisfied.
\end{remark}
The previous lower bound is quite long and includes the term $D(m,k,\overline{\alpha}; R')$ which may be a little bit more difficult to determine. Hence, we prove a little bit weaker but a more simple result. Hence, we first denote
\begin{equation*}
C_1(m,k):=
\begin{cases}
0.0004 k^2+ 0.039 k+0.0004 &\text{if } 3\leq m \leq 36 \\
0.068 k^2+ 2.915 k +0.058 &\text{if } m \geq 37
\end{cases}
\end{equation*}
and then describe the result:
\begin{corollary}
\label{corollary:lowerBound}
Let us use the same definitions and assumptions as in Theorem \ref{thm:lowerBound}. Then we have
\begin{equation*}
\left|\lambda_0+\lambda_1F_p(\alpha_1)+\lambda_2F_p(\alpha_2)+\ldots+\lambda_kF_p(\alpha_k)\right|_p>H^{-C_1(m,k)(\log\log H)^2}
\end{equation*}
for some prime number $p \in R'$.
\end{corollary}
However, in the last two results the exponent of $H$ goes to minus infinity when $H$ goes to infinity. We would like to get an exponent which does not go to minus infinity when $H$ goes to infinity. Hence, as a final main result, we proved that if we consider more than $k\varphi(m)/(k+1)$ residue classes, then under certain conditions we obtain the lower bound we wanted:
\begin{table}[!h]
\centering
\begin{tabular}{|c c c c|}
\hline
$m$ & $A_3(m,k)$ & $B_1(m)$ & $C_2(m,k, \varepsilon)$ \\
\hline
$3\leq m \leq 36$ & $15.480k+16.480$ & $3.275$ & $\frac{50.697 k^2 + 53.972k}{\varepsilon}+3.710k^2-2.538k$ \\
$37\leq m \leq 43$ & $5.368k+6.368$ & $9.912$ & $\frac{53.208 k^2 + 63.120k}{\varepsilon}+10.194k^2-9.435k$ \\
$44\leq m \leq 432$ & $5.172k+6.172$ & $10.523$ & $\frac{54.425 k^2 + 64.948 k}{\varepsilon}+10.779k^2-10.090k$ \\
$433\leq m\leq 10^5$ & $7.176k+8.176$ & $6.612$ & $\frac{34.198 k^2 + 40.810 k}{\varepsilon}+6.636k^2-6.571k$ \\
$10^5< m<4\cdot 10^5$ & $4.422k+5.422$ & $13.852$ & $\frac{61.254 k^2 + 75.106 k}{\varepsilon}+13.853k^2-13.850k$ \\
$4\cdot 10^5\leq m<10^{29}$ & $3.863 k+4.863$ & $18.124$ & $\frac{70.014 k^2 + 88.138 k}{\varepsilon}+19.819k^2-15.254k$ \\
$m \geq 10^{29}$ & $3.087k+4.087$ & $1.127$ & $ \frac{92.333 k^2 + 122.243 k}{\varepsilon}+30.556k^2-28.816k$ \\
\hline
\end{tabular}
\caption{Definitions for $A_3(m,k)$, $B_1(m,k)$ and $C_2(m,k, \varepsilon)$ for different values $m$.}
\label{table:A3B1C2}
\end{table}
\begin{theorem}
\label{thm:lowerBoundMore}
Assume that $m, k, \alpha_1,\ldots, \alpha_k$ and $\lambda_0,\ldots, \lambda_k$ satisfy the same hypothesis as in Theorem \ref{thm:dmGene}. Assume also GRH and that $\varepsilon \in (0,1)$ is a real number, set $R$ consists of primes in arithmetic progressions in at least $(k+\varepsilon)\varphi(m)/(k+1)$ residue classes modulo $m$ and that we have $\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\} \leq H$ where
\begin{equation}
\label{eq:HassumptionEpsilon}
\frac{\log H}{\varepsilon} \geq se^s
\end{equation}
and
\begin{equation}
\label{eq:LowerBoundsForS}
\begin{aligned}
s \geq \max \left\{1866,m^{\varphi(m)}+1, c+1, \max_{1\leq j \leq k}\{|\alpha_j|\}+1, \left(\frac{A_3(m,k)}{\varepsilon}\right)^{1.6}+1 \right\}.
\end{aligned}
\end{equation}
Then there exists a prime
\begin{equation}
\label{eq:intervalEpsilon}
p \in R\cap\left(\log\left(\frac{\log H}{\varepsilon\log\left(\frac{\log H}{\varepsilon}\right)}\right),k\left(\frac{B_1(m)\log H}{\varepsilon\log\left(\frac{\log H}{\varepsilon}\right)}+2\right)\right)
\end{equation}
such that
\begin{equation*}
\left|\lambda_0+\lambda_1F_p(\alpha_1)+\lambda_2F_p(\alpha_2)+\ldots+\lambda_kF_p(\alpha_k)\right|>H^{-\left(\frac{k}{\varepsilon}+1\right)-\frac{C_2(m,k,\varepsilon)\log\log\log H}{\varepsilon \log\log H}}.
\end{equation*}
\end{theorem}
\section{Outline of the proofs}
\label{sec:outline}
In this section, we describe the main idea of the proofs. The idea is to estimate functions $F(\alpha_j)$ using Pad\'e approximations and then obtain the main results applying these approximations. Similar ideas have been used, for example, in articles \cite{EHMS2019,S2020}.
We use Pad\'e approximations to estimate the functions $F(\alpha_j)$ and write
\begin{equation*}
B_{n,\mu,0}(t)F(\alpha_jt)-B_{n,\mu,j}(t)=S_{n,\mu,j}(t)
\end{equation*}
for each $j=1,2\ldots, k$. Further, we write
\begin{equation}
\label{eq:defT}
B_{n,\mu,0}(1)\Lambda_p=\sum_{j=0}^k \lambda_jB_{n,\mu,j}(1)+\sum_{j=1}^k \lambda_jS_{n,\mu,j}(1) \coloneqq T(n, \mu)+\sum_{j=1}^k \lambda_jS_{n,\mu,j}(1).
\end{equation}
Now, let now $R$ be a set of certain primes in arithmetic progressions. It is clearly non-empty in Theorems \ref{thm:limSupCond} and \ref{thm:dmGene} and it is proved to be non-empty in Theorems \ref{thm:lowerBound} and \ref{thm:lowerBoundMore} (see Section \ref{sec:Existence}). If we have $\Lambda_p=0$ for all $p \in R$ (i.e. contradiction to Theorem \ref{thm:limSupCond}), from which it follows that
\begin{equation*}
\left|T(n+1,\mu)\right|_p=\left|\sum_{j=1}^k \lambda_jS_{n+1,\mu,j}(1)\right|_p,
\end{equation*}
or if we have
\begin{equation}
\label{eq:LambdaContra}
\left|B_{n+1,\mu,0}(1)\Lambda_p\right|_p \leq \left|\sum_{j=1}^k \lambda_jS_{n+1,\mu,j}(1)\right|_p
\end{equation}
for all $p \in R$, then estimate
\begin{equation}
\label{eq:Product}
\begin{split}
1=\left|T(n+1, \mu)\right|\prod_{p \in \mathbb{p}} \left|T(n+1, \mu)\right|_p \leq \left|T(n+1, \mu)\right|\prod_{p \in R} \left|T(n+1, \mu)\right|_p \\ \leq (k+1)\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\} \max_{0 \leq i \leq k} \{\left|B_{n+1,\mu,i}(1)\right|\}\cdot \max_{1 \leq i \leq k} \left\{ \prod_{p \in R}\left|S_{n+1,\mu,i}(1)\right|_p\right\}
\end{split}
\end{equation}
holds. In order to prove Theorems \ref{thm:limSupCond}, \ref{thm:lowerBound} and \ref{thm:lowerBoundMore} and Corollary \ref{corollary:lowerBound}, we show that the right-hand side of inequality \eqref{eq:Product} is smaller than $1$ for some integer $n$ when $\mu$ is selected in such a way that $T(n+1,\mu) \neq 0$ (see \cite[Lemma 6.2]{S2020}). This is a contradiction. Hence neither $\Lambda_p =0$ nor inequality \eqref{eq:LambdaContra} cannot hold for all $p \in R$. The first one proves Theorem \ref{thm:limSupCond} and Theorem \ref{thm:dmGene} follows from that.
Now, in order to prove Theorems \ref{thm:lowerBound} and \ref{thm:lowerBoundMore} and Corollary \ref{corollary:lowerBound} we still have to do a little bit more work. Since there must exists $p \in R$ for which inequality \eqref{eq:LambdaContra} cannot hold, we must have
\begin{equation}
\label{eq:TnEnough}
1 \leq \left|T(n+1, \mu)\right|\, \left|T(n+1, \mu)\right|_p=\left|T(n+1, \mu)\right|\, \left|B_{n+1,\mu,0}(1)\Lambda_p\right|_p \leq \left|T(n+1, \mu)\right|\,\left|\Lambda_p\right|_p.
\end{equation}
Hence, in order to find lower bounds for the terms $\left|\Lambda_p\right|_p$, it is sufficient to find and upper bound for the term $\left|T(n, \mu)\right|$.
The estimates for the Pad\'e approximations are derived in Section \ref{sec:Pade}. In Sections \ref{sec:piTheta} and \ref{sec:EstDepend} we prove some results related to primes in arithmetic progressions which are used later in our main proofs. In Section \ref{sec:Existence}, we show that there exists primes in sets \eqref{eq:interval} and \eqref{eq:intervalEpsilon} in Theorems \ref{thm:lowerBound} and \ref{thm:lowerBoundMore}. In Sections \ref{sec:proof1} and \ref{sec:proof2}, we prove Theorems \ref{thm:limSupCond} and \ref{thm:dmGene} respectively. In Sections \ref{sec:Contra} and \ref{sec:Contra2} we show the contradictions that the right-hand side of inequality \eqref{eq:Product} must be smaller than one, first in a setup of Theorem \ref{thm:lowerBound} and then with respect to Theorem \ref{thm:lowerBoundMore}. Finally, in Sections \ref{sec:proof3} and \ref{sec:proof4}, we prove Theorems \ref{thm:lowerBound} and \ref{thm:lowerBoundMore} and Corollary \ref{corollary:lowerBound}.
\section{Preliminary estimates}
\subsection{Estimates for Pad\'e approximations}
\label{sec:Pade}
In this section, we consider Pad\'e approximations for the Euler factorial series and prove estimates for them.
First we give formulas for Pad\'e approximations and then derive some upper bounds for them. The following function is used in the definitions:
\begin{equation}
\label{eq:defSigma}
\sigma_i \coloneqq \sigma_i(n, \overline{\alpha})=(-1)^i\sum_{\substack{i_1+\ldots+i_k=i \\ 0 \leq i_j \leq n}} \binom{n}{i_i}\binom{n}{i_2}\cdots \binom{n}{i_k}\alpha_1^{n-i_i}\alpha_2^{n-i_2}\cdots \alpha_k^{n-i_k}.
\end{equation}
Now we describe the Pad\'e approximations we are going to use:
\begin{theorem}
\cite[Theorem 4.2 and Section 4.2]{S2020}
\label{thm:Pade}
Let $\mu, k, \alpha_i$ be defined as Theorem \ref{thm:lowerBound} and assume that $n \in \mathbb{Z}_+$. Then we have
\begin{equation*}
B_{n,\mu,0}(t)F(\alpha_jt)-B_{n,\mu,j}(t)=S_{n,\mu,j}(t), \quad j=1,2,\ldots, k,
\end{equation*}
where
\begin{equation*}
\begin{split}
& B_{n,\mu,0}(t)=\sum_{i=0}^{kn} \sigma_i\cdot\frac{(kn+\mu)!}{(i+\mu)!}t^{kn-i} \\
& B_{n,\mu,j}(t)=(kn+\mu)!\sum_{N=0}^{kn+\mu-1}t^N\sum_{h=0}^{\min\{kn,N\}}\sigma_{kn-h}\cdot\frac{(N-h)!}{(kn+\mu-h)!}\cdot\alpha_j^{N-h} \\
& S_{n,\mu,j}(t)=(kn+\mu)!n!t^{(k+1)n+\mu}\sum_{h=0}^\infty h!\binom{n+h}{h}\alpha_j^{n+h+\mu}t^h\sum_{i=0}^{kn}\sigma_i\binom{i+\mu+n+h}{i+\mu}\alpha_j^i.
\end{split}
\end{equation*}
\end{theorem}
Next we estimate the approximations. In order to do it, we need the following lemma:
\begin{lemma}
\label{lemma:sigma}
\cite[Lemma 4.1, estimate (15)]{S2020}
Let $k$ be defined as Theorem \ref{thm:lowerBound}, $n\in \mathbb{Z}_+$ and $\sigma_i(n, \overline{\alpha})$ be defined as in \eqref{eq:defSigma}. Then we have
\begin{equation*}
\sum_{i=1}^{kn}\left|\sigma_i(n, \overline{\alpha})\right|t^i \leq \prod_{i=1}^k \left(\left|\alpha_i\right|+t\right)^n.
\end{equation*}
\end{lemma}
Now we are ready to estimate the Pad\'e approximations:
\begin{lemma}
\label{lemma:BSEstimates}
Let $\mu, k, \alpha_i$ be defined as Theorem \ref{thm:lowerBound}, assume that $n \in \mathbb{Z}_+$ and let terms $B_{n,\mu,0}(1), B_{n,\mu,j}(1)$ and $S_{n,\mu,j}(1)$ be as in Theorem \ref{thm:Pade}. Then we have
\begin{equation*}
\begin{split}
& \left|B_{n,\mu,0}(1)\right|\leq (kn)!\binom{kn+\mu}{\mu}\prod_{i=1}^{k}\left(\left|\alpha_i\right|+1\right)^n, \\
& \left|B_{n,\mu,j}(1)\right| \leq (kn+\mu)!\cdot(kn+k)\cdot |\alpha_j|^{k-1} \prod_{i=1}^k \left(\left|\alpha_i\right|+\left|\alpha_j\right|\right)^n \\
&\text{and} \\
& \left|S_{n,\mu,j}(1)\right|_p \leq \left|(kn+\mu)!n!\right|_p\cdot \left(\max\{\left|\alpha_j\right|_p\}\right)^{(k+1)n}.
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
The estimate for the term $B_{n,\mu,0}(1)$ follows from \cite[Section 7]{S2020}. So let us move on to estimate the term $B_{n,\mu,j}(1)$. Using Lemma \ref{lemma:sigma} we can deduce
\begin{align*}
\left|B_{n,\mu,j}(1)\right|&=\left|(kn+\mu)!\sum_{N=0}^{kn+\mu-1}\sum_{h=0}^{\min\{kn,N\}}\sigma_{kn-h}\cdot\frac{(N-h)!}{(kn+\mu-h)!}\cdot\alpha_j^{N-h}\right| \\
&\leq (kn+\mu)!\sum_{N=0}^{kn+\mu-1}\left|\alpha_j\right|^{N-kn}\sum_{h=0}^{\min\{kn,N\}} \left|\sigma_{kn-h}\right|\cdot \left|\alpha_j\right|^{kn-h} \\
& \leq (kn+\mu)!\sum_{N=0}^{kn+\mu-1}\left|\alpha_j\right|^{k-1}\sum_{i=0}^{kn} \left|\sigma_{i}\right|\cdot \left|\alpha_j\right|^{i} \\
&\leq (kn+\mu)!\cdot(kn+k)\cdot |\alpha_j|^{k-1} \prod_{i=1}^k \left(\left|\alpha_i\right|+\left|\alpha_j\right|\right)^n,
\end{align*}
which proves the second case.
Now we have only the term $S_{n,\mu,j}(1)$ left. Keeping in mind that the numbers $\alpha_1,\alpha_2,\ldots, \alpha_k$ are integers and
using the estimate
\begin{equation*}
\left|\sigma_i\right|_p = \left|\sum_{i_1+\ldots+i_k=i} \binom{n}{i_i}\binom{n}{i_2}\cdots \binom{n}{i_k}\alpha_1^{n-i_i}\alpha_2^{n-i_2}\cdots \alpha_k^{n-i_k}\right|_p \leq \left(\max\{\left|\alpha_j\right|_p\}\right)^{kn-i},
\end{equation*}
we get
\begin{align*}
\left|S_{n,\mu,j}(1)\right|_p&=\left|(kn+\mu)!n!\sum_{h=0}^\infty h!\binom{n+h}{h}\alpha_j^{n+h+\mu}\sum_{i=0}^{kn}\sigma_i\binom{i+\mu+n+h}{i+\mu}\alpha_j^i\right|_p \\
&\leq \left|(kn+\mu)!n!\right|_p\cdot \max_{\substack{0\leq h <\infty, \\ 0\leq i \leq kn}}\left| h!\binom{n+h}{h}\binom{i+\mu+n+h}{i+\mu}\right|_p \left(\max\{\left|\alpha_j\right|_p\}\right)^{(k+1)n+\mu} \\
& \leq \left|(kn+\mu)!n!\right|_p\cdot \left(\max\{\left|\alpha_j\right|_p\}\right)^{(k+1)n}.
\end{align*}
Hence we have proved the estimate for the term $S_{n,\mu,j}(1)$ too.
\end{proof}
\subsection{Estimates for the functions $\pi(x;m,a)$ and $\theta(x;m,a)$}
\label{sec:piTheta}
In this section we derive estimates for primes in arithmetic progressions and for related functions. Indeed, we consider the function
\begin{equation*}
\pi(x;m,a):=\sum_{\substack{p\leq x \\ p\equiv a\pmod m }} 1
\end{equation*}
and the function
\begin{equation*}
\theta(x;m,a):=\sum_{\substack{p\leq x \\ p\equiv a\pmod m }}\log{p}.
\end{equation*}
The estimates are later used in Section \ref{sec:EstDepend} to derive estimates for sum sums which are used to prove the main results.
First, we mention two results which are used to prove our estimates for functions $\pi(x;m,a)$ and $\theta(x;m,a)$:
\begin{theorem}
\label{pi}
\cite[Theorem 1]{EHP2022}
Assume the GRH. Let $a$ and $m$ be integers and $x$ be a real number such that $\gcd(a, m) = 1$, $m\geq 3$ and $x\geq m$. Then we have
\begin{multline*}
\left|\pi(x;m,a)-\frac{\mathrm{li}(x)}{\varphi(m)}\right| < \left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x+\left(0.184\log m+8.396\right)\sqrt{x} \\
+\left(6.05\log m+158.745\right)\frac{\sqrt{x}}{\log x}+ \left(5.048\log^2 m+152.085\log{m}+1731.270\right)\frac{\sqrt{x}}{\log^2 x}\\
+\left(0.184\log m+8.250\right)x^{1/4}\log\log{x}+\left(5.254\log^2 m+121.765\log{m}+939.260\right)x^{1/4} \\
+\left(80.768\log^2 m+1753.168\log{m}+11605.056\right)\frac{\sqrt{x}}{\log^3 x}-237.934.
\end{multline*}
\end{theorem}
\begin{lemma}
\label{lemma:varLower}
\cite[Section 4, the first paragraph]{S1943}
If $a \neq 1,2,3,4,6,10,12,18,30$ is a positive integer, then
\begin{equation*}
\varphi(a) > a^{\log 2/\log 3}.
\end{equation*}
\end{lemma}
Now we denote
\begin{equation}
\label{eq:defa7}
a_3(m):=0.184\log m+
\begin{cases}
19.795+\frac{27.414}{\varphi(m)} &\text{if } 3 \leq m \leq 432 \\
8.987+\frac{6.733}{\varphi(m)} &\text{if } 433 \leq m \leq 10^5 \\
8.406+\frac{6.063}{\varphi(m)} &\text{if } 10^5 < m < 4\cdot 10^5 \\
8.400+\frac{6.055}{\varphi(m)} &\text{if } 4\cdot 10^5\leq m <10^{29} \\
8.397+\frac{6.051}{\varphi(m)} & \text{if } m \geq 10^{29}
\end{cases}
\end{equation}
and prove an estimate for $\pi(x;m,a)$ when $x$ is large enough with respect to $m$:
\begin{lemma}
\label{lemma:estPi}
Suppose the same assumptions as in Theorem \ref{pi}. Assume also that $x \geq \max\{m^{\varphi(m)},1865\}$. Then
\begin{equation*}
\left|\pi(x;m,a)-\frac{\mathrm{li}(x)}{\varphi(m)}\right| \leq \left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x +a_3(m)\sqrt{x}-237.934.
\end{equation*}
\end{lemma}
\begin{proof}
Let us first consider case $3 \leq m \leq 10^5$ and $x \leq 10^{11}$. Since $\varphi(m)<m\leq 10^5$ and $x \geq 1865$, by \cite[Theorem 1.9]{BGOR2018} we have
\begin{multline}
\label{eq:piSmallValuesEst}
\left|\pi(x;m,a)-\frac{\mathrm{li}(x)}{\varphi(m)}\right|\leq \frac{\textrm{li}(2)}{\varphi(m)}+2.734\frac{\sqrt{x}}{\log x}
\leq \left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x\\
+\sqrt{x}\left(\frac{\textrm{li}(2)}{\sqrt{1865}\cdot 10^5}-\frac{\log 1865}{8\pi\cdot 10^5}+\frac{2.734}{\log 1865}-\frac{\log 1865}{6 \pi}+\frac{237.934}{\sqrt{1865}}\right)-237.934 \\
<\left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x+5.474\sqrt{x}-237.934.
\end{multline}
Let us now consider the rest of the cases.
The rest of the cases follow easily from Theorem \ref{pi}: Using the theorem and the assumption $x \geq m^{\varphi(m)}$, we can deduce that
\begin{equation}
\label{eq:pivar}
\begin{split}
&\left|\pi(x;m,a)-\frac{\mathrm{li}(x)}{\varphi(m)}\right|< \left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x+\left(0.184\log m+8.396+\frac{6.05}{\varphi(m)} \right. \\
&\quad\left.+\frac{5.048}{\varphi(m)^2}\right)\sqrt{x} +\left(158.745+\frac{152.085}{\varphi(m)}+\frac{80.768}{\varphi(m)^2}\right)\frac{\sqrt{x}}{\log x} \\
&\quad+ \left(1731.270+\frac{1753.168}{\varphi(m)}\right)\frac{\sqrt{x}}{\log^2 x}+11605.056\frac{\sqrt{x}}{\log^3 x} \\
&\quad+\left(\frac{0.184}{\varphi(m)}\log x+8.250\right)x^{1/4}\log\log{x} \\
&\quad+\left(\frac{5.254}{\varphi(m)^2}\log^2 x+\frac{121.765}{\varphi(m)}\log{x}+939.26\right)x^{1/4}-237.934.
\end{split}
\end{equation}
Let us start first consider the case $3 \leq m \leq 432$. We have $\varphi(m) \geq 2$ and $x \geq 10^{11}+1$, since we have already considered the case $x \leq 10^{11}$. Hence, we also have
\begin{equation}
\label{eq:est1865}
\begin{aligned}
&\frac{1}{\log{x}}\leq \frac{1}{\log (10^{11}+1)}, \quad \frac{1}{\log^2{x}}\leq \frac{1}{(\log (10^{11}+1))^2}, \quad \frac{1}{\log^3{x}}\leq \frac{1}{(\log (10^{11}+1))^3}, \\
&x^{1/4}\log{x} \log\log x \leq \frac{(\log (10^{11}+1))\cdot(\log\log (10^{11}+1))}{(10^{11}+1)^{1/4}}\sqrt{x}, \\
& x^{1/4}\log{\log{x}} \leq \frac{\log\log (10^{11}+1)}{(10^{11}+1)^{1/4}}\sqrt{x}, \quad x^{1/4}\log^2 x\leq \frac{(\log (10^{11}+1))^2}{(10^{11}+1)^{1/4}}\sqrt{x}, \\
& x^{1/4}\log x\leq \frac{\log (10^{11}+1)}{(10^{11}+1)^{1/4}}\sqrt{x} \quad\text{and}\quad x^{1/4} \leq \frac{\sqrt{x}}{(10^{11}+1)^{1/4}}.
\end{aligned}
\end{equation}
Thus, using estimate \eqref{eq:pivar}, we have obtained
\begin{multline*}
\left|\pi(x;m,a)-\frac{\mathrm{li}(x)}{\varphi(m)}\right| \leq \left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x \\
+\left(0.184\log m+19.795+\frac{27.414}{\varphi(m)}\right)\sqrt{x}-237.934,
\end{multline*}
if $3 \leq m \leq 433$. Comparing this estimate to the case $x \leq 10^{11}$ (see estimate \eqref{eq:piSmallValuesEst}), we see that the previous estimate applies also to the case $x \leq 10^{11}$.
Let us now move on to the cases $433\leq m\leq 10^5$. By Lemma \ref{lemma:varLower} we have $\varphi(m) >433^{\log 2/\log 3}$. Hence, using estimate \eqref{eq:pivar} and similar estimates as in \eqref{eq:est1865} but where $10^{11}+1$ is replaced with $433^{433^{\log 2/\log 3}}$, we obtain
\begin{multline*}
\left|\pi(x;m,a)-\frac{\mathrm{li}(x)}{\varphi(m)}\right| \leq \left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x \\
+\left(0.184\log m+8.987+\frac{6.733}{\varphi(m)}\right)\sqrt{x}-237.934.
\end{multline*}
Similarly, in the case $10^5 <m <4\cdot 10^{5}$, we can use number $10^5+1$ instead of $433$, in the case $4\cdot10^5 \leq m <10^{29}$ we use $4\cdot 10^5$ and in the case $m \geq 10^{29}$ we use $10^{29}$. Using these estimates, we obtain the wanted results also in the rest of the cases.
\end{proof}
\begin{remark}
In the previous result, the result is divided to different cases with respect to number $m$. These cases come from the cases which we use in Lemma \ref{corollary:theta}.
\end{remark}
Next we prove estimates for function $\theta(x;m,a)$. In order to prove those estimates, we use function
\begin{equation*}
\psi(x;m,a) := \sum_{\substack{p^k \leq x \\ p^k \equiv a \pmod m}} \log p
\end{equation*}
and with notation GRH($m, \text{div}$) we mean that the Generalized Riemann Hypothesis holds for Dirichlet characters modulo $m$ and all moduli dividing $m$.
First, we mention a useful estimate for the function $\psi(x;m,a)$:
\begin{theorem}
\label{psi}
\cite[Theorem 3]{EHP2022} Let $x\geq 2$ be a real number and $m\geq 3$ and $a$, where $\gcd(a, m) = 1$, be integers. Assume GRH($m, \text{div}$). We have
\begin{multline*}
\left|\psi(x;m,a)-\frac{x}{\varphi(m)}\right|< \left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log^2 x+ \left(0.184\log m+8.250\right)\sqrt{x}\log x\\
+(5.314\log m+124.318)\sqrt{x}+(5.048\log^2m+109.573\log m+725.316)\frac{\sqrt{x}}{\log {x}}\\
+\left(2.015\log m+0.5\right)\log x +R(m),
\end{multline*}
where the term $R(m)$ is
\begin{align*}
&R(m)=0.014\sqrt{m}\log{m}+0.034\sqrt{m}+3.679\log m+263.886
\end{align*}
if $3\leq m<4\cdot 10^5$,
\begin{align*}
&R(m)=1.858\log^2{m}+3.679\log m+104.626
\end{align*}
if $4\cdot 10^5\leq m<10^{29}$ and
\begin{align*}
&R(m)=(0.297\log\log m+0.603)\log^2{m}+3.679\log m+104.626
\end{align*}
otherwise.
\end{theorem}
Now we denote
\begin{equation*}
a_4(m):=
\begin{cases}
0.129, &\text{ if } 3\leq m\leq 432 \\
\frac{1}{8\pi\varphi(m)}+1.384, &\text{ if } 433 \leq m \leq 10^5 \\
\frac{1}{8\pi\varphi(m)}+4.120, &\text{ if } 10^5 < m < 4\cdot10^5 \\
\frac{1}{8\pi\varphi(m)}+3.440, &\text{ if } 4\cdot 10^5\leq m<10^{29} \\
\frac{1}{8\pi\varphi(m)}+0.571, &\text{ if } m \geq 10^{29}
\end{cases}.
\end{equation*}
and move on to estimate function $\theta(x;m,a)$.
\begin{lemma}
\label{corollary:theta}
Let $x,m$ and $a$ be as in Theorem \ref{psi} and let also $x \geq \max\{\sqrt{1865},m\}$. Assume that the GRH($m, \text{div}$). Then we have
\begin{equation*}
\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right| \leq a_4(m)\sqrt{x}\log^2 x.
\end{equation*}
\end{lemma}
\begin{proof}
First we notice that as in the third paragraph of the proof of Theorem 1 in \cite{EHP2022}, we have
\begin{equation}
\label{eq:psiEctraTerm}
0\leq \psi(x;m,a)-\theta(x;m,a)<1.4262\sqrt{x}.
\end{equation}
Hence we can use estimate given in Theorem \ref{psi}. Since the formula is quite long and since we also know sharper estimates in some special cases, we would like to simplify the result and we divide the proof to different cases depending on the number $m$.
Let us first consider the case $3 \leq m \leq 72$. First, if $\sqrt{1865}\leq x \leq 10^{10}$, then by \cite[Table 2]{RR1996} we have
\begin{equation*}
\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<1.817557\sqrt{x}\leq \frac{1.817557}{(\log \sqrt{1865})^2}\sqrt{x}\log^2 x<0.129\sqrt{x}\log^2 x.
\end{equation*}
Similarly, if $x \geq 10^{10}$, then by \cite[Corollary 3.3]{D1998} and estimate \eqref{eq:psiEctraTerm}, we get
\begin{multline}
\label{eq:estThetaSmallLarge}
\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<\frac{11}{32\pi}\sqrt{x}\log^2 x+1.4262\sqrt{x}=\left(\frac{11}{32\pi}+\frac{1.4262}{\left(\log(10^{10})\right)^2}\right)\sqrt{x}\log^2 x \\
< 0.113\sqrt{x}\log^2 x.
\end{multline}
Since $0.113<0.129$, we can use the first estimate also in the case $x \geq 10^{10}$.
Let us now consider the case $73 \leq m \leq 10^5$. By \cite[Theorem 1.9]{BGOR2018}
\begin{equation}
\label{est:theta73145Third}
\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<2.072\sqrt{x}\leq \frac{2.072}{(\log 73)^2}\sqrt{x}\log^2 x<0.113\sqrt{x}\log^2 x
\end{equation}
for $x \leq 10^{11}$. Further, if $x \geq 10^{11}$ and $m \leq 432$, then by estimate \eqref{eq:psiEctraTerm} and \cite[Theorem 3.7]{D1998}, we get
\begin{equation}
\label{est:theta73145Fourth}
\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<1.4262\sqrt{x}+\frac{11}{32\pi}\sqrt{x}\log^2 x <0.112\sqrt{x}\log^2 x.
\end{equation}
Hence, in the cases $3 \leq m \leq 432$ we can use the coefficient $0.129$. However, we still have to consider cases $433\leq m$ and $10^{11}<x$ as well as $m >10^5$.
For the rest of the cases we apply Theorem \ref{psi}. Using Theorem \ref{psi} and estimate \eqref{eq:psiEctraTerm} we have
\begin{equation}
\label{eq:psiEst}
\begin{split}
&\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<\left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}+0.184\right)\sqrt{x}\log^2 x \\
&\quad+\left(8.250+5.314+5.048\right)\sqrt{x}\log x+\left(124.318+109.573+1.4262\right)\sqrt{x} \\
&\quad+725.316\frac{\sqrt{x}}{\log x}+2.015\log^2 x+0.5\log x+R(x) \\
&\quad=\left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}+0.184\right)\sqrt{x}\log^2x+18.612\sqrt{x}\log x+235.3172\sqrt{x} \\
&\quad\quad+725.316\frac{\sqrt{x}}{\log x}+2.015\log^2 x+0.5\log x+R(x),
\end{split}
\end{equation}
where $R(x)$ is interpreted by setting $m=x$ to the formulas but using different cases depending on the number $m$, not by the number $x$.
We divide the consideration to four different cases depending on the size of the number $m$.
Let us first consider the case $433 \leq m \leq 10^{5}$ and $x \geq 10^{11}$. Using functions which are decreasing for all $x \geq 10^{11}$, by estimate \eqref{eq:psiEst} we have
\begin{align*}
&\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<\frac{\sqrt{x}\log^2 x}{8\pi \varphi(m)}+\left(\frac{1}{6\pi}+0.184+\frac{18.612+0.014}{\log(10^{11}+1)} \right. \\
&\quad\left.+\frac{235.3172+0.034}{(\log (10^{11}+1))^2}+\frac{725.316}{(\log (10^{11}+1))^3}+\frac{2.015}{\sqrt{10^{11}+1}} \right. \\
&\quad\left. +\frac{0.5+3.679}{\sqrt{10^{11}+1}\log (10^{11}+1)} +\frac{263.886}{\sqrt{10^{11}+1}(\log (10^{11}+1))^2} \right)\sqrt{x}\log^2x \\
&\quad<\left(\frac{1}{8\pi \varphi(m)}+1.384\right)\sqrt{x}\log^2 x.
\end{align*}
Together with estimate \eqref{est:theta73145Third} this leads to an estimate in the case $433 \leq m \leq 10^5$. Even more, using $10^5+1$ instead of $10^{11}+1$, we get the wanted estimate for the case $10^5 <m <4\cdot 10^5$. The cases $4\cdot 10^5\leq m<10^{29}$ and $m \geq 10^{29}$ can be proved similarly but we use the second and the last formulas for $R(x)$ from Theorem \ref{psi} instead of the last one and values $4\cdot10^5$ and $10^{29}$ for $x$.
\end{proof}
In some of the results we are going to prove in the next section, it is used that $x \geq \max\{m^{\varphi(m)}, 1865\}$. Hence we define
\begin{equation}
\label{eq:defa6}
a_5(m):=
\begin{cases}
0.113 &\text{ if } 3\leq m\leq 432 \\
\frac{1}{8\pi\varphi(m)}+0.089 &\text{ if } 433\leq m\leq 10^5 \\
\frac{1}{8\pi\varphi(m)}+0.054 &\text{ if } m > 10^5
\end{cases}
\end{equation}
and prove the following estimate for $\theta(x;m,a)$ when $x \geq \max\{m^{\varphi(m)}, 1865\}$:
\begin{lemma}
\label{corollary:theta2}
Let $x,m$ and $a$ be as in Theorem \ref{psi} and let also $x \geq \max\{m^{\varphi(m)}, 1865\}$. Assume GRH($m, \text{div}$). Then we have
\begin{equation*}
\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<a_5(m)\sqrt{x}\log^2 x.
\end{equation*}
\end{lemma}
\begin{proof}
The claim follows similarly as the results in Lemma \ref{lemma:estPi} and Lemma \ref{corollary:theta}:
Let us first consider the case $3 \leq m \leq 432$ and assume that $3 \leq m \leq 72$. If $1865\leq x \leq 10^{10}$, then by \cite[Table 2]{RR1996} we have
\begin{equation*}
\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<1.817557\sqrt{x}\leq \frac{1.817557}{\log^2 1865}\sqrt{x}\log^2 x<0.033\sqrt{x}\log^2 x.
\end{equation*}
For the case $x \geq 10^{10}$ we can use estimate \eqref{eq:estThetaSmallLarge} which gives $0.113$ and by estimates \eqref{est:theta73145Third} and \eqref{est:theta73145Fourth} this coefficient can be used also in the case $73 \leq m \leq 432$.
Now we move on to the cases $m \geq 433$. Using Theorem \ref{psi}, estimate \eqref{eq:psiEctraTerm} and the assumption $x \geq m^{\varphi(m)}$ we have
\begin{equation}
\label{eq:estTheta2}
\begin{split}
&\left|\theta(x;m,a)-\frac{x}{\varphi(m)}\right|<\left(\frac{1}{8\pi \varphi(m)}+\frac{1}{6\pi}+\frac{0.184}{\varphi(m)}\right)\sqrt{x}\log^2 x \\
&\quad+\left(8.250+\frac{5.314}{\varphi(m)}+\frac{5.048}{\varphi(m)^2}\right)\sqrt{x}\log x+\left(125.7442+\frac{109.573}{\varphi(m)}\right)\sqrt{x} \\
&\quad+725.316\frac{\sqrt{x}}{\log x}+\frac{2.015}{\varphi(m)}\log^2 x+0.5\log x+R\left(\sqrt{x}\right),
\end{split}
\end{equation}
where $R\left(\sqrt{x}\right)$ is interpreted similarly as in the similar step in the proof of Lemma \ref{corollary:theta}.
Let us now assume that $433 \leq m \leq 10^5$. Since by assumptions and Lemma \ref{lemma:varLower} $x \geq m^{\varphi(m)} > m^{m^{\log 2/\log 3}}\geq 433^{433^{\log 2/\log 3}}$, using functions which are decreasing for all $x$ in the consideration, the right-hand side of estimate \eqref{eq:estTheta2} is
\begin{align*}
&<\frac{\sqrt{x}\log^2 x}{8\pi \varphi(m)}+\left(\frac{1}{6\pi}+\frac{0.184}{433^{\log 2/\log 3}}+\frac{8.250+5.314/433^{\log 2/\log 3}+5.048/433^{2\log 2/\log 3}}{433^{\log 2/\log 3}\log 433} \right. \\
&\quad\left.+\frac{125.7442+109.573/433^{\log 2/\log 3}}{(433^{\log 2/\log 3}\log 433)^2}+\frac{725.316}{(433^{\log 2/\log 3}\log 433)^3} \right. \\
&\quad\left.+\frac{0.007}{\left(433^{433^{\log 2/\log 3}}\right)^{1/4} 433^{\log 2/\log 3}\log 433}+\frac{0.034}{\left(433^{433^{\log 2/\log 3}}\right)^{1/4} (433^{\log 2/\log 3}\log 433)^2}\right. \\
&\left.\quad+\frac{2.015}{\sqrt{433^{433^{\log 2/\log 3}}}}+\frac{0.5+3.679/2}{\sqrt{433^{433^{\log 2/\log 3}}}\cdot433^{\log 2/\log 3}\log 433} \right. \\
&\left.+\frac{263.886}{\sqrt{433^{433^{\log 2/\log 3}}}(433^{\log 2/\log 3}\log 433)^2}\right)\sqrt{x}\log^2x \\
&\quad<\left(\frac{1}{8\pi \varphi(m)}+0.089\right)\sqrt{x}\log^2 x.
\end{align*}
Similarly, replacing $433$ with $10^5+1$, we get an estimate for the case $10^5 <m <4\cdot 10^5$. Even more, the estimate for the last case follow similarly when we consider cases $4\cdot10^5 \leq m<10^{29}$ and $m \geq 10^{29}$ using their respective functions $R(\sqrt{x})$.
\end{proof}
\subsection{Estimates for some functions depending on the functions $\pi(x;m,a)$ and $\theta(x;m,a)$}
\label{sec:EstDepend}
In this section, we prove estimates for sum functions closely related to the functions $\pi(x;m,a)$ and $\theta(x;m,a)$. These estimates are used to prove the main results.
First we prove a results concerning to sum running over primes. Please keep in mind that the term $a_2(m)$ is defined in Table \ref{table:a1a2}.
\begin{lemma}
\label{lemma:estimateSumPNLog}
Let $m, a$ and $x$ be as in Theorem \ref{psi} and let $n$ be a positive integer. Assume also that $x \geq \max\{m^{\varphi(m)}, 1865\}$ and that RH and GRH($m,\text{div}$) hold. Then we have
\begin{equation*}
\left|-\sum_{\substack{ p \leq x \\p \equiv a \pmod m}} \frac{n}{p-1}\log p+\frac{n\log x}{\varphi(m)}\right|<a_2(m) n.
\end{equation*}
\end{lemma}
\begin{proof}
By Abel's summation formula we can write
\begin{equation}
\label{eq:BigEst}
\begin{split}
-\sum_{\substack{p \leq x \\p \equiv a \pmod m}}\frac{n}{p-1}\log p&=-\frac{n}{x-1}\sum_{\substack{p \leq x \\p \equiv a \pmod m}} \log p-n\int_{2}^x \frac{1}{(y-1)^2}\sum_{\substack{p \leq y \\p \equiv a \pmod m}} \log p \,dy \\
&= -\frac{n}{x-1}\sum_{\substack{p \leq x \\p \equiv a \pmod m}} \log p-n\int_{2}^{M} \frac{1}{(y-1)^2}\sum_{\substack{p \leq y \\p \equiv a \pmod m}} \log p \,dy \\
&\quad-n\int_{M}^x \frac{1}{(y-1)^2}\sum_{\substack{p \leq y \\p \equiv a \pmod m}} \log p \,dy,
\end{split}
\end{equation}
where $M=\max\{m, \sqrt{1865}\}$.
We consider each of the previous three terms on the last three lines of the equality.
First, by Lemma \ref{corollary:theta2}, the first term is
\begin{multline*}
-\frac{n}{x-1}\left(\frac{x}{\varphi(m)}+a_5(m)\sqrt{x}\log^2 x\right) \\
< -\frac{n}{x-1}\sum_{\substack{p \leq x \\p \equiv a \pmod m}} \log p < -\frac{n}{x-1}\left(\frac{x}{\varphi(m)}-a_5(m)\sqrt{x}\log^2 x\right).
\end{multline*}
Please notice that since all of the terms on the last two lines of computations \eqref{eq:BigEst} are non-positive, absolute value of the largest error term depends on the case
$$
-\frac{a_5(m)n\sqrt{x}\log^2 x}{x-1}.
$$
Similarly, now using Lemma \ref{corollary:theta}, the third term on the right-hand side of equality \eqref{eq:BigEst} is between the values
\begin{equation*}
-n\int_{M}^x \frac{y}{(y-1)^2\cdot \varphi(m)} \,dy \pm n\int_{M}^x \frac{a_4(m)\sqrt{y}\log^2 y}{(y-1)^2} \,dy
\end{equation*}
and we are interested in the case where there is a minus sign in front of the second integral.
The first integral is
\begin{equation*}
\int_{M}^x \frac{y}{(y-1)^2} \,dy =\left[\frac{1}{(1-y)}+\log(y-1)\right]_M^x=\log(x-1)-\log(M-1)+\frac{1}{1-x}-\frac{1}{1-M}
\end{equation*}
and since $M \geq \sqrt{1865}$, the another one is
\begin{align*}
\left| \int_{M}^x \frac{\sqrt{y}\log^2 y}{(y-1)^2} \,dy\right| &< \frac{2.2024}{2}\int_{M}^x \frac{\log^2 y}{y^{1.5}} \,dy \\ &=2.2024\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}-\frac{\log^2 x+4\log x+8}{\sqrt{x}}\right).
\end{align*}
Let us now move on to the second term in the right-hand side of equation \eqref{eq:BigEst}. Under Riemann Hypothesis, by \cite[Theorem 10, (6.5)]{S1976} we have
\begin{equation*}
0\leq \sum_{\substack{ p \leq y \\p \equiv a \pmod m}} \log p \leq \sum_{p \leq y} \log p<y+\frac{\sqrt{y}\log^2 y}{8\pi}
\end{equation*}
for all $y>0$. Hence, in order to estimate the second term in equation \eqref{eq:BigEst}, it is sufficient to compute an integral
\begin{multline*}
\int_2^M \frac{y+\sqrt{y}\log^2 y/(8\pi)}{(y-1)^2} \, dy\leq \int_2^M \left(\frac{y}{(y-1)^2}+\frac{\sqrt{2}\log^2 y}{8\pi y^{1.5}}\right) \, dy \\
=\left[\frac{1}{1-y}+\log(y-1)-\frac{\sqrt{2}}{4\pi}\left(\frac{\log^2 y+4\log y+8}{\sqrt{y}}\right)\right]_{2}^M \\
=-\frac{1}{M-1}+\log(M-1)+1-\frac{\sqrt{2}}{4\pi}\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}-\frac{\log^2 2+4\log 2+8}{\sqrt{2}}\right).
\end{multline*}
Thus we have estimated all of the terms in \eqref{eq:BigEst}.
Putting everything together we get
\begin{align*}
& \left|-\sum_{\substack{ p \leq x \\p \equiv a \pmod m}} \frac{n}{p-1}\log p+\frac{n\log x}{\varphi(m)}\right|<n\left|-\frac{a_4(m)\sqrt{x}\log^2 x}{x-1}+\frac{1}{M-1}-\log(M-1)\right. \\
&\quad \left. -1-\left(\frac{x-1}{x-1}+\log(x-1)-\log x-\log(M-1)-\frac{1}{1-M}\right)\frac{1}{\varphi(m)} \right. \\
&\quad\left. +\frac{\sqrt{2}}{4\pi}\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}-\frac{\log^2 2+4\log 2+8}{\sqrt{2}}\right) \right. \\
&\quad \left. -2.2024a_4(m)\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}-\frac{\log^2 x+4\log x+8}{\sqrt{x}}\right) \right| \\
&\quad<n\left|-\frac{a_4(m)\sqrt{x}\log^2 x}{x-1}+\frac{\varphi(m)-1}{(M-1)\varphi(m)}+\log(M-1)\cdot\frac{1-\varphi(m)}{\varphi(m)}-2\right. \\
&\quad\quad\left. -\frac{\log(x-1)-\log x}{\varphi(m)}+\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}\right)\left(\frac{\sqrt{2}}{4\pi}-2.2024a_4(m)\right) \right. \\
&\quad\quad\left. -\frac{\sqrt{2}}{4\pi}\left(\frac{\log^2 2+4\log 2+8}{\sqrt{2}}\right)+2.2024a_4(m)\left(\frac{\log^2 x+4\log x+8}{\sqrt{x}}\right) \right|.
\end{align*}
Since $-1-\log(x-1)+\log x <0$ for all $x$ under the consideration,
\begin{equation*}
\frac{1}{M-1}-\log(M-1)<0,
\end{equation*}
$\varphi(m) \geq 2$, the function
\begin{equation*}
\frac{\log^2 y+4\log y+8}{\sqrt{y}}
\end{equation*}
is decreasing for all $y>0$, $a_4(m) \geq 0.129$ and $x \geq 1865$, the right-hand side is
\begin{equation}
\label{eq:EstMainTerm}
\begin{split}
<n\left|-\frac{a_5(m)\sqrt{x}\log^2 x}{x-1} -\log M -2-\frac{\sqrt{2}}{4\pi}\left(\frac{\log^2 2+4\log 2+8}{\sqrt{2}}\right)\right. \\
\left. +\left(\frac{\sqrt{2}}{4\pi}-2.2024a_4(m)\right)\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}\right) \right| \\
<\left(\log M+\left(2.2024a_4(m)-\frac{\sqrt{2}}{4\pi}\right)\left(\frac{\log^2 M+4\log M+8}{\sqrt{M}}\right)\right. \\
\left.+\frac{a_5(m)\sqrt{x}\log^2 x}{x-1}+2.896\right)n.
\end{split}
\end{equation}
We are almost ready, we still want to simplify the last line in estimate \eqref{eq:EstMainTerm} a little bit. Let us consider the last line in \eqref{eq:EstMainTerm} in different cases depending on the size of $m$.
In each of the cases we apply the facts that the functions
\begin{equation*}
\frac{\log^2 y+4\log y+8}{\sqrt{y}} \quad\text{and}\quad \frac{\sqrt{y}\log^2 y}{y-1}
\end{equation*}
are decreasing for all $y>0$.
First we consider the case $3 \leq m \leq 432$. Since $M \geq \max\{m,\sqrt{1865}\}\geq \sqrt{1865}$ and $x \geq 1865$, the last line in \eqref{eq:EstMainTerm} is
\begin{multline*}
<n(\log\left(\max\{m,\sqrt{1865}\}\right)+2.896) \\
+n\left(2.2024\cdot0.129-\frac{\sqrt{2}}{4\pi}\right)\frac{\log^2 \sqrt{1865}+4\log \sqrt{1865}+8}{1865^{1/4}}+\frac{0.113\cdot\sqrt{1865}\log^2 1865}{1865-1}n \\
<\left(\log\left(\max\{m,\sqrt{1865}\}\right)+4.017\right)n.
\end{multline*}
Next we consider the case $433\leq m\leq 10^5$. Then $M \geq m \geq 433$ and by Lemma \ref{lemma:varLower} we have $x \geq m^{\varphi(m)}> 433^{433^{\log 2/\log 3}}$. Hence, the last line in \eqref{eq:EstMainTerm} is
\begin{multline*}
<n(\log m+2.896)+n\left(2.2024\left( \frac{1}{8\pi\varphi(m)}+1.384\right)-\frac{\sqrt{2}}{4\pi}\right)\frac{\log^2 433+4\log 433+8}{\sqrt{433}} \\
+\frac{\sqrt{433^{433^{\log 2/\log 3}}}\log^2\left(433^{433^{\log 2/\log 3}}\right)}{433^{433^{\log 2/\log 3}}-1}\left(\frac{1}{8\pi\varphi(m)}+0.089\right)n \\
<\left(\log m+\frac{0.292}{\varphi(m)}+12.647\right)n.
\end{multline*}
The estimates for the rest of the cases follow similarly using the respective lower bounds in each case.
\end{proof}
Next we define
\begin{equation*}
a_6(m):=
\begin{cases}
27.640+\frac{39.717}{\varphi(m)} &\text{if } 3 \leq m \leq 36 \\
0.001+\frac{1.501}{\varphi(m)} &\text{if } m \geq 37
\end{cases}
\end{equation*}
and estimate a term which depends on $p$-adic valuation of factorials:
\begin{lemma}
\label{lemma:Log}
Let $m$ and $a$ be defined as in Theorem \ref{psi} and $x$ as in Lemma \ref{lemma:estimateSumPNLog}. Assume also GRH. Further, let $n \geq x$ be a positive integer.
Then we have
\begin{equation*}
\left|\log\left(\prod_{\substack{p \leq x \\ p \equiv a \pmod m}} \left|n!\right|_p\right)+\frac{n\log x}{\varphi(m)}\right|< a_2(m)n+\frac{x\log n}{\varphi(m)\log x}+\frac{x}{\varphi(m)}+\frac{a_6(m)n}{\log n}.
\end{equation*}
\end{lemma}
\begin{proof}
We follow the same steps as in the proof of Lemma 9 in \cite{EHMS2019}. First, we modify the first term on the left-hand side of the claim and then derive the right-hand side using Lemma \ref{corollary:theta}.
First, we write the the first term on the left-hand side as
\begin{equation}
\label{eq:asSum}
\log\left(\prod_{\substack{p \leq x \\p \equiv a \pmod m}} \left|n!\right|_p\right)=\sum_{\substack{p \leq x \\ p \equiv a \pmod m}} \log |n!|_p.
\end{equation}
As in \cite[Proof of Lemma 9]{EHMS2019}, by a well known fact we have
\begin{equation*}
p^{-\frac{n}{p-1}} \leq \left|n!\right|_p \leq p^{-\frac{n}{p-1}+\frac{\log n}{\log p}+1}
\end{equation*}
and hence we can estimate the right-hand side of equation \eqref{eq:asSum} with sums
\begin{equation}
\label{eq:sumNThree}
-\sum_{\substack{ p \leq x \\p \equiv a \pmod m}} \frac{n}{p-1}\log p \quad\text{and} \sum_{\substack{ p \leq x \\p \equiv a \pmod m}} \left(\log n+\log p\right).
\end{equation}
We consider these sums separately.
First we notice that the first sum in \eqref{eq:sumNThree} can be estimated with Lemma \ref{lemma:estimateSumPNLog}. Hence we can move to the rest of the terms in \eqref{eq:sumNThree}.
By Lemma \ref{lemma:estPi} and Lemma \ref{corollary:theta2} we have
\begin{multline}
\label{eq:logSumSecondTerm}
\left|\sum_{\substack{ p \leq x \\p \equiv a \pmod m}} \left(\log n+\log p\right)\right|<\log n\left(\frac{\textrm{li}(x)}{\varphi(m)}+\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x \right. \\
\left.\vphantom{\frac{\textrm{li}(n)}{\varphi(m)}}+a_3(m)\sqrt{x}-237.934\right)+\frac{x}{\varphi(m)}+a_5(m)\sqrt{x}\log^2 x.
\end{multline}
Further, by \cite[Lemma 5.9]{BGOR2018} we have
\begin{equation}
\label{eq:li}
\textrm{li}(x)=\textrm{Li}(x)+\textrm{li}(2)<\frac{x}{\log x}+\frac{3x}{2\log^2 x}+\textrm{li}(2)
\end{equation}
for all $x \geq 1865$. Hence we have estimated the second sum in \eqref{eq:sumNThree}.
Let us now combine our estimates. Using Lemma \ref{lemma:estimateSumPNLog} and estimates \eqref{eq:logSumSecondTerm} and \eqref{eq:li} for the sums in \eqref{eq:sumNThree} we have
\begin{align*}
&\left|\log\left(\prod_{\substack{p \leq x \\ p \equiv a \pmod m}} \left|n!\right|_p\right)+\frac{n\log x}{\varphi(m)}\right|<a_2(m)n+\frac{x\log n}{\varphi(m)\log x}+\frac{x}{\varphi(m)}+\frac{3x\log n}{2\varphi(m)\log^2 x} \\
&\quad+\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x \cdot \log n+a_5(m)\sqrt{x}\log^2 x\\
&\quad +a_3(m)\sqrt{x}\log n+\left(\frac{\textrm{li}(2)}{\varphi(m)}-237.934\right)\log n.
\end{align*}
Since $x \leq n$, the function $x/\log^2 x$ is increasing for all $x$ under the consideration and we have $\textrm{li}(2)/\varphi(m)-237.934<0$ for all $m$, the right-hand side is
\begin{multline}
\label{eq:a3XLong}
\leq a_2(m)n+\frac{x\log n}{\varphi(m)\log x}+\frac{x}{\varphi(m)} \\
+\frac{3n}{2\varphi(m)\log n}+\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}+a_5(m)\right)\sqrt{n}\log^2 n+a_3(m)\sqrt{n}\log n.
\end{multline}
Next we would like to simplify the last line in estimate \eqref{eq:a3XLong}.
Keeping in mind definitions \eqref{eq:defa7} and \eqref{eq:defa6} the terms $a_3(m)$ and $a_5(m)$, we get different estimates for the last line in \eqref{eq:a3XLong}. Since $n \geq \max \{m^{\varphi(m)},1865\}$, for $m \in [3, 36]$ we get
\begin{multline*}
<\frac{n}{\log n}\left(\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}+0.113+\frac{0.184+27.414/\log 1865}{\varphi(m)}+\frac{19.795}{\log{1865}}\right)\frac{\log^3 1865}{\sqrt{1865}} \right. \\
\left. +\frac{3}{2\varphi(m)}\right) <\left(27.640+\frac{39.717}{\varphi(m)}\right)\frac{n}{\log n}.
\end{multline*}
Further, for all $m \geq 37$, the last line in \eqref{eq:a3XLong} is
\begin{multline*}
<\frac{n}{\log n}\left(\frac{3}{2\varphi(m)}+\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}+0.113+\frac{0.184+27.414/\log (37^{37^{\log 2/\log 3}})}{\varphi(m)} \right.\right. \\
\left.\left.+\frac{19.795}{\log (37^{37^{\log 2/\log 3}})}\right)\frac{\log^3 (37^{37^{\log 2/\log 3}})}{\sqrt{37^{37^{\log 2/\log 3}}}} \right) <\left(0.001+\frac{1.501}{\varphi(m)}\right)\frac{n}{\log n}.
\end{multline*}
\end{proof}
\begin{remark}
\label{remark:Log}
Since $|q|_p=1$ for all primes $p>q$, for all $x>n$ we have
\begin{equation*}
\log\left(\prod_{\substack{p \in \mathbb{P} \\ p \equiv a \pmod m}} \left|n!\right|_p\right)= \log\left(\prod_{\substack{p \leq x \\ p \equiv a \pmod m}} \left|n!\right|_p\right)=\log\left(\prod_{\substack{p \leq n \\ p \equiv a \pmod m}} \left|n!\right|_p\right).
\end{equation*}
\end{remark}
\section{Sets \eqref{eq:interval} and \eqref{eq:intervalEpsilon} are non-empty}
\label{sec:Existence}
In this section we prove that sets \eqref{eq:interval} and \eqref{eq:intervalEpsilon} are non-empty.
Let us start with set \eqref{eq:interval}:
\begin{lemma}
\label{lemma:exits1}
Let us use the same definitions and assumptions as in Theorem \ref{thm:lowerBound}. Then there is at least one prime in set \eqref{eq:interval}.
\end{lemma}
\begin{proof}
Since $k\geq 1$ and because of assumption \eqref{eq:BoundH}, it is sufficient to show that there is a prime in
\begin{equation*}
R \cap \left[2,2\max\left\{1866, m^{\varphi(m)}+1\right\}+2\right),
\end{equation*}
where $R$ is defined as in Theorem \ref{thm:dmGene}.
Let us denote $x:=2\max\left\{1866, m^{\varphi(m)}+1\right\}+2$. We divide the proof to different cases depending on the numbers $m$ and $x$.
Let us first consider the case $3 \leq m \leq 10^5$ and $x \leq 10^{11}$. Then by \cite[Theorem 1.9 and Lemma 5.8]{BGOR2018}
\begin{equation}
\label{eq:piRightInterval}
\pi(x; m, a)> \frac{\textrm{Li}(x)}{\varphi(m)}-2.734\frac{\sqrt{x}}{\log x}>\frac{x}{\varphi(m)\log x}-2.734\frac{\sqrt{x}}{\log x}.
\end{equation}
First, if $m\leq 6$, then the right-hand side is
\begin{equation*}
\frac{x}{4\log x}-2.734\frac{\sqrt{x}}{\log x}>1
\end{equation*}
for all $x \geq 3734$. Further, if $m \geq 7$, then
by Lemma \ref{lemma:varLower} and computing values $\varphi(m)$, where $m=10, 12, 18, 30$, we get $x \geq 2m^{\varphi(m)}+4\geq 2m^3+4>(3.734m)^2$. Hence, the right-hand side of inequality \eqref{eq:piRightInterval} is at least one meaning that there are at least one prime in the wanted set.
Let us now consider the case $3 \leq m \leq 10^5$ and $x>10^{11}$. By \cite[Corollary 1.7 and Lemma 5.8]{BGOR2018}
\begin{equation*}
\pi(x; m, a)> \frac{\textrm{Li}(x)}{\varphi(m)}-0.027\frac{\sqrt{x}}{\log^2 x}>\frac{\sqrt{x}}{\log^2 x}\left(\frac{\log x}{\varphi(m)}-0.027\right)>\frac{\sqrt{x}}{\log^2 x}(\log m-0.027)>1
\end{equation*}
for all $x \geq \max\{10^{11}, m^{\varphi(m)}\}$. Hence there are at least one prime in set \eqref{eq:interval} also in this case.
Let us now consider the case $m >10^5$. By Lemma \ref{lemma:varLower} we have
$$
x>m^{\varphi(m)}>m^{m^{\log 2/\log 3}}>m^{1400}.
$$
Further, by Lemma \ref{lemma:estPi}, \cite[Lemma 5.8]{BGOR2018} and since $\varphi(m)<m$, we have
\begin{align*}
&\pi(x; m, a)> \frac{\textrm{li}(x)}{\varphi(m)}-\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x-a_3(m)\sqrt{x} \\
& \quad>\frac{x}{\varphi(m)\log x }-\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}\right)\sqrt{x}\log x-\left(0.184\log m+8.406+\frac{6.063}{\varphi(m)}\right)\sqrt{x}\\
&\quad >\frac{\sqrt{x}\log x}{\varphi(m)}\left(\frac{\sqrt{x}}{10^{3492}\log^2 x}-\frac{1}{8\pi}\right)+\frac{\sqrt{x}}{\varphi(m)}\left(\frac{\sqrt{x}}{10^{3495}\log x}-6.063\right) \\
&\quad\quad+\sqrt{x}\log x\left(\frac{m^{699}}{10^{3487}\log^2(m^{1400})}-\frac{1}{6\pi}\right)+\sqrt{x}\log m\left(\frac{m^{699}}{10^{3490}\cdot1400(\log m)^2}-0.184\right) \\
&\quad\quad +\sqrt{x}\left(\frac{m^{699}}{10^{3489}\cdot1400\log m}-8.406\right).
\end{align*}
Since $x> (10^5)^{1400}$ and $m>10^5$, the right-hand side is greater than one and the claim is proved.
\end{proof}
Let us now prove that set \eqref{eq:intervalEpsilon} is non-empty:
\begin{lemma}
\label{lemma:exists2}
Let us use the same definitions and assumptions as in Theorem \ref{thm:lowerBoundMore}. Then there is at least one prime in set \eqref{eq:intervalEpsilon}.
\end{lemma}
\begin{proof}
Since $k \geq 1$ and $B_1(m) >1$ and because of assumptions \eqref{eq:LowerBoundsForS}, it is sufficient to show that there is a prime in
$$
R\cap\left(\log x,x\right),
$$
where $R$ is defined as in Theorem \ref{thm:lowerBoundMore} and
$$
x:=\frac{\max\left\{1866e^{1866},\left(m^{\varphi(m)}+1\right)e^{m^{\varphi(m)}+1}\right\}}{\log\left(\max\left\{1866e^{1866},\left(m^{\varphi(m)}+1\right)e^{m^{\varphi(m)}+1}\right\}\right)}.
$$
Again, we divide the proof to different cases depending on the sizes of $m$ and $x$.
Let us first consider the case $3 \leq m \leq 10^5$ and $\log x \leq 10^{11}$. Please notice that $x>10^{800}$. Hence, by \cite[Corollary 1.7, Theorem 1.9, Lemmas 5.8 and 5.9]{BGOR2018}, we have
\begin{equation}
\label{eq:B1PiSum}
\begin{split}
&\pi\left(x; m, a\right)-\pi\left(\log x; m, a\right) \\
&\quad > \frac{\textrm{Li}(x)}{\varphi(m)}-0.027\frac{x}{\log^2 x}-\frac{\textrm{Li}(\log x)}{\varphi(m)}-2.734\frac{\sqrt{\log x}}{\log\log x} \\
&\quad >\frac{x}{\varphi(m)\log x}-0.027\frac{x}{\log^2 x}-\frac{\log x}{\varphi(m)\log \log x}-\frac{3\log x}{2\varphi(m)(\log\log x)^2}-2.734\frac{\sqrt{\log x}}{\log\log x} \\
&\quad >\frac{\log x}{\varphi(m) \log\log x}\left(\frac{x\log\log x}{10^{804}\log^2 x}-1\right)+\frac{\log x}{\varphi(m)(\log\log x)^2}\left(\frac{x (\log\log x)^2}{10^{794}(\log x)^2}-\frac{3}{2}\right) \\
&\quad\quad +\frac{x}{\log^2 x}\left(\frac{0.014\log x}{\varphi(m)}-0.027\right)+\frac{\sqrt{\log x}}{\log\log x}\left(\frac{0.447x\log\log x}{\varphi(m)(\log x)^{1.5}}-2.734\right).
\end{split}
\end{equation}
Due to the size of $x$, the second last line in the right-hand side of the inequality is positive. Let us consider the last line of the right-hand side of inequality \eqref{eq:B1PiSum}
First, if $3\leq m \leq 6$, then the last line is
\begin{equation*}
>\frac{x}{\log^2 x}\left(\frac{0.014\log x}{4}-0.027\right)+\frac{\sqrt{\log x}}{\log\log x}\left(\frac{0.447x\log\log x}{4(\log x)^{1.5}}-2.734\right)>1
\end{equation*}
for all $x >10^{800}$.
Let us now assume that $7 \leq m \leq 10^5$. Similarly as in the second paragraph of the proof of Lemma \ref{lemma:exits1}, we can deduce that
$$
x>\frac{m^{\varphi(m)}e^{m^{\varphi(m)}}}{\log(m^{\varphi(m)}e^{m^{\varphi(m)}})}>m^{\varphi(m)} \geq m^3.
$$
Further, since we also have $\varphi(m)<m$, the last line of the right-hand side of inequality \eqref{eq:B1PiSum} is
\begin{equation*}
>\frac{x}{\log^2 x}\left(0.014\log m-0.027\right)+\frac{\sqrt{\log x}}{\log\log x}\left(\frac{0.447m^2\log\log m^2}{(\log m^2)^{1.5}}-2.734\right)>1
\end{equation*}
for $m \geq 7$ and $x>10^{800}$. Hence, we have proved that for $3 \leq m \leq 10^5$ and $\log x \leq 10^{11}$ there always is at least one prime in the wanted set.
Let us now consider the case $3 \leq m \leq 10^5$ and $\log x > 10^{11}$. By \cite[Corollary 1.7, Lemmas 5.8 and 5.9]{BGOR2018}, we have
\begin{align*}
&\pi\left(x; m, a\right)-\pi\left(\log x; m, a\right) \\
&\quad > \frac{\textrm{Li}(x)}{\varphi(m)}-0.027\frac{x}{\log^2 x}-\frac{\textrm{Li}(\log x)}{\varphi(m)}-0.027\frac{\log x}{(\log\log x)^2} \\
&\quad >\frac{x}{\varphi(m)\log x}-0.027\frac{x}{\log^2 x}-\frac{\log x}{\varphi(m)\log \log x}-\frac{3\log x}{2\varphi(m)(\log\log x)^2}-0.027\frac{\log x}{(\log\log x)^2}\\
&\quad >\frac{\log x}{\varphi(m) \log\log x}\left(\frac{x\log\log x}{10^{804}\log^2 x}-1\right)+\frac{x}{\log^2 x}\left(\frac{0.014\log x}{\varphi(m)}-0.027\right) \\
&\quad\quad +\frac{\log x}{(\log\log x)^2}\left(\frac{0.342x (\log\log x)^2}{\varphi(m) (\log x)^2}-1.527\right).
\end{align*}
Similarly as in the previous case, we can first deduce that there is at least one prime number when $m \leq 6$ and then that there also is such a number when $m \geq 7$.
Now we have the case $m>10^5$ left. Again, we clearly have $x>10^{800}$ and by Lemma \ref{lemma:varLower} we have $x>m^{\varphi(m)}>m^{1400}$.
By Lemma \ref{lemma:estPi}, \cite[Lemmas 5.8 and 5.9]{BGOR2018} and since $\varphi(m)<m$, we get
\begin{align*}
& \pi(x; m, a)-\pi(\log x; m, a)> \frac{\textrm{li}(x)}{\varphi(m)}-\frac{\textrm{li}(\log x)}{\varphi(m)}\\
&\quad -\left(\frac{1}{8\pi\varphi(m)}+\frac{1}{6\pi}\right)\left(\sqrt{x}\log x+\sqrt{\log x}\log\log x\right)-a_3(m)(\sqrt{x}+\sqrt{\log x}) \\
&\quad >\frac{\log x}{\varphi(m) \log\log x}\left(\frac{x\log\log x}{10^{804}\log^2 x}-1\right)+\frac{\log x}{\varphi(m)(\log\log x)^2}\left(\frac{x (\log\log x)^2}{10^{794}(\log x)^2}-\frac{3}{2}\right) \\
&\quad\quad +\frac{\sqrt{x}\log x}{\varphi(m)}\left(\frac{\sqrt{x}}{10^{394}\log^2 x}-\frac{1}{8\pi}\right)+\frac{\sqrt{\log x}\log\log x}{\varphi(m)}\left(\frac{x}{10^{795}(\log x)^{1.5}\log\log x}-\frac{1}{8\pi}\right) \\
&\quad\quad +\frac{\sqrt{x}}{\varphi(m)}\left(\frac{\sqrt{x}}{10^{395}\log x}-6.733\right)+\frac{\sqrt{\log x}}{\varphi(m)}\left(\frac{x}{10^{794}(\log x)^{1.5}}-6.733\right) \\
&\quad\quad +\sqrt{x}\log x\left(\frac{m^{699}}{10^{3487}(\log(m^{1400}))^2}-\frac{1}{6\pi}\right)\\
&\quad\quad +\sqrt{\log x}\log\log x\left(\frac{m^{1399}}{10^{6988}(\log m^{1400})^{1.5}\log\log m^{1400}}-\frac{1}{6\pi}\right) \\
&\quad\quad +\sqrt{x}\log m\left(\frac{m^{699}}{10^{3490}\cdot1400(\log m)^2}-0.184\right) \\
&\quad\quad +\sqrt{\log x}\log m\left(\frac{m^{1399}}{10^{6988}\log m (\log m^{1400})^{1.5}}-0.184\right) \\
&\quad\quad +\sqrt{x}\left(\frac{m^{699}}{10^{3489}\cdot1400\log m}-8.987\right)+\sqrt{\log x}\left(\frac{m^{1399}}{10^{6987}(\log m^{1400})^{1.5}}-8.987\right)>1.
\end{align*}
Hence, there again exists at least one prime in set \eqref{eq:intervalEpsilon}.
\end{proof}
\section{Proof of Theorem \ref{thm:limSupCond}}
\label{sec:proof1}
In this section, we prove Theorem \ref{thm:limSupCond}.
\begin{proof}[Proof of Theorem \ref{thm:limSupCond}]
Let us assume contrary, indeed that for all $p \in R$ we have $\Lambda_p =0$. As in Section \ref{sec:outline}, we obtain
\begin{equation*}
1 \leq (k+1)\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\} \max_{0 \leq i \leq k} \{\left|B_{n,\mu,i}(1)\right|\}\cdot \max_{1 \leq i \leq k} \left\{ \prod_{p \in R}\left|S_{n,\mu,i}(1)\right|_p\right\}.
\end{equation*}
By Lemma \ref{lemma:BSEstimates} the right-hand side is
\begin{align*}
&\leq (k+1)\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\}(kn+\mu)!\cdot(kn+k) \left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)^{k-1} \\
&\quad\cdot\left(\prod_{i=1}^k \left(\left|\alpha_i\right|+\max_{1\leq j \leq k}\{|\alpha_j|\}\right)^n\right)\prod_{p \in R}\left(\left|(kn+\mu)!n!\right|_p \left(\max_{1 \leq i \leq k}\{\left|\alpha_i\right|_p\}\right)^{(k+1)n}\right) \\
&\quad \leq (k+1)\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\}\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)^{k-1}c^n(kn+k)!\cdot(kn+k)\prod_{p \in R}\left|(kn)!n!\right|_p,
\end{align*}
where $c$ is defined as in \eqref{def:c1}. Thus, if condition \eqref{eq:limSupCond} holds, then the right-hand side of the previous inequality is smaller than one when $n$ is large enough. This is a contradiction and hence we cannot have $\Lambda_p =0$ for all $p \in R$.
\end{proof}
\section{Proof of Theorem \ref{thm:dmGene}}
\label{sec:proof2}
In this section we prove Theorem \ref{thm:dmGene}.
\begin{proof}
[Proof of Theorem \ref{thm:dmGene}]
First of all, by Theorem \ref{thm:limSupCond} it is sufficient to show that the limit superior of the formula
\begin{equation}
\label{eq:toMinusInf}
n\log c(\overline{\alpha}; R)+\log(kn+k)+\log((kn+k)!)+\log\left(\prod_{p \in R} \left|(nk)! \cdot n!\right|_p\right)
\end{equation}
is minus infinity. Hence, we consider the sum in \eqref{eq:toMinusInf}.
First we consider the second and third term in sum \eqref{eq:toMinusInf}. The second term can be estimated with
\begin{equation}
\label{eq:Logknk}
\log(kn+k)=\log(n+1)+\log k< \log n+\log k+\frac{1}{n},
\end{equation}
for all $n >0$. By the previous estimate and by Stirling's formula (see e.g. \cite[formula 6.1.38]{AS1964}) we have
\begin{align*}
\log((kn+k)!)&=\left(kn+k\right)\log(kn+k)-(kn+k)+O(\log n) \\
&=kn\log n+kn\left(\log k-1\right)+O(\log n).
\end{align*}
Hence, we have estimated the second and the third term in sum \eqref{eq:toMinusInf}.
Using the estimates obtained in the previous paragraph, Lemma \ref{lemma:Log} and Remark \ref{remark:Log}, sum \eqref{eq:toMinusInf} is
\begin{align*}
&= n\log c(\overline{\alpha}; R)+kn\log n+kn\left(\log k-1\right)+O(\log n)\\
&\quad+\sum_{j=1}^{k\varphi(m)/(k+1)}\left(\log\left(\prod_{p \in \overline{h}_j \cap \mathbb{P}} \left|(nk)!\right|_p\right)+\log\left(\prod_{p \in \overline{h}_j \cap \mathbb{P}} \left|n!\right|_p\right)\right) \\
&\quad< kn\log n+n\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right) \\
&\quad\quad-\frac{k^2 n\log n}{k+1}-\frac{k n\log n}{k+1}+O\left(\frac{n}{\log n}\right) \\
&\quad=n\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)+O\left(\frac{n}{\log n}\right).
\end{align*}
Using upper bound \eqref{eq:cUpper} for the number $c(\overline{\alpha}; R)$, the right-hand side of the previous inequality goes to $-\infty$ when $n$ goes to infinity. By Theorem \ref{thm:limSupCond} the claim follows.
\end{proof}
\begin{remark}
It is essential to use number $c$ in \eqref{def:c1} instead of number $c_2$ in Theorem \cite[Theorem 3.1]{S2020} since the later one is always at least one and the upper bound obtained in Theorem \ref{thm:dmGene} is smaller than one.
\end{remark}
\section{Preliminaries and proofs for Theorem \ref{thm:lowerBound} and Corollary \ref{corollary:lowerBound}}
\label{sec:FirstLower}
In this section, we prove Theorem \ref{thm:lowerBound} and Corollary \ref{corollary:lowerBound}. The section is divided to two parts: First we derive the contradiction that the right-hand side of inequality \eqref{eq:Product} is smaller than one (see Section \ref{sec:outline}) as well as estimates for the number $n$ which gives the contradiction. After that, we are ready to prove Theorem \ref{thm:lowerBound} and Corollary \ref{corollary:lowerBound}.
\subsection{Contradiction and estimates for number $n$}
\label{sec:Contra}
In this section, we show that the right-hand side of inequality \eqref{eq:Product} is smaller than one. Further, we also estimate the number $n$ which gives the mentioned contradiction.
First, we define
\begin{align*}
N_1(a)&:=a\left(\log c(\overline{\alpha}; R')+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)+\frac{a_6(m)k\varphi(m)a}{\log a} \\
&\quad+2.5\log a+\log(k+1)+2.5\log k+\log H+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)\\
&\quad+\log\sqrt{2\pi}+1+k\left(a_2(m)\varphi(m)+2\right)+\frac{a_6(m)k\varphi(m)}{\log a}+\frac{19}{12a}.
\end{align*}
Now we prove the contradiction keeping in mind that the term $c(\overline{\alpha}; R)$ is defined in \eqref{def:c1} and the term $a_2(m)$ in Table \ref{table:a1a2}:
\begin{lemma}
\label{lemma:contra}
Let $k,m,\lambda_0,\ldots, \lambda_k,$ be defined as in Theorem \ref{thm:dmGene}. Further, let $\alpha_1, \ldots, \alpha_k$ and $R'=R\cap[2, k(n+2)]$ be defined as in Theorem \ref{thm:lowerBound}. Assume GRH and that $c(\overline{\alpha}; R')$ satisfies bound \eqref{eq:cUpper}. Further, let us define
\begin{equation}
\label{eq:nBound}
n:=\max\left\{ a \in \mathbb{Z}: N_1(a) \geq 0\right\}
\end{equation}
and assume that
\begin{equation}
\label{eq:HBound}
\log H \geq -\left(\log c(\overline{\alpha}; R')+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)\cdot\max \left\{1866, m^{\varphi(m)}+1 \right\}.
\end{equation}
Then, we have
\begin{equation*}
(k+1)\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\} \max_{0 \leq i \leq k} \{\left|B_{n+1,\mu,i}(1)\right|\}\cdot \max_{1 \leq i \leq k}\left\{ \prod_{p \in R\cap [2,k(n+2)]}\left|S_{n+1,\mu,i}(1)\right|_p\right\}<1.
\end{equation*}
\end{lemma}
\begin{proof}
We would like to estimate the left-hand side of the claim. Hence, we first notice that due to assumptions \eqref{eq:nBound} and \eqref{eq:HBound} we must have
\begin{equation*}
n> \frac{-\log H}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}-1\geq \max\{1865, m^{\varphi(m)}\}.
\end{equation*}
Thus we can apply Lemma \ref{lemma:Log} in our estimates.
Now we estimate the left-hand side of the claim. First, we notice that by Lemma \ref{lemma:exits1} there is at least one prime in the set $R'=R\cap[2, k(n+2)]$. As in the proof of Theorem \ref{thm:limSupCond}, using Lemma \ref{lemma:BSEstimates} and keeping in mind $\mu \in \{0,1,\ldots, k\}$, we have
\begin{align*}
& (k+1)\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\} \max_{0 \leq i \leq k} \{\left|B_{n+1,\mu,i}(1)\right|\}\cdot \max_{1 \leq i \leq k} \left\{ \prod_{p \in R\cap[2, k(n+2)]}\left|S_{n+1,\mu,i}(1)\right|_p\right\} \\
&\quad \leq (k+1)H\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)^{k-1}c^{n+1}\left(k(n+1)+\mu\right)!\cdot\left(k(n+1)+k\right) \\
&\quad\quad \left(\prod_{p \in R\cap [2,k(n+2)]}\left|(k(n+1)+\mu)!\right|_p\right)\cdot \left(\prod_{p \in R\cap [2,k(n+2)]}\left|(n+1)!\right|_p\right).
\end{align*}
Similarly as in the proof of Theorem \ref{thm:dmGene} and especially in estimate \eqref{eq:Logknk} and by Stirling's formula (see e.g. \cite[formula 6.1.38]{AS1964}), the logarithm of the right-hand side can be estimated as
\begin{equation*}
\begin{split}
<\log(k+1)+\log H+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)+(n+1)\log c+\log k+\log(n+1) \\
+\frac{1}{n+1}+ \left(k(n+1)+\mu+0.5\right)\log(k(n+1)+\mu)-k(n+1)-\mu\\+\log\sqrt{2\pi}+\frac{1}{12(k(n+1)+\mu)}\\
+\log\left(\prod_{p \in R\cap[2,k(n+2)]}\left|(k(n+1)+\mu)!\right|_p\right)+\log\left(\prod_{p \in R\cap[2,k(n+2)]} \left|(n+1)!\right|_p\right).
\end{split}
\end{equation*}
Our goal is to show that the previous estimate is at most zero. Since $R\cap[2,k(n+2)]$ contains primes in arithmetic progressions up to height $k(n+2)$ and $0 \leq \mu \leq k$, by Lemma \ref{lemma:Log} and Remark \ref{remark:Log} the right-hand side is
\begin{align*}
&<\log(k+1)+\log H+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)+(n+1)\log c+\log k+\log (n+1) \\
&\quad+\frac{13}{12(n+1)}+\left(k(n+1)+\mu+0.5\right)\log(k(n+1)+\mu)-k(n+1)-\mu+\log\sqrt{2\pi}\\
&\quad+\frac{k\varphi(m)}{k+1}\left( -\frac{(k(n+1)+\mu)\log (k(n+1)+\mu)}{\varphi(m)}+\left(a_2(m)+\frac{2}{\varphi(m)}\right)(k(n+1)+\mu)\right.\\
&\quad\left.+a_6(m)\frac{k(n+1)+\mu}{\log (k(n+1)+\mu)}-\frac{(n+1)\log( n+1)}{\varphi(m)}+\left(a_2(m)+\frac{2}{\varphi(m)}\right)(n+1)\right.\\
&\quad\left.+\frac{a_6(m)(n+1)}{\log (n+1)}\right) \\
&\quad < (n+1)\left(\log c(\overline{\alpha}; R')+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)+\frac{a_6(m)k\varphi(m)(n+1)}{\log(n+1)} \\
&\quad\quad+2.5\log(n+1)+\log(k+1)+2.5\log k+\log H+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)\\
&\quad\quad+\log\sqrt{2\pi}+1+k\left(a_2(m)\varphi(m)+2\right)+\frac{a_6(m)k\varphi(m)}{\log (n+1)}+\frac{19}{12(n+1)} \\
&\quad=N_1(n+1).
\end{align*}
Because of assumption \eqref{eq:nBound}, the right-hand side is smaller than zero.
\end{proof}
\begin{remark}
The assumption \eqref{eq:cUpper} is essential in order to find number $n$ such that $N_1(n+1)<0$.
\end{remark}
In the previous lemma, number $n$ is given quite a quite complicated way. In order to apply the result, we would like to give a simplified upper bound for number $n$. We have included more assumptions for number $H$ in this simplified version to obtain a simpler result. In the result, please keep in mind that $A_1(m,k)$ is given in \eqref{def:A1mk}.
\begin{lemma}
\label{lemma:UpperBoundsN}
Let us use the same definitions and assumptions as in Lemma \ref{lemma:contra} including that $n$ is defined as in \eqref{eq:nBound}.
Assume also that $H$ satisfies the bounds given in \eqref{eq:BoundH}.
Then we have
\begin{equation*}
n< -\frac{A_1(m,k)n/\log n+\log H}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}
\end{equation*}
and
\begin{equation*}
n< -\frac{2\log H}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}.
\end{equation*}
\end{lemma}
\begin{proof}
The idea is to prove bounds for number $n$ using the term $N_1(n)$: By assumption \eqref{eq:nBound} we have $0 \leq N_1(n)$. We derive an upper found of form
\begin{equation}
\label{eq:Amk2}
\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right) n+A_1(m,k)\frac{n}{\log n}+\log H,
\end{equation}
for the term $N_1(n)$. Using this upper bound and keeping the inequality $0 \leq N_1(n)$ in mind, we find the wanted upper bounds.
Let us now derive the wanted upper bound for the term $N_1(n)$. As in the first paragraph of the proof of Lemma \ref{lemma:contra}, we can deduce that $n>m^{\varphi(m)}$ and $n>s \geq \max_{1\leq j \leq k}\{|\alpha_j|\}$ and we also have $\log k<\log (k+1)<k$ for all $k \geq 1$. Hence, the term $N_1(n)$ can be written as
\begin{align*}
N_1(n)&=n\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)+\frac{a_6(m)k\varphi(m)n}{\log n} \\
&\quad+2.5\log n+\log(k+1)+2.5\log k+\log H+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)\\
&\quad+\log\sqrt{2\pi}+1+k\left(a_2(m)\varphi(m)+2\right)+\frac{a_6(m)k\varphi(m)}{\log n}+\frac{19}{12n} \\
&<n\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)+\frac{n}{\log n}\left(\vphantom{\frac{2.5\log^2 n}{n}} a_6(m)k\varphi(m)\right. \\
&\quad\left. +\frac{(k+1.5)\log^2 n}{n}+\frac{ka_2(m)\log^2 n}{n\log m}+\frac{\left(5.5k+\log\sqrt{2\pi}+1\right)\log n}{n}\right. \\
&\quad\left. +\frac{ka_6(m)\log n}{n\log m} +\frac{19\log n}{12n^2}\right)+\log H.
\end{align*}
Next we derive the coefficient for $n/\log n$ using different cases depending on the size of $m$.
Let us start with the case $m \in [3, 36]$. We can set $n=1865$, $\log m=\log 3$ and $\varphi(m)=2$ when $\varphi(m)$ is in the denominator. This leads to an estimate
\begin{multline*}
N_1(n)<n\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right) \\
+\left(27.640k\varphi(m)+40.160k+0.054\right)\frac{n}{\log n}+\log H.
\end{multline*}
Let us now move to the case $m \in [37,432]$. We divide this to two sub-cases: $m \leq 43$ and $m \geq 44$. In the first case in $a_2(m)$ we have $\log \sqrt{1865}$ and in the second case we have $\log m$. In the case $m \leq 43$, we can set $\log m=\log 37$ and by Lemma \ref{lemma:varLower} $n=37^{37^{\log 2/\log 3}}$ and $\varphi(m)=37^{\log 2/\log 3}$ when $\varphi(m)$ is in the denominator. This leads to an estimate
\begin{multline*}
N_1(n)<n\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right) \\
+\left(0.001\varphi(m)+1.502\right)\frac{kn}{\log n}+\log H.
\end{multline*}
Setting $\log m=\log 44$, $n=44^{44^{\log 2/\log 3}}$ and $\varphi(m)=44^{\log 2/\log 3}$ when $\varphi(m)$ is in the denominator in the second case, we get the same estimate as above.
Further, we get the same upper bound also in the case $m \in [433,10^5]$. Since the functions $a_2(m)$ and $a_6(m)$ are decreasing for all $m \geq 433$, this estimate applies also for the cases $m>10^5$. Hence, we have proved that $0 \leq N_1(n)$ is smaller than the formula in \eqref{eq:Amk2} and the first wanted upper bound follows.
Let us now move the second bound. We have obtained
\begin{equation}
\label{eq:A1Positive}
-\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)+\frac{A_1(m,k)}{\log n}\right) n<\log H.
\end{equation}
Due to the assumption \eqref{eq:BoundH}, the left-hand side is positive. Dividing the inequality by the coefficient of $n$ and setting
$$
\log n=-\frac{2A_1(m,k)}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}
$$
we get the second result.
\end{proof}
\begin{remark}
In the case $m \geq 37$, there is no term which does not depend on the number $k$ in $A_1(m,k)$. The reason for this is that it is included to the term $k$ using three decimal accuracy.
\end{remark}
\begin{remark}
It is actually sufficient to assume that
$$
\frac{\log H}{-\left(\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)} \geq e^{-\frac{A_1(m,k)}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}}+1
$$
in order to have the left-hand side of \eqref{eq:A1Positive} to be positive. However, an exact lower bound would lead the left-hand side to be arbitrary close to zero. To avoid this problem and to make computations easier, a larger lower bound was selected.
\end{remark}
\subsection{Proofs for Theorem \ref{thm:lowerBound} and Corollary \ref{corollary:lowerBound}}
\label{sec:proof3}
In this section, we prove $p$-adic lower bounds for linear forms in the Euler's factorial series.
First, we prove a useful lemma regarding to the lower bound for the term $\log H$. In the proof, $W(x)$ denotes the Lambert $W$ function meaning that $W(x)e^{W(x)}=x$ and $W(x)$ corresponds to the principal value.
\begin{lemma}
\label{lemma:xex}
Assume that $\log H \geq xe^{y/x}+x$, where $H$, $x$ and $y \geq 1$ are positive real numbers. Then
$$
\frac{y}{2\log\log H}\leq x \leq \frac{\log H}{2}.
$$
\end{lemma}
\begin{proof}
Let us prove the claim by assuming the contrary: $x < y/(2\log\log H)$ or $x > \log H/2$.
Let us first consider the first case. First we notice that the function $xe^{y/x}+x$ obtains its minimum at $x=y/\left(W(1/e)+1\right)$, where $W$ is the Lambert $W$ function. Hence, we have
\begin{equation}
\label{eq:HLowerA1}
\log H \geq xe^{y/x}+x\geq \frac{y}{W\left(1/e\right)+1}\left(e^{W(1/e)+1}+1\right)>3.591y.
\end{equation}
Thus, we also have
$$
\frac{y}{2\log\log H}<\frac{y}{2\log(3.591y)}<y/\left(W(1/e)+1\right) \approx 0.782y,
$$
Also, since the function $xe^{y/x}+x$ is decreasing for all $x<y/\left(W(1/e)+1\right)$, we have
$$
xe^{y/x}+x > \frac{ye^{2\log\log H}}{2\log\log H}+\frac{y}{2\log\log H}> \frac{\left(\log H\right)^{2}}{\log\log H}>\log H,
$$
for all $H>e$. This is a contradiction.
In the second case we have
$$
xe^{y/x}+x>\frac{\log H\cdot e^0}{2}+\frac{\log H}{2}=\log H,
$$
which is again a contradiction. Hence, the claim holds.
\end{proof}
\begin{remark}
It is possible to prove a little bit sharper bounds in Lemma \ref{lemma:xex} formulating the bounds with the Lambert $W$ function instead of logarithms. However, in order to keep the results relatively simple, logarithms where decided to be used.
\end{remark}
Now we are ready to move Theorem \ref{thm:lowerBound}:
\begin{proof}[Proof of Theorem \ref{thm:lowerBound}]
First of all, by Lemma \ref{lemma:exits1}, there is at least one prime number in the set $R\cap [2,k(n+2)]$. Further, because of Lemma \ref{lemma:contra}, there exists a number $n$ such that the right-hand side of estimate \eqref{eq:Product} is smaller than one. On the other hand, in estimate \eqref{eq:Product} we noticed that the same term is at least one if there is no number $p \in R\cap [2,k(n+2)]$ such that
\begin{equation}
\label{eq:BLarge}
\left|B_{n+1,\mu,0}(1)\Lambda_p\right|_p > \left|\sum_{j=1}^k \lambda_jS_{n+1,\mu,j}(1)\right|_p.
\end{equation}
This is a contradiction and hence there must be a prime number $p \in R\cap [2,k(n+2)]$ such that inequality \eqref{eq:BLarge} holds.
We consider this prime number $p$. First we prove that $p$ is in the set \eqref{eq:interval} and then that the wanted lower bound is satisfied.
Trivially we must have $p \geq 2$. Further, by Lemma \ref{lemma:UpperBoundsN}, we have
\begin{equation*}
k(n+2)<-\frac{2k\log H}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}+2k.
\end{equation*}
Hence, number $p$ is in the wanted set and we can move on to find a $p$-adic lower bound for the linear form.
We only have to find a lower bound for the term $\left|\Lambda_p\right|_p$. By estimate \eqref{eq:TnEnough} it is sufficient to find an upper bound for the term $\left|T(n+1, \mu)\right|$ so we will consider this term.
By definition \eqref{eq:defT} of the term $T(n+1, \mu)$, Lemma \ref{lemma:BSEstimates} and Stirling's formula (see e.g. \cite[formula 6.1.38]{AS1964}) we have
\begin{multline*}
\left|T(n+1, \mu)\right| \leq (k+1) H \max_{1 \leq j \leq k} \left|B_{n+1,\mu, j}(1)\right| \\
\leq (k+1)H\sqrt{2\pi} \cdot e^{\frac{1}{12k(n+1)}-k}\left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)^{k-1}\cdot \left(k(n+2)\right)^{k+1.5} \\
\cdot e^{k(n+1)\left(\log 2+\log\left(k(n+2)\right)-1+\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)\right)}.
\end{multline*}
Applying the fact $\log(a+1)<\log a+1/a$, the logarithm of the right-hand side is
\begin{multline*}
\log H+(k+2.5)\log k+\frac{1}{k}+\log\sqrt{2\pi}+\frac{1}{12k(n+1)}-k\\
+(k-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)\left(1+k(n+1)\right)+(k+1.5)\left(\log n+\frac{2}{n}\right)\\
+k(n+1)\left(\log 2+\log k+\log n+\frac{2}{n}-1\right).
\end{multline*}
Using the facts $\log k<k$ and $k \geq 1$ and assumption \eqref{eq:BoundH} meaning that $n> \max_{1\leq j \leq k}\{|\alpha_j|\}$, the previous one can be estimated as
\begin{multline}
\label{eq:almostUpper}
<\log H+kn\log n+n\left(k(k-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)+k\left(k+\log 2-1\right)\right) \\
+\log n\left(k^2+2k+0.5+\frac{(2k+2.5) k+\log\sqrt{2\pi}+1+k\log 2}{\log n}+\frac{4k+37/12}{n\log n}\right).
\end{multline}
By Lemma \ref{lemma:UpperBoundsN}, we can estimate
\begin{equation}
\label{eq:estnLogn}
n\log n< -\frac{A_1(m,k)n+\log H\cdot \log n}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}
\end{equation}
and
\begin{equation*}
n< -\frac{2\log H}{\log c(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)}
\end{equation*}
and hence estimate \eqref{eq:almostUpper} is
\begin{equation}
\label{eq:DSum}
\begin{split}
&<\log H+k\log H\cdot \frac{\log\log H+\log 2-\log D}{D}+\frac{kA_1(m,k)\log H}{D^2}+\frac{kA_1(m,k)^2n/\log n}{D^2} \\
&\quad+\left(\frac{\log H}{D}+\frac{A_1(m,k)n/\log n}{D}\right)\left(k(k-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)+k\left(k+\log 2-1\right)\right) \\
&\quad+\left(\log\log H+\log (2/D)\right)\left(k^2+2k+0.5+\frac{(2k+2.5) k+\log\sqrt{2\pi}+1+k\log 2}{\log n}\right. \\
&\quad\left.+\frac{4k+37/12}{n\log n}\right) \\
&<\quad \log H\cdot\left(\frac{k}{D}\log\log H+1+\frac{k\left(\log 2-\log D\right)}{D}+\frac{kA_1(m,k)}{D^2}\right. \\
&\quad\quad\left.\frac{k(k-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)+k\left(k+\log 2-1\right)}{D}+\frac{2kA_1(m,k)^2}{D^3\left(\log\log H+\log (2/D)\right)} \right. \\
&\quad\quad\left. +\frac{2A_1(m,k)\left(k(k-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)+k\left(k+\log 2-1\right)\right)}{D^2\left(\log\log H+\log (2/D)\right)}\right. \\
&\quad\quad\left. +\frac{\log\log H+\log(2/D)}{\log H}\cdot\left(\frac{(2k+2.5) k+\log\sqrt{2\pi}+1+k\log 2}{\log n}\right.\right. \\
&\quad\left.\left.+k^2+2k+0.5+\frac{4k+37/12}{n\log n}\right)\right),
\end{split}
\end{equation}
where $D:=D(m,k,\overline{\alpha}; R')=-\left(\log c(\overline{\alpha}; R')+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)$.
Next we simplify the previous estimate. Our goal is to write it in a form
\begin{equation}
\label{eq:ABH}
A\log H\cdot\log\log H+B,
\end{equation}
where $A$ and $B$ do not depend on number $H$ but may depend on other terms such as number $m$.
Since by assumption \eqref{eq:BoundH} we have
$$
\log H\geq De^{2A_1(m,k)/D}+D
$$
and
$$\log\log H-\log D>\max\left\{1,\log\left(\max_{i\leq j\leq k}\left\{|\alpha_j|\right\}\right)\right\},
$$
by Lemma \ref{lemma:xex} the terms inside the brackets in the right-hand side of inequality \eqref{eq:DSum} are
\begin{align*}
&<\frac{k^2}{D}\log\log H+1-\frac{k^2\log D}{D}+\frac{2k^2+3k\log 2-2k}{D}+\frac{2k^2A_1(m,k)}{D^2} \\
&\quad +\frac{\log\log H+\log\log\log H+\log 2}{\log H}\cdot\left(\frac{(2k+2.5) k+\log\sqrt{2\pi}+1+k\log 2}{\log n}\right. \\
&\quad\left.+k^2+2k+0.5+\frac{4k+37/12}{n\log n}\right).
\end{align*}
The claim follows when we do the following substitutions to the last two lines of the previous estimate: In every case, by inequality \eqref{eq:HLowerA1}, we can estimate $\log H \geq 7.182A_1(m,k)>7.182A_1(m,1)$. Further,
in the case $m \in [3,36]$, we set $n=1865$ and $\varphi(m)=2$. In the case $m \geq 37$, by Lemma \ref{lemma:varLower} we set $n=37^{37^{\log 2/\log 3}}$ and $\varphi(m)=37^{\log 2/\log 3}$.
These lead to the wanted estimates.
\end{proof}
\begin{remark}
At the end of the fourth paragraph of the previous proof, we want to minimize the coefficients of the terms having terms $\log H$ multiplied by something which does not go to zero when $H$ or $n$ goes to infinity. Even more, we also want to keep the results relatively simple. Hence, in some of the cases we have used estimate \eqref{eq:estnLogn} once, twice or not at all depending on would it improve the coefficients or not.
\end{remark}
As a last part of this section, let us prove Corollary \ref{corollary:lowerBound}:
\begin{proof}[Proof of Corollary \ref{corollary:lowerBound}]
As in the previous proof, it is sufficient to estimate the term $\log |T(n+1,\mu)|$.
By the proof of Theorem \ref{thm:lowerBound}, $\log |T(n+1,\mu)|$ is smaller than $\log H$ multiplied by
\begin{equation}
\label{eq:DMAx}
<\frac{k^2}{D}\log\log H-\frac{k^2\log D}{D}+\frac{2k^2+3k\log 2-2k}{D}+\frac{2k^2A_1(m,k)}{D^2}+A_2(m,k).
\end{equation}
Further, we notice that $-\log D/D\leq 0$ if $D\geq 1$ and the function $-\log D/D$ is decreasing when $D<1$. By assumption \eqref{eq:BoundH} and Lemma \ref{lemma:xex}, inequality \eqref{eq:DMAx} is
\begin{multline*}
<(\log\log H)^2\left(\frac{3k^2}{A_1(m,k)}+\frac{k^2 \log\log\log H}{A_1(m,k)\log\log H}\right. \\
\left.+\frac{2k^2+3k\log 2-2k}{A_1(m,k)\log\log H}+\frac{A_2(m,k)}{(\log\log H)^2}\right)
\end{multline*}
Again, by \eqref{eq:HLowerA1} we can deduce that $\log H \geq 7.182A_1(m,1)$. Further, the term $A_1(m,k)$ in the denominators is at least $(27.640\cdot2 + 40.160)k$ if $m \in [3,36]$ and at least $(0.001\cdot37^{\log2/\log3} + 1.502)k$ if $m \geq 37$. Hence, the claim is proved.
\end{proof}
\section{Preliminaries and proof of Theorem \ref{thm:lowerBoundMore}}
In this section, we prove Theorem \ref{thm:lowerBoundMore}, indeed that if we consider $(k+\varepsilon)\varphi(m)/(k+1)$ different residue classes, then we find a $p$-adic lower bound for a certain linear form. The proof follows similarly as in Section \ref{sec:FirstLower}: First we prove a contradiction that inequality \eqref{eq:Product} must be smaller than one under certain assumptions, then estimate such number $n$ and finally, prove Theorem \ref{thm:lowerBoundMore}.
\subsection{Contradiction and estimates for number $n$}
\label{sec:Contra2}
Again, we would like to show a contradiction to estimate \eqref{eq:Product} and to prove simpler estimates for number $n$ appearing in Lemma \ref{lemma:contra2}.
First, let us define
\begin{align*}
N_2(a)&:=-\varepsilon a\log a+(k+1)a\log\log a+a\left(\log c-k+\log k+2(k+1)(a_2(m)\varphi(m)+1)\right)\\
&\quad+\frac{2a_6(m)\varphi(m)a(k+1)}{\log a} +\frac{2\log^2 a}{\log\log a}+(k+3.5)\log a+\frac{\log k\cdot\log a}{\log\log a}+k\log\log a \\
&\quad+\log(k+1)+2.5\log k+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)+2\left(a_2(m)\varphi(m)+1\right)k \\
&\quad+\log\sqrt{2\pi}+1+\log H+\frac{2a_6(m)\varphi(m)k}{\log a}+\frac{\log a}{a\log\log a}+\frac{29}{12a}.
\end{align*}
Then we can prove the contradiction:
\begin{lemma}
\label{lemma:contra2}
Assume that $m, k, \alpha_1,\ldots, \alpha_k$ and $\lambda_0,\ldots, \lambda_k$ satisfy the same hypothesis as in Theorem \ref{thm:dmGene} and let $R$ be defined as in Theorem \ref{thm:lowerBoundMore}. Further, let also $\varepsilon \in (0,1)$ and $H$ be real numbers such that $H \geq \max_{0 \leq i \leq k}\{|\lambda_i|\}$ and
$$
\frac{\log H}{\varepsilon} \geq se^s,
$$
where $s \geq \max\{m+1,1866\}$. Even more, assume GRH and define
\begin{equation}
\label{eq:nBoundEpsilon}
n:=\max\left\{ a \in \mathbb{Z}: N_2(a) \geq 0\right\} .
\end{equation}
Then, we have
\begin{equation*}
(k+1)\max_{0 \leq i\leq k} \{\left|\lambda_i\right|\} \max_{0 \leq i \leq k} \{\left|B_{n+1,\mu,i}(1)\right|\}\cdot \max_{1 \leq i \leq k}\left\{ \prod_{p \in R\cap (\log(n+1),k(n+2)]}\left|S_{n+1,\mu,i}(1)\right|_p\right\}<1.
\end{equation*}
\end{lemma}
\begin{proof}
We approach the proof similarly as in the proof of Lemma \ref{lemma:contra}. First, by Lemma \ref{lemma:exists2} there is at least one prime in $R\cap (\log(n+1),k(n+2)]$. Further, we keep in mind that the primes under the consideration are between $\log (n+1)$ and $k(n+2)$. Even more, we also notice that from definition \eqref{eq:nBoundEpsilon} it follows that $\varepsilon (n+1) \log (n+1) > \log H$ and due to assumption \eqref{eq:HassumptionEpsilon} this means that $\log (n+1)> s > \max\{m,10^{29}\}$. Hence, we can apply Lemma \ref{lemma:Log} and Remark \ref{remark:Log} and deduce that the logarithm of the left-hand side of the claim is
\begin{align*}
&<\log(k+1)+\log H+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)+(n+1)\log c+\log k+\log (n+1) \\
&\quad+\frac{1}{n+1}+\left(k(n+1)+\mu+0.5\right)\log(k(n+1)+\mu)-k(n+1)-\mu+\log\sqrt{2\pi}\\
&\quad+\frac{1}{12(k(n+1)+\mu)}+\frac{(k+\varepsilon)\varphi(m)}{k+1}\left( -\frac{(k(n+1)+\mu)\log (k(n+1)+\mu)}{\varphi(m)} \right. \\
&\quad \left.+\left(2a_2(m)+\frac{2}{\varphi(m)}\right)(k(n+1)+\mu)+\frac{2a_6(m)(k(n+1)+\mu)}{\log(k(n+1)+\mu)} \right. \\
&\quad \left.+\frac{(k(n+1)+\mu)\log\log (n+1)}{\varphi(m)}+\frac{\log (n+1)\cdot \log(k(n+1)+\mu)}{\varphi(m)\log\log (n+1)}+\frac{2\log (n+1)}{\varphi(m)} \right. \\
&\quad \left.-\frac{(n+1)\log (n+1)}{\varphi(m)}+\left(2a_2(m)+\frac{2}{\varphi(m)}\right)(n+1)+\frac{2a_6(m)(n+1)}{\log (n+1)} \right. \\
&\quad \left.+\frac{(n+1)\log\log (n+1)}{\varphi(m)}+\frac{\log^2 (n+1)}{\varphi(m)\log\log (n+1)} \right).
\end{align*}
Further, using inequalities $0 \leq \mu \leq k$, \eqref{eq:Logknk} and $k(n+1)+\mu \geq n+1$ for all $n$ under the consideration and $0<\varepsilon<1$, the previous estimate is
\begin{align*}
& <-\varepsilon (n+1)\log (n+1)+(k+1)(n+1)\log\log (n+1)+\frac{2a_6(m)\varphi(m)(n+1)(k+1)}{\log (n+1)} \\
&\quad+(n+1)\left(\log c-k+\log k+2(k+1)(a_2(m)\varphi(m)+1)\right)+\frac{2\log^2 (n+1)}{\log\log (n+1)}\\
&\quad+(k+3.5)\log (n+1)+\frac{\log k\cdot\log (n+1)}{\log\log (n+1)}+k\log\log (n+1) \\
&\quad+\log(k+1)+2.5\log k+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)+2\left(a_2(m)\varphi(m)+1\right)k \\
&\quad+\log\sqrt{2\pi}+1+\log H+\frac{2a_6(m)\varphi(m)k}{\log (n+1)}+\frac{\log (n+1)}{(n+1)\log\log (n+1)}+\frac{29}{12(n+1)} \\
&\quad= N_2(n+1).
\end{align*}
Because of the definition \eqref{eq:nBoundEpsilon}, the previous estimate is smaller than zero and the claim follows.
\end{proof}
Again, we would like to provide a simpler estimate for number $n$. Hence, we first introduce the following lemma:
\begin{lemma}
\cite[Lemma 3.3]{EHMS2019M}
\label{lemma:Inversez}
If $r \geq e$ is a real number, $z(y)$ is the inverse of the function $y(z)=z\log z$ and $y\geq re^r$, then
$$
z(y) \leq \left(1+\frac{\log r}{r}\right) \frac{y}{\log y}.
$$
\end{lemma}
Now we define $B_2(m)$ (see Table \ref{table:B2}) and keep in mind that numbers $A_3(m,k)$ and $B_1(m)$ are given in Table \ref{table:A3B1C2}.
\begin{table}[!h]
\centering
\begin{tabular}{|c c|}
\hline
$m$ & $B_2(m)$ \\
\hline
$3\leq m \leq 36$ & $0.013$ \\
$37\leq m \leq 43$ & $0.180$ \\
$44\leq m \leq 432$ & $0.201$ \\
$433\leq m\leq 10^5$ & $0.080$ \\
$10^5< m<4\cdot 10^5$ & $0.327$ \\
$4\cdot 10^5\leq m<10^{29}$ & $0.511$ \\
$m \geq 10^{29}$ & $1.127$ \\
\hline
\end{tabular}
\caption{Definition for $B_2(m)$ for different values $m$.}
\label{table:B2}
\end{table}
Next, we prove estimates for number $n$:
\begin{lemma}
\label{lemma:nUpperEpsilon}
Let us use the same definitions and assumptions as in Lemma \ref{lemma:contra2}. Further, let also assume that $H$ satisfy the same assumptions as in Theorem \ref{thm:lowerBoundMore}.
Then
\begin{equation}
\label{eq:nEstUpperEpsilon}
n\log n < \frac{A_3(m,k)}{\varepsilon }n\log\log n+\frac{\log H}{\varepsilon }, \quad n< \frac{B_1(m)\log H}{\varepsilon\log\left(\frac{\log H}{\varepsilon}\right)} \quad\text{and} \quad n <B_2(m)\log H.
\end{equation}
\end{lemma}
\begin{proof}
By assumption \eqref{eq:nBoundEpsilon} we have $0 \leq N_2(n)$. We derive an upper found of form
\begin{equation}
\label{eq:Amk3}
-\varepsilon n\log n+A_3(m,k)n\log\log n+\log H,
\end{equation}
for the term $N_2(n)$. Using this upper bound and keeping the inequality $0 \leq N_2(n)$ in mind, we find the wanted upper bounds.
The term $N_2(n)$ can be written in the form
\begin{align*}
N_2(n) &= -\varepsilon n\log n+n\log\log n\left(k+1+\frac{\log c-k+\log k+2(k+1)(a_2(m)\varphi(m)+1)}{\log\log n}\right.\\
&\quad\left.+\frac{2a_6(m)\varphi(m)(k+1)}{\log n\cdot\log\log n}+\frac{2\log^2 n}{n\left(\log\log n\right)^2}+\frac{(k+3.5)\log n}{n\log\log n}+\frac{\log k\cdot\log n}{n\left(\log\log n\right)^2}+\frac{k}{n}\right. \\
&\quad\left.+\frac{\log(k+1)+2.5\log k+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right)+2\left(a_2(m)\varphi(m)+1\right)k}{n\log\log n}\right. \\
&\quad\left.+\frac{\log\sqrt{2\pi}+1}{n\log\log n}+\frac{2a_6(m)\varphi(m)k}{n\log n\cdot\log\log n}+\frac{\log n}{\left(n\log\log n\right)^2}+\frac{29}{12n^2\log\log n} \right)+\log H.
\end{align*}
Further, since by assumption \eqref{eq:nBoundEpsilon} we have $N_2(n+1) <0$, similarly as in the first paragraph in the proof of Lemma \ref{lemma:contra2} we can deduce that $\log(n+1)>s$. Hence, we have
$$
\log n > s-1 \geq \max\left\{m^{\varphi(m)}, c, \max_{1\leq j \leq k}\{|\alpha_j|\}\right\},
$$
and also $\log (k+1) <k$. Thus the term $N_2(n)$ is
\begin{equation}
\label{eq:NnUpper}
\begin{aligned}
&< -\varepsilon n\log n+n\log\log n\left(k+2+\frac{2(k+1)a_2(m)}{\log m}+\frac{2(k+1)}{\log\log n}+\frac{2a_6(m)(k+1)}{\log n\cdot\log m}\right.\\
&\quad\left.+\frac{2\log^2 n}{n\left(\log\log n\right)^2}+\frac{(k+3.5)\log n}{n\log\log n}+\frac{k\log n}{n\left(\log\log n\right)^2}+\frac{2k}{n}+\frac{2a_2(m)k}{n\log m}\right. \\
&\quad\left.+\frac{5.5k+\log\sqrt{2\pi}+1}{n\log\log n}+\frac{2a_6(m)k}{n\log n\cdot\log m}+\frac{\log n}{\left(n\log\log n\right)^2}+\frac{29}{12n^2\log\log n} \right)+\log H.
\end{aligned}
\end{equation}
Since the terms $a_2(m)$ and $a_6(m)$ are defined differently for different values of $m$, we estimate $N_2(n)$ differently depending on the size of $m$.
Let us first consider the case $m \in [3, 432]$ in three sub-cases: $m \in [3, 36]$, $m \in [37, 43]$ and $m \in [44, 432]$. In the first case, we have $a_2(m)=\log\sqrt{1865}+4.017$, $a_6(m)=27.640+39.717/\varphi(m)$ and we can set $\log n =1865$, $\varphi(m)=2$ and $\log m=\log 3$ in estimate \eqref{eq:NnUpper}. These lead to an estimate
\begin{equation*}
N_2(n) <-\varepsilon n\log n+\left(16.480 + 15.480k\right)n\log\log n+\log H.
\end{equation*}
Using similar idea but $a_6(m)=0.001+1.501/\varphi(m)$, $\log n=37^{37^{\log 2/\log 3}}$, $\varphi(m)=37^{\log 2/\log 3}$ and $\log m=\log 37$, in the case $m \in [37, 43]$ we get
\begin{equation*}
N_2(n) <-\varepsilon n\log n+\left(6.368 + 5.368 k\right)n\log\log n+\log H.
\end{equation*}
Similarly, in the case $m \in [44, 432]$ we obtain
\begin{equation*}
N_2(n) <-\varepsilon n\log n+\left(6.172 + 5.172 k\right)n\log\log n+\log H.
\end{equation*}
Using similar idea, in the case $m \in [433, 10^5]$ we get
\begin{equation*}
N_2(n) <-\varepsilon n\log n+\left(8.176 + 7.176 k\right)n\log\log n+\log H.
\end{equation*}
Further, in the case $10^5 <m <4\cdot 10^5$ we obtain
\begin{equation*}
N_2(n) <-\varepsilon n\log n+\left(5.422 + 4.422 k\right)n\log\log n+\log H,
\end{equation*}
in the case $4\cdot 10^5\leq m < 10^{29}$
\begin{equation*}
N_2(n) <-\varepsilon n\log n+\left(4.863 + 3.863 k\right)n\log\log n+\log H
\end{equation*}
and in the case $m \geq 10^{29}$
\begin{equation*}
N_2(n) <-\varepsilon n\log n+\left(4.087 + 3.087k\right)n\log\log n+\log H.
\end{equation*}
Thus, we have found the wanted upper bounds of form \eqref{eq:Amk3} for the term $N_2(n)$.
Since
\begin{equation*}
0 \leq N_2(n) \leq -\varepsilon n\log n+A_3(m,k)n\log\log n+\log H,
\end{equation*}
we have proved the first bound in \eqref{eq:nEstUpperEpsilon} and we also have
\begin{equation}
\label{eq:A2lower}
\varepsilon n\log n\left(1-A_3(m,k)\frac{\log\log n}{\varepsilon\log n}\right) \leq \log H.
\end{equation}
Further, since $\log\log n/\log n$ is decreasing for all $\log n \geq e$ and by assumption \eqref{eq:LowerBoundsForS} we have
$$
\log n > \left(\frac{A_3(m,k)}{\varepsilon}\right)^{1.6}>A_3(m,1)^{1.6}\geq 7.174^{1.6},
$$
we can deduce that
$$
1-A_3(m,k)\frac{\log\log n}{\varepsilon\log n}>1-\frac{1.6\log\frac{A_3(m,k)}{\varepsilon}}{\left(A_3(m,k)/\varepsilon\right)^{0.6}}>0.
$$
Thus we have obtained
\begin{equation}
\label{eq:Lefty}
\frac{\left(A_3(m,k)/\varepsilon\right)^{0.6}}{\left(A_3(m,k)/\varepsilon\right)^{0.6}-1.6\log(A_3(m,k)/\varepsilon)}\cdot\log H\geq \varepsilon n\log n \geq \varepsilon re^{r},
\end{equation}
where
\begin{equation*}
r=\max\left\{1865, m^{\varphi(m)}, \left(\frac{A_3(m,k)}{\varepsilon}\right)^{1.6}\right\}
\end{equation*}
by assumptions \eqref{eq:LowerBoundsForS}.
Let now $z$ be as in Lemma \ref{lemma:Inversez}. Hence, by applying Lemma \ref{lemma:Inversez}, setting $y$ be the left-hand side of inequality \eqref{eq:Lefty} divided by $\varepsilon$ and keeping in mind that the $z(y)$ in Lemma \ref{lemma:Inversez} is increasing, we get
\begin{equation}
\label{eq:nlogelogy}
\begin{split}
n = z\left(n \log n \right)\leq z\left(\frac{\left(A_3(m,k)/\varepsilon\right)^{0.6}}{\left(A_3(m,k)/\varepsilon\right)^{0.6}-1.6\log(A_3(m,k)/\varepsilon)}\cdot\frac{\log H}{\varepsilon}\right) \\
< \left(1+\frac{\log r}{r}\right)\cdot\frac{\left(A_3(m,k)/\varepsilon\right)^{0.6}}{\left(A_3(m,k)/\varepsilon\right)^{0.6}-1.6\log(A_3(m,k)/\varepsilon)}\cdot\frac{\log H/\varepsilon}{\log\left((\log H)/\varepsilon\right)}.
\end{split}
\end{equation}
Next we prove the wanted estimates containing the terms $B_1(m)$ and $B_2(m)$ using estimate \eqref{eq:nlogelogy}.
Let us now prove the case containing the term $B_1(m)$. In order to prove the wanted estimate, we recognise the followings: The functions $\log r/r$ and $x^{0.6}/(x^{0.6}-1.6\log x)$ are decreasing and positive for all values $r, x$ under the consideration. Hence, we can use estimate $A_3(m,k)/\varepsilon>A_3(m,1)$ in the second term of the product in the last line of estimate \eqref{eq:nlogelogy}. For number $r$ we use the following: In the case $m \in [3, 36]$ we apply $r=1865$ and in the other cases $r=m^{m^{\log2/\log 3 }}$, where we set number $m$ be the lower bound of the interval under the consideration.
These replacements prove the second estimate in \eqref{eq:nEstUpperEpsilon}.
Let us now prove the last estimate in \eqref{eq:nEstUpperEpsilon}. Now, using the assumption
$$
\log H/\varepsilon>\left(A_3(m,k)/\varepsilon\right)^{1.6}e^{\left(A_3(m,k)/\varepsilon\right)^{1.6}},
$$
the right-hand side of estimate \eqref{eq:nlogelogy} is
\begin{multline*}
<\frac{\left(A_3(m,k)/\varepsilon\right)^{1.6}}{\left(\left(A_3(m,k)/\varepsilon\right)^{0.6}-1.6\log(A_3(m,k)/\varepsilon)\right)\left(\left(A_3(m,k)/\varepsilon\right)^{1.6}+1.6\log(A_3(m,k)/\varepsilon)\right)}\\
\cdot\left(1+\frac{\log r}{r}\right)\cdot\frac{\log H}{A_3(m,k)}.
\end{multline*}
Since the function in the first line is a decreasing function of $A_3(m,k)/\varepsilon$ for all possible values of $A_3(m,k)/\varepsilon$, we can again use the estimate $A_3(m,k)/\varepsilon>A_3(m,1)$. Further, we can also use the same estimates for $r$ as in the previous case. This proves the last estimate in \eqref{eq:nEstUpperEpsilon}.
\end{proof}
\begin{remark}
For the left-hand side in \eqref{eq:A2lower} to be positive, it is enough to assume that $s \geq -\frac{A_3(m,k)}{\varepsilon}W_{-1}\left(-\frac{\varepsilon}{A_3(m,k)}\right)+1$ instead of the last assumption in \eqref{eq:LowerBoundsForS}. However, this would lead to more complicated computations. Hence, in order to keep the computations relatively simple, we have done a more lenient but a simpler assumption in \eqref{eq:LowerBoundsForS}.
\end{remark}
\begin{remark}
The exponent $1.6$ of $A_3(m,k)/\varepsilon$ in assumptions \eqref{eq:LowerBoundsForS} is selected in such a way that it is the largest exponent $r$ with one decimal accuracy that
$$
\frac{rx\log x}{x^r}<1
$$
for all $x \geq 7.174$.
\end{remark}
\subsection{Proof of Theorem \ref{thm:lowerBoundMore}}
\label{sec:proof4}
In this section, we prove Theorem \ref{thm:lowerBoundMore}:
\begin{proof}[Proof of Theorem \ref{thm:lowerBoundMore}]
Because of Lemmas \ref{lemma:exists2}, \ref{lemma:contra2} and estimate \eqref{eq:Product} we must have
$$
\left|B_{n+1,\mu,0}(1)\Lambda_p\right|_p > \left|\sum_{j=1}^k \lambda_jS_{n+1,\mu,j}(1)\right|_p
$$
for some $p \in R\cap (\log(n+1),k(n+2)]$. We consider this prime number $p$. First we prove that $p$ is in set \eqref{eq:intervalEpsilon} and then that the wanted lower bound is satisfied.
Due to the second estimate in \eqref{eq:nEstUpperEpsilon}, number $p$ is smaller than the end of the interval in \eqref{eq:intervalEpsilon}. Even more, if
\begin{equation}
\label{eq:wronglower}
n+1 \leq \frac{\log H}{\varepsilon\log\left(\frac{\log H}{\varepsilon}\right)}< \frac{\log H}{\varepsilon\log\log H},
\end{equation}
then
\begin{equation}
\label{eq:lowern1}
\log(n+1)(n+1) < \frac{\log H}{\varepsilon\log\log H}\log\left(\frac{\log H}{\varepsilon\log\log H}\right)=\frac{\log H}{\varepsilon}\left(1-\frac{\log\left(\varepsilon\log\log H\right)}{\log\log H}\right).
\end{equation}
Further, by assumptions \eqref{eq:HassumptionEpsilon} and \eqref{eq:LowerBoundsForS} and since $\varepsilon \in (0,1)$ we have
\begin{equation*}
\log\log H>\varepsilon\log \log H>\varepsilon \log\left(\varepsilon/\varepsilon^{1.6}e^{\varepsilon^{-1.6}}\right)>\varepsilon^{-0.6}>1.
\end{equation*}
Hence, the right-hand side of estimate \eqref{eq:lowern1} is smaller than $\log H/\varepsilon$ if inequality \eqref{eq:wronglower} holds. However, this is a contradiction with the first paragraph of the proof of Lemma \ref{lemma:contra2}. Hence, inequality \eqref{eq:wronglower} cannot hold and hence $\log(n+1)$ is greater than the lower bound of interval in \eqref{eq:intervalEpsilon}. Thus $p$ is in the wanted set.
As in the proof of Theorem \ref{thm:lowerBoundMore},we only have to find a lower bound for the term $\left|\Lambda_p\right|_p$. By estimate \eqref{eq:TnEnough} it is sufficient to find an upper bound for the term $\left|T(n+1, \mu)\right|$.
Similarly as in the proof of Theorem \ref{thm:lowerBoundMore}, we get
\begin{multline*}
\log \left|T(n+1,\mu)\right| <\log H+(k+2.5)\log k+\frac{1}{k}+\log\sqrt{2\pi}+\frac{1}{12k(n+1)}-k \\
+(k-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)\left(1+k(n+1)\right)
+(k+1.5)\left(\log n+\frac{2}{n}\right) \\
+k(n+1)\left(\log 2+\log k+\log n+\frac{2}{n}-1\right).
\end{multline*}
Now, applying Lemma \ref{lemma:nUpperEpsilon} for $n \log n$ and $n$ and using assumptions \eqref{eq:LowerBoundsForS}, $k \geq 1$ and the facts that $\log k <k$ and $\log H/\varepsilon>\log H$, the previous estimate is
\begin{align*}
&< \left(\frac{k}{\varepsilon}+1\right)\log H+n\log\log n\left(\frac{A_3(m,k)k}{\varepsilon}+\frac{(k-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)k}{\log\log n}\right.\\
&\quad\left.+\frac{(2k+1.5)\log n}{n\log\log n}+\frac{k\left(\log k+\log 2+1\right)}{\log\log n}+\frac{1}{kn\log\log n}+\frac{4k+37/12}{n^2\log\log n}\right. \\
&\quad\left.+ \frac{(2k+2.5)\log k+\log\sqrt{2\pi}+k\log 2-2k+(k^2-1)\log \left(\max_{1 \leq j \leq k} \left\{|\alpha_j|\right\}\right)}{n\log\log n}\right) \\
&\quad<\left(\frac{k}{\varepsilon}+1\right)\log H+\frac{B_1(m)\log H}{\varepsilon\log\log H}\cdot\log\log \left(B_2(m)\log H\right)\left(\frac{A_3(m,k)k}{\varepsilon}\right.\\
&\quad\quad\left.+k^2-k+\frac{k^2}{n}+\frac{k\left(k+\log 2+1\right)}{\log\log n}+\frac{3.5k\log n}{n\log\log n}\right. \\
&\quad\quad\left.+\frac{(2k+\log 2+\log\sqrt{2\pi}+1.5)k}{n\log\log n}+\frac{85k}{12n^2\log\log n} \right).
\end{align*}
Further, since $B_2(m)\log H<\log H$ when $m <10^{29}$ and also $\log\log B_2(m)<0$ when $m \geq 10^{29}$, we can replace the term $\log\log\left(B_2(m)\log H\right)$ with $\log\log\log H$.
Again, we simplify this expression using different cases depending on the numbers $m$ and $n$.
All functions depending on number $n$ in the right-hand side of the previous inequality are decreasing for all $n>e$. The claim follows when we use the same bounds for number $\log n$ that were used in the proof of Lemma \ref{lemma:Inversez} to prove the first estimate in \eqref{eq:nEstUpperEpsilon}.
\end{proof}
\begin{remark}
In the previous proof we have changed the constant terms which do not depend on the term $k$ to constant terms which depend on the term $k$ since with three decimal accuracy this does not change the coefficient of term $k$ but removes the need to add a new constant term to the result.
\end{remark}
\section*{Acknowledgements}
I would like to thank Dr. Louna Sepp\"al\"a for her support during the research process.
\bibliographystyle{abbrv}
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Capitán Confederación (Captain Confederacy en el inglés original) es un comic book de historia alternativa creado por Will Shetterly y Vince Stone que se publicó entre 1986 y 1992. Cuenta la historia de un superhéroe creado con fines de propaganda, en un mundo en el que los Estados Confederados de América ganó su independencia.
Trayectoria editorial
John M. Ford escribió tres números de la primera serie, y escribió una parte del número 10 "Driving Norte".
Trama
La primera línea argumental, publicada en doce números por SteelDragon Press, cuenta cómo el primer Capitán Confederación, un hombre blanco, se une a la rebelión contra su país. El argumento de la segunda, publicado en cuatro tomos por Epic Comics, se centra en la lucha por controlar la política de los países de América del Norte, en una conferencia mundial de superhéroes en Free Louisiana. La nueva Capitán Confederación es una mujer afroamericana, embarazada con el bebé del capitán anterior.
Enlaces externos
Captain Confederacy en la "Grand Comics Database" (inglés)
Captain Confederacy en "Comic Book DB" (inglés)
Héroes de Marvel Comics
Ucronía en el cómic
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\section{Introduction}
Artificial atoms employ controlled coherent spin-photon interfaces to transfer quantum states between long-lived stationary spins and flying photons.
Three central requirements for a scalable spin-photon interface are a long spin coherence time, efficient spin-photon coupling, and operation at telecommunication wavelengths~\cite{ruf_quantum_2021}.
However, current material platforms fail to meet these requirements at once~\cite{zaporski_ideal_2023, pompili_realization_2021, bhaskar_experimental_2020}.
A central challenge is to enhance the naturally weak coherent radiative emission rate while suppressing other excited-state decoherence processes.
In particular, the modified local density of optical states in a cavity can increase the radiative emission fraction $\beta$ into a desired mode while suppressing emission into other modes:
\begin{equation} \label{eq:beta}
\beta = \frac{(1+F_\text{P}) \gamma_\text{R}}{(1+F_\text{P})\gamma_\text{R} + \gamma_0},
\end{equation}
where $\gamma_R$ is the radiative rate for the transition of interest and $\gamma_0$ encompasses the rates for other radiative and non-radiative transitions.
$F_\text{P}$ is the cavity Purcell factor, which in the case of perfect cavity-atom coupling is defined as~\cite{purcell_spontaneous_1995}
\begin{equation} \label{eq:purcellmain}
F_\text{P}=\frac{3\lambda^3}{4\pi^2}\frac{Q}{V},
\end{equation}
with $\lambda$ the wavelength in the material, $Q$ the quality factor, and $V$ the effective mode volume of the cavity. To enable efficient collection of the light emitted from the cavity, it is
required that the cavity far-field emission is matched to the optical mode of interest --- such as the mode of an optical fiber. This light collection is quantified using the coupling efficiency $\eta$.
The net collection efficiency $\beta\eta$ defines the performance of a quantum network built with such devices.
Accommodating both high $Q/V$ and high $\eta$ in a single device is a nontrivial design challenge that depends largely on the materials, the fabrication, and the operation wavelengths of the artificial atom of choice.
These challenges have so far resulted in weak and small-scale spin-photon coupling for current leading artificial atom platforms~\cite{ruf_quantum_2021}.
\begin{figure*}[htbp]
\centering
\includegraphics[width=\linewidth]{Figs/motivation_main.pdf}
\caption{\textbf{Description of the system.} a) Illustration of the system under study, consisting of optimized 2D photonic crystal cavities coupled to single G-centers in silicon. b) The G-center consists of two substitutional carbon atoms (blue spheres) and a silicon interstitial atom (gray spheres). c) The O-band radiative transition occurs between singlet states 1 and 2 and can be excited via above-band excitation. The system features an additional triplet state (3). d) Spectrum showing the photoluminescence (PL) of a single cavity-coupled G-center with a zero phonon line (ZPL) around 1279~nm.}
\label{fig:motivation}
\end{figure*}
Recently, there has been a resurgence of interest in silicon as a host material for single artificial atoms operating in the telecommunication bands~\cite{bergeron_silicon-integrated_2020, baron_detection_2021, durand_broad_2021, higginbottom_optical_2022}.
Initial reports have focused on the G-center~\cite{hollenbach_engineering_2020, redjem_single_2020}, the T-center~\cite{higginbottom_optical_2022, deabreu2022waveguide}, and the W-center~\cite{baron_detection_2021}, and include the first optical observation of an isolated spin in silicon~\cite{higginbottom_optical_2022}, and the first isolation of single artificial atoms in silicon waveguides and their spectral programming~\cite{prabhu_individually_2022}.
These demonstrations, combined with the experimentally reported 30-min-long spin coherence times in ionized donors in $^{28}$Si~\cite{saeedi_room-temperature_2013-1} and the success of silicon microelectronics and photonics~\cite{vivien_handbook_2016}, make this technology compelling for large-scale quantum information processing.
However, although high $Q/V$ and $\eta$ optical cavities have recently been shown in silicon at a large scale~\cite{panuski2022full}, their coupling to single artificial atoms remains a challenge.
Here, we report on the inverse design of high $Q/V$, $\eta$-optimized photonic crystal cavities, and demonstrate cavity-enhanced interaction of light with single artificial atoms at telecommunication wavelengths in silicon.
\section{Results}
Our device, illustrated in Fig.~\ref{fig:motivation}a, consists of single G-centers coupled to inverse-designed 2D photonic crystal cavities.
The G-center is a quantum emitter formed by two substitutional carbon atoms and a silicon interstitial (Fig.~\ref{fig:motivation}b), and features a zero phonon line (ZPL) transition at 970~meV (1279~nm) in the telecommunications O-band along with a spin triplet metastable state~\cite{udvarhelyi_identification_2021} (Figs.~\ref{fig:motivation}c, d).
Our cavities were designed following our previous work~\cite{panuski2022full} to simultaneously achieve a target $Q/V$ while optimizing for vertical coupling $\eta$ by matching the emission to a narrow numerical aperture in the O-band.
Fig.~\ref{fig:cav_ff} shows one of our cavity designs, including its near-field cavity mode (Fig.~\ref{fig:cav_ff}a) and its far-field scattering profile (Fig.~\ref{fig:cav_ff}b), with more than 70\% of the emitted power simulated to radiate into an objective NA of 0.55.
More information on the optimization can be found in Methods, and the results of the cavity optimization in SI Section~\ref{sec:optimization} and Fig.~\ref{sfig:optimization}.
The fabrication of our device follows our previous work~\cite{prabhu_individually_2022} with the addition of an underetch step, and is described in Methods.
\begin{figure*}[htbp]
{\centering
\includegraphics[width=\linewidth]{Figs/cav_main.pdf}}
\caption{\textbf{Optimized cavities.} Simulated a) near-field electric field amplitude and b) scattered far-field power for the 2D photonic crystal cavity design used in this work. c) Scanning electron micrograph showing one of our optimized photonic crystal cavities. d) Confocal PL 2D scan showing the reflectivity map (blue scale) overlaid with a PL map (red and green) of the same area, highlighting the PL emission from the cavity center.}
\label{fig:cav_ff}
\end{figure*}
The measurements on our device were performed with a setup consisting of a home-built cryogenic confocal microscope featuring temperature and CO$_2$ gas control, and optimized for visible light excitation and infrared collection into a single-mode fiber (details in SI Section~\ref{sec:setup}).
A scanning electron micrograph of a representative cavity in our chip is shown in Fig.~\ref{fig:cav_ff}c. Its reflectivity was characterized in cross-polarization~\cite{altug_polarization_2005, panuski2022full,deabreu2022waveguide} (details in SI Section~\ref{sec:cross}). Fig.~\ref{fig:cav_ff}d shows a photoluminescence (PL) 2D scan of one of our systems, where the cavity-coupled artificial atom is evidenced by the color-labeled IR emission in the cavity center upon excitation with green light. The spectral signature of the PL, shown in Fig.~\ref{fig:motivation}d, features a ZPL centered at around 1279~nm and thus aligns
with the previously reported G-center ZPLs~\cite{hollenbach_engineering_2020, baron_detection_2021, prabhu_individually_2022}.
For the cavities of interest, we measure quality factors of $\sim3700$ and 2100, and center wavelengths around 1279~nm, near the G-center ZPL.
Fig.~\ref{fig:coupling} shows the experimental results confirming the presence of single artificial atoms in our photonic crystal cavities. The coupling between the atom and the cavity results in an enhancement of the atom single-photon emission. We observe linearly polarized PL emission (Fig.~\ref{fig:coupling}a) from our system, indicative of coupling through the expected transverse electric cavity mode.
Measuring the PL saturation under increasing excitation power yields that of a two-level emitter model (see Fig.~\ref{fig:coupling}b and SI Section~\ref{sec:pulsed}).
We further confirmed the addressing of a single artificial atom by demonstrating single-photon emission via a Hanbury-Brown-Twiss (HBT) experiment.
Our second-order autocorrelation results (Fig.~\ref{fig:coupling}c) show excellent antibunching with a fitted $g^{(2)}(0)$ value of $0.03 \substack{+0.07 \\ -0.03}$ without background correction (details in SI Section~\ref{sec:g2}).
This value is nearly an order of magnitude lower than the rest of the literature (see SI Table~\ref{tab:comparison}), and indicates high-purity single-photon emission.
The bunching near $\pm 10$~ns delay conforms with the presence of a third dark state, which has been attributed to a metastable triplet state~\cite{udvarhelyi_identification_2021}.
These measurements demonstrate the presence of a single G-center in our cavity.
To confirm the cavity enhancement of our single artificial atom, we detuned the cavity resonance wavelength from the G-center ZPL using two different methods, i.e. thermal and gas tuning.
In the thermal tuning experiments, starting with a cryostat temperature of 4~K, we brought the temperature up to 24~K and consequently shifted the cavity away from our G-center ZPL. This effect is visible in Fig.~\ref{fig:coupling}d, where the pink (orange) curves show the cavity and ZPL profiles before (after) the temperature increase. Different detunings $\delta_{\mathrm{t}}$ and $\delta_{\mathrm{t}}^{\prime}$, defined as the difference between the cavity and ZPL wavelengths, are therefore achieved at 4~K and 24~K, respectively.
While a significant cavity wavelength shift occurs, we also observe a much smaller ZPL shift, not shown in the figure. Therefore, this plot shall be indicative only of the relative shift between the cavity and ZPL wavelength. A more detailed discussion about the figure can be found in SI Section~\ref{sec:Tuningofcavity}a.
We note that temperatures below 30~K have been reported to not affect G-center ensembles~\cite{beaufils_optical_2018, chartrand_highly_2018}.
We observe an intensity enhancement of $6.078\pm0.218$ with a cavity $Q$ of $\sim 3700$ and a mode volume of $V<1(\lambda/n)^3$.
To validate that the intensity enhancement does not originate from the temperature change induced by thermal cavity tuning, we performed additional experiments using gas tuning.
We injected CO$_2$ gas into the cryogenic sample chamber to coat the cavity with solid CO$_2$, followed by selective gas sublimation using a $532$~nm continuos wave (CW) laser.
A further description of the process can be found in SI Section~\ref{sec:Tuningofcavity}b.
Analogously to the thermal tuning case, Fig.~\ref{fig:coupling}e shows the cavity reflectivity and G-center ZPL now under gas tuning for two different detunings $\delta_{\mathrm{g}}$ and $\delta_{\mathrm{g}}^{\prime}$. We calculate an enhancement in the PL emission of $3.837\pm0.074$ with a cavity $Q$ of $\sim 2100$ and a mode volume of $V<1(\lambda/n)^3$. Also in this case, we observe a ZPL shift (not shown in the figure). More details are to be found in SI Section~\ref{sec:Tuningofcavity}b.
Ultimately, we measured the excited state lifetime of our artificial atoms under both gas and thermal tuning using a 0.5~ns pulsed laser at 532~nm for all cavity detunings (see SI Section~\ref{sec:pulsed} for details about these measurements).
Fig.~\ref{fig:coupling}f shows our measured lifetimes and the emission rates for all of our experiments.
We do not observe a statistically significant lifetime modification even under a clear cavity enhancement of the emission rates above 6x.
\begin{figure*
\centering
\includegraphics[width=\linewidth]{Figs/results_main.pdf}
\caption{\textbf{Cavity-enhanced single-photon emission.} a) Polarization plot of the PL emission from our system, which matches that of an electric dipole.
b) PL counts at increasing CW excitation powers. Saturation matching that of a two-level system is observed.
c) The second-order autocorrelation function, which yields $g^{(2)}(0) \approx 0$, demonstrating high-purity single-photon emission. d) Cavity-atom coupling is demonstrated by changing the sample temperature from 4~K (pink) to 24~K (orange) to spectrally tune a cavity with respect to the G-center ZPL, which results in a reduction of the PL magnitude. e) This effect is confirmed with a second cavity-atom system on the same chip before (green) and after (blue) gas detuning. f) PL counts and lifetimes for both systems under the measured cavity-atom detuning magnitudes.}
\label{fig:coupling}
\end{figure*}
The quantum efficiency (QE) of any silicon color center is one of the central unanswered questions in the field.
Recent reports have estimated the QE of a single G-center to be above 1\% from waveguide-coupled counts~\cite{prabhu_individually_2022, komza_indistinguishable_2022}, and $<$~10\% for ensembles coupled to separate cavities~\cite{lefaucher2023cavity}.
To gain a valid estimate of the QE of the G-center, an experiment varying the coupling rate between the same single artificial atom and a cavity was required.
Our measurements allow us to extract such a value.
Using the derivation described in SI Section~\ref{sec:qe}~\cite{lefaucher2023cavity}, the literature value of the Debye-Waller factor $F_\text{DW}=0.15$~\cite{beaufils_optical_2018}, and our measured values of off-resonance lifetime $\tau_\text{off}=6.09\pm0.25$~ns and count rate enhancement for thermal tuning,
we obtain a quantum efficiency bounded as $\text{QE}~<~18\%$.
\section{Discussion}
We show, for the first time, strong enhancement of the quantum emission of individual artificial atoms coupled to silicon nanocavities.
A central requirement for the scalability of our system is localized spatial and spectral alignment of both many cavities and many atoms to a common global frequency.
The spatial alignment of the cavity and atom can be achieved by making use of the recently reported localized implantation of single G- and W-centers~\cite{hollenbach_wafer-scale_2022}.
Silicon artificial atoms can be spectrally aligned using the recently reported non-volatile optical tuning for G-centers~\cite{prabhu_individually_2022} or methods used in other artificial atom systems such as tuning via electric fields~\cite{anderson_electrical_2019-2}, or mechanical strain~\cite{wan_large-scale_2020-2}.
Cavity tuning via local thermal oxidation of silicon has been achieved on a large scale~\cite{panuski2022full}, and a similar method could be used to align large arrays of cavity-atom systems at room or cryogenic temperatures.
Our approach directly applies to other silicon artificial atoms, such as the T-center, which would enable direct access to a spin outside of a metastable state~\cite{higginbottom_optical_2022}.
A hypothesis was recently raised regarding the possibility of two different physical systems being reported as G-centers~\cite{baron_single_2022}.
We believe that our work provides conclusive evidence for this claim.
Table~\ref{tab:comparison} in SI Section~\ref{sec:comparison} compares the reported experimental results for single G-center labeled artificial atoms in silicon, and shows two clear clusters.
The first group comprises the single emitter reports in Refs.~\cite{hollenbach_engineering_2020, baron_single_2022, prabhu_individually_2022, komza_indistinguishable_2022}, and aligns with G-center ensemble work~\cite{lefaucher2023cavity}. These studies show a ZPL centered around 1279~nm and a narrow inhomogeneous linewidth <~1.1~nm, a QE between 1 and 18\%, and a short excited state lifetime <~10~ns that does not change significantly under Purcell enhancement, confirming the QE magnitude.
The second group comprises Refs.~\cite{redjem_single_2020, redjem_all-silicon_2023}, and features a shorter ZPL centered around 1270~nm and a larger inhomogeneous linewidth of 9.1~nm, a QE$\sim$50\%, and a longer excited state lifetime >~30~ns, which changes significantly under Purcell enhancement and thus qualitatively aligns with the estimated QE.
Our measurements align with the first system, i.e. the originally reported G-centers, and provide the first upper bound for the QE of single G-centers, previously estimated to be between 1\%~\cite{prabhu_individually_2022, komza_indistinguishable_2022} and 10\% for ensembles~\cite{lefaucher2023cavity}.
More information on this comparison can be found in SI Section~\ref{sec:comparison}.
We conclude that our work highlights the need for further theoretical and experimental investigation regarding the creation process and the photophysics of G-center-like artificial atoms in silicon platforms.
\section{Conclusion}
We showed cavity-enhanced single artificial atoms in silicon by integrating single G-centers into inverse-designed photonic crystal nanocavities.
We demonstrated an intensity enhancement of $\sim$6x with a cavity featuring a $Q/V>3700$, which yielded the highest purity single-photon emission for silicon color centers in the literature, and the first bound to the QE of single G-centers of $<18$\%.
Our demonstration lays the groundwork for efficient spin-photon interfaces at telecommunication wavelengths for large-scale quantum information processing in silicon.
\section*{Methods}
\subsection{Sample fabrication}
The fabrication process follows~\cite{redjem_single_2020}, starting from a commercial SOI wafer with $220$~nm silicon on $2$~\textmu m silicon dioxide.
Cleaved chips from this wafer were implanted with $^{12}$C with a dose of $5\times10^{13}$~ions/cm$^{2}$ and $36$~keV energy, and subsequently annealed at $1000~^{\circ}$C for $20$~s to form G-centers in the silicon layer.
The samples were then processed by a foundry (Applied Nanotools) for electron beam patterning and etching, resulting in through-etched silicon cavities with SiO$_2$ bottom cladding and air as top cladding.
The silicon etching was performed using inductively coupled plasma reactive ion etching with SF$_6$-C$_4$F$_8$ mixed-gas.
As a final step, the samples were under-etched in a $49\%$ solution of hydrofluoric acid for $2$~min and dried using a critical point dryer.
\subsection{Cavity far-field optimization}
\label{methods:cav_ff}
Traditional photonic crystal cavity optimization aims to cancel radiative loss to enhance quality factor $Q$, which also reduces collection efficiency.
To avoid this, we incorporate the far-field collection efficiency $\eta$ to the optimization objective function alongside maximizing $Q$ and minimizing mode volume $V$~\cite{panuski2022full} (Fig.~\ref{sfig:optimization}).
This process is implemented using the open-source package \texttt{Legume}~\cite{Minkov_inverse_2020} which maps the problem of cavity design onto efficient and auto-differentiable guided mode expansion (GME) for gradient-based optimization.
In practice, we observe quality factors much lower than the $Q \sim \mathcal{O}(10^6)$ result expected from both simulation (Fig.~\ref{sfig:optimization}) and previous statistical studies on thousands of photonic crystals designed for $\sim$1550~nm operation under the same optimization method~\cite{panuski2022full}.
We attribute this disparity to the high carbon doping density used to produce cavity-coupled G-centers with sufficient probability. Reducing the doping density or applying localized doping~\cite{hollenbach_engineering_2020} could play a role in recovering performance closer to intrinsic silicon.
Applying large-scale characterization techniques~\cite{sutula_2022_largescale} to locate ideal emitters and fabricate cavities around these positions could enhance the yield of coupled emitters in the case of reduced doping density.
\section*{Notes}
During the preparation of this manuscript, we became aware of a manuscript reporting on a cavity-coupled single silicon color center~\cite{redjem_all-silicon_2023} and a second manuscript reporting on single rare-earth ions in silicon cavities~\cite{gritsch_purcell_2023}.
\section*{Acknowledgements}
The authors acknowledge Kevin C. Chen, Chao Li, Hugo Larocque and Mohamed ElKabbash for helpful discussions.
C.E-H. and L.D. acknowledge funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreements No.896401 and 840393.
M.P. acknowledges funding from the National Science Foundation (NSF) Convergence Accelerator Program under grant No.OIA-2040695 and Harvard MURI under grant No.W911NF-15-1-0548.
I.C. acknowledges funding from the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program and NSF award DMR-1747426.
M.C. acknowledges support from MIT Claude E. Shannon award.
D.E. acknowledges support from the NSF RAISE TAQS program.
This material is based on research sponsored by the Air Force Research Laboratory (AFRL), under agreement number FA8750-20-2-1007.
The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.
The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory (AFRL), or the U.S. Government.
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\section{Introduction}
To evaluate the effectiveness of a testing technique for software systems, various approaches can be employed. A natural and well-known approach to assess the effectiveness of a test suite generated by a testing technique is to measure the defect detection rate when applying a generated test suite to a System Under Test (SUT). As such, an experimental SUT that represents a real-world system containing real defects from the past software development process can be useful here. Alternatively, the mutation testing technique can be applied by introducing artificial defects into the code of an experimental SUT using defined mutation operators \cite{siami2008sufficient, offutt2011mutation}. Additionally, a defect injection technique, which can be considered to be a more general variant, can be employed. In defect injection, defects are introduced into an experimental SUT and various technical possibilities can be used.
Measuring the defect detection rate can be used to determine the effectiveness of the Combinatorial or Constrained Interaction Testing \cite{nie2011survey,CombConsTBestoun} or Path-based Testing \cite{bures2015pctgen} techniques. As a typical example, one can examine the strength of test cases generated by the Combinatorial Interaction Testing (CIT) algorithm using a mutation testing technique. As an experimental SUT, an open-source software system can be selected. Then, various mutants are created from the source code by a set of mutation operators. Subsequently, the generated test cases are assessed in the experimental SUT and it is determined whether the test case can detect a defect introduced into SUT by a mutation operator. These types of experiments are typically run in multiple series with various sets of mutants and test cases to obtain convincing evidence regarding the effectiveness of the generated test cases \cite{offutt2011mutation}.
The approach can be generalized as illustrated in Figure \ref{fig:defect_introduction_process}.
\begin{figure}[htbp]
\centerline{\includegraphics[width=7cm]{defect_introduction_process.png}}
\caption{Defect introduction process to an experimental SUT.}
\label{fig:defect_introduction_process}
\end{figure}
In this paper, we present a new open-source benchmark testbed to support defect injection testing. The testbed is available to the community and can be used to evaluate various testing techniques. In this approach, we do not insist on defined mutation operators. The goal of the testbed is to provide a complement to the classical mutation testing approach for evaluating the effectiveness of test cases.
In contrast to the established classical code mutation operators, various complex software defects can be introduced into the code, especially defects caused by a misunderstanding of the SUT design specification or requirements during the development process. The practical use case of the presented testbed is to provide researchers with a complementary option to the mutation testing technique to be able to simulate a broader spectrum of possible software defects during experiments. The testbed is, hence, a complement to mutation testing rather a replacement of mutation testing via a defect injection approach. As we show later in Section \ref{sec:background_state_of_the_art}, both approaches have certain advantages and disadvantages. Hence, both approaches can be combined to provide the best objective measurement of the effectiveness of a testing technique.
The rest of this paper is organized as follows. Section \ref{sec:background_state_of_the_art} discusses the background in more depth and analyzes the state of the art. Section \ref{sec:testbed_description} describes the presented testbed from various viewpoints, including the system scope, implementation details, available automated tests, mechanism of insertion of artificial defects and process of evaluating the effectiveness of the examined testing techniques. Section \ref{sec:discussion_and_possible_limits} discusses the presented concept and also analyzes its possible limits. The last section concludes the paper.
\section{Background and State of The Art}
\label{sec:background_state_of_the_art}
As mentioned previously, a common practice to evaluate a set of test cases generated by an algorithm is to assess the defect detection rate of the test cases in an experimental SUT that contains defects. In this general approach, several aspects have to be maintained to give the technical possibility of conducting a well-defined and objective experiment. The following bullet-points address three common aspects in this direction:
\begin{enumerate}
\item The defects presented in the experimental SUT simulate real defects in software projects.
\item It is possible to create a set of various instances of an experimental SUT with different sets of injected defects to examine the testing technique for a reliable sample of situations.
\item The experimental SUT has to support effective automated evaluation of the examined test cases. Hence, the experiments can be repeated with different sets of defects in an effective manner to assess more extensive sets of situations.
\end{enumerate}
Table \ref{tab:comparison_of_defect_introduction_techniques} presents an analysis of these aspects for three possibilities of artificial defect introduction within an experimental SUT. These possibilities are as follows: (1) using real project defects, (2) mutation testing, and (3) defect injection. Defect injection, here, is a generalized method in which we do not employ standard source code mutation operators. In fact, it is difficult to reach a clear understanding of an objective approach from all three discussed options when considering all the advantages and disadvantages presented in Table \ref{tab:comparison_of_defect_introduction_techniques}. Instead, it is worthwhile to consider a combination of the presented approaches to increase the reliability of the experiments.
\begin{table*}
\caption{Brief comparison of artificial defect introduction types to an experimental SUT}\label{tab:comparison_of_defect_introduction_techniques}
\begin{centering}
\begin{tabular}{|p{1.8cm}|p{4.4cm}|p{4.4cm}|p{4.5cm}|}
\hline
\textbf{Discussed}&\multicolumn{3}{|c|}{\textbf{Defect introduction method}} \\
\cline{2-4}
\textbf{aspects} & \textbf{\textit{Historic defects}}& \textbf{\textit{Mutation testing}}& \textbf{\textit{Defect injection}} \\
\hline
The objectivity of the defects$^{\mathrm{a}}$ & The defects correspond to a real software project; however, the used sample of defects can be limited, which can restrict the objectivity of the experiment to only one particular experience-based case. &
Various combinations of mutation operators can be selected. This approach allows the flexible mixing of various defects made by the programmer. More complex defects caused by a misunderstanding of the specification can be simulated by a set of mutation operators.
& More complex simulated defects are not limited to a defined set of mutation operators. Additionally, it might be difficult to prove that an artificially elaborated defect is likely to occur in the real software development process.
\\
\hline
Ease to create instances$^{\mathrm{b}}$ & In some cases, creating multiple instances might be challenging, as there are a limited number of defects from the past software development process.
& Technically, creating new mutants is straightforward, and the number of various created SUT instances is practically unlimited. &
If a set of artificially elaborated defects is limited, then the possible number of instances of experimental SUTs that can be configured is limited.
\\
\hline
Test automation coverage$^{\mathrm{c}}$& \multicolumn{3}{|p{13.8cm}|}{Test automation options are not influenced by a particular defect introduction method; automated testability is rather influenced by the structure and coding standards employed in an experimental SUT} \\
\hline
\multicolumn{4}{l}{$^{\mathrm{a}}$How the introduced defects are realistic in comparison to real current software development process}\\
\multicolumn{4}{l}{$^{\mathrm{b}}$How easy is it to create an extensive set of various configurations of an experimental SUT with different inserted defects}\\
\multicolumn{4}{l}{$^{\mathrm{c}}$How easy is it to cover an experimental SUT by automated tests that help to evaluate whether the examined test scenarios detect an inserted defect}
\end{tabular}
\end{centering}
\end{table*}
Among the discussed approaches, mutation testing can be considered to be the most established approach, originating in the late 70s \cite{demillo1979program}. On the technical level, this approach depends on a particular programming language. However, code mutations have been performed for major programming languages. As an example, the Mujava system \cite{ma2005mujava} is used for the Java programming language and MuCPP \cite{delgado2017assessment} is used for C++. Here, for a particular program code mutation, a set of established operators is defined \cite{siami2008sufficient, offutt2011mutation}. While these operators are useful, there are concerns in the literature about the relation of the code mutants to real software defects and types of software defects that are difficult to express using various mutants \cite{papadakis2019mutation,gopinath2014mutations}. To overcome this problem, various approaches have been considered in the literature -- for instance, the construction of more complex mutants \cite{papadakis2019mutation}. However, the mutation testing approach might still meet its limit when trying to insert certain types of complex defects that may be caused by a misunderstanding of the design specification \cite{gopinath2014mutations}. Generally, the similarity of mutants to real defects varies in empirical experiments \cite{gopinath2014mutations,andrews2005mutation}.
The defect injection method can be seen as a more general method than mutation testing to insert artificial defects into experimental software. In this process, various techniques at any software level can be used to insert defects, e.g., \cite{cotroneo2012experimental,kooli2014survey}. As an alternative to mutation testing and artificial defect injection, a number of experiments have also been conducted using real defects from past software projects, e.g., \cite{bures2018tapir}. Here, comparing those different approaches is challenging because the objectivity of the experiment in which we evaluate the effectiveness of the testing techniques strongly depends on the testing technique, the characteristics of the software used as a benchmark, and how realistic the inserted defects are compared to real defects. Moreover, the characteristics of the software defects might also change with changes in the development styles, the usage of integrated development environments and the best practices of programming. To this end, in this paper, we suggest applying a benchmark testbed as a complement to the mutation testing approach.
\section{Testbed Description}
\label{sec:testbed_description}
To create a benchmark testbed for the evaluation of the effectiveness of the test technique, we designed and implemented the University Information System Testbed (TbUIS)\footnote{https://projects.kiv.zcu.cz/tbuis/}. The testbed is an open-source testbed that can be used to evaluate any test technique. The TbUIS system is a three-layered web application that uses a relational database as a persistent data storage and object-relational mapping (ORM) layer.
The system supports the artificial defect injection approach, as discussed in Table \ref{tab:comparison_of_defect_introduction_techniques} (column \textit{Defect injection}). A special module allows the creation of defect clones of the system by introducing defects from a catalog of predefined defect types as well as creating customized artificial defects to be inserted into the SUT. As a demonstration and for a quick start for experiments\color{black}, a set of 28 already assembled defect clones are available for testbed users.
In this section, we describe the following aspects of the TbUIS: (A) the scope of the system and its use cases, (B) the implementation and technical details, (C) the available automated tests to be employed in the experiments, (D) the mechanism for introducing artificial defects in the system and (E) the test case effectiveness evaluation process, including the logging mechanism used in the evaluations.
\subsection{Scope and Use Cases of the TbUIS}
\label{sec:scope_and_use_cases}
The TbUIS is a fictional university study information system that supports a study agenda related to students' enrolment in courses, management of exams and related processes. The standard actors of the system are students and lecturers.
The whole system can be summarized into 21 general, high-level use cases. Five use cases are related to a user who is not logged in. Another two use cases are common for lecturers and students and cover the login mechanism and user settings. The students' part is then defined by five use cases and the lecturers' part by nine separate use cases.
The graphical layout of the user interface (UI) of the system is kept relatively simple and compact, considering the goal of the system, which is to evaluate testing techniques as well as the need to cover the application by reliable front-end (FE) based automated tests to support this process (introduced later in Section \ref{sec:automated_tests}). An example of the TbUIS user interface is presented in Figure \ref{fig:ui_sample}.
\begin{figure}
\centerline{\includegraphics[width=8.5cm]{ui_sample.png}}
\caption{Example of TbUIS user interface---lecturer's view.}
\label{fig:ui_sample}
\end{figure}
Regarding the process flow, the possible states and functions of the system are documented in the UML Activity Diagram schema in the Oxygen\footnote{http://still.felk.cvut.cz/oxygen/} \cite{bures2015pctgen} application and are available in the Oxygen project format as well as the SVG graphical format. This model of the current version of the TbUIS is composed of 119 different states and 164 transitions among them.
\subsection{Implementation and Technical Details}
Technically, the TbUIS is a layered web-based application implemented in J2EE with Java Server Pages (JSP) and Spring. As the ORM layer, Hibernate is used. For the implementation of the UI, Bootstrap is used.
In the user interface of the TbUIS, all the common basic types of control for web elements (e.g., menus, buttons, check boxes, selections, modal windows, etc.) are used. Each element (including rows in tables) has its own unique ID attribute to ease the creation of FE-based functional automated tests.
The extent of the TbUIS source code is documented in Table \ref{tab:source_code_size}. Here, the number of source files, size of source code files in kilobytes and number of lines of code (LOC) are presented separately for back-end code in Java as well as for UI code in JSP. Unit tests as well as functional automated tests are not included in these statistics.
\begin{table}
\begin{center}
\caption{Size of TbUIS source code}
\begin{tabular}{|r|c|c|c|}\hline
& Number of files & Size of files [KB] & LOC \\ \hline\hline
Java & 87 & 340 & 8550 \\ \hline
JSP & 18 & 94 & 1550 \\ \hline\hline
total & 105 & 434 & 10100 \\ \hline
\end{tabular}
\label{tab:source_code_size}
\end{center}
\end{table}
Any important activity in the TbUIS testbed is reported in detailed application logs implemented by Log4J2. Because of the logging framework configuration options, the user can customize the level of detail and the output stream of the log. The application logs are also extended by the activation information of the inserted artificial defects (further discussed in Section \ref{sec:artificial_defects}) and can be paired with the logs of available functional automated tests (further discussed in Section \ref{sec:automated_tests}).
\subsection{Automated tests}
\label{sec:automated_tests}
TbUIS is strongly covered by various types of automated tests that have the following two goals:
\begin{enumerate}
\item To ensure that the system (before the introduction of controlled artificial defects used to evaluate the effectiveness of testing techniques) is largely free of other defects and
\item To support the process of evaluating the effectiveness of the testing techniques by executing the defined test cases that are to be examined in the system via automated tests,
\end{enumerate}
Two types of tests are available as extra modules for the TbUIS testbed:
\begin{enumerate}
\item \textbf{Unit tests} implemented in the JUnit framework, which test individual methods of the system and the basic sequences of methods calls on the technical level.
\item \textbf{FE-based functional tests}, which simulate users' tests accessing the system UI. These tests are written in Java with the Selenium Web Driver API, currently version 3.141.59. The tests are structured using the PageObject pattern, which significantly decreases their maintenance and allows future extensions of the test set, as independently verified \cite{bures2015model}.
\end{enumerate}
Regarding the coverage level, in the current version of the TbUIS, the line coverage of the available unit tests is greater than 85\%.
The FE-based functional tests cover all of the processes, as documented in the process flow schema created in the Oxygen application (introduced above in Section \ref{sec:scope_and_use_cases}).
To determine the expected test results of the FE-based functional tests, the Oracle module is implemented and is thoroughly tested using a special set of unit tests.
Table \ref{tab:source_codes_of_tests} provides insight into the extent of the implemented automated tests. The number of source code files, their size in kilobytes and the number of lines of code (LOC) are presented. The FE-based functional tests for TbUIS employ several modules of reusable objects and support code (in Table \ref{tab:source_codes_of_tests} denoted as \textit{Shared libraries for FE-based functional tests}). These modules are also covered by their own set of unit tests.
\begin{table}
\begin{center}
\caption{Extent of source code of automated tests}
\begin{tabular}{|p{2.5cm}|c|c|c|}\hline
& Number of files & Size of files [KB] & LOC \\ \hline\hline
Unit tests for TbUIS code& \hfill{}\lower1.8mm\hbox{33}\hspace*{7mm} & \hfill{}\lower1.8mm\hbox{277}\hspace*{8mm} & \hfill{}\lower1.8mm\hbox{6945}\hspace*{0.5mm} \\ \hline
Shared libraries for FE-based functional tests & \hfill{}\lower3.0mm\hbox{101}\hspace*{7mm} & \hfill{}\lower3.0mm\hbox{452}\hspace*{8mm} & \hfill{}\lower3.0mm\hbox{14707}\hspace*{0.5mm} \\ \hline
Unit tests for shared libraries for FE-based functional tests & \hfill{}\lower3.0mm\hbox{30}\hspace*{7mm} & \hfill{}\lower3.0mm\hbox{97}\hspace*{8mm} & \hfill{}\lower3.0mm\hbox{2649}\hspace*{0.5mm} \\ \hline
FE-based functional tests for TbUIS & \hfill{}\lower1.8mm\hbox{81}\hspace*{7mm} & \hfill{}\lower1.8mm\hbox{248}\hspace*{8mm} & \hfill{}\lower1.8mm\hbox{7530}\hspace*{0.5mm} \\ \hline\hline
total & \hfill{}245\hspace*{7mm} & \hfill{}1024\hspace*{8mm} & \hfill{}31831\hspace*{0.5mm} \\ \hline
\end{tabular}
\label{tab:source_codes_of_tests}
\end{center}
\end{table}
Compared to size of the source code of the TbUIS (see Table \ref{tab:source_code_size}), the extent of the automated tests measured in terms of LOC is approximately three times higher.
The FE-based functional automated tests are divided into several types, covering various technical and user aspects of the TbUIS:
\begin{itemize}
\item \textbf{Atomic tests} that are verifying if elements of application UI are correctly rendered and filled with correct data
\item \textbf{Process tests} that are exercising individual processes in the TbUIS (e.g. enrolling a course or assigning a grade to the student)
\item \textbf{Negative tests} that are testing boundary conditions and correct handling of wrong input data
\end{itemize}
Atomic types of tests are also orchestrated as parts of the process tests. The test scripts are organized into building blocks that allow the automated composition of an automated end-to-end test via a defined path-based test scenario (the details are presented in Section \ref{sec:test_case_effectiveness_evaluation}).
The numbers of tests in the individual categories with their numbers of asserts and average runtime are presented in Table \ref{tab:testing_scale}. The runtimes were measured using the following configuration: Intel i5 1.6 GHz, 16 GB RAM, MS Windows 10pro operating system, Apache Tomcat 9.0 application server and MySQL database. The database and web and application servers were installed on the same workstation, and the automated tests were run on the same computer.
\begin{table}
\begin{center}
\caption{Types of FE-based functional automated tests}
\begin{tabular}{|l|p{1.2cm}|p{1.2cm}|p{1.2cm}|}\hline
& Number of tests & Number of asserts & Elapsed time [sec] \\ \hline\hline
Atomic tests & \hfill{}890\hspace*{4mm} & \hfill{}2702\hspace*{4mm} & \hfill{}780\hspace*{4mm} \\ \hline
Process tests & \hfill{}64\hspace*{4mm} & \hfill{}2351\hspace*{4mm} & \hfill{}1477\hspace*{4mm} \\ \hline
Negative tests & \hfill{}29\hspace*{4mm} & \hfill{}52\hspace*{4mm} & \hfill{}50\hspace*{4mm} \\ \hline\hline
total & \hfill{}983\hspace*{4mm} & \hfill{}5105\hspace*{4mm} & \hfill{}2307\hspace*{4mm} \\ \hline
\end{tabular}
\label{tab:testing_scale}
\end{center}
\end{table}
The automated atomic tests cover 100\% of all active and passive elements composing the user interface of the TbUIS. As active elements, we consider user control elements (e.g., text boxes, drop-down menus, links, etc.) and fields that display data loaded from the database or are taken from the runtime memory of the application. Each of the active elements is tested at least by one atomic test.
FE-based automated functional tests can be easily run from a special application, TestRunner, which provides its own user interface in which particular tests to run can be selected. The TestRunner application can be downloaded from the project web page.
The extent of the building blocks of the FE-based automated functional tests introduced in this section allows the effective composition of automated tests for the path-based test scenarios to be evaluated in the testbed. The relevant part of these blocks can also be used to evaluate the combinatorial or constrained interaction testing test sets.
\subsection{Introduction of Artificial Defects}
\label{sec:artificial_defects}
Artificial defects are introduced into the TbUIS by the error seeder module, which conducts the following process:
\begin{enumerate}
\item The error seeder takes the baseline TbUIS code, which is considered free of defects (which is verified by the thorough automated tests introduced in Section \ref{sec:automated_tests}).
\item Based on the artificial defect specification, the error seeder assembles the source code of a defect clone of the TbUIS.
\item Then, the defect clone of the TbUIS system is built and deployed to a testing environment.
\end{enumerate}
Artificial defect specification defines a set of artificial defects that are to be introduced into the TbUIS code. Predefined catalogue defect types are available as well as the possibility to define custom artificial defects. The catalogue of defect types is available on the project web page.
Each artificial defect inserted into the TbUIS code is accompanied by a logging mechanism that records information if and when the defect has been activated by a test. The main purpose of this information is to support the evaluation of the effectiveness of the testing techniques. The defect activation logs can be paired with the logs of available automated tests to give reliable sources of information, which artificial defect were detected by which test cases.
In the current version of the TbUIS testbeds, a set of 27 artificial defects of various types from the above catalogue is available for initial experiments and are accompanied by detailed information making their application easy\footnote{https://projects.kiv.zcu.cz/tbuis/web/page/download}.
As mentioned above, for further experiments and to evaluate the effectiveness of the testing techniques, more various defect clones can be created and compiled from available artificial defects, and also, based on the well-documented examples in the source code, the user can implement their own artificial defects.
\subsection{Test Case Effectiveness Evaluation Process}
\label{sec:test_case_effectiveness_evaluation}
As introduced above, in principle, the effectiveness of various testing techniques can be evaluated in the TbUIS testbed. In the following section, we focus on two major representatives, path-based techniques and combinatorial/constrained interaction testing techniques.
The parts of the TbUIS testbed related to the evaluation of path-based testing techniques are summarized in Figure \ref{fig:testbed_parts_paths}. The inputs and outputs of the process are depicted by yellow boxes.
\begin{figure}
\centerline{\includegraphics[width=9cm]{testbed_parts__paths.png}}
\caption{TbUIS parts for evaluation of path-based testing techniques.}
\label{fig:testbed_parts_paths}
\end{figure}
The input to the process is a \textit{path-based test set}, whose effectiveness is going to be evaluated. The test cases in this test set have to correspond to an available \textit{TbUIS process model} (unless we intentionally created invalid paths-based test cases in the experiment). Using predefined building blocks from the \textit{FE-based functional automated test scripts} (introduced in Section \ref{sec:automated_tests}), the \textit{process test builder} chains these building blocks as instructed by the input path-based test cases to produce \textit{assembled FE automated tests}, which represent individual path-based test cases. For each of the path-based test cases at the input, a corresponding automated FE test is created.
The second input of the process is \textit{artificial defect specification}, which can be created via predefined \textit{catalogue defect types} or \textit{own defects} defined in the SUT. To create an \textit{defect clone} of the TbUIS with the specified defects, \textit{Error seeder} takes the specification of the defects and inserts them into the code of the Baseline UIS specification. Then, the defect clone is built as a running system instance.
At this stage, experimental evaluation of the path-based test set can be performed (an example is provided on the project web pages). Automated FE tests corresponding to the input path-based test cases are run in the defect clone, and the results are reported to the \textit{test report}, which can be evaluated. The information from the test report can also be paired with detailed \textit{application logs} to obtain more context information about the activated defects.
The schema for evaluating combinatorial or constrained testing test cases slightly differs, but the general principle remains the same.
In this type of evaluation, we do not compile FE automated tests to correspond to path-based test cases; instead, we can use
\begin{enumerate}
\item available automated FE-based functional tests covering all active elements and processes in the TbUIS,
\item available unit tests available together with the TbUIS code, or
\item combinations of both types of tests (the automated tests available to the TbUIS were introduced in Section \ref{sec:automated_tests}).
\end{enumerate}
Input data combinations to be exercised in the testbed can be entered into the available automated tests via the standardized DataProvider interface of the JUnit framework.
\section{Discussion and Possible Limits}
\label{sec:discussion_and_possible_limits}
Like other alternative artificial defect introduction approaches discussed in this paper, namely, using real defects from a previous software project and code mutation, the approach taken in the proposed testbed has certain advantages and disadvantages. We summarize these advantages and disadvantages in this section.
Regarding the possible complexity of the artificial defects introduced into an experimental SUT, the proposed approach does not limit an artificial defect to a set of mutation operators or a conditionally switched block of code. Instead, the defect clone can be built with the changes made in several different places in the source code, which allows high flexibility in simulating complex defects.
Concern whether the introduced defects represent typical defects that are being made during real software projects can be raised. This responsibility in experiments is up to the researchers and testing practitioners. Typical defects might vary between various software architectures, development styles, programming languages, business domains, and even decades when the empirical observations are made. Hence, the testbed provides a general possibility to create different types of defects and defect clones, and the decision is up to the testbed user.
In the proposed concept, the artificial defects are selected from a pre-defined set, which might limit the generalization of experiment results. This potential limit can be solved by the addition of more artificial defects as well as the correct interpretation of the results of the experiments.
Also, certain defects might be easier to detect than other defects, which may impact the results of the experiments \cite{papadakis2016threats}. However, this concern can be raised generally for any defect injection technique and shall be mitigated by correct interpretation of the results of the experiments.
Another concern is that the system is artificially created; however, the use cases and processes in the SUT are similar to real-world study information systems. The more important factor here is the selection of artificial defects that are representative of real-world projects. In the presented testbed, this selection is enabled by the possible introduction of more complex defects via the described mechanism of the defect clones.
Also, the size of the TbUIS system might limit its potential applicability as a benchmark for larger software systems. We are going to mitigate this concern by further evolution and extensions of the TbUIS.
\section{Conclusion}
In evaluating the effectiveness of testing techniques based on the measurement of the defect number that the test cases produced by these techniques detect in an experimental system, the established mutation testing approach can be accompanied by an alternative allowing the insertion of more complex defects caused by a misunderstanding of the design specification or other causes. We describe such an alternative in this paper: the presented TbUIS testbed, which is available as an open-source application and comprises a fictional university information system. The TbUIS testbed gives its user a mechanism to introduce artificial defects, including those from a predefined catalogue of possible defects, an extensive set of unit and FE-based functional automated tests, which can be used to examine test cases in the system, and a logging mechanism, which allows the collection of the data regarding which defects were activated by the examined test cases. Together with a good level of code and system documentation, the open structure of the TbUIS testbed eases its employment as a benchmark system to be used in the evaluation of path-based and combinatorial/constrained interaction testing techniques.
\section*{Acknowledgment}
This work was supported by the European structural and investment funds (ESIF) project CZ.02.1.01/0.0/0.0/17\_048/0007267 (InteCom)---Intelligent Components of Advanced Technologies for the Pilsen metropolitan area. Work package WP1.3: Methods and processes for control software safety assurance. The authors acknowledge the support of the OP VVV funded project CZ.02.1.01/0.0/0.0/16\_019/0000765 "Research Center for Informatics".
\bibliographystyle{IEEEtran}
\section{Introduction}
To evaluate the effectiveness of a testing technique for software systems, various approaches can be employed. A natural and well-known approach to assess the effectiveness of a test suite generated by a testing technique is to measure the defect detection rate when applying a generated test suite to a System Under Test (SUT). As such, an experimental SUT that represents a real-world system containing real defects from the past software development process can be useful here. Alternatively, the mutation testing technique can be applied by introducing artificial defects into the code of an experimental SUT using defined mutation operators \cite{siami2008sufficient, offutt2011mutation}. Additionally, a defect injection technique, which can be considered to be a more general variant, can be employed. In defect injection, defects are introduced into an experimental SUT and various technical possibilities can be used.
Measuring the defect detection rate can be used to determine the effectiveness of the Combinatorial or Constrained Interaction Testing \cite{nie2011survey,CombConsTBestoun} or Path-based Testing \cite{bures2015pctgen} techniques. As a typical example, one can examine the strength of test cases generated by the Combinatorial Interaction Testing (CIT) algorithm using a mutation testing technique. As an experimental SUT, an open-source software system can be selected. Then, various mutants are created from the source code by a set of mutation operators. Subsequently, the generated test cases are assessed in the experimental SUT and it is determined whether the test case can detect a defect introduced into SUT by a mutation operator. These types of experiments are typically run in multiple series with various sets of mutants and test cases to obtain convincing evidence regarding the effectiveness of the generated test cases \cite{offutt2011mutation}.
The approach can be generalized as illustrated in Figure \ref{fig:defect_introduction_process}.
\begin{figure}[htbp]
\centerline{\includegraphics[width=7cm]{defect_introduction_process.png}}
\caption{Defect introduction process to an experimental SUT.}
\label{fig:defect_introduction_process}
\end{figure}
In this paper, we present a new open-source benchmark testbed to support defect injection testing. The testbed is available to the community and can be used to evaluate various testing techniques. In this approach, we do not insist on defined mutation operators. The goal of the testbed is to provide a complement to the classical mutation testing approach for evaluating the effectiveness of test cases.
In contrast to the established classical code mutation operators, various complex software defects can be introduced into the code, especially defects caused by a misunderstanding of the SUT design specification or requirements during the development process. The practical use case of the presented testbed is to provide researchers with a complementary option to the mutation testing technique to be able to simulate a broader spectrum of possible software defects during experiments. The testbed is, hence, a complement to mutation testing rather a replacement of mutation testing via a defect injection approach. As we show later in Section \ref{sec:background_state_of_the_art}, both approaches have certain advantages and disadvantages. Hence, both approaches can be combined to provide the best objective measurement of the effectiveness of a testing technique.
The rest of this paper is organized as follows. Section \ref{sec:background_state_of_the_art} discusses the background in more depth and analyzes the state of the art. Section \ref{sec:testbed_description} describes the presented testbed from various viewpoints, including the system scope, implementation details, available automated tests, mechanism of insertion of artificial defects and process of evaluating the effectiveness of the examined testing techniques. Section \ref{sec:discussion_and_possible_limits} discusses the presented concept and also analyzes its possible limits. The last section concludes the paper.
\section{Background and State of The Art}
\label{sec:background_state_of_the_art}
As mentioned previously, a common practice to evaluate a set of test cases generated by an algorithm is to assess the defect detection rate of the test cases in an experimental SUT that contains defects. In this general approach, several aspects have to be maintained to give the technical possibility of conducting a well-defined and objective experiment. The following bullet-points address three common aspects in this direction:
\begin{enumerate}
\item The defects presented in the experimental SUT simulate real defects in software projects.
\item It is possible to create a set of various instances of an experimental SUT with different sets of injected defects to examine the testing technique for a reliable sample of situations.
\item The experimental SUT has to support effective automated evaluation of the examined test cases. Hence, the experiments can be repeated with different sets of defects in an effective manner to assess more extensive sets of situations.
\end{enumerate}
Table \ref{tab:comparison_of_defect_introduction_techniques} presents an analysis of these aspects for three possibilities of artificial defect introduction within an experimental SUT. These possibilities are as follows: (1) using real project defects, (2) mutation testing, and (3) defect injection. Defect injection, here, is a generalized method in which we do not employ standard source code mutation operators. In fact, it is difficult to reach a clear understanding of an objective approach from all three discussed options when considering all the advantages and disadvantages presented in Table \ref{tab:comparison_of_defect_introduction_techniques}. Instead, it is worthwhile to consider a combination of the presented approaches to increase the reliability of the experiments.
\begin{table*}
\caption{Brief comparison of artificial defect introduction types to an experimental SUT}\label{tab:comparison_of_defect_introduction_techniques}
\begin{centering}
\begin{tabular}{|p{1.8cm}|p{4.4cm}|p{4.4cm}|p{4.5cm}|}
\hline
\textbf{Discussed}&\multicolumn{3}{|c|}{\textbf{Defect introduction method}} \\
\cline{2-4}
\textbf{aspects} & \textbf{\textit{Historic defects}}& \textbf{\textit{Mutation testing}}& \textbf{\textit{Defect injection}} \\
\hline
The objectivity of the defects$^{\mathrm{a}}$ & The defects correspond to a real software project; however, the used sample of defects can be limited, which can restrict the objectivity of the experiment to only one particular experience-based case. &
Various combinations of mutation operators can be selected. This approach allows the flexible mixing of various defects made by the programmer. More complex defects caused by a misunderstanding of the specification can be simulated by a set of mutation operators.
& More complex simulated defects are not limited to a defined set of mutation operators. Additionally, it might be difficult to prove that an artificially elaborated defect is likely to occur in the real software development process.
\\
\hline
Ease to create instances$^{\mathrm{b}}$ & In some cases, creating multiple instances might be challenging, as there are a limited number of defects from the past software development process.
& Technically, creating new mutants is straightforward, and the number of various created SUT instances is practically unlimited. &
If a set of artificially elaborated defects is limited, then the possible number of instances of experimental SUTs that can be configured is limited.
\\
\hline
Test automation coverage$^{\mathrm{c}}$& \multicolumn{3}{|p{13.8cm}|}{Test automation options are not influenced by a particular defect introduction method; automated testability is rather influenced by the structure and coding standards employed in an experimental SUT} \\
\hline
\multicolumn{4}{l}{$^{\mathrm{a}}$How the introduced defects are realistic in comparison to real current software development process}\\
\multicolumn{4}{l}{$^{\mathrm{b}}$How easy is it to create an extensive set of various configurations of an experimental SUT with different inserted defects}\\
\multicolumn{4}{l}{$^{\mathrm{c}}$How easy is it to cover an experimental SUT by automated tests that help to evaluate whether the examined test scenarios detect an inserted defect}
\end{tabular}
\end{centering}
\end{table*}
Among the discussed approaches, mutation testing can be considered to be the most established approach, originating in the late 70s \cite{demillo1979program}. On the technical level, this approach depends on a particular programming language. However, code mutations have been performed for major programming languages. As an example, the Mujava system \cite{ma2005mujava} is used for the Java programming language and MuCPP \cite{delgado2017assessment} is used for C++. Here, for a particular program code mutation, a set of established operators is defined \cite{siami2008sufficient, offutt2011mutation}. While these operators are useful, there are concerns in the literature about the relation of the code mutants to real software defects and types of software defects that are difficult to express using various mutants \cite{papadakis2019mutation,gopinath2014mutations}. To overcome this problem, various approaches have been considered in the literature -- for instance, the construction of more complex mutants \cite{papadakis2019mutation}. However, the mutation testing approach might still meet its limit when trying to insert certain types of complex defects that may be caused by a misunderstanding of the design specification \cite{gopinath2014mutations}. Generally, the similarity of mutants to real defects varies in empirical experiments \cite{gopinath2014mutations,andrews2005mutation}.
The defect injection method can be seen as a more general method than mutation testing to insert artificial defects into experimental software. In this process, various techniques at any software level can be used to insert defects, e.g., \cite{cotroneo2012experimental,kooli2014survey}. As an alternative to mutation testing and artificial defect injection, a number of experiments have also been conducted using real defects from past software projects, e.g., \cite{bures2018tapir}. Here, comparing those different approaches is challenging because the objectivity of the experiment in which we evaluate the effectiveness of the testing techniques strongly depends on the testing technique, the characteristics of the software used as a benchmark, and how realistic the inserted defects are compared to real defects. Moreover, the characteristics of the software defects might also change with changes in the development styles, the usage of integrated development environments and the best practices of programming. To this end, in this paper, we suggest applying a benchmark testbed as a complement to the mutation testing approach.
\section{Testbed Description}
\label{sec:testbed_description}
To create a benchmark testbed for the evaluation of the effectiveness of the test technique, we designed and implemented the University Information System Testbed (TbUIS)\footnote{https://projects.kiv.zcu.cz/tbuis/}. The testbed is an open-source testbed that can be used to evaluate any test technique. The TbUIS system is a three-layered web application that uses a relational database as a persistent data storage and object-relational mapping (ORM) layer.
The system supports the artificial defect injection approach, as discussed in Table \ref{tab:comparison_of_defect_introduction_techniques} (column \textit{Defect injection}). A special module allows the creation of defect clones of the system by introducing defects from a catalog of predefined defect types as well as creating customized artificial defects to be inserted into the SUT. As a demonstration and for a quick start for experiments\color{black}, a set of 28 already assembled defect clones are available for testbed users.
In this section, we describe the following aspects of the TbUIS: (A) the scope of the system and its use cases, (B) the implementation and technical details, (C) the available automated tests to be employed in the experiments, (D) the mechanism for introducing artificial defects in the system and (E) the test case effectiveness evaluation process, including the logging mechanism used in the evaluations.
\subsection{Scope and Use Cases of the TbUIS}
\label{sec:scope_and_use_cases}
The TbUIS is a fictional university study information system that supports a study agenda related to students' enrolment in courses, management of exams and related processes. The standard actors of the system are students and lecturers.
The whole system can be summarized into 21 general, high-level use cases. Five use cases are related to a user who is not logged in. Another two use cases are common for lecturers and students and cover the login mechanism and user settings. The students' part is then defined by five use cases and the lecturers' part by nine separate use cases.
The graphical layout of the user interface (UI) of the system is kept relatively simple and compact, considering the goal of the system, which is to evaluate testing techniques as well as the need to cover the application by reliable front-end (FE) based automated tests to support this process (introduced later in Section \ref{sec:automated_tests}). An example of the TbUIS user interface is presented in Figure \ref{fig:ui_sample}.
\begin{figure}
\centerline{\includegraphics[width=8.5cm]{ui_sample.png}}
\caption{Example of TbUIS user interface---lecturer's view.}
\label{fig:ui_sample}
\end{figure}
Regarding the process flow, the possible states and functions of the system are documented in the UML Activity Diagram schema in the Oxygen\footnote{http://still.felk.cvut.cz/oxygen/} \cite{bures2015pctgen} application and are available in the Oxygen project format as well as the SVG graphical format. This model of the current version of the TbUIS is composed of 119 different states and 164 transitions among them.
\subsection{Implementation and Technical Details}
Technically, the TbUIS is a layered web-based application implemented in J2EE with Java Server Pages (JSP) and Spring. As the ORM layer, Hibernate is used. For the implementation of the UI, Bootstrap is used.
In the user interface of the TbUIS, all the common basic types of control for web elements (e.g., menus, buttons, check boxes, selections, modal windows, etc.) are used. Each element (including rows in tables) has its own unique ID attribute to ease the creation of FE-based functional automated tests.
The extent of the TbUIS source code is documented in Table \ref{tab:source_code_size}. Here, the number of source files, size of source code files in kilobytes and number of lines of code (LOC) are presented separately for back-end code in Java as well as for UI code in JSP. Unit tests as well as functional automated tests are not included in these statistics.
\begin{table}
\begin{center}
\caption{Size of TbUIS source code}
\begin{tabular}{|r|c|c|c|}\hline
& Number of files & Size of files [KB] & LOC \\ \hline\hline
Java & 87 & 340 & 8550 \\ \hline
JSP & 18 & 94 & 1550 \\ \hline\hline
total & 105 & 434 & 10100 \\ \hline
\end{tabular}
\label{tab:source_code_size}
\end{center}
\end{table}
Any important activity in the TbUIS testbed is reported in detailed application logs implemented by Log4J2. Because of the logging framework configuration options, the user can customize the level of detail and the output stream of the log. The application logs are also extended by the activation information of the inserted artificial defects (further discussed in Section \ref{sec:artificial_defects}) and can be paired with the logs of available functional automated tests (further discussed in Section \ref{sec:automated_tests}).
\subsection{Automated tests}
\label{sec:automated_tests}
TbUIS is strongly covered by various types of automated tests that have the following two goals:
\begin{enumerate}
\item To ensure that the system (before the introduction of controlled artificial defects used to evaluate the effectiveness of testing techniques) is largely free of other defects and
\item To support the process of evaluating the effectiveness of the testing techniques by executing the defined test cases that are to be examined in the system via automated tests,
\end{enumerate}
Two types of tests are available as extra modules for the TbUIS testbed:
\begin{enumerate}
\item \textbf{Unit tests} implemented in the JUnit framework, which test individual methods of the system and the basic sequences of methods calls on the technical level.
\item \textbf{FE-based functional tests}, which simulate users' tests accessing the system UI. These tests are written in Java with the Selenium Web Driver API, currently version 3.141.59. The tests are structured using the PageObject pattern, which significantly decreases their maintenance and allows future extensions of the test set, as independently verified \cite{bures2015model}.
\end{enumerate}
Regarding the coverage level, in the current version of the TbUIS, the line coverage of the available unit tests is greater than 85\%.
The FE-based functional tests cover all of the processes, as documented in the process flow schema created in the Oxygen application (introduced above in Section \ref{sec:scope_and_use_cases}).
To determine the expected test results of the FE-based functional tests, the Oracle module is implemented and is thoroughly tested using a special set of unit tests.
Table \ref{tab:source_codes_of_tests} provides insight into the extent of the implemented automated tests. The number of source code files, their size in kilobytes and the number of lines of code (LOC) are presented. The FE-based functional tests for TbUIS employ several modules of reusable objects and support code (in Table \ref{tab:source_codes_of_tests} denoted as \textit{Shared libraries for FE-based functional tests}). These modules are also covered by their own set of unit tests.
\begin{table}
\begin{center}
\caption{Extent of source code of automated tests}
\begin{tabular}{|p{2.5cm}|c|c|c|}\hline
& Number of files & Size of files [KB] & LOC \\ \hline\hline
Unit tests for TbUIS code& \hfill{}\lower1.8mm\hbox{33}\hspace*{7mm} & \hfill{}\lower1.8mm\hbox{277}\hspace*{8mm} & \hfill{}\lower1.8mm\hbox{6945}\hspace*{0.5mm} \\ \hline
Shared libraries for FE-based functional tests & \hfill{}\lower3.0mm\hbox{101}\hspace*{7mm} & \hfill{}\lower3.0mm\hbox{452}\hspace*{8mm} & \hfill{}\lower3.0mm\hbox{14707}\hspace*{0.5mm} \\ \hline
Unit tests for shared libraries for FE-based functional tests & \hfill{}\lower3.0mm\hbox{30}\hspace*{7mm} & \hfill{}\lower3.0mm\hbox{97}\hspace*{8mm} & \hfill{}\lower3.0mm\hbox{2649}\hspace*{0.5mm} \\ \hline
FE-based functional tests for TbUIS & \hfill{}\lower1.8mm\hbox{81}\hspace*{7mm} & \hfill{}\lower1.8mm\hbox{248}\hspace*{8mm} & \hfill{}\lower1.8mm\hbox{7530}\hspace*{0.5mm} \\ \hline\hline
total & \hfill{}245\hspace*{7mm} & \hfill{}1024\hspace*{8mm} & \hfill{}31831\hspace*{0.5mm} \\ \hline
\end{tabular}
\label{tab:source_codes_of_tests}
\end{center}
\end{table}
Compared to size of the source code of the TbUIS (see Table \ref{tab:source_code_size}), the extent of the automated tests measured in terms of LOC is approximately three times higher.
The FE-based functional automated tests are divided into several types, covering various technical and user aspects of the TbUIS:
\begin{itemize}
\item \textbf{Atomic tests} that are verifying if elements of application UI are correctly rendered and filled with correct data
\item \textbf{Process tests} that are exercising individual processes in the TbUIS (e.g. enrolling a course or assigning a grade to the student)
\item \textbf{Negative tests} that are testing boundary conditions and correct handling of wrong input data
\end{itemize}
Atomic types of tests are also orchestrated as parts of the process tests. The test scripts are organized into building blocks that allow the automated composition of an automated end-to-end test via a defined path-based test scenario (the details are presented in Section \ref{sec:test_case_effectiveness_evaluation}).
The numbers of tests in the individual categories with their numbers of asserts and average runtime are presented in Table \ref{tab:testing_scale}. The runtimes were measured using the following configuration: Intel i5 1.6 GHz, 16 GB RAM, MS Windows 10pro operating system, Apache Tomcat 9.0 application server and MySQL database. The database and web and application servers were installed on the same workstation, and the automated tests were run on the same computer.
\begin{table}
\begin{center}
\caption{Types of FE-based functional automated tests}
\begin{tabular}{|l|p{1.2cm}|p{1.2cm}|p{1.2cm}|}\hline
& Number of tests & Number of asserts & Elapsed time [sec] \\ \hline\hline
Atomic tests & \hfill{}890\hspace*{4mm} & \hfill{}2702\hspace*{4mm} & \hfill{}780\hspace*{4mm} \\ \hline
Process tests & \hfill{}64\hspace*{4mm} & \hfill{}2351\hspace*{4mm} & \hfill{}1477\hspace*{4mm} \\ \hline
Negative tests & \hfill{}29\hspace*{4mm} & \hfill{}52\hspace*{4mm} & \hfill{}50\hspace*{4mm} \\ \hline\hline
total & \hfill{}983\hspace*{4mm} & \hfill{}5105\hspace*{4mm} & \hfill{}2307\hspace*{4mm} \\ \hline
\end{tabular}
\label{tab:testing_scale}
\end{center}
\end{table}
The automated atomic tests cover 100\% of all active and passive elements composing the user interface of the TbUIS. As active elements, we consider user control elements (e.g., text boxes, drop-down menus, links, etc.) and fields that display data loaded from the database or are taken from the runtime memory of the application. Each of the active elements is tested at least by one atomic test.
FE-based automated functional tests can be easily run from a special application, TestRunner, which provides its own user interface in which particular tests to run can be selected. The TestRunner application can be downloaded from the project web page.
The extent of the building blocks of the FE-based automated functional tests introduced in this section allows the effective composition of automated tests for the path-based test scenarios to be evaluated in the testbed. The relevant part of these blocks can also be used to evaluate the combinatorial or constrained interaction testing test sets.
\subsection{Introduction of Artificial Defects}
\label{sec:artificial_defects}
Artificial defects are introduced into the TbUIS by the error seeder module, which conducts the following process:
\begin{enumerate}
\item The error seeder takes the baseline TbUIS code, which is considered free of defects (which is verified by the thorough automated tests introduced in Section \ref{sec:automated_tests}).
\item Based on the artificial defect specification, the error seeder assembles the source code of a defect clone of the TbUIS.
\item Then, the defect clone of the TbUIS system is built and deployed to a testing environment.
\end{enumerate}
Artificial defect specification defines a set of artificial defects that are to be introduced into the TbUIS code. Predefined catalogue defect types are available as well as the possibility to define custom artificial defects. The catalogue of defect types is available on the project web page.
Each artificial defect inserted into the TbUIS code is accompanied by a logging mechanism that records information if and when the defect has been activated by a test. The main purpose of this information is to support the evaluation of the effectiveness of the testing techniques. The defect activation logs can be paired with the logs of available automated tests to give reliable sources of information, which artificial defect were detected by which test cases.
In the current version of the TbUIS testbeds, a set of 27 artificial defects of various types from the above catalogue is available for initial experiments and are accompanied by detailed information making their application easy\footnote{https://projects.kiv.zcu.cz/tbuis/web/page/download}.
As mentioned above, for further experiments and to evaluate the effectiveness of the testing techniques, more various defect clones can be created and compiled from available artificial defects, and also, based on the well-documented examples in the source code, the user can implement their own artificial defects.
\subsection{Test Case Effectiveness Evaluation Process}
\label{sec:test_case_effectiveness_evaluation}
As introduced above, in principle, the effectiveness of various testing techniques can be evaluated in the TbUIS testbed. In the following section, we focus on two major representatives, path-based techniques and combinatorial/constrained interaction testing techniques.
The parts of the TbUIS testbed related to the evaluation of path-based testing techniques are summarized in Figure \ref{fig:testbed_parts_paths}. The inputs and outputs of the process are depicted by yellow boxes.
\begin{figure}
\centerline{\includegraphics[width=9cm]{testbed_parts__paths.png}}
\caption{TbUIS parts for evaluation of path-based testing techniques.}
\label{fig:testbed_parts_paths}
\end{figure}
The input to the process is a \textit{path-based test set}, whose effectiveness is going to be evaluated. The test cases in this test set have to correspond to an available \textit{TbUIS process model} (unless we intentionally created invalid paths-based test cases in the experiment). Using predefined building blocks from the \textit{FE-based functional automated test scripts} (introduced in Section \ref{sec:automated_tests}), the \textit{process test builder} chains these building blocks as instructed by the input path-based test cases to produce \textit{assembled FE automated tests}, which represent individual path-based test cases. For each of the path-based test cases at the input, a corresponding automated FE test is created.
The second input of the process is \textit{artificial defect specification}, which can be created via predefined \textit{catalogue defect types} or \textit{own defects} defined in the SUT. To create an \textit{defect clone} of the TbUIS with the specified defects, \textit{Error seeder} takes the specification of the defects and inserts them into the code of the Baseline UIS specification. Then, the defect clone is built as a running system instance.
At this stage, experimental evaluation of the path-based test set can be performed (an example is provided on the project web pages). Automated FE tests corresponding to the input path-based test cases are run in the defect clone, and the results are reported to the \textit{test report}, which can be evaluated. The information from the test report can also be paired with detailed \textit{application logs} to obtain more context information about the activated defects.
The schema for evaluating combinatorial or constrained testing test cases slightly differs, but the general principle remains the same.
In this type of evaluation, we do not compile FE automated tests to correspond to path-based test cases; instead, we can use
\begin{enumerate}
\item available automated FE-based functional tests covering all active elements and processes in the TbUIS,
\item available unit tests available together with the TbUIS code, or
\item combinations of both types of tests (the automated tests available to the TbUIS were introduced in Section \ref{sec:automated_tests}).
\end{enumerate}
Input data combinations to be exercised in the testbed can be entered into the available automated tests via the standardized DataProvider interface of the JUnit framework.
\section{Discussion and Possible Limits}
\label{sec:discussion_and_possible_limits}
Like other alternative artificial defect introduction approaches discussed in this paper, namely, using real defects from a previous software project and code mutation, the approach taken in the proposed testbed has certain advantages and disadvantages. We summarize these advantages and disadvantages in this section.
Regarding the possible complexity of the artificial defects introduced into an experimental SUT, the proposed approach does not limit an artificial defect to a set of mutation operators or a conditionally switched block of code. Instead, the defect clone can be built with the changes made in several different places in the source code, which allows high flexibility in simulating complex defects.
Concern whether the introduced defects represent typical defects that are being made during real software projects can be raised. This responsibility in experiments is up to the researchers and testing practitioners. Typical defects might vary between various software architectures, development styles, programming languages, business domains, and even decades when the empirical observations are made. Hence, the testbed provides a general possibility to create different types of defects and defect clones, and the decision is up to the testbed user.
In the proposed concept, the artificial defects are selected from a pre-defined set, which might limit the generalization of experiment results. This potential limit can be solved by the addition of more artificial defects as well as the correct interpretation of the results of the experiments.
Also, certain defects might be easier to detect than other defects, which may impact the results of the experiments \cite{papadakis2016threats}. However, this concern can be raised generally for any defect injection technique and shall be mitigated by correct interpretation of the results of the experiments.
Another concern is that the system is artificially created; however, the use cases and processes in the SUT are similar to real-world study information systems. The more important factor here is the selection of artificial defects that are representative of real-world projects. In the presented testbed, this selection is enabled by the possible introduction of more complex defects via the described mechanism of the defect clones.
Also, the size of the TbUIS system might limit its potential applicability as a benchmark for larger software systems. We are going to mitigate this concern by further evolution and extensions of the TbUIS.
\section{Conclusion}
In evaluating the effectiveness of testing techniques based on the measurement of the defect number that the test cases produced by these techniques detect in an experimental system, the established mutation testing approach can be accompanied by an alternative allowing the insertion of more complex defects caused by a misunderstanding of the design specification or other causes. We describe such an alternative in this paper: the presented TbUIS testbed, which is available as an open-source application and comprises a fictional university information system. The TbUIS testbed gives its user a mechanism to introduce artificial defects, including those from a predefined catalogue of possible defects, an extensive set of unit and FE-based functional automated tests, which can be used to examine test cases in the system, and a logging mechanism, which allows the collection of the data regarding which defects were activated by the examined test cases. Together with a good level of code and system documentation, the open structure of the TbUIS testbed eases its employment as a benchmark system to be used in the evaluation of path-based and combinatorial/constrained interaction testing techniques.
\section*{Acknowledgment}
This work was supported by the European structural and investment funds (ESIF) project CZ.02.1.01/0.0/0.0/17\_048/0007267 (InteCom)---Intelligent Components of Advanced Technologies for the Pilsen metropolitan area. Work package WP1.3: Methods and processes for control software safety assurance. The authors acknowledge the support of the OP VVV funded project CZ.02.1.01/0.0/0.0/16\_019/0000765 "Research Center for Informatics".
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 483
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Q: Stop page refreshing when closing documents in SharePoint 2010 We had previously used the following to stop the page refreshing when closing a document. This works fine in Sharepoint 2007 but doesn't work in 2010. Does anyone know the SP 2010 equivalent
g_varSkipRefreshOnFocus = true;
from
http://mattknott.com/content/blog/2009/08/Stop_DispEx_Redirecting.html
A: You need to put the code in after core.js and before you start your code. Will work in a content editor webpart.
<script type="text/javascript">
g_varSkipRefreshOnFocus = true;
</script>
for testing you just add the content editor web part.
A: You could also put it in windows load, did not work in document ready for me...
$(window).load(function () {
console.log("g_varSkipRefreshOnFocus set to true!");
g_varSkipRefreshOnFocus = true;
});
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,150
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5 things to watch for when the New York Knicks play the Orlando Magic
Andy Villamarzo
Guns, greyhounds and privacy dominate constitution hearing
Guns, greyhounds, and privacy.
Two of the three most commonly- and passionately-discussed topics at Monday's first public hearing for the Florida Constitution Revision Commission's proposed amendments weren't even among the 37 active propositions.
Yet dozens of speakers at the Maxwell C. King Center in Melbourne Monday spoke of their desire to see a Florida Constitution amendment banning assault weapons, a proposal that initially irked Chairman Carlos Beruff when League ofWomen Voters of Florida President Pamela Goodman first urged its consideration. But Beruff recovered, and expressed more tolerance and patience listening to numerous successors to Goodman on the topic.
Many of those speakers, and many others, also urged the commission to keep dead a proposal from Commissioner John Stemberger that would revise the state's privacy guarantee in a way many of the speakers said was a clear attack on women's rights to chose abortion.
Stemberger is president and general counsel of the anti-abortion group Florida Family Policy Center. He listened quietly and did not address the opposition to his Proposal 22, which was voted down by the commission's judicial committee, but still could be revised by the full commission.
Since neither was on the agenda, none of the more than 220 registered speakers spoke in favor of assault weapons nor Proposal 22 on Monday, and the 15 commissioners who attended, with a few rare exceptions, just listened and said nothing.
That was not the case with greyhounds, subject of Proposal 67 from state Sen. Tom Lee of Thonotosassa. Dozens of speakers, including children, spoke about the horrors they had heard about or seen involving the lives of racing greyhound dogs, and they urged the commission to put the proposal on the ballot.
"It's shameful that our state provides strong anti-cruelty laws other dogs, but allows greyhound racing dogs to suffer and die," said Janet Winikoff, director of education for the Human Society of Vero Beach, and a board member of the Florida Associations of Animal Welfare Associations.
Numerous representatives of the industry dispute the claims of dog abuse, contending that, as businessmen, they could not possibly succeed if they did not take good care of the dogs, and arguing that thousands of jobs were on the line.
"We have to fight to save these people's jobs, including mine," said Frank McCarron, owner of Seminole Animal Supply.
The high stakes led to high levels of animosity, with shouts of "Lies!" against one speaker, and exchanges of insults as speakers passed each other heading to and from the microphones.
There was that level of passion for a handful of other issues, including the upstart effort to get an assault weapon ban into the constitution, and to protect the privacy rights.
Also drawing powerful emotional support was Proposition 96, which would bring so-called "Marsy's Law" provisions into Florida to protect the victims of crime, with, among other things, notifications of when their attackers are released from jail or prison. Several victims of violent crime, including women stalked and haunted in later years by their attackers, pleaded for its support.
Proposal 88 to offer "bills of rights" to nursing home residents got mixed responses, as did proposals to address local elections, Proposals 13 and 43, and a handful of more specifically-targeted proposals dealing with items ranging from the hiring of security in courtrooms to Bar Association membership requirements.
Much of the audience took on a progressive political attitude, salted in part by a large press conference held prior to the meeting, involving the League of Women Voters, the National Organization of Women, and Planned Parenthood, among others, who all then went inside and signed up to speak.
The strongest oppositions came to such things as Proposition 4 to allow for state funding of religious schools, decried as a proposition that would tear down of the state's wall between church and state; and the strongest support came for such things as Proposition 91, banning oil and gas drilling off the Florida coast.
. Johnsonassault weaponsCarlos BeruffFlorida Constitution Revision CommissionJohn StembergerLeague of Women VotersMarsy's LawMelbournePlanned ParenthoodProposal 13Proposal 22Proposal 43Proposal 67Proposal 88Proposal 91Proposal 96public hearingTom Lee
Scott Powers is an Orlando-based political journalist with 30+ years' experience, mostly at newspapers such as the Orlando Sentinel and the Columbus Dispatch. He covers local, state and federal politics and space news across much of Central Florida. His career earned numerous journalism awards for stories ranging from the Space Shuttle Columbia disaster to presidential elections to misplaced nuclear waste. He and his wife Connie have three grown children. Besides them, he's into mystery and suspense books and movies, rock, blues, basketball, baseball, writing unpublished novels, and being amused. Email him at scott@floridapolitics.com.
Marsy's Law amendment leaves plenty for Legislature to figure out
Marsy's Law launches another round of victims' rights ads for Amendment 6
Ron DeSantis vows grand jury to investigate Tallahassee corruption, Andrew Gillum
Marsy's Law group offers crime victims' pleas in new Amendment 6 ads
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,921
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Q: Change mouse velocity with pyautogui I am trying to automate a game using pyautogui but I was trying to change the mouse velocity but I couldn't find a way so is there a way? I mean like to look behind you and thanks in forward.
A: A modified example from the Automate the boring stuff
import pyautogui
for i in range(5, 0, -1):
pyautogui.moveTo(100, 100, duration=0.10*i)
pyautogui.moveTo(200, 100, duration=0.10*i)
pyautogui.moveTo(200, 200, duration=0.10*i)
pyautogui.moveTo(100, 200, duration=0.10*i)
There is duration parameter in the moveTo() function that you can use to control the speed of the mouse. The unit of duration is seconds.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 543
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Isaac Hopkins (9 November 1870 – 25 October 1913) was an Australian cricketer. He played one first-class cricket match for Victoria in 1903.
See also
List of Victoria first-class cricketers
References
External links
1870 births
1913 deaths
Australian cricketers
Victoria cricketers
Cricketers from Melbourne
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,399
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/*
* person.h
*
* Created on: 09.05.2014
* Author: trifon
*/
#ifndef PERSON_H_
#define PERSON_H_
#include <iostream>
using namespace std;
#include "printable.h"
const int EGN_SIZE = 11;
class Person : public Printable {
char* name;
char id[EGN_SIZE];
public:
// голяма четворка
Person(char const* ="", char const* ="0");
Person(Person const&);
Person& operator=(Person const&);
virtual ~Person();
// селектори
char const* getName() const { return name; }
char const* getID() const { return id; }
// мутатори
void setName(char const*);
// извеждане
virtual void print(ostream& = cout) const;
void prettyPrint() const;
private:
void copy(Person const&);
void clean();
protected:
void setID(char const*);
};
ostream& operator<<(ostream&, Person const&);
#endif /* PERSON_H_ */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 47
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Az alábbi lista Ázsia állatkertjeit tartalmazza, országonként betűrendben:
Afganisztán
Kabul Zoo, Kabul
Azerbajdzsán
Baku Zoo, Baku
Banglades
Bangabandhu Sheikh Mujib Safari Park, Gazipur
Chittagong Zoo
Comilla Zoo
Dhaka Zoo
Dulhazra Safari Park
Gazipur Borendra Park
Khulna Zoo
Nijhum Dhip Park
Rajshahi Zoo
Rangpur Zoo
Bhután
Motithang Takin Preserve, Motithang
Dél-Korea
Seoul Grand Park Zoo, Gwacheon
Seoul Children's Zoo, Szöul
Everland Zoo-topia, Jongin
Dalseong Park, Tegu
Egyesült Arab Emírségek
Al Ain Zoo, El-Ajn
Arabian Wildlife Centre, Sharjah
Dubai Zoo, Dubaj
Emirates Park Zoo, Abu Dzabi
Észak-Korea
Korea Central Zoo, Phenjan
Fülöp-szigetek
Albay Park and Wildlife, Legazpi, Albay
Animal Island (zoo), Binakayan, Kawit, Cavite
Animal Wonderland, Star City, Pasay
Ark Avilon Zoo, Frontera Verde, Ortigas Avenue, Pasig
Ave Maria Sanctuary and Park, Carcar, Cebu
Avilon Zoo, Rodriguez, Rizal
Baluarte Zoo, Vigan, Ilocos Sur
Birds International, Quezon City
Botolan Wildlife Farm, San Juan, Botolan, Zambales
Calauit Safari Park, Calauit Island, Palawan
Cavite City Zoological and Botanical Park, Sampaguita Road, Seabreeze Subdivision, Santa Cruz, Cavite City, Cavite
Cebu City Zoo and Conservation Zoo, Calunasan, Cebu, Cebu
Corregidor Aviary and Theme Park, Corregidor Island, Cavite
Crocolandia Foundation, Biasong, Talisay, Cebu
Davao Crocodile Park, Diversion Highway, Ma-a, Davao
D'Family Park Mini-Zoo, Talamban, Cebu, Cebu
Eden Nature Park and Resort, Toril, Davao
Father Tropa's Spaceship 2000 Zoo, Zamboanguita, Negros Oriental
Laguna Wildlife Park and Rescue Center, La Vista Pansol Complex, Pansol, Calamba, Laguna
Lombija Wildlife Park and Heritage Resort, Napandong, Nueva Valencia, Guimarães
Lungsod Kalikasan, Quezon City
Maasin Zoo, Maasin, Southern Leyte
Malabon Zoo and Aquarium, Governor Pascual Street, Potrero, Malabon
Manila Orchidarium and Butterfly Pavilion, Rizal Park, Manila
Manila Zoological and Botanical Garden, M. Adriatico Street, Malate, Manila
Mari-it Wildlife Conservation Park, Lambunao, Iloilo
Maze Park and Resort Mini-Zoo, Mimbalot Buru-un, Iligan, Lanao del Norte
Negros Forests & Ecological Foundation, South Capitol Road, Bacolod, Negros Occidental
Ninoy Aquino Parks & Wildlife Center, Diliman, Quezon City
Palawan Butterfly Garden, Santa Monica, Puerto Princesa, Palawan
Palawan Wildlife Rescue and Conservation Center, Irawan, Puerto Princesa, Palawan
ParadiZoo, Mendez, Cavite
Paradise Reptile Zoo, Puerto Galera, Oriental Mindoro
Philippine Eagle Center, Malagos, Davao
Philippine Tarsier and Wildlife Sanctuary, Corella, Bohol
RACSO'S Woodland Mini-Hotel and Wildlife Resort, Rizal-Tuguisan, Guimbal, Iloilo
Residence Inn Mini-Zoo, Barrio Neogan, Tagaytay, Cavite
San Fernando Mini-Zoo and Butterfly Sanctuary, San Fernando, La Union
Sagbayan Peak Tarsier Sanctuary and Butterfly Dome, Sagbayan, Bohol
Silangang Nayon Mini-Zoo, Pagbilao, Quezon City
Silliman University Center for Tropical Conservation Studies (also known as the A.Y. Reyes Zoological and Botanical Garden), Ipil Street, Daro, Dumaguete
Silliman University Marine Laboratory, Bantayan, Dumaguete
WIN Rescue Center, Subic Bay Freeport Zone, Zambales
Zoo Paradise of the World, Zamboanguita, Negros Oriental
Zoobic Safari, Subic, Zambales
Zoocobia Fun Zoo, Clark Freeport Zone, Angeles, Pampanga
Grúzia
Tbilisi Zoo
Szuhumi Majomtenyészet
India
Aizawl Zoo, Aizawl, Mizoram
Alipore Zoological Gardens, Kalkutta, Nyugat-Bengál
Allen Forest Zoo, Kánpur, Uttar Prades
Amirthi Zoological Park, Vellore, Tamilnádu
Arignar Anna Zoological Park (Vandalur Zoo), Csennai, Tamilnádu
Assam State Zoo-cum-Botanical Garden, Gauháti, Asszám
Bannerghatta National Park, Bengaluru
Bhiwani Zoo, Harijána
Birsa Deer Park (Kalamati Birsa Mrig Vihar), Ráncsí
Black Buck Breeding Centre, Pipli Mini Zoo, Kurukshetra, Harijána
ChattBir Zoo, Zirakpur, Pandzsáb
Chennai Snake Park Trust, Csennai, Tamilnádu
Chinkara Breeding Centre Kairu, Bhiwaninear Bahal, Bhivándi, Harijána
Crocodile Breeding Centre, Kurukshetra, Bhaur Saidan
Gopalpur Zoo, Gopalpur, Himácsal Prades
Gulab Bagh and Zoo, Udaipur, Rádzsasztán
Hisar Deer Park, Harijána
Indira Gandhi Zoological Park, Visákhapatnam, Ándhra Prades
Indore Zoo, Indaur, Madhja Prades
Jaipur Zoo, Dzsaipur, Rádzsasztán
Jawaharlal Nehru Biological Park, Bokaro Steel City
Jhargram Zoo, Jhargram, Nyugat-Bengál
Jijamata Udyaan, Mumbai, Mahárástra
Kanan Pendari Zoo, Bilaspur, Cshattíszgarh
Kankaria, Ahmadábád, Gudzsarát
Kanpur Zoo, Kánpur, Uttar Prades
Lucknow Zoo, Lakhnau, Uttar Prades
Madras Crocodile Bank Trust, Csennai, Tamilnádu
Maitri Bagh, Bhilainagar, Cshattíszgarh
Marble Palace zoo, Kalkutta, Nyugat-Bengál
Mysore Zoo, Maiszúr, Karnátaka
Nandankanan Zoological Park, Bhuvanesvar, Orisza
National Zoological Park, Delhi
Nehru Zoological Park, Haidarábád, Telangána
Padmaja Naidu Himalayan Zoological Park, Dardzsiling, Nyugat-Bengál
Parassinikkadavu Snake Park
Peacock & Chinkara Breading Centre, Jhabua, Rewari district, Harijána
Pheasant Breeding Centre, Berwala, Panchkula district, Harijána
Pheasant Breeding Centre Morni, Panchkula district, Harijána
Pt. G.B. Pant High Altitude Zoo, Nainital, Uttarakhand
Rajiv Gandhi Zoological Park, Púna, Mahárástra
Ranchi Zoo (Bhagwan Birsa Munda Biological Park), Ráncsí, Dzshárkhand
Rohtak Zoo, Harijána
Sakkarbaug Zoological Garden, Dzsúnágarh, Gudzsarát
Sanjay Gandhi Jaivik Udyan, Patna, Bihár
Sarthana Zoo, Szúrat, Gudzsarát
Sayaji Baug Zoo, Vadodara, Gudzsarát
Sipahijola Wildlife Sanctuary, Tripura
Sri Venkateswara Zoological Park, Tirupati, Ándhra Prades
Tata Steel Zoological Park, (Jubilee Park) Dzsamsedpur, Dzshárkhand
Thim Park, Dzsamsedpur, Dzshárkhand
Thiruvananthapuram Zoo, Trivandrum, Kerala
Thrissur Zoo, Trisúr, Kerala
Vulture Conservation and Breeding Centre, Pinjore, Harijána
Indonézia
Batu Secret Zoo, Batu, East Java
Bali Bird Park, Gianyar, Bali
Bali Zoo, Gianyar, Bali
Bandung Zoo, Bandung, West Java
Surabaya Zoo, Surabaya, East Java
Gembira Loka Zoo, Yogyakarta
Ragunan Zoo, Jakarta
Taman Safari, Bogor, West Java
Taman Safari II, Pasuruan, East Java
Taman Safari III Bali Safari and Marine Park, Gianyar, Bali
Maharani Zoo & Goa, Lamongan, East Java
Medan Zoo, Medan, Észak-Szumátra
Irak
Baghdad Zoo
Irán
Vakil Abad Zoo, Meshed
Amol Zoo
Bandar Abbas Zoo
Darabad Museum of Wildlife, Teherán
Isfahan Zoo
Tehran Zoological Garden, Teherán
Baghlarbaghy Zoo, Tebriz
Izrael
Arena Tropical World, Herzliya
Beer Sheba Municipal Zoological Garden, Beersheba
Carmel Hai-Bar Nature Reserve, Haifa
Children's Zoo, Sa'ad
Educational Zoo of the Haifa Biological Institute, Haifa
Hai Kef, Risón Lecijon
Hai Park, Kirjat Mockín
Hamat-Gader Crocodile Farm, Terrarium and Mini Zoo, Hamat Gader
I. Meier Segals Garden for Zoological Research, Tel-Aviv
Jerusalem Bird Observatory, Jeruzsálem
Monkey Park, Kfar Daniel
Nahariya Zoo-Botanical Garden, Naharija
Nir-David Australian Animal Park, Nir David
Petah Tikva Zoo, Petah Tikva
Tel-Aviv Bird Park, Tel-Aviv
Tisch Family Biblical Zoological Gardens, Jeruzsálem
Zoological Center of Tel Aviv-Ramat Gan, Ramat Gan
Yotvata Hai-Bar Nature Reserve, Yotvata
Japán
Higashiyama Zoo and Botanical Gardens, Nagoja
Japan Monkey Centre, Inuyama, Aicsi
Okazaki Higashi Park Zoo, Okazaki, Aicsi
Toyohashi Zoo & Botanical Park, Tojohasi, Aicsi
Toyota City Kuragaike-Park, Toyota, Aicsi
Akita Omoriyama Zoo, Akita, Akita
Chiba Zoological Park, Csiba, Csiba
Ichihara Elephant Kingdom, Ichihara, Csiba
Ichikawa Zoological & Botanical Garden, Ichikawa, Csiba
Mother Farm, Futtsu, Csiba
Tobe Zool. Park of Ehime Prefecture, Iyo, Ehime
Sabae Nishiyama Park Zoo, Sabae, Fukui
Fukuoka Municipal Zoo and Botanical Garden, Fukuoka
Itozu no Mori Zoological Park, Kitakyūshū
Kurume City Bird Center, Kurume, Fukuoka
Ōmuta Zoo, Ōmuta, Fukuoka
Uminonakamichi Seaside Park Zoological Garden, Fukuoka
Gunma Safari Park, Tomioka, Gunma
Kiryugaoka Zoo, Kiryu, Gunma
Fukuyama City Zoo, Fukuyama, Hirosima
Hiroshima City Asa Zoological Park, Hirosima
Asahiyama Zoo, Asahikawa, Hokkaidó
Kushiro Zoo, Kushiro, Hokkaidó
Noboribetsu Bear Park, Noboribetsu, Hokkaidó
Obihiro Zoo, Obihiro, Hokkaidó
Sapporo Maruyama Zoo, Szapporo
Awaji Farm Park England Hill, Minamiawaji, Hjógo
Himeji Central Park, Himeji, Hjógo
Himeji City Zoo, Himeji, Hjógo
Kobe Kachoen, Kóbe
Oji Zoo, Kóbe
Hitachi Kamine Zooligical Garden, Hitachi, Ibaraki
Ishikawa Zoo, Nomi, Isikava prefektúra
Morioka Zoological Park, Morioka, Ivate
Amami Islands Botanical Garden, Kagosima
Hirakawa Zoological Park, Kagosima
Kanazawa Zoological Gardens, Jokohama
Nogeyama Zoo, Jokohama
Odawara Zoo, Odavara, Kanagava
Yokohama Zoo (Zoorasia), Jokohama
Yumemigasaki Zoological Park, Kawasaki, Kanagava
Noichi Zoological Park of Kōchi Prefecture, Komi District, Kócsi
Wanpark Kōchi Animal Land, Kōchi, Kócsi
Cuddly Dominion, Aso, Kumamoto
Kumamoto City Zool. & Bot. Gardens, Kumamoto, Kumamoto
Kyoto Municipal Zoo, Kiotó
Yagiyama Zoological Park, Szendai
Miyazaki City Phenix Zoo, Miyazaki, Mijazaki
Iida City Zoo, Iida, Nagano
Nagano Chausuyama Zoo, Nagano, Nagano
Omachi Alpine Museum, Omachi, Nagano
Suzaka Zoo, Suzaka, Nagano
Nagasaki Biopark, Saikai, Nagaszaki
Sasebo Zoological Park and Botanical Garden, Sasebo, Nagaszaki
Kyushu African Lion Safari, Usa, Óita
Takasakiyama Natural Zoo, Ōita, Óita
Ikeda Zoo, Okayama, Okajama
Great Ape Research institute, Hayashibara, Tamano, Okajama
Neo Park Okinawa, Nago, Okinava
Okinawa Kodomo Future Zone, Okinava
Kashihara City Insectary Museum, Kashiwara, Oszaka
Misaki Koen, Sennan, Oszaka
Satsukiyama Zoo, Ikeda, Oszaka
Tennoji Zoo, Oszaka
Miyazawako Nakayoshi Zoo, Hannō, Saitama
Saitama Children's Zoo, Higashimatsuyama, Saitama
Saitama Omiya Park Zoo, Szaitama, Saitama
Sayama Chikosan Park Children Zoo, Sayama, Saitama
Tobu Zoological Park, Minamisaitama District, Saitama
Matsue Vogel Park, Macue, Simane
Atagawa Tropical & Alligator Garden, Kamo District, Sizuoka
Fuji Safari Park, Susono, Sizuoka
Hamamatsu Municipal Zoo, Hamamatsu
iZoo, Kawazu, Sizuoka
Izu Biopark, Kamo District, Sizuoka
Izu Cactus Park, Ito, Sizuoka
Mishima City Park Rakujuen, Mishima, Sizuoka
Shizuoka Municipal Nihondaira Zoo, Shizuoka, Sizuoka
Nasu Animal Kingdom, Nasu, Tocsigi
Utsunomiya Zoo, Utsunomiya, Tocsigi
Tokushima Municipal Zoo, Tokushima, Tokusima
Edogawa City Natural Zoo, Edogawa, Tokyo
Hamura Zoological Park, Hamura, Tokyo
Inogashira Park Zoo, Tokió
Ōshima Park Zoo, Izu Ōshima
Tama Zoo, Tokió
Ueno Állatkert, Ueno, Tokyo
Takaoka Kojo ParkZoo, Takaoka, Tojama
Toyama Municipal Family Park Zoo, Toyama, Tojama
Adventure World,, Shirahama, Wakayama
Wakayama Park Zoo, Wakayama, Wakayama
Akiyoshidai Safari Land, Mine, Jamagucsi
Shunan Municipal Tokuyama Zoo, Shūnan, Jamagucsi
Ube Tokiwa Park, Ube, Jamagucsi
Kofu Yuki Park Zoo, Kofu, Yamanashi
Kambodzsa
Bayap Zoo
Kampot Zoo
Koh Kong Safari World
Phnom Tamao Wildlife Rescue Centre
Katar
Doha Zoo, Doha
Kazahsztán
Almaty Zoo, Almati
Karaganda Zoo, Karagandi
Shymkent Zoo, Shymkent
Temirtau Aquapark of Children's Park, Temirtau
Kína
Badaling Safari World
Beijing Zoo
Bifengxia Wild Animal Park
Chongqing Zoo
Chengdu Zoo
Dalian Forest Zoo
Hangzhou Zoo
Harbin Northern Forest Zoo
Guangzhou Zoo
Guangzhou Panyu Chime-long Night Zoo
Guangzhou Xiangjian Safari Park
Hangzhou Zoo
Jinan Safari Park
Kunming Zoo
Nanning Zoo
Nanjing Hongshan Forest Zoo
Qingdao Forest Wildlife World
Qingdao Zoo
Shanghai Zoo
Shanghai Wild Animal Park
Shenzhen Safari Park
Shijiazhuang Zoo
Suzhou Zoo
Tianjin Zoo
Edward Youde Aviary, Hong Kong Park
Hong Kong Wetland Park
Hong Kong Zoological and Botanical Gardens
Kadoorie Farm and Botanic Garden
Lai Chi Kok Zoo (zárva)
Ocean Park Hong Kong
Bifengxia Giant Panda Base
Chengdu Research Base of Giant Panda Breeding
Wolong Giant Panda Breeding Center, Szecsuan
Kuvait
Kuwait Zoo, Kuvaitváros
Al Omariya
Makaó
Two Dragon Throat Public Garden
Malajzia
Kuala Lumpur-i Madárpark, Kuala Lumpur
Zoo Melaka, Ayer Keroh, Malacca
National Zoo of Malaysia (Zoo Negara), Ulu Klang, Kuala Lumpur
Zoo Taiping, Taiping, Perak
Lok Kawi Wildlife Park, Lok Kawi, Sabah
Zoo Johor, Johor Bahru, Johor
Zoo Terengganu, Kemaman, Terengganu
Sunway Lagoon Wildlife Park, Petaling Jaya, Selangor
Sunway Petting Zoo, Subang Jaya, Selangor
Lost World Petting Zoo, Ipoh, Perak
Kuala Krai mini zoo, Kuala Krai, Kelantan
Butterfly & Reptiles Sanctuary, Malacca
Kuala Lumpur Butterfly Park, Kuala Lumpur
Penang Butterfly Farm, Teluk Bahang, Pinang
Bukit Jambul Orchid, Hibiscus and Reptile Farm, Pinang
Taman Teruntum Mini Zoo, Taman Teruntum, Pahang
Kuala Lumpur Deer Park, Kuala Lumpur
Danga Bay Mini Zoo, Danga Bay, Johor
Afamosa Animal World Safari, Alor Gajah, Malacca
Langkawi Bird Paradise, Langkawi, Kedah
Snake and Reptile Farm, Perlis
Langkawi Crocodile Farm, Langkawi, Kedah
Ayer Keroh Crocodile Farm, Ayer Keroh, Malacca
Jong Crocodile Farm, Kuching, Sarawak
Sandakan Crocodile Farm, Sandakan, Sabah
Mianmar
Yadanabon Zoological Gardens, Mandalaj
Naypyitaw Zoo, Nepjida
Nepál
Central Zoo, Jawalakhel
Örményország
Yerevan Zoo, Jereván
Pakisztán
Bahria Town Zoos, Lahor
Citi Housing Zoos, Gudzsranvála
Bahawalpur Zoo, Bahawalpur, Punjab
Faisalabad Zoo Park, Fajszalábád
Hyderabad Zoo, Hiderábád, Szindh
Murghzar Zoo, Iszlámábád, Capital Territory
Karachi Zoo, Karacsi, Szindh
Lahore Zoo, Lahor, Punjab
Landhi Korangi Zoo, Karacsi, Szindh
Multan zoo, Multán
Peshawar Zoo, Pesavar, Haibar-Pahtúnhva
Rawalpindi Zoo, Ravalpindi
Wildlife Park, Rahim Yar Khan, Punjab
Jungle World (korábban Jungle Kingdom), Ravalpindi, Punjab
Kund Park, Nowshera, Haibar-Pahtúnhva
Lake View Park, Iszlámábád, Capital Territory
Jallo Wildlife Park, Lahor, Punjab
Laal Sunhara Safari Park,(National Park Laal Sunhara Bahawalpur), Bahawalpur, Punjab
Lahore Zoo Safari, (korábban Lahore Wildlife Park), (Woodland Wildlife Park), Lahor, Punjab
Lohi Bher Wildlife Park, Ravalpindi, Punjab
Murree Wildlife Park,(Murree National Park) Marree, Punjab
Karachi Safari Park, Karacsi, Sindh
Changa Manga Vulture Center, Lahor, Punjab
Dhodial Pheasantry, Mansehra, Haibar-Pahtúnhva
Karachi Walkthrough Aviary, Karacsi, Sindh
Lahore Walkthrough Aviary, Lahor, Punjab
Lake View Park Aviary, Iszlámábád, Capital Territory
Lakki Marwat Crane Center, Lakki Marwat, Haibar-Pahtúnhva
Saidpur Hatchery, Iszlámábád, Capital Territory
Clifton Fish Aquarium, Karacsi, Sindh
Karachi Municipal Aquarium, Karacsi, Sindh
Landhi Korangi Aquarium, Karacsi, Sindh
Attock Wildlife Park, Attock, Punjab
Bahawalnagar Wildlife Park, Bahawalnagar, Punjab
Bhagat Wildlife Park, Toba Tek Singh, Punjab
Changa Manga Wildlife Park, Lahor, Punjab
Dera Ghazi Khan Wildlife Park, Dera Ghazi Khan, Punjab
Gatwala Wildlife Park, Fajszalábád, Punjab
Jallo Wildlife Park, Lahor, Punjab
Kamalia Wildlife Park, Toba Tek Singh, Punjab
Lalazar Wildlife Park, Abbotábád, Haibar-Pahtúnhva
Perowal Wildlife Park, Khanewal, Punjab
Rahim Yar Khan Wildlife Park, Rahim Yar Khan, Punjab
Sulemanki Wildlife Park, Okara, Punjab
Vehari Wildlife Park, Vehari, Punjab
Woodland Wildlife Park (Lahore Zoo Safari), Lahor, Punjab
Changa Manga Breeding Center, Lahor, Punjab
Faisalabad Breeding Center, Fajszalábád, Punjab
Hawke's Bay/Sandspit Turtle Hatchery, Karacsi, Sindh
Jallo Breeding Center, Lahor, Punjab
Rawat Breeding Center, Ravalpindi, Punjab
Palesztina
Gaza Zoo, Gáza (zárva)
Qalqilya Zoo, Ciszjordánia
Srí Lanka
Ahungalla Animal Park
Dehiwala Zoo
Pinnawala Elephant Orphanage
Pinnawala Elephant Safari
Pinnawela Open Zoo
Hambantota Safari Zoo
Wagolla Zoo
Battaramulla Mini Zoo
Szingapúr
Jurong Bird Park
Jurong Reptile Park (zárva)
Night Safari, Szingapúr
River Safari
Singapore Crocodile Farm (zárva)
Singapore Zoo
Tajvan
Taipei Zoo, Tajpej
Kaohsiung Zoo
Green World Ecological Farm
1-Lan Wild Animal Rescue Centre
Far East Animal Farm
Feng-Hung-Ku Bird Park
Hsinchu City Zoo
Leefoo Zoo
National Museum of Marine Biology/Aquarium
Ocean World, Yehliu Wanli
Pintung Rescue Centre
Thaiföld
Chiang Mai Night Safari
Chiang Mai Zoo
Crocodiles Farm and Elephant Theme Show Sampran, Nakorn Pathom
Samutprakarn Crocodile Farm and Zoo
Dusit Zoo
Chaiyaphum Star Tiger Zoo
Elephant Nature Park, Csiangmaj
Khao Kheow Open Zoo
Khao Suan Kwang Zoo, Khonken
Khao Prathap Chang Wildlife Breeding Center, Ratchaburi
Lopburi Zoo
Nakhon Ratchasima Zoo
Nong Nooch Tropical Botanical Garden
Pata Zoo, Bangkok
Phuket Zoo, Phuket
Phuket bird park, Phuket
Safari Park Kanchanaburi, Kancsanaburi
Safari World
Songkhla Zoo
Sriracha Tiger Zoo, Chonburi
The Million Years Park and Pattaya Crocodiles Farm
Türkmenisztán
Ashgabat Zoo, Aşgabat
Üzbegisztán
Tashkent Zoo, Taskent
Termez Zoo, Termez
Vietnám
Saigon Zoo and Botanical Gardens, Ho Si Minh-város
Ha Noi zoo, Hanoi
Ázsia
|
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Tür an Tür ist ein deutscher Fernsehfilm von Matthias Steurer aus dem Jahr 2013, der im Auftrag für Das Erste produziert wurde. In den Hauptrollen agieren Thekla Carola Wied und Tanja Wedhorn, in tragenden Rollen Uwe Friedrichsen und Bernhard Schir.
Das Erste schrieb zum Start des Films: "Matthias Steurer, Regisseur von hintersinnigen Unterhaltungsfilmen wie 'Zimtstern und Halbmond', erzählt in 'Tür an Tür' von der Freundschaft zweier in jeder Hinsicht gegensätzlicher Frauen, die von Thekla Carola Wied und Tanja Wedhorn verkörpert werden. Dabei gelingen ihm einige pointierte Seitenhiebe auf modernes Großstadt- und Beziehungsleben und auf die Rollenklischees von Alt und Jung." "Humorvoll und einfühlsam" erzähle der Film "die Geschichte einer ungewöhnlichen Frauenfreundschaft".
Handlung
Die Architektin Sophie Mehnert, die gerade eine sehr schöne Altbauwohnung in Oldenburg angemietet hat, ist guter Dinge, dass der Chefarzt Martin Ahlers dort mit ihr einziehen wird. Ahlers ist allerdings noch verheiratet. Mit Verweis auf seine kleine Tochter hat er die angeblich kurz bevorstehende Trennung immer wieder hinausgezögert. Am selben Tag erhält Sophie dann auch noch eine für sie erfreuliche Festanstellung in dem Architekturbüro, in dem sie bisher nur lose mitgearbeitet hat.
In der Sophies Wohnung gegenüberliegenden Wohnung lebt Hannah Weller, eine sehr eigensinnige ältere Dame. Sie ist herzkrank und gehbehindert. Hannah entschließt sich an einem Pilotprojekt für Senioren teilzunehmen, das in der Lage ist Daten auszuwerten und so zu überwachen, wie es ihr gesundheitlich geht, ob sie ihre Medikamente eingenommen hat oder ob sie sich in irgendeiner Gefahrensituation befindet. Gleichzeitig eignet Hannah sich in einem begleitenden Kurs Computerkenntnisse an. Dort lernt sie auch den Rentner Friedrich Seliger kennen, der ihren spitzen Bemerkungen geduldige Gelassenheit entgegensetzt.
Nachdem Sophie einen Streit mit Martin hatte, bittet sie Hannah, sie kurz in ihre Wohnung zu lassen, um in ihrer Tasche nach ihrem Schlüssel zu suchen. Dabei verliert sie einen Stick, den Hannah nach Sophies Fortgang findet. Durch ihre neu erworbenen Kenntnisse kann sie sich in Sophies E-Mail-Konto einloggen. Sie tut das allerdings nur, weil sie spürt, dass da etwas im Argen liegt zwischen Sophie und dem Mann, mit dem zusammen sie die Wohnung beziehen wollte und ist sich sicher, dass dieser zu jener Sorte von Männern gehört, die ihren Worten keine Taten folgen lassen. In der folgenden Zeit versucht sie Sophie zum Nachdenken zu bringen, indem sie ihr erzählt, ihre Tochter sei mit einem Mann liiert, der ihr weismache, dass er sich von seiner Familie trennen werde, was aber wahrscheinlich nie passieren werde. Und auf Sophies Einwand, vielleicht sei es ja die große Liebe, ja für sie vielleicht, für ihn ganz sicher nicht, sonst wäre er ja bei ihr.
Hannah entschließt sich, nachdem sie im Internet über Martin Ahlers recherchiert hat, bei ihm als Patientin vorzusprechen. Sie konfrontiert auch den Neurologen mit den Problemen, die ihre Tochter angeblich habe, und setzt dem Arzt damit ganz schön zu. Nur wenig später lässt er Sophie wissen, dass er seine Trennungspläne erst einmal zurückstellen werde, da die ganze Situation ihn zur Zeit zu sehr belaste. Er werde sich irgendwann von seiner Frau trennen, vielleicht in drei Monaten, vielleicht aber auch erst in drei Jahren, er wisse es einfach nicht, aber er liebe sie und wolle sie nicht verlieren. Nach diesem Gespräch unterhält sich Sophie mit Hannah. Hannah versteht es sehr gut, Sophie zum Nachdenken zu bringen.
Durch einen Zufall kommt es heraus, dass Hannah sich in Sophies E-Mail-Konto eingeloggt hat und dass sie gar keine Tochter hat. Sophie ist unglaublich wütend und will nichts mehr mit Hannah zu tun haben. Mit einiger Mühe gelingt es der alten Dame jedoch, Sophie dazu zu bringen, ihr wenigstens fünf Minuten zuzuhören. Hannah erzählt, dass sie gewollt habe, dass es Sophie nicht so ergehe, wie ihr selbst. Vor vierzig Jahren sei sie in der gleichen Situation gewesen wie Sophie heute, sie habe gewartet, jahrelang, gern habe sie Kinder haben wollen, er habe jedoch kein weiteres gewollt, solange seine Verhältnisse nicht geklärt seien. Also habe sie kein Kind bekommen. Nachdem er sich dann schließlich entschieden habe, bei seiner Familie zu bleiben, habe sie nie mehr geheiratet und habe immer allein gelebt. Als Sophie fragt, was sie jetzt machen solle, meint Hannah, Geschichten könnten durchaus unterschiedlich ausgehen, vielleicht erfülle sich bei ihr ja das, was sie sich erhoffe.
Nachdem Sophie einen Schlussstrich unter die Beziehung mit Martin gezogen hat, kann sie endlich wieder frei arbeiten und legt ihrem Chef den Entwurf eines Modells vor, dessen Konzept darauf gründet, dass alte und junge Menschen zwar zusammen leben, aber gleichzeitig auch ihre jeweils eigene Privatsphäre wahren können. Sophies Chef ist davon äußerst angetan, nachdem ihre zuvor abgelieferten Entwürfe ihn schon daran zweifeln ließen, sich mit Sophies Einstellung richtig entschieden zu haben. Zwischen Hannah und Sophie kommt es zu einer Aussprache und anschließenden Freundschaft. Den nahenden Weihnachtsabend wollen die beiden Frauen zusammen verbringen. Sophie hat schon eine große Tanne besorgt, die mit echten Kerzen geschmückt werden soll.
Produktion
Produktionsnotizen, Hintergrund
Tür an Tür wurde vom 15. November bis zum 22. Dezember 2011 in Oldenburg und Umgebung gedreht. Für den Film zeichnete die Cinecentrum Hannover Film- und Fernsehproduktion verantwortlich. Die Redaktion für die ARD Degeto lag bei Katja Kirchen, für den Norddeutscher Rundfunk|NDR bei Daniela Mussgiller.
Uwe Friedrichsen ist in diesem Film in seiner letzten Filmrolle zu sehen.
Veröffentlichung
Der Film wurde erstmals am 13. Dezember 2013 im Programm der ARD Das Erste ausgestrahlt. Zuvor war er schon auf dem Internationalen Filmfest Oldenburg vorgestellt worden.
Rezeption
Einschaltquoten
Bei der Erstausstrahlung wurde Tür an Tür von 4,50 Millionen Zuschauern eingeschaltet bei einem Marktanteil von 15 Prozent. Bei der Wiederholung des Filmes schauten 3,96 Millionen Zuschauer zu, der Marktanteil lag bei 15,6 Prozent.
Kritik
Die Kritiker der Fernsehzeitschrift TV Spielfilm schrieben: "Die Akteurinnen reden oft laut vor sich hin, um Dinge zu erklären, die die Dramödie auch filmisch hätte darstellen können. Interessant: das computergestützte Seniorenbetreuungssystem." Der Film erhielt für Humor und Spannung je einen von drei möglichen Punkten und eine mittlere Wertung, indem der Daumen zur Seite zeigte. Das Fazit lautete: "Von Frau zu Frau: nachdenklich bis betulich"
Rainer Tittelbach gab dem Film auf seiner Seite tittelbach.tv drei von sechs möglichen Sternen und fasste seine Bewertung folgendermaßen zusammen: "Die eine ist grantig, bärbeißig und alt, die andere freundlich, optimistisch und jung. Allein sind sie bald beide – und sie kommen sich näher; schließlich wohnen sie 'Tür an Tür'. Alleinsein im Alter, ein gutes Thema für den ARD-Freitagsfilm. Auch gegen die Botschaft, Technik und Internet sind okay, doch direkte Kommunikation ist echter, ehrlicher, besser, gibt es nichts einzuwenden. Nur leider verzichtet 'Tür an Tür' in seiner Machart auf jene Authentizität, die die Story vorweihnachtlich anpreist. Die Situationen wirken nicht aus dem Leben gegriffen, sondern ausgedacht und 'gemacht'. Der Film gibt Beispiele vom Leben, aber er erzählt nicht. Das alles spiegelt sich prägnant in der Verkleidung der fehlbesetzten Thekla Carola Wied." Wied wirke "von Anfang an verkleidet, ihr Spiel, ihre Maske, vor allem aber ihre Gehbehinderung" hätten "mehr von einem Comedy-Sketch als einem Fernsehfilm mit realistischem Anspruch – die 69-Jährige" sei "eine Fehlbesetzung", sie sei zu jung für die Verkörperung einer 78-jährigen.
Tilmann P. Gangloff nahm sich des Films für evangelisch.de an und meinte, geschickt verknüpfe Drehbuchautorin Nina Bohlmann zwei im Grunde gänzlich unterschiedliche Geschichten und lasse auf diese Weise eine dritte entstehen. Auch hier führe Matthias Steurer die beiden Hauptdarstellerinnen zu sehenswerten Leistungen, wobei Thekla Carola Wied allerdings dank Hannahs Bosheit die mit Abstand besten Dialoge habe.
Weblinks
Tür an Tür vollständiger Film in der ARD-Mediathek (verfügbar bis 4. Juli 2020, 14:30 Uhr)
Einzelnachweise
Filmtitel 2013
Deutscher Film
Fernsehfilm
Weihnachtsfilm
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 1,688
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credit
Transcribed from the 1915 Jarrold & Sons edition by David Price, email
ccx074@pglaf.org
[Picture: Reproduced from Borrow's original manuscript of "Glendower's
Mansion."]
WELSH POEMS
AND BALLADS
BY
GEORGE BORROW
[Picture: Decorative graphic, Sans Peur et san Reproche]
WITH AN INTRODUCTION BY
ERNEST RHYS
* * * * *
LONDON
JARROLD & SONS
MCMXV
* * * * *
TO
THOMAS J. WISE,
Bibliophile, Bibliographer and
Good Borrovian
(at whose instance
this Norfolk Budget
of Welsh Verse
was brought
together).
CONTENTS
Page
Introduction 9
Glendower's Mansion. By Iolo Goch. Borrow MSS. 27
Ode to the Comet. By Iolo Goch. Borrow MSS. 33
Ode to Glendower. By Iolo Goch. Borrow MSS 39
"Here's the Life I've sighed for long." By Iolo Goch. 45
"Wild Wales"
The Prophecy of Taliesin. "Targum" 49
The History of Taliesin. "Targum" 53
The Mist. By Dafydd ab Gwilym. "Wild Wales" 59
The Cuckoo's Song in Meiron. By Lewis Morris o Fon. 63
Borrow MSS.
The Snow on Eira. "Wild Wales" 69
The Invitation. By Goronwy Owen. "Targum" 73
The Pedigree of the Muse. Goronwy Owen. Borrow MSS. 79
The Harp. Goronwy Owen. Borrow MSS 87
Epigram on a Miser. "Targum" 91
Griffith ap Nicholas. By Gwilym ab Ieuan Hen. "Wild 95
Wales"
Riches and Poverty. By Twm o'r Nant. "Wild Wales" 99
The Perishing World. By Elis Wynn. "The Sleeping Bard" 109
Death the Great. By Elis Wynn. "The Sleeping Bard" 115
The Heavy Heart. By Elis Wynn. "The Sleeping Bard" 121
Ryce of Twyn. By Dafydd Nanmor 129
Llywelyn. By Dafydd Benfras. "Quarterly Review" 133
Plynlimmon. By Lewis Glyn Cothi. 137
QUATRAINS AND STRAY STANZAS FROM "WILD WALES."
I. Chester Ale. By Sion Tudor 141
II. Englyn: Dinas Bran. By Roger Cyffyn. "Gone, 141
gone are thy Gates"
III. Madoc's Epitaph 142
IV. Epitaph on Elizabeth Williams 142
V. The Last Journey. By Huw Morus 142
VI. The Four and Twenty Measures. By Edward Price 143
VII. Mona: By Robert Lleiaf 143
VIII. Mona: Englyn. From "Y Greal" 143
IX. Eryri 143
X. Eryri. From Goronwy Owen 144
XI. Ellen. From Goronwy Owen 144
XII. Mon. By Robin Ddu 144
XIII. Mon. From Huw Goch 144
XIV. Lewis Morris of Mon. By Goronwy Owen 145
XV. The Grave of Beli 145
XVI. The Garden. By Gwilym Du 145
XVII. The Satirist. From Gruffydd Hiraethog 146
XVIII. On Gruffydd Hiraethog. By Wm. Lleyn 146
XIX. Llangollen Ale. (George Borrow) 146
XX. Tom Evans. By Twm Tai 147
XXI. The Waterfall 147
XXII. Dafydd Gam. Attributed to Owain Glendower 147
XXIII. Llawdden. By Lewis Meredith 148
XXIV. Twm o'r Nant 148
XXV. Severn and Wye 148
XXVI. Glamorgan. By D. ab Gwilym 148
XXVII. Dafydd ab Gwilym. From Iolo Goch 149
XXVIII. The Yew Tree. After Gruffydd Gryg 149
XXIX. Hu Gadarn. By Iolo Goch 150
XXX. Earth in Earth. Epitaph 150
XXXI. God's Better than All 151
XXXII. The Sun in Glamorgan. By Dafydd ab Gwilym 153
ADDITIONAL POEMS FROM THE "QUARTERLY REVIEW."
I. The Age of Owen Glendower 157
II. The Spider 158
III. The Seven Drunkards 159
Sir Rhys ap Thomas. Borrow MSS. 163
Hiraeth. Borrow MSS. 167
Pwll Cheres: The Vortex of Menai. Borrow MSS. 171
The Mountain Snow 175
Carolan's Lament. "Targum" 179
Epigrams by Carolan. "Targum" 183
The Delights of Finn Mac Coul. "Targum" 187
Icolmcill. "Targum" 191
The Dying Bard. "Targum" 195
The Song of Deirdra. Borrow MSS. 201
The Wild Wine. "Wild Wales" 205
INTRODUCTION.
IN a collection of unedited odds and ends from Borrow's papers bearing
upon Wales, and dating from various periods of his career, there is one
insignificant-looking sheet on whose back some lines are pencilled,
beginning "The mountain snow." They are reproduced in the text, but
deserve notice here because of the evidence they bring of Borrow's
long-continued Welsh obsession and his long practice as a Welsh
translator. Apparently they date from the time when he was writing
"Lavengro," since the other side of the leaf contains a draft in ink of
the preface to that book. Other sheets of blue foolscap in the same
bundle--folded small for the pocket--are devoted to unnumbered chapters
of "Wild Wales." Yet another scrap, from a much earlier period, is so
closely packed in a microscopic hand that it reminds one at a first
glance of the painfully minute script of the Bronte sisters in their
earliest attempts. Its matter is only a footnote on the Celts, Gaels and
Cymry, and its substance often reappears in later pages; but other items
both in the early script of a fine minuscule, and in the later bold,
untidy scrawl, serve to carry on the Welsh account, with references to
Pwll Cheres and Goronwy Owen; and the upshot of them all goes to show
that Borrow, whether he was at Norwich or in London, was not only a stout
Celtophile, but much inclined, early and late, to be a Welsh idolater.
And since the days when the monks of the Priory at Carmarthen wrote the
"Black Book" in a noble script, I suppose no copyist ever took more pains
than Borrow did in his early years in transcribing the lines of the Welsh
poets, as the _facsimile_ page given in this volume can tell.
Of the bards and rhymers that he attempted in English, he gave most care
to translating Iolo Goch, four of whose odes open the present collection.
He was tempted to dilate on Iolo, or "Edward the Red," because of that
poet's association with Owen Glendower, a hero in whose exploits he
greatly delighted. The tribute to Owen in "Wild Wales" is, or should be,
familiar enough to Borrovians. In Chapter XXIII. there is an account of
the landmark which Borrow calls "Mont Glyndwr" (though I have never heard
it so called in my Welsh wanderings); while in Chapter LXVI. a
description of the other mount at Sycharth accompanies a translation of
the Ode by Iolo, which in a slightly different earlier text is printed on
page eight. It was after repeating these lines, Borrow tells us, that he
exclaimed, "How much more happy, innocent and holy" he was in the days of
his boyhood, when he translated the ode, than "at the present time." And
then, covering his face with his hands, he wept "like a child." If one
re-reads the ode in the light of this confession, one observes that there
is a strong vein of personal feeling about its lines, and a certain
pilgrim strain in its opening, which would lend themselves readily to
Borrow's mood and the idea, never far away from his thoughts, that in his
wanderings he too was a bard doing "Clera." It need hardly be said that
he was wrong in estimating Iolo's age as "upwards of a hundred years,"
when the ode was written. In other details of the poem he is more
picturesque than literal; but the English copy of the Welsh sketch is in
essentials near enough for all ordinary purposes; and the achievement in
a boy of eighteen, living at Norwich, far from Wales, is an extraordinary
one. The sort of error that he fell into was a very natural one to
occur; for instance, misled by his mere dictionary knowledge, he omits
the reference to St. Patrick's clock-tower and the cloisters of
Westminster. The words "Kloystr Wesmestr," only lead in one text to the
line, "A cloister of festivities," and in the other to the yet freer
rendering--"muster the merry pleasures all." Again, the original has no
mention of "Usquebaugh," though the Shrewsbury ale is in order. In
medieval Wales, I may add, the bragget mentioned in these lines was made
by mixing ale with mead, and spicing the mixture--a decidedly heady
liquor, one gathers, when it was kept awhile.
Iolo Goch, like the greater--indeed one may say the greatest Welsh poet,
Dafydd ab Gwilym, used a form of verse in his odes which it is not easy
to imitate or follow in English, keeping all its subtle graces and
assonances. It is termed the "Cywydd," which may be taken to signify a
verse in which the words are well knit and finely co-ordinated; or, as
Sir John Rhys puts it, "elegantly, artistically put together." The
verse, it should be said, is written in couplets, and the lines are
required also to follow a definite symphonic pattern. Try for example
Dafydd's lines, which Borrow has translated (see page 59), upon the mist.
In Welsh they run:
"Och! it 'niwlen felen-fawr
_Na throet_ ti, _na therit_ awr:
Casul _yr_ aw_yr_ ddu-lwyd,
Carth_en_ annib_en_ ia_wn_ wyd,
Mwg ellylldan o annwn,
_Abid_ teg ar _y byd_ hwn.
Fal _tarth_ uffern-_barth_ ffwrn-bell;
_Mwg_ y _byd_ yn _magu_ o _bell_."
The second and last of these verses well show the use of what is called
the "cynghanedd" or consonancy of echoing syllables required in the
cywydd metre. Borrow, in getting his own rhyme, rather loses the force
of the original. For instance, he omits the "awyr ddu-lwyd" in verse
three--the air black-grey--and he spoils in expanding the idea of the
verse--"carthen anniben," etc. Here the Welsh poet suggests that the
mist is an endless cloth, woven perpetually in space. The packed lines
of the cywydd, and the concreteness of the imagery, set the translator,
however, a hard task. Borrow, in the "Wild Wales" version, omits the
opening of the poem, whose last lines lead up to the apostrophe; but the
MS. has enabled Mr. Wise to complete it in his Bibliography. More
literally, the Welsh might be rendered thus:--
"Before I had gone a step of the way,
I no longer saw a place in the land:
Neither birchclad cliff, nor coast;
Neither hill's-breast, mountain-side, nor sea."
Then it is he turns in his humorous rage:
"Och! confound thee, great yellow thing,
That neither turns lighter, nor clears a bit;
Black-grey chasuble of the air;
An endless woven clout, thou art!"
Borrow's difficulty in attacking the Welsh of a poet so rapid and easy
and light-footed, was that of a Zeppelin in pursuit of a Farman. He was
over-weighted from the start. His early awkwardness in verse, his
rhetoric learnt from the artificial style of the generation before him,
were in his way. Iolo Goch was much nearer to him, with the admiring
inventory of a chieftain's house, than was the art of the poet of the
leaves, the birch-grove and the love-tryst.
But as time went on Borrow returned on his old steps, and he took up some
of his former handiwork, and smoothed away some of its crudities. Mr.
Wise, indeed, maintains that the Borrow of 1826 was a much less finished
verseman than the Borrow of 1854-60; and his Bibliography illustrates
some of the changes made for the better in Borrow's verse. Thus, in one
Norse ballad, he changes "gore" into "blood," and we remark in many lines
an attempt to get at a more natural style in verse. The account of "The
Sleeping Bard" in the Bibliography, shows that the improvement in
Borrow's craftsmanship went on after 1860, in which year the book was
printed at Yarmouth (a very limited edition, 250 copies at 5_s._ a copy).
For instance, in the poem, "Death the Great," the seventh stanza ran
originally:
"The song and dance afford, I _ween_,
Relief from _spleen_, and sorrow's grave;
How very strange there is no _dance_
Nor tune of _France_, from Death can save."
In 1871 the four lines were recast as follows:--
"The song and dance can drive, they _say_,
The spleen _away_, and humour's grave;
Why hast thou not devised, O _France_
Some tune and _dance_ from Death to save?"
Here again, we see, he purges his poetic diction, and turns "I ween" into
"they say." It is remarkable that in translating these lines by Elis
Wynn he is not content to get the end-rhymes only, but accepts to the
full the difficulty of following the Welsh in the interned rhymes
throughout--as shown by the words italicised.
In his interesting account of "George Borrow and his Circle," Mr. Shorter
quotes a letter from Professor Cowell to a Norwich correspondent, Mr.
James Hooper, which betrays some disappointment over Borrow's Welsh
interest at the close of his life. Cowell had been inspired by "Wild
Wales" to learn Welsh, and even nursed a wish to do so under Borrow
himself. He found his way to Oulton Hall one autumn day, and its
master--now an old man close on eighty--opened the door in person. The
ardent visitor talked to him of Ab Gwilym, but his interest was languid;
and even the news that the Honourable Cymmrodorion were about to publish
the poems of Iolo Goch did not rouse him. Cowell himself, it may be
added, afterwards wrote an excellent appreciation of Ab Gwilym in the
Transactions of the same society. In his letter, Cowell speaks of
Borrow's carelessness as a translator, and declares the very
title--"Visions of the Sleeping Bard"--to be wrong; it should be, not the
"Sleeping Bard," but the "Bard Sleep." However, in this case, Borrow's
instinct was truer than his critic's. For "Cwsg" is used as a
noun-adjective by Elis Wynn; and the latest translator of the book--Mr.
Gwyneddon Davies {17}--adopts the same title precisely.
Borrow's record as a Welsh translator would not be complete without a
page or so of his version of the prose text of the same work. Elis Wynn,
I may explain, was, after the tale-writers of the Mabinogion, the best
author of Welsh narrative prose that the language possesses. He was at
once idiomatic and exact in style. He knew how to get the golden
epithet; his diction was bold and biblical, his vocabulary could be at
times startling and Rabelaisean. Borrow's efficiency in rendering him
may be tested by a couple of passages. The first takes us to the City of
Destruction and its streets:--
"'What are those streets called,' said I. 'Each is called,' he
replied, 'by the name of the princess who governs it: the first is
the street of Pride, the middle one the street of Pleasure, and the
nearest, the street of Lucre.' 'Pray, tell me,' said I, 'who are
dwelling in these streets? What is the language which they speak?
What are the tenets which they hold? To what nation do they belong?'
'Many,' said he, 'of every language, faith and nation under the sun
are living in each of those vast streets below; and there are many in
each of the three streets alternately, and everyone as near as
possible to the gate; and they frequently remove, unable to tarry
long in the one, from the great love they bear to the princess of
some other street; and the old fox looks slyly on, permitting
everyone to love his choice, or all three if he pleases, for then he
is most sure of him.'
"'Come nearer to them,' said the angel, and hurried with me
downwards, shrouded in his impenetrable veil, through much noxious
vapour which was rising from the city; presently, we descended in the
street of Pride, upon a spacious mansion open at the top, whose
windows had been dashed out by dogs and crows, and whose owners had
departed to England or France, to seek there for what they could have
obtained much easier at home; thus, instead of the good, old,
charitable, domestic family of yore, there were none at present but
owls, crows, or chequered magpies, whose hooting, cawing, and
chattering were excellent comments on the practices of the present
owners. There were in that street myriads of such abandoned palaces,
which might have been, had it not been for Pride, the resorts of the
best, as of yore, places of refuge for the weak, schools of peace and
of every kind of goodness; and blessings to thousands of small houses
around."
This comes from the first of the Three Dreams, that of the World; and a
further quotation from the same dream-book touches what is Borrow's
high-water mark as a translator:--
"Thereupon we turned our faces from the great city of Perdition, and
went up to the other little city. In going along, I could see at the
upper end of the streets many turning half-way from the temptations
of the gates of Perdition and seeking for the gate of Life; but
whether it was that they failed to find it, or grew tired upon the
way, I could not see that any went through, except one sorrowful
faced man, who ran forward resolutely, while thousands on each side
of him were calling him fool, some scoffing him, others threatening
him, and his friends laying hold upon him, and entreating him not to
take a step by which he would lose the whole world at once. 'I only
lose,' said he, 'a very small portion of it, and if I should lose the
whole, pray what loss is it? For what is there in the world so
desirable, unless a man should desire deceit, and violence, and
misery, and wretchedness, giddiness and distraction? Contentment and
tranquillity,' said he, 'constitute the happiness of man; but in your
city there are no such things to be found. Because who is there here
content with his station? Higher, higher! is what everyone
endeavours to be in the street of Pride. Give, give us a little
more, says everyone in the street of Lucre. Sweet, sweet, pray give
me some more of it, is the cry of everyone in the street of Pleasure.
"'And as for tranquillity, where is it? and who obtains it? If you
be a great man, flattery and envy are killing you. If you be poor,
everyone is trampling upon and despising you. After having become an
inventor, if you exalt your head and seek for praise, you will be
called a boaster and a coxcomb. If you lead a godly life and resort
to the Church and the altar, you will be called a hypocrite. If you
do not, then you are an infidel or a heretic. If you be merry, you
will be called a buffoon. If you are silent, you will be called a
morose wretch. If you follow honesty, you are nothing but a simple
fool. If you go neat, you are proud; if not, a swine. If you are
smooth speaking, then you are false, or a trifler without meaning.
If you are rough, you are an arrogant, disagreeable devil. Behold
the world that you magnify!' said he; 'pray take my share of it.'"
In the foregoing extract Borrow makes a few obvious errors. For
instance, he turns the Welsh word "dyfeiswr" into "inventor," whereas the
sense here implies a schemer, or intriguer (the last is the rendering
adopted by Mr. Gwyneddon Davies), and the translation suffers a
corresponding lapse in the same clause. But on the whole Borrow's
rendering is good of its kind, and it gains by its freedom at times, as
in the page where he turns "dwylla o'th arian a'th hoedl hefyd," into
"chouses you of your money and your life."
The fact is, Borrow was vital in prose, while the shackles of verse often
weighed on him. It was only in mid-career that he learnt to move at all
easily in them--how much more easily we should not have known had not Mr.
Wise, with his bibliographical intrepidity, set about printing for his
own library some of the unpublished matter. In the light of those green
quartos, Borrow is seen to be a translator of more force than grace, who
generally contrived to give a flavour of his own to whatever he touched.
Because of the subtleties of the prosody, he was rather less effective in
dealing with Welsh and Celtic than with Norse and Gothic verse. But he
managed to create an English that was undoubtedly rare in his day, and is
now unique because the Borrovian accent is in it, and the masculine voice
of Borrow--like the cry of Vidrik in the ballad--is unmistakable. He
knew the art of giving a name to things; and, again like Vidrik, who
called his sword "mimmering," and his shield "skrepping," this Cornish
East Anglian, who dabbled in gipsy lore and learnt Welsh, made his
weapons part of himself, whether they consisted of his pen, his
portentous umbrella, or his father's silver-handled blade:--
"Thou'st decked old chiefs of Cornwall's land
To face the fiend with thee they dared;
Thou prov'dst a Tirfing in their hand,
Which victory gave whene'er 'twas bared.
"Though Cornwall's moors 'twas ne'er my lot
To view, in Eastern Anglia born,
Yet I her sons' rude strength have got,
And feel of death their fearless scorn."
Little need be added about the various sources of the following text.
The first three poems are from a quarto MS. owned by Mr. Gurney of
Norwich, who has kindly lent it to the publishers. Its title runs:
_poems_.
By IOLO GOCH;
With a Metrical English Translation.
Some former owner has pencilled below, "By Mr. Borrer of Norwich"
(_sic_). From Mr. Wise's green quartos, already referred to, or from
MSS. in his library, come the two Goronwy Owen poems, "The Pedigree of
the Muse," and "The Harp." Also Lewis Morris the Elder's lines, "The
Cuckoo's Song in Meirion," or Merion, according to Borrow. The Epigrams
by Carolan and "Song of Deirdra" are Irish items from the same source;
while "Pwll Cheres, the Vortex of Menai," and "The Mountain Snow," are
two Welsh ones, which have not, I believe, been printed in any other
form. The familiar pages of "Wild Wales," and the less-known volume,
"Targum," account for the bulk of the remaining poems and fragments;
while Borrow's "Quarterly Review" article on Welsh Poetry (January, 1861)
provides us with four more translations. The versions are printed with
all their faults on their head; and if he put a whiting into a
fresh-water fish-pond (in the Ode on Sycharth, original text), or mistook
a saint for a secular detail, the collector of his works will be glad to
have the plain evidence under his hand, and will not wonder a bit the
less at the boyish achievement of this East-country Celt. It remains to
be said that, being Borrow, he was duly astonished at himself, and under
the Sycharth poem wrote in Welsh a footnote which runs in effect: "The
English translation is the work of George Borrow, an English lad of the
City of Norwich, who has never been in Wales, and has never in all his
life heard a word of Welsh from man or woman."
GLENDOWER'S MANSION.
IOLO GOCH was a celebrated Bard of North Wales, and flourished about the
end of the fourteenth and commencement of the fifteenth century. He was
the contemporary of the celebrated Owain Glendower, and one of the most
devoted and not the least effectual of his partisans; for by his songs he
kindled the spirit of his countrymen against the English, and by his
praises of Glendower increased their pre-existing enthusiasm for that
chieftain. The present poem was composed some years previous to the
insurrection of Glendower against Henry the Fourth, and describes with
the utmost possible minuteness his place of residence at Sycharth, to
which place Iolo, after receiving frequent invitations from its owner,
repaired to reside in his old age.
A PROMISE has been made by me
Twice of a journey unto thee;
His promises let every man
Perform, as far as e'er he can.
Easy is done the thing that's sweet,
And sweet this journey is and meet;
I've vow'd to Owain's court to go,
To keep that vow no harm will do;
And thither straight I'll take the way,
A happy thought, and there I'll stay,
Respect and honor whilst I live
With him united to receive.
My Chief of long-lin'd ancestry,
Can harbour sons of poesy.
To hear the sweet Muse singing bold
A fine thing is when one is old;
And to the Castle I will hie,
There's none to match it 'neath the sky;
It is a Baron's stately court,
Where bards for sumptuous fare resort.
The Lord and star of powis land,
He granteth every just demand.
Its likeness now I will draw out:
Water surrounds it in a moat;
Stately's the palace with wide door,
Reach'd by a bridge the blue lake o'er;
It is of buildings coupled fair,
Coupled is every couple there;
A quadrate structure tall it is,
A cloister of festivities.
Conjointly are the angles bound;
In the whole place no flaw is found.
Structures in contact meet the eye
Grottoways, on the hill on high.
Into each other fasten'd, they
The form of a hard knot display.
There dwells the Chief, we all extoll,
In fair wood house on a light knoll.
Upon four wooden columns proud
Mounteth his mansion to the cloud.
Each column's thick, and firmly bas'd,
And upon each a loft is plac'd.
In these four lofts, which coupled stand,
Repose at night the minstrel band:
These four lofts, nests of luxury
Partition'd, form eight prettily.
Tiled is the roof, on each house top
Chimneys, where smoke is bred, tower up.
Nine halls in form consimilar,
And wardrobes nine to each there are,
Wardrobes well stock'd with linen white
Equal to shops of London quite.
A church there is, a cross which has,
And chapels neatly paned with glass.
All houses are contained in this,
An orchard, vineyard 'tis of bliss.
Beside the Castle, 'bove all praise,
Within a park the red deer graze.
A coney park the Chief can boast,
Of ploughs and noble steeds a host;
Meads, where for hay the fresh grass grows,
Cornfields which hedges trim enclose;
Mill a perennial stream upon,
And pigeon tower fram'd of stone;
A fish pond deep and dark to see,
To cast nets in when need there be;
And in that pond there is no lack
Of noble whitings and of jack.
Three boards he keeps, his birds abound,
Peacocks and cranes are seen around.
All that his household-wants demand
Is order'd straight by his command:
Ale he imports from Shrewsbury far,
Glorious his beer and bragget are.
All drinks he keeps, bread white of look,
And in his kitchen toils his cook.
His castle is the minstrels' home,
You'll find them there whene'er you come.
Of all her sex his wife's the best,
Her wine and mead make life thrice blest.
She's scion of a knightly tree,
She's dignified, she's kind and free;
His bairns come to me pair by pair,
O what a nest of chieftains fair!
There difficult it is to catch
A sight of either bolt or latch;
The porter's place there none will fill--
There handsels shall be given still,
And ne'er shall thirst and hunger rude
In Sycharth venture to intrude.
The noblest Welshman, lion for might,
The Lake possesses, his by right,
And 'midst of that fair water plac'd,
The Castle, by each pleasure grac'd.
ODE TO THE COMET.
Which appeared in the Month of March, A.D. 1402.
By
IOLO GOCH.
THIS piece appears to have been written at the period when Glendower had
nearly attained the
summit of his greatness; the insurrection which he commenced in
September, 1400, by sacking and burning the town of Ruthin, having
hitherto sustained no check whatever. In the present poem his bard hails
the appearance of the Comet as a divine prognostic of the eventual
success of the Welsh Hero, and of his elevation to the throne of Britain.
'BOUT the stars' nature and their hue
Much has been said, both false and true;
They're wondrous through their countenance--
Signs to us in the blue expanse.
The first that came, to merit praise,
Was that great star of splendid rays,
From a fair country seen of old
High in the East, a mark of gold;
Conveying to the sons of Earth
News of the King of glory's birth.
In the advantage I had share,
Though some to doubt the event will dare,
That Christ was born from Mary maid,
A merciful and timely aid,
With his veins' blood to save on high
The righteous from the enemy.
The second, a right glorious lamp,
Of yore went over Uther's camp.
There as it flam'd distinct in view
Merddin amongst the warrior crew
Standing, with tears of anguish, thought
Of the dire act on Emrys wrought, {34}
And he caus'd Uther back to turn,
The victory o'er the foe to earn;
From anger to revenge to spring
Is with the frank a common thing.
Arthur the generous, bold and good,
Was by that comet understood.
Man to be cherish'd well and long,
Foretold through ancient Bardic song:
With ashen shafted lance's thrust
He shed his foe's blood on the dust.
The third to Gwynedd's hills was born
By time and tempest-fury worn,
Similar to the rest it came,
In origin and look the same,
Powerfully lustrous, yellow, red
Both, both as to its beam and head.
The wicked far about and near
Enquire of me, who feel no fear,
For where it comes there luck shall fall,
What means the hot and starry ball?
I know and can expound aright
The meaning of the thing of light:
To the son of the prophecy
Its ray doth steel or fire imply;
There has not been for long, long time
A fitting star to Gwynedd's clime,
Except the star this year appearing,
Intelligence unto us bearing;
Gem to denote we're reconcil'd
At length with God the undefil'd.
How beauteous is that present sheen,
Of the excessive heat the queen;
A fire upmounting 'fore our face,
Shining on us God's bounteous grace;
For where they sank shall rise once more
The diadem and laws of yore.
'Tis high 'bove Mona in the skies,
In the angelic squadron's eyes;
A golden pillar hangs it there,
A waxen column of the air.
We a fair gift shall gain ere long,
Either a pope or Sovereign strong;
A King, who wine and mead will give,
From Gwynedd's land we shall receive;
The Lord shall cease incens'd to be,
And happy times cause Gwynedd see,
Fame to obtain by dint of sword,
Till be fulfill'd the olden word.
ODE TO GLENDOWER
After His Disappearance.
By
IOLO GOCH.
FORTUNE having turned against Glendower, he fought many unsuccessful
battles, in which all his sons perished, bravely maintaining the cause of
their father. His adherents being either slaughtered or dispirited, the
Welsh Chieftain retired into concealment--but where, no mortal at the
present day can assert with certainty, but it is believed that he died of
grief and disappointment in the year 1415, at the house of his daughter,
the wife of Sir John Scudamore, of Monington in Herefordshire. The fall
of Glendower was a bitter mortification to the Bards, whom he had so long
feasted in the watery valley {39} from which he derived his surname; many
poetical compositions are still preserved, written with the view of
reviving the hopes of his dispirited friends. Amongst these the
following by Iolo Goch is perhaps the most remarkable. He hints that the
Chieftain has repaired to Rome, from which he will return with a warrant
under the seal of the Pope, to take possession of his right. Then he
flings out a surmise that he has travelled to the Holy Sepulchre, and
will re-appear, with a Danish and Irish fleet to back his cause.
Notwithstanding the little regard paid to truth and probability in this
piece, and notwithstanding its strange metaphors and obscure allusions,
it displays marks of no ordinary poetic talent, and is a convincing proof
that the fire and genius of the author had not deserted him at fourscore,
to which advanced age he had attained when he wrote it.
TALL man, whom Harry loves but ill,
Thou'st had reverses, breath'st thou still?
If so, with fire-spear seek the fray,
Come, and thy target broad display.
From land of Rome, which glory's light
Environs, come in armour dight,
With writ, which bears the blest impression
Of Peter's seal, to take possession.
Big Bull! from eastern climates speed,
Bursting each gate would thee impede.
Flash from thy face shall fiery rays,
On thee shall all with reverence gaze.
Fair Eagle! earl of trenchant brand!
Betake thee to the Lochlin land,
Whose sovereign on his buckler square,
Sign of success, is wont to bear
Three lions blue, through fire to see
Like azure, and steel-fetters three.
We'll trust, far casting black despair,
Hence in the peacock, hog and bear!
For O the three shall soon unite,
A dread host in the hour of fight.
Launch forth seven ships, do not delay,
Launch forth seven hundred, tall and gay;
From the far north, at Mona's pray'r,
To verdant Eirin's shore repair.
To seek O'Neil must be thy task,
And at his hand assistance ask;
Ere feast of John we shall not fail
To hear a rising of the Gael:
Through the wild waste to Dublin town
Shall come a leader of renown.
Prepare a fleet with stout hearts mann'd
From Irishmen's dear native land.
Come thou who did'st by treachery fall,
Where'er thou art my soul is all.
Yellow and red, before a feast,
The colours are, the Erse love best,
Deck with the same, their hearts to win,
The banner old of Llywellin.
Call Britain's host (may woe betide
England for treachery!) to thy side;
Come to our land, tough steel, and o'er
The islands rule, an Emperor;
A fire ignite on shore of Mon
Staunch Eagle! ere an hour be flown.
The castles break, retreats of care,
Conquer of Caer Ludd's dogs the lair!
Mona's gold horn! the Normans smite,
Kill the mole and his men outright:
A prophecy there stands from old,
That numerous battles thou shalt hold;
Where'er thou'st opportunity
Fight the tame Lion furiously;
Fierce shall thy hands' work prove, I trow,
Dying and dead shall Merwyg strow;
War shall my Chief through summer wage,
That the wheel turn, my life I'll gage;
Like to the burst of Derri's stream
The onset of his war shall seem.
With Mona's flag through Iaithon's glen
Shall march a host of armed men:
Nine fights he'll wage and then have done,
Successful in them every one.
Come heir of Cadwallader blest,
And thy sire's land from robbers wrest:
Take thou the portion that's thine own,
Us from the chains 'neath which we groan.
HERE'S THE LIFE I'VE SIGH'D FOR LONG.
By
IOLO GOCH.
HERE'S the life I've sigh'd for long:
Abash'd is now the Saxon throng,
And Britons have a British lord
Whose emblem is the conquering sword;
There's none I trow but knows him well
The hero of the watery dell.
Owain of bloody spear in field,
Owain his country's strongest shield;
A sovereign bright in grandeur drest,
Whose frown affrights the bravest breast.
Let from the world upsoar on high
A voice of splendid prophecy!
All praise to him who forth doth stand
To 'venge his injured native land!
Of him, of him a lay I'll frame
Shall bear through countless years his name:
In him are blended portents three,
Their glories blended sung shall be:
There's Owain, meteor of the glen,
The head of princely generous men;
Owain, the lord of trenchant steel,
Who makes the hostile squadrons reel;
Owain besides, of warlike look,
A conqueror who no stay will brook;
Hail to the lion leader gay,
Marshaller of Griffith's war array;
The scourger of the flattering race,
For them a dagger has his face;
Each traitor false he loves to smite,
A lion is he for deeds of might;
Soon may he tear, like lion grim,
All the Lloegrians limb from limb!
May God and Rome's blest father high
Deck him in surest panoply!
Hail to the valiant carnager,
Worthy three diadems to bear!
Hail to the valley's belted King!
Hail to the widely conquering,
The liberal, hospitable, kind,
Trusty and keen as steel refined!
Vigorous of form he nations bows,
Whilst from his breast-plate bounty flows.
Of Horsa's seed on hill and plain
Four hundred thousand he has slain.
The cope-stone of our nation's he,
In him our weal, our all we see;
Though calm he looks his plans when breeding,
Yet oaks he'd break his clans when leading.
Hail to this partisan of war,
This bursting meteor flaming far!
Where'er he wends Saint Peter guard him,
And may the Lord five lives award him!
THE PROPHECY {49a} OF TALIESIN.
_From the Ancient British_.
WITHIN my mind
I hold books confin'd,
Of Europa's land all the mighty lore;
O God of heaven high!
With how many a bitter sigh,
I my prophecy upon Troy's line {49b} pour:
A serpent coiling,
And with fury boiling,
From Germany coming with arm'd wings spread,
Shall Britain fair subdue
From the Lochlin ocean blue,
To where Severn rolls in her spacious bed.
And British men
Shall be captives then
To strangers from Saxonia's strand;
From God they shall not swerve,
They their language shall preserve,
But except wild Wales, they shall lose their land.
THE HISTORY OF TALIESIN.
_From The Ancient British_.
TALIESIN was a foundling, discovered in his infancy lying in a coracle,
on a salmon-weir, in the domain of Elphin, a prince of North Wales, who
became his patron. During his life he arrogated to himself a
supernatural descent and understanding, and for at least a thousand years
after his death he was regarded by the descendants of the ancient Britons
in the character of a prophet or something more. The poems which he
produced procured for him the title of "Bardic King;" they display much
that is vigorous and original, but are disfigured by mysticism and
extravagant metaphor; one of the most spirited of them is the following,
which the author calls his "Hanes" or history.
THE head Bard's place I hold
To Elphin, chieftain bold;
The country of my birth
Was the Cherubs' land of mirth;
I from the prophet John
The name of Merddin won;
And now the Monarchs all
Me Taliesin call.
I with my Lord and God
On the highest places trod,
When Lucifer down fell
With his army into hell.
I know each little star
Which twinkles near and far;
And I know the Milky Way
Where I tarried many a day.
My inspiration's {54a} flame
From Cridwen's cauldron came;
Nine months was I in gloom
In Sorceress Cridwen's womb;
Though late a child--I'm now
The Bard of splendid brow; {54b}
When roar'd the deluge dark,
I with Noah trod the Ark.
By the sleeping man I stood
When the rib grew flesh and blood.
To Moses strength I gave
Through Jordan's holy wave;
The thrilling tongue was I
To Enoch and Elie;
I hung the cross upon,
Where died the . . . (_only son_)
A chair of little rest
'Bove the Zodiac I prest,
Which doth ever, in a sphere,
Through three elements career;
I've sojourn'd in Gwynfryn,
In the halls of Cynfelyn;
To the King the harp I play'd,
Who Lochlyn's sceptre sway'd.
With the Israelites of yore
I endur'd a hunger sore;
In Africa I stray'd
Ere was Rome's foundation laid;
Now hither I have hied
With the race of Troy to bide;
In the firmament I've been
With Mary Magdalen.
I work'd as mason-lord
When Nimrod's pile up-soar'd;
I mark'd the dread rebound
When its ruins struck the ground;
When stroke to victory on
The men of Macedon,
The bloody flag before
The heroic King I bore.
I saw the end with horror
Of Sodom and Gomorrah!
And with this very eye
Have seen the . . . (_end of Troy_;)
I till the judgment day
Upon the earth shall stray:
None knows for certainty
Whether fish or flesh I be.
THE MIST.
A TRYSTE with Morfydd true I made,
'Twas not the first, in greenwood glade,
In hope to make her flee with me;
But useless all, as you will see.
I went betimes, lest she should grieve,
Then came a mist at close of eve;
Wide o'er the path by which I passed,
Its mantle dim and murk it cast.
That mist ascending met the sky,
Forcing the daylight from my eye.
I scarce had strayed a furlong's space
When of all things I lost the trace.
Where was the grove and waving grain?
Where was the mountain, hill and main?
O ho! thou villain mist, O ho!
What plea hast thou to plague me so!
I scarcely know a scurril name,
But dearly thou deserv'st the same;
Thou exhalation from the deep
Unknown, where ugly spirits keep!
Thou smoke from hellish stews uphurl'd
To mock and mortify the world!
Thou spider-web of giant race,
Spun out and spread through airy space!
Avaunt, thou filthy, clammy thing,
Of sorry rain the source and spring!
Moist blanket dripping misery down,
Loathed alike by land and town!
Thou watery monster, wan to see,
Intruding 'twixt the sun and me,
To rob me of my blessed right,
To turn my day to dismal night.
Parent of thieves and patron best,
They brave pursuit within thy breast!
Mostly from thee its merciless snow
Grim January doth glean, I trow.
Pass off with speed, thou prowler pale,
Holding along o'er hill and dale,
Spilling a noxious spittle round,
Spoiling the fairies' sporting ground!
Move off to hell, mysterious haze;
Wherein deceitful meteors blaze;
Thou wild of vapour, vast, o'ergrown,
Huge as the ocean of unknown.
Before me all afright and fear,
Above me darkness dense and drear.
My way at weary length I found
Into a swaggy willow ground,
Where staring in each nook there stood
Of wry-mouthed elves a wrathful brood.
Full oft I sunk in that false soil,
My legs were lamed with length of toil.
However hard the case may be,
No meetings more in mist for me.
THE CUCKOO'S SONG IN MERION.
_From the Welsh of Lewis Morris_.
THOUGH it has been my fate to see
Of gallant countries many a one;
Good ale, and those that drank it free,
And wine in streams that seemed to run;
The best of beer, the best of cheer,
Allotted are to Merion.
The swarthy ox will drag his chain,
At man's commandment that is done;
His furrow break through earth with pain,
Up hill and hillock toiling on;
Yet with more skill draw hearts at will
The maids of county Merion.
Merry the life, it must be owned,
Upon the hills of Merion;
Though chill and drear the prospect round,
Delight and joy are not unknown;
O who would e'er expect to hear
'Mid mountain bogs the cuckoo's tone?
O who display a mien full fair,
A wonder each to look upon?
And who in every household care
Defy compare below the sun?
And who make mad each sprightly lad?
The maids of county Merion.
O fair the salmon in the flood,
That over golden sands doth run;
And fair the thrush in his abode,
That spreads his wings in gladsome fun;
More beauteous look, if truth be spoke,
The maids of county Merion.
Dear to the little birdies wild
Their freedom in the forest lone;
Dear to the little sucking child
The nurse's breast it hangs upon;
Though long I wait, I ne'er can state
How dear to me is Merion.
Sweet in the house the Telyn's {64} strings
In love and joy where kindred wone;
While each in turn a stanza sings,
No sordid themes e'er touched upon;
Full sweet in sound the hearth around
The maidens' song of Merion.
And though my body here it be
Travelling the countries up and down;
Tasting delights of land and sea,
True pleasure seems my heart to shun;
Alas! there's need home, home to speed--
My soul it is in Merion.
THE SNOW ON EIRA.
COLD is the snow on Snowdon's brow,
It makes the air so chill;
For cold, I trow, there is no snow
Like that of Snowdon's hill.
A hill most chill is Snowdon's hill,
And wintry is his brow;
From Snowdon's hill the breezes chill
Can freeze the very snow.
THE INVITATION.
_By Goronwy Owen_.
_From the Cambrian British_.
[Sent from Northolt, in the year 1745, to William Parry, Deputy
Comptroller of the Mint.]
PARRY, of all my friends the best,
Thou who thy Maker cherishest,
Thou who regard'st me so sincere,
And who to me art no less dear;
Kind friend, in London since thou art,
To love thee's not my wisest part;
This separation's hard to bear:
To love thee not far better were.
But wilt thou not from London town
Journey some day to Northolt down,
Song to obtain, O sweet reward,
And walk the garden of the Bard?--
But thy employ, the year throughout,
Is wandering the White Tower about,
Moulding and stamping coin with care,
The farthing small and shilling fair.
Let for a month thy Mint lie still,
Covetous be not, little Will;
Fly from the birth-place of the smoke,
Nor in that wicked city choke;
O come, though money's charms be strong,
And if thou come I'll give thee song,
A draught of water, hap what may,
Pure air to make thy spirits gay,
And welcome from an honest heart,
That's free from every guileful art.
I'll promise--fain thy face I'd see--
Yet something more, sweet friend, to thee:
The poet's cwrw {74} thou shalt prove,
In talk with him the garden rove,
Where in each leaf thou shalt behold
The Almighty's wonders manifold;
And every flower, in verity,
Shall unto thee show visibly,
In every fibre of its frame,
His deep design, who made the same.--
A thousand flowers stand here around,
With glorious brightness some are crown'd:
How beauteous art thou, lily fair!
With thee no silver can compare:
I'll not forget thy dress outshone
The pomp of regal Solomon.
I write the friend, I love so well,
No sounding verse his heart to swell.
The fragile flowerets of the plain
Can rival human triumphs vain.
I liken to a floweret's fate
The fleeting joys of mortal state;
The flower so glorious seen to-day
To-morrow dying fades away;
An end has soon the flowery clan,
And soon arrives the end of man;
The fairest floweret, ever known,
Would fade when cheerful summer's flown;
Then hither haste, ere turns the wheel!
Old age doth on these flowers steal;
Though pass'd two-thirds of autumn-time,
Of summer temperature's the clime;
The garden shows no sickliness,
The weather old age vanquishes,
The leaves are greenly glorious still--
But friend! grow old they must and will.
The rose, at edge of winter now,
Doth fade with all its summer glow;
Old are become the roses all,
Decline to age we also shall;
And with this prayer I'll end my lay,
Amen, with me, O Parry say;
To us be rest from all annoy,
And a robust old age of joy;
May we, ere pangs of death we know,
Back to our native Mona go;
May pleasant days us there await,
United and inseparate!
And the dread hour, when God shall please
To bid our mutual journey cease,
May Christ, who reigns in heaven above,
Receive us to his breast of love!
THE PEDIGREE OF THE MUSE.
_From Goronwy Owen_.
OLD Homer, Grecian bard divine,
He Muses had, the tuneful Nine,
Of Goddesses a lovely quire,
Full like to Jove their heavenly Sire;
But their inventing song and strain
Is but a minstrel vision vain,
Nor in their birth, so proud and high,
I ween is more reality.
One Muse there was and one alone,
No fabled lustre round her shone,
With this fair girl the maiden band
Of Homer unconnected stand.
A different birth I claim for her,
Far older she than Jupiter;
The youths of heaven felt her power
In heavenly residence of yore;
And from her dwelling blest may she
To a vile man propitious be.
Grant to me, Lord, of her a share,
That I to sing her praise may dare.
Better thy help it were to gain
Than thousand, thousand tongues obtain.
I'll tell ye where a strain was sung
Ere in its orb earth's bullet swung,
Ere ocean had obtain'd its doors
Which hold confin'd its watery stores,
And of the world th' Almighty made
The firm foundation yet was laid.
When at the word th' Almighty said
The heaven above abroad was spread,
The morning stars in beauty bold,
Arose a concert high to hold.
Yes, yes, the beauteous morning train
Arose to sing a triumph strain.
When ended was the work sublime
They rose to sing a second time.
Thousands of heaven's brightest powers
Assembled from their azure bowers.
The sons of heaven unitedly
Pour'd out a hymn of harmony.
Completed is thy work, O God;
Wise are the courses by Thee trod,
Master of all Eternity.
O who is great and wise like Thee?
No organ's voice in sacred fane
E'er rivall'd that celestial strain;
A million accents all divine,
But different all, therein combine.
Of angel voices the accord
Downward pierc'd and upward soar'd.
The wandering stars who heard the strain
Into their orbits leapt again.
And louder, louder as it peal'd,
The arch of heaven shook and reel'd.
Down from the heaven's lofty blue
To this low world the accents flew,
In Paradise's blissful bound.
Our Father Adam heard the sound;
Delighted man's first father hears
The praise and music of the spheres;
To imitate the strain he tries,
And soon succeeds in gallant guise.
Delighted was his Eva dear
His good and pleasant song to hear;
Eva sang, so fair of feature;
Adam sang, tall noble creature.
Both sang from their green retreat
To God until the hour of heat.
From five past noon descanted they
Till disappeared the orb of day.
Young Abel's song was clear and mild,
And free from bursts of passion wild;
But fiercely harsh the ditty rang
Which Cain, red-handed ruffian, sang.
The gentle Muse you'll never find
United to a cruel mind;
The Almighty God this gift bestows
On breasts alone where virtue glows.
A thing of ancient date is song,
A muse to Moses did belong;
A muse--a sample of its power
He gave when quitting Egypt's shore.
A hundred sang, and with renown,
Ere we arrive at David down;
He sang like heaven's minstrel prime,
And harmony compos'd sublime.
'Twas he who framed the blessed psalms,
To souls distrest those sovereign balms;
He also many a deathless air
Produc'd from harp and dulcimer;
Mov'd with his hand the Muse along,
That hand so fair and yet so strong.
Soon as the blush of morn appear'd,
The anointed poet's voice was heard:
"Awake, my harp," so sang the King,
"A sweet and fitting song to sing;
Glory I'll give with tongue and chord,
Glory and praise to heaven's Lord."
His like ne'er was, and ne'er will be,
For music and for minstrelsy.
A Muse, and wondrous sweet its tone,
There was again to Solomon.
He sang in Judah's brightest days
A wondrous song, the lay of lays.
His Rose of Sharon all must love,
The lily and the hawthorn grove.
To his effusion sweet belongs
A station next to David's songs.
The offspring of a pious Muse
The Almighty God will not refuse,
Showing his loving kindness clear
To us his lowly children here.
In halls of heaven so bright and sheen
The power of song is great, I ween;
When there above in mighty quire
With us shall join heaven's host entire,
The one high God to glorify,
Commingle then shall earth and sky.
O what a blest employ to raise
Our voices in our Maker's praise!
Let's learn, my friends, the fitting song,
To sing it we may hope ere long
Above in courts where angels be,
Above where all is harmony,
And ne'er shall cease our anthem then
Of Holy, Holy Praise. Amen.
THE HARP.
_From Goronwy Owen_.
THE harp to every one is dear
Who hateth vice, and all things evil;
Hail to its gentle voice so clear,
Its gentle voice affrights the Devil!
The Devil can not the Minstrel quell--
He by the Minstrel is confounded;
From Saul was cast the spirit fell,
When David's harp melodious sounded.
EPIGRAM.
On a Miser who had built a stately Mansion.
_From the Cambrian British_.
OF every pleasure is thy mansion void;
To ruin-heaps may soon its walls decline.
O heavens, that one poor fire's but employ'd,
One poor fire only for thy chimneys nine!
Towering white chimneys--kitchen cold and drear--
Chimneys of vanity and empty show--
Chimneys unwarm'd, unsoil'd throughout the year--
Fain would I heatless chimneys overthrow.
Plague on huge chimneys, say I, huge and neat,
Which ne'er one spark of genial warmth announce;
Ignite some straw, thou dealer in deceit--
Straw of starv'd growth--and make a fire for once!
The wretch a palace built, whereon to gaze,
And sighing, shivering there around to stray;
To give a penny would the niggard craze,
And worse than bane he hates the minstrel's lay.
GRIFFITH AP NICHOLAS.
By Gwilym ab Ieuan Hen.
GRIFFITH AP NICHOLAS, who like thee
For wealth and power and majesty!
Which most abound, I cannot say,
On either side of Towy gay,
From hence to where it meets the brine,
Trees or stately towers of thine?
The chair of judgment thou didst gain,
But not to deal in judgments vain--
To thee upon thy judgment chair
From near and far do crowds repair;
But though betwixt the weak and strong
No questions rose of right and wrong,
The strong and weak to thee would hie;
The strong to do thee injury,
And to the weak thou wine wouldst deal
And wouldst trip up the mighty heel.
A lion unto the lofty thou,
A lamb unto the weak and low.
Much thou resemblest Nudd of yore,
Surpassing all who went before;
Like him thou'rt fam'd for bravery,
For noble birth and high degree.
Hail, captain of Kilgarran's hold!
Lieutenant of Carmarthen old!
Hail chieftain, Cambria's choicest boast!
Hail Justice, at the Saxon's cost!
Seven castles high confess thy sway,
Seven palaces thy hands obey.
Against my chief, with envy fired,
Three dukes and judges two conspired,
But thou a dauntless front did'st show,
And to retreat they were not slow.
O, with what gratitude is heard
From mouth of thine the whispered word;
The deepest pools in rivers found
In summer are of softest sound;
The sage concealeth what he knows,
A deal of talk no wisdom shows;
The sage is silent as the grave,
Whilst of his lips the fool is slave;
Thy smile doth every joy impart,
Of faith a fountain is thy heart;
Thy hand is strong, thine eye is keen,
Thy head o'er every head is seen.
RICHES AND POVERTY.
By Twm o'r Nant.
_Enter_ Captain Poverty.
O RICHES,--thy figure is charming and bright,
And to speak in thy praise all the world doth delight,
But I'm a poor fellow all tatter'd and torn,
Whom all the world treateth with insult and scorn.
Riches.
However mistaken the judgment may be
Of the world which is never from ignorance free,
The parts we must play, which to us are assign'd,
According as God has enlighten'd our mind.
Of elements four did our Master create,
The earth and all in it with skill the most great;
Need I the world's four materials declare--
Are they not water, fire, earth, and air?
Too wise was the mighty Creator to frame
A world from one element, water or flame;
The one is full moist and the other full hot,
And a world made of either were useless, I wot.
And if it had all of mere earth been compos'd,
And no water nor fire been within it enclos'd,
It could ne'er have produc'd for a huge multitude
Of all kinds of living things suitable food.
And if God what was wanted had not fully known,
But created the world of these three things alone,
How would any creature the heaven beneath,
Without the blest air have been able to breathe?
Thus all things created, the God of all grace,
Of four prime materials, each good in its place.
The work of His hands, when completed, He view'd,
And saw and pronounc'd that 'twas seemly and good.
Poverty.
In the marvellous things, which to me thou hast told
The wisdom of God I most clearly behold,
And did He not also make man of the same
Materials He us'd when the world He did frame?
Riches.
Creation is all, as the sages agree,
Of the elements four in man's body that be;
Water's the blood, and fire is the nature
Which prompts generation in every creature.
The earth is the flesh which with beauty is rife,
The air is the breath, without which is no life;
So man must be always accounted the same
As the substances four which exist in his frame.
And as in their creation distinction there's none
'Twixt man and the world, so the Infinite One
Unto man a clear wisdom did bounteously give
The nature of everything to perceive.
Poverty.
But one thing to me passing strange doth appear:
Since the wisdom of man is so bright and so clear,
How comes there such jarring and warring to be
In the world betwixt Riches and Poverty?
Riches.
That point we'll discuss without passion or fear,
With the aim of instructing the listeners here;
And haply some few who instruction require
May profit derive like the bee from the briar.
Man as thou knowest, in his generation
Is a type of the world and of all the creation;
Difference there's none in the manner of birth
'Twixt the lowliest hinds and the lords of the earth.
The world which the same thing as man we account
In one place is sea, in another is mount;
A part of it rock, and a part of it dale--
God's wisdom has made every place to avail.
There exist precious treasures of every kind
Profoundly in earth's quiet bosom enshrin'd;
There's searching about them, and ever has been,
And by some they are found, and by some never seen.
With wonderful wisdom the Lord God on high
Has contriv'd the two lights which exist in the sky;
The sun's hot as fire, and its ray bright as gold,
But the moon's ever pale, and by nature is cold.
The sun, which resembles a huge world of fire,
Would burn up full quickly creation entire
Save the moon with its temp'rament cool did assuage
Of its brighter companion the fury and rage.
Now I beg you the sun and the moon to behold,
The one that's so bright, and the other so cold,
And say if two things in creation there be
Better emblems of Riches and Poverty.
Poverty.
In manner most brief, yet convincing and clear,
You have told the whole truth to my wond'ring ear,
And I see that 'twas God, who in all things is fair,
Has assign'd us the forms, in this world which we bear.
In the sight of the world doth the wealthy man seem
Like the sun which doth warm everything with its beam;
Whilst the poor needy wight with his pitiable case
Resembles the moon which doth chill with its face.
Riches.
You know that full oft, in their course as they run,
An eclipse cometh over the moon or the sun;
Certain hills of the earth with their summits of pride
The face of the one from the other do hide.
The sun doth uplift his magnificent head,
And illumines the moon, which were otherwise dead,
Even as Wealth from its station on high,
Giveth work and provision to Poverty.
Poverty.
I know, and the thought mighty sorrow instils,
The sins of the world are the terrible hills
An eclipse which do cause, or a dread obscuration,
To one or another in every vocation.
Riches.
It is true that God gives unto each from his birth
Some task to perform whilst he wends upon earth,
But He gives correspondent wisdom and force
To the weight of the task, and the length of the course.
[_Exit_.
Poverty.
I hope there are some, who 'twixt me and the youth
Have heard this discourse, whose sole aim is the truth,
Will see and acknowledge, as homeward they plod,
Each thing is arrang'd by the wisdom of God.
THE PERISHING WORLD.
[From "The Sleeping Bard," by Elis Wynn.]
O MAN, upon this building gaze,
The mansion of the human race,
The world terrestrial see!
Its Architect's the King on high,
Who ne'er was born and ne'er will die--
The blest Divinity.
The world, its wall, its starlights all,
Its stores, where'er they lie,
Its wondrous brute variety,
Its reptiles, fish, and birds that fly,
And cannot number'd be,
The God above, to show His love,
Did give, O man, to thee.
For man, for man, whom He did plan,
God caus'd arise
This edifice,
Equal to heaven in all but size,
Beneath the sun so fair;
Then it He view'd, and that 'twas good
For man, He was aware.
Man only sought to know at first
Evil, and of the thing accursed
Obtain a sample small.
The sample grew a giantess,
'Tis easy from her size to guess
The whole her prey will fall.
Cellar and turret high,
Through hell's dark treachery,
Now reeling, rocking, terribly,
In swooning pangs appear;
The orchards round, are only found
Vile sedge and weeds to bear;
The roof gives way, more, more each day,
The walls too, spite
Of all their might,
Have frightful cracks down all their height,
Which coming ruin show;
The dragons tell, that danger fell,
Now lurks the house below.
O man! this building fair and proud,
From its foundation to the cloud,
Is all in dangerous plight;
Beneath thee quakes and shakes the ground;
'Tis all, e'en down to hell's profound,
A bog that scares the sight.
The sin man wrought, the deluge brought,
And without fail
A fiery gale,
Before which everything shall quail,
His deeds shall waken now;
Worse evermore, till all is o'er,
Thy case, O world, shall grow.
There's one place free yet, man for thee,
Where mercies reign;
A place to which thou may'st attain.
Seek there a residence to gain
Lest thou in caverns howl;
For save thou there shalt quick repair,
Woe to thy wretched soul!
Towards yon building turn your face!
Too strong by far is yonder place
To lose the victory.
'Tis better than the reeling world;
For all the ills by hell up-hurl'd
It has a remedy.
Sublime it braves the wildest waves;
It is a refuge place
Impregnable to Belial's race,
With stones, emitting vivid rays,
Above its stately porch;
Itself, and those therein, compose
The universal Church.
Though slaves of sin we long have been,
With faith sincere
We shall win pardon there;
Then in let's press, O brethren dear,
And claim our dignity!
By doing so, we saints below
And saints on high shall be.
DEATH THE GREAT.
[From "The Sleeping Bard," by Elis Wynn.]
LEAVE land and house we must some day,
For human sway not long doth bide;
Leave pleasures and festivities,
And pedigrees, our boast and pride.
Leave strength and loveliness of mien,
Wit sharp and keen, experience dear;
Leave learning deep, and much-lov'd friends,
And all that tends our life to cheer.
From Death then is there no relief?
That ruthless thief and murderer fell,
Who to his shambles beareth down
All, all we own, and us as well.
Ye monied men, ye who would fain
Your wealth retain eternally,
How brave 'twould be a sum to raise,
And the good grace of Death to buy!
How brave! ye who with beauty beam,
On rank supreme who fix your mind,
Should ye your captivations muster,
And with their lustre King Death blind.
O ye who are of foot most light,
Who are in the height now of your spring,
Fly, fly, and ye will make us gape,
If ye can scape Death's cruel fling.
The song and dance afford, I ween,
Relief from spleen and sorrow's grave;
How very strange there is no dance,
Nor tune of France, from Death can save!
Ye travellers of sea and land,
Who know each strand below the sky;
Declare if ye have seen a place
Where Adam's race can Death defy!
Ye scholars, and ye lawyer crowds,
Who are as gods reputed wise;
Can ye from all the lore ye know,
'Gainst death bestow some good advice?
The world, the flesh, and Devil, compose
The direst foes of mortals poor;
But take good heed of Death the Great,
From the Lost Gate, Destruction o'er.
'Tis not worth while of Death to prate,
Of his Lost Gate and courts so wide;
But O reflect! it much imports,
Of the two courts in which ye're tried.
It here can little signify
If the street high we cross, or low;
Each lofty thought doth rise, be sure,
The soul to lure to deepest woe.
But by the wall that's ne'er re-pass'd,
To gripe thee fast when Death prepares,
Heed, heed thy steps, for thou may'st mourn
The slightest turn for endless years.
When opes the door, and swiftly hence
To its residence eternal flies
The soul, it matters much, which side
Of the gulf wide its journey lies.
Deep penitence, amended life,
A bosom rife of zeal and faith,
Can help to man alone impart,
Against the smart and sting of Death.
These things to thee seem worthless now,
But not so low will they appear
When thou art come, O thoughtless friend!
Just to the end of thy career.
Thou'lt deem, when thou hast done with earth,
These things of worth unspeakable,
Beside the gulf so black and drear,
The gulf of Fear, 'twixt Heaven and Hell.
THE HEAVY HEART.
[From "The Sleeping Bard," by Elis Wynn.]
HEAVY'S the heart with wandering below,
And with seeing the things in the country of woe;
Seeing lost men and the fiendish race,
In their very horrible prison place;
Seeing that the end of the crooked track
Is a flaming lake
Where dragon and snake
With rage are swelling.
I'd not, o'er a thousand worlds to reign,
Behold again,
Though safe from pain,
The infernal dwelling.
Heavy's my heart, whilst so vividly
The place is yet in my memory;
To see so many, to me well known,
Thither unwittingly sinking down.
To-day a hell-dog is yesterday's man,
And he has no plan,
But others to trepan
To Hell's dismal revels.
When he reached the pit he a fiend became,
In face and in frame,
And in mind the same
As the very devils.
Heavy's the heart with viewing the bed,
Where sin has the meed it has merited;
What frightful taunts from forked tongue,
On gentle and simple there are flung!
The ghastliness of the damned things to state,
Or the pains to relate
Which will ne'er abate
But increase for ever,
No power have I, nor others I wot:
Words cannot be got;
The shapes and the spot
Can be pictured never.
Heavy's the heart, as none will deny,
At losing one's friend, or the maid of one's eye;
At losing one's freedom, one's land or wealth;
At losing one's fame, or alas! one's health;
At losing leisure; at losing ease;
At losing peace
And all things that please
The heaven under.
At losing memory, beauty and grace,
Heart-heaviness,
For a little space
Can cause no wonder.
Heavy's the heart of man when first
He awakes from his worldly dream accursed;
Fain would he be freed from his awful load
Of sin, and be reconciled with his God;
When he feels for pleasures and luxuries
Disgust arise,
From the agonies
Of the ferment unruly,
Through which he becomes regenerate,
Of Christ the mate,
From his sinful state
Springing blithe and holy.
Heavy's the heart of the best of mankind,
Upon the bed of death reclined;
In mind and body ill at ease,
Betwixt remorse and the disease,
Vext by sharp pangs and dreading more.
O mortal poor!
O dreadful hour!
Horrors surround him!
To the end of the vain world he has won;
And dark and dun
The Eternal One
Beholds beyond him.
Heavy's the heart, the pressure below,
Of all the griefs I have mentioned now;
But were they together all met in a mass,
There's one grief still would all surpass;
Hope frees from each woe, while we this side
Of the wall abide--
At every tide
'Tis an outlet cranny.
But there's a grief beyond the bier;
Hope will ne'er
Its victims cheer,
That cheers so many.
Heavy's the heart therewith that's fraught;
How heavy is mine at merely the thought!
Our worldly woes, however hard,
Are trifles when with that compared:
That woe--which is not known here--that woe
The lost ones know,
And undergo
In the nether regions;
How wretched the man who, exil'd to Hell,
In Hell must dwell,
And curse and yell
With the Hellish legions!
At nought, that may ever betide thee, fret
If at Hell thou art not arrived yet;
But thither, I rede thee, in mind repair
Full oft, and observantly wander there;
Musing intense, after reading me,
Of the flaming sea,
Will speedily thee
Convert by appalling.
Frequent remembrance of the black deep
Thy soul will keep,
Thou erring sheep,
From thither falling.
RYCE OF TWYN.
["I'll bet a guinea that however clever a fellow you may be, you never
sang anything in praise of your landlord's housekeeping equal to what
Dafydd Nanmor sang in praise of that of Ryce of Twyn four hundred years
ago."]
FOR Ryce if hundred thousands plough'd,
The lands around his fair abode;
Did vines of thousand vineyards bleed,
Still corn and wine great Ryce would need;
If all the earth had bread's sweet savour,
And water all had cyder's flavour,
Three roaring feasts in Ryce's hall
Would swallow earth and ocean all.
LLYWELYN.
By Dafydd Benfras.
LLYWELYN of the potent hand oft wrought
Trouble upon the kings and consternation;
When he with the Lloegrian monarch fought,
Whose cry was "Devastation!"
Forward impetuously his squadrons ran;
Great was the tumult ere the shout began;
Proud was the hero of his reeking glaive,
Proud of their numbers were his followers brave.
O then were heard resounding o'er the fields
The clash of faulchions and the crash of shields!
Many the wounds in yonder fight receiv'd!
Many the warriors of their lives bereaved!
The battle rages till our foes recoil
Behind the Dike which Offa built with toil,
Bloody their foreheads, gash'd with many a blow,
Blood streaming down their quaking knees below.
Llywelyn, we as our high chief obey,
To fair Porth Ysgewin extends his sway;
For regal virtues and for princely line
He towers above imperial Constantine.
PLYNLIMMON.
By Lewis Glyn Cothi.
FROM high Plynlimmon's shaggy side
Three streams in three directions glide,
To thousands at their mouth who tarry
Honey, gold and mead they carry.
Flow also from Plynlimmon high
Three streams of generosity; {137}
The first, a noble stream indeed,
Like rills of Mona runs with mead;
The second bears from vineyards thick
Wine to the feeble and the sick;
The third, till time shall be no more,
Mingled with gold shall silver pour.
QUATRAINS AND STRAY STANZAS FROM "WILD WALES."
I.
CHESTER ale, Chester ale! I could ne'er get it down,
'Tis made of ground-ivy, of dirt, and of bran,
'Tis as thick as a river below a huge town!
'Tis not lap for a dog, far less drink for a man.
II.
Gone, gone are thy gates, Dinas Bran on the height!
Thy warders are blood-crows and ravens, I trow;
Now no one will wend from the field of the fight
To the fortress on high, save the raven and crow.
III.
Here, after sailing far, I, Madoc, lie,
Of Owain Gwynedd lawful progeny:
The verdant land had little charms for me;
From earliest youth I loved the dark-blue sea.
God in his head the Muse instill'd,
And from his head the world he fill'd.
IV. EPITAPH ON ELIZABETH WILLIAMS.
Though thou art gone to dwelling cold,
To lie in mould for many a year,
Thou shalt, at length, from earthy bed,
Uplift thy head to blissful sphere.
V. THE LAST JOURNEY. From Huw Morus.
Now to my rest I hurry away,
To the world which lasts for ever and aye,
To Paradise, the beautiful place,
Trusting alone in the Lord of Grace.
VI. THE FOUR AND TWENTY MEASURES. From Edward Price.
I've read the master-pieces great
Of languages no less than eight,
But ne'er have found a woof of song
So strict as that of Cambria's tongue.
VII. MONA. By Robert Lleiaf.
Av i dir Mon, er dwr Menai,
Tros y traeth, ond aros trai.
I will go to the land of Mona, notwithstanding the water of the Menai,
across the sand, without waiting for the ebb.
VIII. MONA. From "Y Greal."
I got up in Mona as soon as 'twas light,
At nine in old Chester my breakfast I took;
In Ireland I dined, and in Mona, ere night,
By the turf fire sat, in my own ingle nook.
IX. ERYRI.
Easy to say, "Behold Eryri!"
But difficult to reach its head;
Easy for him whose hopes are cheery
To bid the wretch be comforted.
X. ERYRI. From Goronwy Owen.
Ail i'r ar ael Eryri,
Cyfartal hoewal a hi.
The brow of Snowdon shall be levelled with the ground, and the eddying
waters shall murmur round it.
XI. ELLEN. From Goronwy Owen.
Ellen, my darling,
Who liest in the churchyard of Walton.
XII. MON. From the Ode by Robin Ddu.
Bread of the wholesomest is found
In my mother-land of Anglesey;
Friendly bounteous men abound
In Penmynnydd of Anglesey. . . .
Twelve sober men the muses woo,
Twelve sober men in Anglesey,
Dwelling at home, like patriots true,
In reverence for Anglesey. . .
Though Arvon graduate bards can boast,
Yet more canst thou, O Anglesey.
XIII. MON. From Huw Goch.
Brodir, gnawd ynddi prydydd;
Heb ganu ni bu ni bydd.
A hospitable country, in which a poet is a thing of course. It has never
been and will never be without song.
XIV. LEWIS MORRIS OF MON. From Goronwy Owen.
"As long as Bardic lore shall last, science and learning be cherished,
the language and blood of the Britons undefiled, song be heard on
Parnassus, heaven and earth be in existence, foam be on the surge, and
water in the river, the name of Lewis of Mon shall be held in grateful
remembrance."
XV. THE GRAVE OF BELI.
Who lies 'neath the cairn on the headland hoar,
His hand yet holding his broad claymore,
Is it Beli, the son of Benlli Gawr?
XVI. THE GARDEN. From Gwilym Du o Eifion.
In a garden the first of our race was deceived;
In a garden the promise of grace he received;
In a garden was Jesus betray'd to His doom;
In a garden His body was laid in the tomb.
XVII. THE SATIRIST. From Gruffydd Hiraethog.
He who satire loves to sing,
On himself will satire bring.
XVIII. ON GRUFFYDD HIRAETHOG. From William Lleyn.
In Eden's grove from Adam's mouth
Upsprang a muse of noble growth;
So from thy grave, O poet wise,
Cross Consonancy's boughs shall rise.
XIX. LLANGOLLEN ALE. (George Borrow).
Llangollen's brown ale is with malt and hop rife;
'Tis good; but don't quaff it from evening till dawn;
For too much of that ale will incline you to strife;
Too much of that ale has caused knives to be drawn.
XX. TOM EVANS _alias_ Twm o'r Nant. By Twm Tai.
Tom Evan's the lad for hunting up songs,
Tom Evan to whom the best learning belongs;
Betwixt his two pasteboards he verses has got,
Sufficient to fill the whole country, I wot.
XXI. ENGLYN ON A WATERFALL.
Foaming and frothing from mountainous height,
Roaring like thunder the Rhyadr falls;
Though its silvery splendour the eye may delight,
Its fury the heart of the bravest appals.
XXII. DAVID GAM. Attributed to Owain Glyndower.
Shouldst thou a little red man descry
Asking about his dwelling fair,
Tell him it under the bank doth lie,
And its brow the mark of the coal doth bear.
XXIII. LLAWDDEN. From Lewis Meredith.
Whilst fair Machynlleth decks thy quiet plain,
Conjoined with it shall Lawdden's name remain.
XXIV. TWM O'R NANT.
Tom O Nant is a nickname I've got,
My name's Thomas Edwards, I wot.
XXV. SEVERN AND WYE.
O pleasantly do glide along the Severn and the Wye;
But Rheidol's rough, and yet he's held by all in honour high.
XXVI. GLAMORGAN. From Dafydd ab Gwilym.
If every strand oppression strong
Should arm against the son of song,
The weary wight would find, I ween,
A welcome in Glamorgan green.
XXVII. DAFYDD AB GWILYM. From Iolo Goch (?).
To Heaven's high peace let him depart,
And with him go the minstrel art.
XXVIII. TO THE YEW TREE on the Grave of Dafydd ab Gwilym at Ystrad Flur.
After Gruffydd Grug.
Thou noble tree; who shelt'rest kind
The dead man's house from winter's wind:
May lightnings never lay thee low,
Nor archer cut from thee his bow;
Nor Crispin peel thee pegs to frame,
But may thou ever bloom the same,
A noble tree the grave to guard
Of Cambria's most illustrious bard!
O tree of yew, which here I spy,
By Ystrad Flur's blest monast'ry,
Beneath thee lies, by cold Death bound,
The tongue for sweetness once renown'd.
* * *
Better for thee thy boughs to wave,
Though scath'd, above Ab Gwilym's grave,
Than stand in pristine glory drest
Where some ignobler bard doth rest;
I'd rather hear a taunting rhyme
From one who'll live through endless time,
Than hear my praises chanted loud
By poets of the vulgar crowd.
XXIX. HU GADARN. From Iolo Goch.
The Mighty Hu who lives for ever,
Of mead and wine to men the giver,
The emperor of land and sea,
And of all things that living be,
Did hold a plough with his good hand,
Soon as the Deluge left the land,
To show to men both strong and weak,
The haughty-hearted and the meek,
Of all the arts the heaven below
The noblest is to guide the plough.
XXX. EPITAPH.
Thou earth from earth reflect with anxious mind
That earth to earth must quickly be consigned,
And earth in earth must lie entranced, enthralled,
Till earth from earth to judgment shall be called.
XXXI. GOD'S BETTER THAN ALL.
By Vicar Pritchard of Llandovery.
GOD'S better than heaven or aught therein,
Than the earth or aught we there can win,
Better than the world or its wealth to me--
God's better than all that is or can be.
Better than father, than mother, than nurse,
Better than riches, oft proving a curse,
Better than Martha or Mary even--
Better by far is the God of heaven.
If God for thy portion thou hast ta'en
There's Christ to support thee in every pain,
The world to respect thee thou wilt gain,
To fear the fiend and all his train.
Of the best of portions thou choice didst make
When thou the high God to thyself didst take,
A portion which none from thy grasp can rend
Whilst the sun and the moon on their course shall wend.
When the sun grows dark and the moon turns red,
When the stars shall drop and millions dread,
When the earth shall vanish with its pomps in fire,
Thy portion still shall remain entire.
Then let not thy heart though distressed, complain!
A hold on thy portion firm maintain.
Thou didst choose the best portion, again I say--
Resign it not till thy dying day.
XXXII. THE SUN IN GLAMORGAN.
From Dafydd ab Gwilym.
EACH morn, benign of countenance,
Upon Glamorgan's pennon glance!
Each afternoon in beauty clear
Above my own dear bounds appear!
Bright outline of a blessed clime,
Again, though sunk, arise sublime--
Upon my errand, swift repair,
And unto green Glamorgan bear
Good days and terms of courtesy
From my dear country and from me!
Move round--but need I thee command?--
Its chalk-white halls, which cheerful stand--
Pleasant thy own pavilions too--
Its fields and orchards fair to view.
O, pleasant is thy task and high
In radiant warmth to roam the sky,
To keep from ill that kindly ground,
Its meads and farms, where mead is found,
A land whose commons live content,
Where each man's lot is excellent.
Where hosts to hail thee shall upstand,
Where lads are bold and lasses bland;
A land I oft from hill that's high
Have gazed upon with raptur'd eye;
Where maids are trained in virtue's school,
Where duteous wives spin dainty wool;
A country with each gift supplied,
Confronting Cornwall's cliffs of pride.
ADDITIONAL POEMS FROM THE "QUARTERLY REVIEW."
I. THE AGE OF OWEN GLENDOWER.
ONE thousand four hundred, no less and no more,
Was the date of the rising of Owen Glendower;
Till fifteen were added with courage ne'er cold
Liv'd Owen, though latterly Owen was old.
II. THE SPIDER.
FROM out its womb it weaves with care
Its web beneath the roof;
Its wintry web it spreadeth there--
Wires of ice its woof.
And doth it weave against the wall
Thin ropes of ice on high?
And must its little liver all
The wondrous stuff supply?
III. THE SEVEN DRUNKARDS.
O WHERE are there seven beneath the sky
Who with these seven for thirst can vie?
But the best for good ale these seven among
Are the jolly divine and the son of song.
SIR RHYS AP THOMAS.
"Great Rice of Wales."
Y BRENIN biau'r ynys,
Ond sy o ran i Syr Rys.
The King owns all the island wide
Except the part where Rice doth bide.
* * *
Y Brenin biau'r ynys;
A chyriau Frank, a chorf Rys.
The King owns all the island wide,
A part of France, and Rice beside.
_Rhys Nanmor a'i Kant_.
HIRAETH. {167}
A Short Elegy.
" . . . An old bard, who wrote a short elegy on the death of the
governor of -- and his dame, and who says that he himself was fading
with longing on their account."
--_Borrow MS._
LONGING for them doth fade my cheek;
He was a man, and she was meek;
A lion was he, she full of glee;
He handsome was, she fair to see.
A wondrous concord here was view'd;
He was wise, and she was good;
He liberal was, she kind of mood;
To heaven he went, she him pursued.
PWLL CHERES: THE VORTEX OF MENAI.
PWLL CHERES, the dread whirlpool of Menai,
Twisteth the waves, as if a knot should tie:
A hideous howling hollow, an abyss
Enough to scare the heart is Pwll Cheres.
THE MOUNTAIN SNOW.
THE mountain snow: the stag doth fly,
The wind about the roofs doth sigh.
Love cannot in concealment lie.
The mountain snow: the grove is dark,
The raven black; the hound doth bark.
God keep you from all evil work.
The mountain snow: the crust is sound;
The wind doth twist the reeds around.
Where ignorance is, no grace is found.
CAROLAN'S LAMENT.
_From the Irish_.
THE arts of Greece, Rome, and of Eirin's fair earth,
If at my sole command they this moment were all,
I'd give, though I'm fully aware of their worth,
Could they back from the dead my lost Mary recall.
I'm distrest every noon, now I sit down alone,
And at morn, now with me she arises no more:
With no woman alive after thee would I wive,
Could I flocks and herds gain, and of gold a bright store.
Awhile in green Eirin so pleasant I dwelt,
With her nobles I drank to whom music was dear;
Then left to myself, O how mournful I felt
At the close of my life, with no partner to cheer.
My sole joy and my comfort wast thou 'neath the sun,
Dark gloom, now I'm reft of thee, filleth my mind;
I shall know no more happiness now thou art gone,
O my Mary, of wit and of manners refin'd.
EPIGRAMS BY CAROLAN.
_On Friars_.
WOULD'ST thou on good terms with friars live,
Ever be humble and admiring;
All they ask of thee freely give,
And in return be nought requiring.
_On a Surly Butler_,
_who had refused him admission to the cellar_.
O Dermod Flynn, it grieveth me
Thou keepest not Hell's portal;
As long as thou should'st porter be,
Thou would'st admit no mortal.
THE DELIGHTS OF FINN MAC COUL. {187}
_From the Ancient Irish_.
FINN MAC COUL 'mongst his joys did number
To hark to the boom of the dusky hills;
By the wild cascade to be lull'd to slumber,
Which Cuan Na Seilg with its roaring fills.
He lov'd the noise when storms were blowing,
And billows with billows fought furiously;
Of Magh Maom's kine the ceaseless lowing,
And deep from the glen the calves' feeble cry;
The noise of the chase from Slieve Crott pealing,
The hum from the bushes Slieve Cua below,
The voice of the gull o'er the breakers wheeling,
The vulture's scream, over the sea flying slow;
The mariners' song from the distant haven,
The strain from the hill of the pack so free,
From Cnuic Nan Gall the croak of the raven,
The voice from Slieve Mis of the streamlets three;
Young Oscar's voice, to the chase proceeding,
The howl of the dogs, of the deer in quest.
But to recline where the cattle were feeding
That was the delight which pleas'd him best.
TO ICOLMCILL.
_From the Gaelic of MacIntyre_.
ON Icolmcill may blessings pour!
It is the island blest of yore;
Mull's sister-twin in the wild main,
Owning the sway of high Mac-Lean;
The sacred spot, whose fair renown
To many a distant land has flown,
And which receives in courteous way
All, all who thither chance to stray.
There in the grave are many a King
And duine-wassel {191} slumbering;
And bodies, once of giant strength,
Beneath the earth are stretch'd at length;
It is the fate of mortals all
To ashes fine and dust to fall;
I've hope in Christ, for sins who died,
He has their souls beatified.
Now full twelve hundred years, and more,
On dusky wing have flitted o'er,
Since that high morn when Columb grey
Its wall's foundation-stone did lay;
Images still therein remain
And death-memorials carv'd with pain;
Of good hewn stone from top to base,
It shows to Time a dauntless face.
A man this day the pulpit fill'd,
Whose sermon brain and bosom thrill'd,
And all the listening crowd I heard
Praising the mouth which it proffer'd.
Since death has seiz'd on Columb Cill,
And Mull may not possess him still,
There's joy throughout its heathery lands,
In Columb's place that Dougal stands.
THE DYING BARD.
_From the Gaelic_.
O FOR to hear the hunter's tread
With his spear and his dogs the hills among;
In my aged cheek youth flushes red
When the noise of the chase arises strong.
Awakes in my bones the marrow whene'er
I hark to the distant shout and bay;
When peals in my ear, "We've kill'd the deer"--
To the hill-tops boundeth my soul away;
I see the slug-hound tall and gaunt,
Which follow'd me, early and late, so true;
The hills, which it was my delight to haunt,
And the rocks, which rang to my loud halloo.
I see Scoir Eild by the side of the glen,
Where the cuckoo calleth so blithe in May,
And Gorval of pines, renown'd 'mongst men
For the elk and the roe which bound and play.
I see the cave, which receiv'd our feet
So kindly oft from the gloom of night,
Where the blazing tree with its genial heat
Within our bosoms awak'd delight.
On the flesh of the deer we fed our fill--
Our drink was the Treigh, our music its wave;
Though the ghost shriek'd shrill, and bellow'd the hill,
'Twas pleasant, I trow, in that lonely cave.
I see Benn Ard of form so fair,
Of a thousand hills the Monarch proud;
On his side the wild deer make their lair,
His head's the eternal couch of the cloud.
But vision of joy, and art thou flown?
Return for a moment's space, I pray,--
Thou dost not hear--ohone, ohone,--
Hills of my love, farewell for aye.
Farewell, ye youths, so bold and free,
And fare ye well, ye maids divine!
No more I can see ye--yours is the glee
Of the summer, the gloom of the winter mine.
At noon-tide carry me into the sun,
To the bank by the side of the wandering stream,
To rest the shamrock and daisy upon,
And then will return of my youth the dream.
Place ye by my side my harp and shell,
And the shield my fathers in battle bore;
Ye halls, where Oisin and Daoul {197} dwell,
Unclose--for at eve I shall be no more.
THE SONG OF DEIRDRA.
FAREWELL, grey Albyn, much loved land,
I ne'er shall see thy hills again;
Upon those hills I oft would stand
And view the chase sweep o'er the plain.
'Twas pleasant from their tops, I ween,
To see the stag that bounding ran;
And all the rout of hunters keen,
The sons of Usna in the van.
The chiefs of Albyn feasted high,
Amidst them Usna's children shone;
And Nasa kissed in secrecy
The daughter fair of high Dundron.
To her a milk-white doe he sent,
With little fawn that frisked and played,
And once to visit her he went,
As home from Inverness he strayed.
The news was scarcely brought to me
When jealous rage inflamed my mind;
I took my boat and rushed to sea,
For death, for speedy death, inclined.
But swiftly swimming at my stern
Came Ainlie bold and Ardan tall;
Those faithful striplings made me turn
And brought me back to Nasa's hall.
Then thrice he swore upon his arms,
His burnished arms, the foeman's bane,
That he would never wake alarms
In this fond breast of mine again.
Dundron's fair daughter also swore,
And called to witness earth and sky,
That since his love for her was o'er
A maiden she would live and die.
Ah, did she know that slain in fight,
He wets with gore the Irish hill,
How great would be her moan this night,
But greater far would mine be still.
THE WILD WINE.
_From the Gaelic of MacIntyre_.
THE wild wine of nature,
Honey-like in its taste,
The genial, fair, thin element
Filtering through the sands,
Which is sweeter than cinnamon,
And is well-known to us hunters.
O, that eternal, healing draught,
Which comes from under the earth,
Which contains abundance of good
And costs no money!
* * * * *
PRINTED BY JARROLD AND SONS, LTD.,
NORWICH, ENGLAND
FOOTNOTES.
{17} "The Visions of the Sleeping Bard:" Being Ellis Wynne's
"Gweledigaethau y Bardd Cwsg," Translated by Robert Gwyneddon Davies.
Carnarvon (Welsh Publishing Co., Ltd.), 1909.
{34} Emrys, King of Britain, lying sick at Canterbury, a Saxon of the
name of Eppa disguised himself as a religious person, and pretending to
be versed in medicine, obtained admission to the Monarch and administered
to him a poisoned draught, of which he died.
{39} Glyndwr signifies watery valley.
{49a} Written in the fifth century.
{49b} The British, like many other nations, whose early history is
involved in obscurity, claim a Trojan descent.
{54a} Awen, or poetic genius, which he is said to have imbibed in his
childhood, whilst employed in watching the cauldron of the Sorceress
Cridwen.
{54b} I was but a child, but am now Taliesin,--Taliesin signifies: brow
of brightness.
{64} The harp.
{74} Ale.
{137} The "streams of generosity" were those of Dafydd ab Thomas Vychan.
(See "Wild Wales," chap. lxxxviii.)--_Ed._
{167} "What is _hiraeth_? Hiraeth is longing, the mourning, consuming
feeling which one experiences for the loss of a beloved object."--_G.B._
{187} The personage who figures in the splendid forgeries of MacPherson
under the name of Fingal.
{191} The Gaelic word for nobleman.
{197} Ancient bards, to whose mansion, in the clouds, the speaker hopes
that his spirit will be received.
***
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Grande maestro dal 1989, con la caduta dell'URSS ha rappresentato la federazione bielorussa fino al 1998, quando è emigrato in Israele, ottenendone la cittadinanza.
Protagonista in diverse edizioni del campionato del mondo, nell'edizione 2012 è stato lo sfidante di Viswanathan Anand per il titolo mondiale.
Ha partecipato a undici edizioni delle Olimpiadi degli scacchi, conquistando l'oro nel 1990 con la nazionale sovietica.
Nella lista FIDE di novembre 2013 ha ottenuto il suo record personale nel punteggio Elo, raggiungendo quota 2777 punti (7º al mondo, 1º tra i giocatori israeliani).
Carriera
Tra i risultati di torneo, le vittorie al torneo interzonale di Manila 1990, ai torneo di Wijk aan Zee 1992, tornei di Biel 1993 (interzonale) e 2005, Dos Hermanas e Pamplona 1994, Belgrado e Debrecen 1995, Polanica-Zdrój 1998, open di Cannes 2002, Tilburg e Vienna 1996.
Nel 2008 ha vinto due argenti alle Olimpiadi di Dresda, una con la squadra e una individuale, come prima scacchiera.
Nel maggio del 2009 ha vinto la "ACP World Rapid Cup" di Odessa, superando nell'ordine Vüqar Həşimov, Dmitrij Jakovenko e nella finale Pëtr Svidler.
Nel dicembre del 2009 ha vinto, battendo nella finale agli spareggi lampo Ruslan Ponomarёv, la Coppa del Mondo di scacchi 2009.
Nel maggio del 2012 ha disputato a Mosca contro Viswanathan Anand il match per il campionato del mondo, dal quale è uscito sconfitto per 8,5 a 7,5.
Nell'ottobre del 2012 vince a pari merito con Veselin Topalov e Şəhriyar Məmmədyarov il Fide Grand Prix di Londra.
Nel giugno del 2013 vince a Mosca il forte torneo Tal Memorial superando di mezzo punto Magnus Carlsen.
Nell'ottobre del 2013 vince a pari merito con Fabiano Caruana il Fide Grand Prix di Parigi.
Nell'ottobre del 2014 vince a pari merito con Fabiano Caruana il Fide Grand Prix di Baku.
Note
Altri progetti
Collegamenti esterni
Scacchisti bielorussi
Scacchisti sovietici
Vincitori di medaglie alle Olimpiadi degli scacchi
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{"url":"https:\/\/eyebulb.com\/what-is-the-standard-deviation-of-randn\/","text":"# What is the standard deviation of randn?\n\n## What is the standard deviation of randn?\n\nEssentially, Numpy random randn generates normally distributed numbers from a normal distribution that has a mean of 0 and a standard deviation of 1.\n\n### What is the range of randn in Matlab?\n\nFrom the definition of normal distribution, the \u2018range\u2019 is from -\\infty to \\infty. In your case, 0.2 is the standard deviation. ^ +1 \u2013 randn() generates numbers all along the real line, just with very very little probability beyond +\/- 3*sigma.\n\nHow do you generate random numbers with mean and standard deviation in Matlab?\n\nr = normrnd( mu , sigma ) generates a random number from the normal distribution with mean parameter mu and standard deviation parameter sigma . r = normrnd( mu , sigma , sz1,\u2026,szN ) generates an array of normal random numbers, where sz1,\u2026,szN indicates the size of each dimension.\n\nWhat is the range of normal distribution?\n\nThe area under each curve is one but the scaling of the X axis is different. Note, however, that the areas to the left of the dashed line are the same. The BMI distribution ranges from 11 to 47, while the standardized normal distribution, Z, ranges from -3 to 3.\n\n## How do you define a range of numbers in Matlab?\n\ny = range( X ) returns the difference between the maximum and minimum values of sample data in X .\n\n1. If X is a vector, then range(X) is the range of the values in X .\n2. If X is a matrix, then range(X) is a row vector containing the range of each column in X .\n\n### How do you generate random numbers with mean and standard deviation?\n\nGenerate random number by given mean and standard deviation\n\n1. Firstly, you need to type your needed mean and standard deviation into two empty cells, here, I select A1 and A2.\n2. Then in cell B3, type this formula =NORMINV(RAND(),$B$1,$B$2), and drag the fill handle to the range you need.\n\nHow do you generate data from mean and standard deviation?\n\nYou can generate standard normal random variables with the Box-Mueller method. Then to transform that to have mean mu and standard deviation sigma, multiply your samples by sigma and add mu. I.e. for each z from the standard normal, return mu + sigma*z.\n\nHow to find random variables with mean and variance?\n\nThe general theory of random variables states that if x is a random variable whose mean is and variance is , then the random variable, y, defined by where a and b are constants, has mean and variance You can apply this concept to get a sample of normally distributed random numbers with mean 500 and variance 25.\n\n## How to create a random variable in MATLAB?\n\nrandn in matlab produces normal distributed random variables W with zero mean and unit variance. To change the mean and variance to be the random variable X (with custom mean and variance), follow this equation: X = mean + standard_deviation*W Please be aware of that standard_deviation is square root of variance.\n\n### When to use RNG instead of randn in MATLAB?\n\nAlways use the rng function (rather than the rand or randn functions) to specify the settings of the random number generator. For more information, see Replace Discouraged Syntaxes of rand and randn. Create a 3-by-2-by-3 array of random numbers.\n\nWhat kind of array does the randn function generate?\n\nThe randnfunction generates arrays of random numbers whose elements are normally distributed with mean 0, variance, and standard deviation.\n\n12\/11\/2019","date":"2023-03-23 13:54:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.820241391658783, \"perplexity\": 582.1775861318674}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945168.36\/warc\/CC-MAIN-20230323132026-20230323162026-00232.warc.gz\"}"}
| null | null |
What It Took These Four Women To Get Into Robotics
Originally Posted on ReadWrite
By: Lauren Orsini
Lead photo: Cynthia Breazeal
When it comes to gender ratio, engineering is one of the most highly imbalanced fields around. Not only are few women engineers working today, far fewer female college students are entering science and technology-related fields compared to their male colleagues.
And one of the fields where women are most strongly outnumbered is one that most needs diverse perspectives: robotics. Last Sunday was International Women's Day, an event for celebrating the progress women have made toward gender equality. It seemed like a good moment to speak with some of the world's leading female roboticists about their experiences.
Melonee Wise: "It's a people issue"
Wise is the CEO of Fetch Robotics and a veteran roboticist. Her contributions to platforms like Turtlebot and humanoid robots like PR2 and UBR-1 precede her. Currently, her company is working on a pair of industrial robots to be marketed to logistics companies.
Female CEOs historically encounter more difficulties acquiring funding for their companies, but that hasn't been the case for Wise. Fetch robotics just acquired $3 million in funding.
"I think sometimes people talk about being a woman in a field full of men like it's a handicap and that's insulting sometimes because I was born this way," she said. "Other times, however, it can be very uplifting to be a positive role model for young women who want to go into robotics. [My presence] encourages young engineers and tells them, 'Hey women have been successful, just apply yourself and work hard and barriers will move out of the way.'"
At Fetch, twenty percent of employees are female—or two out of the ten engineers, though the division is about to change now that Fetch is doubling its staff. Wise said the key to hiring candidates, men and women alike, can depend on subtle cues from a company:
When I was at Willow Garage, we went to a women's event semiannually. One of the years we went, Steve Cousins gave a nice talk about Willow Garage. Somebody asked, "Is Willow Garage hiring women?" and we were both shocked. Why wouldn't we be hiring women? Even working there I never thought of it as an issue. It made us reflect on how the company was advertised. Most of the stuff on the website showed an overwhelming number of men. We were shocked this sent the message to women that they couldn't apply. It's a two sided problem—on the one side, women sometimes feel out of place, but also some women self select. It's a people issue, not a women issue.
Louise Poubel: "Give the girls the toys"
Today, the biggest barriers standing between robots and regular people are complexity and price. Louise Poubel works on the robotics simulator Gazebo, a graphical user interface (GUI), at the Open Source Robotics Foundation, with the aim of knocking down both barriers.
"Robots should be made focusing on everybody, not just a few people who can afford it. That's why I like open source," she said. "I think tech should be easy to use, and giving people more tools to start with can lower barriers. GUIs are very user friendly for people being introduced to something for the first time."
Poubel, who grew up in Brazil, got her engineering degree in Japan and masters degrees in Poland and France, is a truly global citizen. Her award-winning graduate project, in which her team programmed a Nao robot to respond to human movement, emphasizes her belief that robots should be programmed to cater to all sorts of people.
She reflects:
When I was a little girl, I had as much access to computers as my brother, but my father refused to give me a remote-control car even though he gave my brother one. It always made me kind of sad, but it didn't change much in the way i grew up. It's important to let kids know it's ok for them to pursue anything they like. Give the girls the toys, whether they like computers or remote-control cars.
Cynthia Breazeal: "The work reflects your life"
Ever since her MIT days, Cynthia Breazeal has been building robots that not only serve to assist people, but also to be sensitive to their feelings. Her earliest projects, fluffy Leonardo and Disney-eyed Kismet, are designed to emote and respond to human emotions. Today, she is known as the mother of personal robotics.
Breazeal's latest project raised $2.2 million on Indiegogo to bring that sort of emotive warmth to a commercial robot. The result, Jibo, is like a helpful, ambulatory iPad for the home.
"I think in many ways, the work reflects your life and what matters to you and your lived experience," said Breazeal. "Some of mine comes from being a mom and having kids, and thinking about social robots in context relevant to families. I think there's tremendous good a robot can do in the family."
In Breazeal's experience—validated by her successful crowdfunding campaign—social robots appeal to many demographics and groups. This is where she believes the gender and ethnic diversity of creators are key to helpful social robots.
"No matter who you are as an engineer or scientist, you can't help but bring what matters to you into the work," she said.
In order to encourage women and minorities pursuing robotics, Breazeal lectures, engages with the media, judges robotics competitions, and is involved in the Boston Museum of Science.
"I think it would be very unfortunate if a woman were recognized by gender first and work second, but when reputation leads, it can inspire critical mass," she said. "Having girls grow up with role models serves an important function. They do need to see people who look like them."
Tessa Lau: "Robots should be for everyone"
As Chief Robot Whisperer for Savioke, Tessa Lau doesn't just engineer robots. She analyzes and optimizes the ways they can do the most good for the most people.
"I think robots should be for everyone," she said. "You read a lot in media about the gender gap in technology. The problem is that many tech companies built by the white male demographic. It's really important, as we design our robots, to make sure they appeal to everyone—people with disabilities, minorities, people who are disadvantaged. Diversity is one of the qualities I look for when i hire."
In the absence of more diverse roboticists, Lau says she works hard to test her robots with a diverse group of people. Currently, Savioke is working on an unnamed hotel delivery bot designed to deliver items to hotel guests' rooms.
"We found out that women had specific set of concerns," she said. "If strange men were delivering the items, they were worried, but with robots, they were OK. Robots can make people feel more safe, and it's not something you'd know if you weren't looking for it. Disadvantaged groups can offer a viewpoint we would have never considered."
Lau, who was a software engineer at IBM for 11 years before becoming a roboticist, has worked hard to correct the heavily male skew in technology fields. She helped organize IBM's Grace Hopper conference for women in computing, serves on the board for the Committee on the Status of Women in Computing Research, and employs a female robotics intern.
She put it this way to me:
I'm very aware of being a woman in technology. I know that I'm a minority. I think it's awesome to have more people and more kinds of people because it increases the quality of products available to the world. In my particular case, I'm excited there are more women coming into technology, and I'm happy to serve as a role model.
Summer/Fall 2016
The 1st Newsletter
Ann Quiroz Gates Wins the Nico Habermann Award Explore Resources for Women Undergrads in Science, Engineering
|
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Q: How to read collection data from Mongodb and publish into kafka topic periodically in spring boot I need to read my mongo DB table data periodically and publish it into a Kafka topic using spring boot. I have created a collection in Mongo DB and inserted a few records in Mongo DB. Further, I want to read the data from Mongo DB periodically and need to publish those table data in Kafka's topic using spring boot. I'm very new to spring batch scheduler. Can you please suggest me an idea to achieve this?
Thanks in advance.
A: What you are talking about is more relevant to Spring Integration: https://spring.io/projects/spring-integration#overview
So, you configure a MongoDbMessageSource with a Poller to read collection periodically.
And then you have service-activator based on the KafkaProducerMessageHandler to damp data into a Kafka topic.
See more in docs:
https://docs.spring.io/spring-integration/docs/5.3.2.RELEASE/reference/html/mongodb.html#mongodb
https://docs.spring.io/spring-integration/docs/5.4.0-M3/reference/html/kafka.html#kafka
Not sure though how to do that with Spring Batch...
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|
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According to the Oxford English Dictionary, the definition of autumn is when "the countryside is ablaze with colour".A quick look out of the window on a sunny day will confirm how beautiful nature can be. But a blast of hurricane force winds, and we're faced with the colourful leaves on the ground!
It's time to welcome autumn and all it brings, with mad weather from blustery gusts and 80mph winds, to sunshine and showers, all in the same week.
Putting your garden to bed for the winter is a time to take stock of the plants and flowers, take care of garden furniture and ornaments, and make a wish list for Christmas.
Feed your compost bin with fallen leaves, food peelings and grass cuttings – layer them and turn it all weekly with a fork to get air into the mixture, and you'll have genuinely organic rich compost to mix into the soil in the spring.
Weed your borders and trim the edges of your lawns. If you don't need all the seed heads, leave some for the birds for the winter.
Make sure your pots and containers have enough water, plants that keep their leaves all year need regular water and mustn't dry out.
Vegetable plots need digging over to break the soil down, especially if your garden has clay based soil. Pot your herbs for the winter – especially parsley, chives and mint. They'll do well on a sunny windowsill until the spring and will keep your kitchen smelling fragrant.
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