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https://en.wikipedia.org/wiki/Mbalizi%20Road
Mbalizi Road is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,662 people in the ward, from 6,045 in 2012. Neighborhoods The ward has 4 neighborhoods. Kabisa Kisoki Mwasyoge Sabasaba References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Mwakibete
Mwakibete is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 25,700 people in the ward, from 23,319 in 2012. Neighborhoods The ward has 7 neighborhoods. Bomba mbili Itongo Ivumwe Ng'osi Nyibuko Shewa Viwandani References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Mwansekwa
Mwansekwa is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 1,987 people in the ward, from 1,803 in 2012. Neighborhoods The ward has 4 neighborhoods. Ilembo Luwala Mengo Mwanzumbo References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Mwansanga
Mwansanga, also known as Mwasanga, is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 1,042 people in the ward, from 945 in 2012. Neighborhoods The ward has 2 neighborhoods; Nduguya, and Isoso. References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Nonde
Nonde is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 2,742 people in the ward, from 2,488 in 2012. Neighborhoods The ward has 4 neighborhoods. Mbwile A Mbwile B Mwalingo Nonde References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Nsalaga
Nsalaga is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 20,933 people in the ward, from 18,993 in 2012. Neighborhoods The ward has 7 neighborhoods. Igamba Itezi Mashariki Itezi Mlimani Kibonde Nyasi Majengo mapya Nsalaga Ntundu References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Nsoho
Nsoho is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 2,005 people in the ward, from 1,819 in 2012. Neighborhoods The ward has 4 neighborhoods. Idunda Kilabuni Mbeya Peak Nsoho References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Sinde%20%28Mbeya%29
Sinde (ward) is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,730 people in the ward, from 7,014 in 2012. Neighborhoods The ward has 4 neighborhoods. Ilolo Kati Janibichi Kagwina Sinde A References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Sisimba
Sisimba is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 4,532 people in the ward, from 4,112 in 2012. The Sokoine Stadium is located within the Sisimba ward. Neighborhoods The ward has 6 neighborhoods. Jakaranda A Jakaranda B Soko Kuu TANESCO Uzunguni A Uzunguni B References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Tembela
Tembela is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 2,572 people in the ward, from 2,334 in 2012. The TAZARA Railway run through the northern part of the ward. Neighborhoods The ward has 2 neighborhoods; Reli, and Tembela. References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Uyole
Uyole is an administrative ward in the Mbeya Urban district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,722 people in the ward, from 11,543 in 2012. Neighborhoods The ward has 4 neighborhoods. Hasanga Ibara Iwambala Utukuyu References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Igale
Igale is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,973 people in the ward, from 10,864 in 2012. Villages and hamlets The ward has 5 villages, and 40 hamlets. Horongo Ibolelo Ileya Isusa Kiwanjani Laini Lutali Njombo Nsongole Igale Ihova Ilindi Ing'oli Ipembati Isonso Isyesye Itete Iyunga Itaga Isanzi A Isanzi B Ivumo Izuo Kiwanjani Mbibi Mpongota Yeriko Izumbwe I Itala Kilambo Kiwiga Lusungo Luzwiwi Mlima Baruti Mwashala Nyula Segela Ujamaa Shongo Ikeja Isela Kawetele Matula Mbushi Nkuyu References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Ihango
Ihango is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,194 people in the ward, from 8,342 in 2012. Villages and hamlets The ward has 6 villages, and 49 hamlets. Haporoto Ibomani Fikeni Gezaulole Soweto Igoma Itiliwindi Shihokwa Mwawayo Sokolo Idimi Isanga A Fatahilo Uzunguni Makungulu Majengo Inyala Kijiweni Kidungu Isengo Jangwani Mwabowo Nonde Isyagunga Iyunga Itigi Ituta Galilaya Impomu Malagala Shiwe Impombombo Ilambo Linga Iwanza Songambele Ntangano Itete Ilolezya Itete 'B' Ileya Itaka Itiliwindi Uzunguni A Uzunguni B Isangati A Isangati B Izagati Mbeye Ishinda Sibempe Iwindi Ipinda Ipinda Simwambalafu References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Ijombe
Ijombe is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,225 people in the ward, from 10,185 in 2012. Villages and hamlets The ward has 6 villages, and 39 hamlets. Iwalanje Ibowola Ilembo Isyema Iwangwa Majengo Njohole Sogeza Ifiga Ijombe Ilanji A Ilanji B Ilowe A Ilowe B Mafyeko Mawe Shilongo Ntangano Igalama Ikeka Iwanda Majengo Nsheto A Nsheto B Nsongwi Mantanji Izumbwe Kijiweni Mecco Moshi Mpakani Hatwelo Halembo Ing'anda Mahambi Mangoto Masoko Mbowe Nsongwi juu Halanzi Ilangali Ilembo Mwatezi Simambwe Soweto References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Ikukwa
Ikukwa is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,639 people in the ward, from 6,024 in 2012. Villages and hamlets The ward has 2 villages, and 24 hamlets. Ikukwa Ikukwa Itende Kati Itende juu Jua kali Kariakoo Kiwanja Mahonza Mbuwi Mdonya Shongo Ujamaa Ukwaheri Unguja Zongo Simboya Inyala Itombi Kulasini Magomeni Maji moto Manzese Ostabay Shokwa Simboya Wanga References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Ilembo
Ilembo is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,440 people in the ward, from 17,391 in 2012. Villages and hamlets The ward has 10 villages, and 48 hamlets. Ilembo Igwila Ijembe Ilembo madukani Itala Mafune Manzigula Masangati Dimbwe Ihombe Isumbi Nsalala Sheyo Mwala Halonje Iringa Isyasya Iyuli Mwala Shuleni Ndanga Pikwi Mbawi Igamba Isela Mbawi Mbagala Itewe Iwonde Kesalia Ujunjulu Iyunga Iyunga "A" Iyunga "B" Mbozi Msena Italazya Ibula Ilaga Kanona Lyoto Mbana Shigamba II Itale Mpaza Msimishe Shigamba Mwakasita Mantenga Mwakasita Ng'ambi Shilungwe Shilanga Igambo shuleni Itigi Luswaya Majengo Mwanda Shilanga ofisini References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Ilungu
Ilungu is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,330 people in the ward, from 12,095 in 2012. Villages and hamlets The ward has 7 villages, and 52 hamlets. Kikondo CCM Dodoma Mjimwema Mpakani Ndwila Pambamoto Shango Chamasengo Ilangali Itiwa Mjimwema Mnyinga Shango Nyalwela Ikuha Isanga Isyonje A Isyonje B Itete Katumba Kumbulu Loleza Mwanjembe Nyalwela Ngole Igalama Mabande Muungano Mwambanga Ngole chini Ngole juu Nzumba A Nzumba B Mwela Idumbwe Isongole Kamficheni Kilimani Kitulo Nsengo Ifupa Ifupa A Ifupa B Loleza A Loleza B Loleza kati Maendeleo Majenje Matwitwi Ngwenyu Voya Mashese Ileje Isanga Magombati Makunguru Mashese kati Mji mwema References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Inyala
Inyala is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,621 people in the ward, from 10,544 in 2012. Villages and hamlets The ward has 7 villages, and 35 hamlets. Inyala Hamwenje Hamwenje B Inyala Kolya Relini Sisiyunje Tuyombo Shamwengo Ipogoro Itondwe Myela Utulivu Imezu Inzawa Isanga Magoye Masementi Masyeto Sawa Iyawaya Galiaya B Galilaya Kijiweni Madizini Vijana Makwenje Ibohola Makwenje Makwenje B Mlowo Mlowo B Syite Darajani Darajani Iduda Iganjo Imezu Mjini Iwanga Mwashoma Inolo Mwashoma References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Isuto
Isuto is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,383 people in the ward, from 13,958 in 2012. Villages and hamlets The ward has 8 villages, and 75 hamlets. Mlowo Ihanda Inganzo Mbuga Mlowo A Mlowo B Mlowo C Nyondo Samora Tozi A Tozi B Itete Chombezi Igagala Itete A Itete B Itete C Mwanjelwa A Mwanjelwa B Mwanjelwa C Nsonga Shinzingo Iwaga Iwalanje Masoko Mjele Mkuyuni Mlima Nyundu Mwendo Ngala Nsalala Shelela Idiwili Azimio Idiwili A Idiwili B Itaga Lutengano Mtakuja Shilungu A Shilungu B Shisonta Ileya Ileya kati Kaloleni Matipu Mkuyuni Muungano Mwanjelwa Shisonta A Shisonta B Shisonta C Isuto Igosya Ihenga A Ihenga B Ilizya Isuto A Isuto B Lywayo Mapinduzi A Mapinduzi B Mporoto Mtakuja Njiapanda Shitete Halungu Ilanga Shitete A Shitete B Yona A Yona B Yona C Ilindi Honde Igunda Ilindi A Ilindi B Mkuyuni Mlingotini Sanyesya Shilungu Sogea Ilala References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Iwiji
Iwiji is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,973 people in the ward, from 15,056 in 2012. Villages and hamlets The ward has 3 villages, and 24 hamlets. Iwiji Iwiji Mabula Magole Mtukula Ntinga Soweto Vimetu Isende Isende Kalashi Luhuma Masala Nachingwengwe Shihola Shinandala Izumbwe II Chawama Chilanzi Hayende Ikese Ileya A Ileya B Kafule Lupamba Sayuma Songwe References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Iwindi
Iwindi is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 20,276 people in the ward, from 18,397 in 2012. Villages and hamlets The ward has 10 villages, and 57 hamlets. Iwindi Igawilo Ihobha Ijombe Ipipa Isanga Iwindi Nshala Usahandeshe Igonyamu Gezatulolane Igonyamu Lyanda mpalala Mazoha Nsalafu Nsega Sahandeshe Inolo Iduji Inolo Kasalia Isangala Igagu Igosa Isangala kati Iwe Madindika Mande Nzinga Itimu Igombe Itimu Msituni Mwambale Mwansita Njiapanda Maganjo Barabarani Maganjo Makatani Mbilwa Mjimwema Mlimani Mwampalala Iwuzi Lusungo Majengo Masebe Mwambale Nsenga Sibudwa Mwaselela Isanga B Iwanga Mwaselela Shizi Tazama Mwashiwawala 2Mina Lusungo Mwashiwawala Nsongole Relini Nsambya Ilindi A Ilindi B Nsambya References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Iyunga%20Mapinduzi
Iyunga Mapinduzi is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,130 people in the ward, from 7,377 in 2012. Villages and hamlets The ward has 5 villages, and 36 hamlets. Isangati Isangati A Isangati B Isangati C Izuo B Izuo C Izuo Izuo A Mpenye Ndyeki Ntumba Igowe Igowe Isonganya Lusungo Iyunga Mapinduzi Iyunga A Iyunga B Matenga Mpande A Mpande B Mwasanga A Mwasanga B Ntyanya Shuwa Igagu Ilanga A Ilanga B Ilanga C Isonso Lupembe Nsangano Shuwa A Shuwa B Shuwa C Madugu Madugu "A" Madugu "B" Shiwele "A" Shiwele "B" Zola "A" Zola "B" References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Mshewe
Mshewe is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,278 people in the ward, from 11,241 in 2012. Villages and hamlets The ward has 5 villages, and 40 hamlets. Mshewe Chang'ombe Chapaulenje Ihanga Ihanga mpakani Ijombe Itotowe Maula Mpalule Mshewaje Mshewe Kati Ilota Ilota Maporomoko Mpona A Mpona B Muvwa Chang'ombe Forest Ijombe Ivomo Kafupa Kagera Lutengano Mtakuja Mwembeni Mwembesongwa Tononoka Njelenje Ileya Ilolo Iwola Kizota Mpinza Mpunguruma Njela Ujombe Mapogoro Forest Ibojo Majengo Mbuyuni Mkonge Nandala Soweto References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Santilya
Santilya is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 19,373 people in the ward, from 17,578 in 2012. Villages and hamlets The ward has 11 villages, and 62 hamlets. Santilya Ibungu Ivugula Iyungwe Izosya Mantanji Mtyeti Soweto Sanje Ilindi Magao Mantenga Sanje A Sanje B Santwinji Swaya Iswago Hazumbi Inyala A Inyala B Itambila Lusungo Nsongole Ntenga Ntete Shikulusi Sindyanga Mpande Idunda A Idunda B Mpande A Mpande B Jojo Horongo Ilewe A Ilewe B Ilomba Itundu A Itundu B Ivugula Iwale Jojo Mwanjelwa Nsheha Ilembo Nsheha Sukamawela Itizi Itizi "A" Itizi "B" Itizi "C" Ruanda Isuwa Ivimbizya A Ivimbizya B Ntole Shipongo A Shipongo B Mpande Idunda A Idunda B Mpande A Mpande B Isongole Isongole A Isongole B Shizyangule Ugaya A Ugaya B Masyeta Masyeta "A" Masyeta "B" Masyeta "C" References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Ulenje
Ulenje is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,655 people in the ward, from 6,946 in 2012. Villages and hamlets The ward has 6 villages, and 8 hamlets. Ihango Ihango Ilembo Malonji Togwa Itala Ibula Igwila Itala A Itala B Itandu A Itandu B Kagera Majengo Nyeta A Nyeta B Mbonile Ilala Kalaja Kamficheni Mwakibete Nyakonde Tegela Togwa Mkuyuni Igala Mdwadwa Mkuyuni Natela Wanging'ombe Ulenje Barazani Ikeka A Ikeka B Imaji Isyonje Kiwanjani Magharibi A Magharibi B Makabichi Mansamu Wambishe Hafurwe Ikulila A Ikulila B Ikulila C Magoye Nsonya Shipinga Wambishe References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Utengule%20Usongwe
Utengule Usongwe is an administrative ward in the Mbeya Rural district of the Mbeya Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 46,236 people in the ward, from 41,952 in 2012. Villages and hamlets The ward has 5 villages, and 38 hamlets. Magulula Ilonjelo Lena - Mtakuja Mfinga Muungano Nengelesa Ujora Mpolo Ihanga 'A' Ihanga 'B' Ihanga 'C' Mahango 'A Mahango 'B' Mahango 'C' Muungano Ijumbi Itambo Mpolo Kajunjumele Lyanumbusi Mahango Mswiswi Majojolo Marawatu Misufini Senganinjala Shuleni Ugandilwa Simike Mapula 'A' Mapula 'B' Mapululu Miambeni Mianzini Shuleni Tengatenga Wambilo Utengule Usangu Iduya A Iduya B Jemedari Ubajulie - Mbela Ujola Ulyankha Wimbwa References Wards of Mbeya Region
https://en.wikipedia.org/wiki/Jim%20Propp
James Gary Propp is a professor of mathematics at the University of Massachusetts Lowell. Education and career In high school, Propp was one of the national winners of the United States of America Mathematical Olympiad (USAMO), and an alumnus of the Hampshire College Summer Studies in Mathematics. Propp obtained his AB in mathematics in 1982 at Harvard. After advanced study at Cambridge, he obtained his PhD from the University of California at Berkeley. He has held professorships at seven universities, including Harvard, MIT, the University of Wisconsin, and the University of Massachusetts Lowell. Mathematical research Propp is the co-editor of the book Microsurveys in Discrete Probability (1998) and has written more than fifty journal articles on game theory, combinatorics and probability, and recreational mathematics. He lectures extensively and has served on the Mathematical Olympiad Committee of the Mathematical Association of America, which sponsors the USAMO. In the early 90s Propp lived in Boston and later in Arlington, Massachusetts. In 1996, Propp and David Wilson invented coupling from the past, a method for sampling from the stationary distribution of a Markov chain among Markov chain Monte Carlo (MCMC) algorithms. Contrary to many MCMC algorithms, coupling from the past gives in principle a perfect sample from the stationary distribution. His papers have discussed the use of surcomplex numbers in game theory; the solution to the counting of alternating sign matrices; and occurrences of Grandi's series as an Euler characteristic of infinite-dimensional real projective space. Other contributions Propp was a member of the National Puzzlers' League under the nom Aesop. He was recruited for the organisation by colleague Henri Picciotto, cruciverbalist and co-author of the league's first cryptic crossword collection. Propp is the creator of the "Self-Referential Aptitude Test", a humorous multiple-choice test in which all questions except the last make self-references to their own answers. It was created in the early 1990s for a puzzlers' party. Propp is the author of Tuscanini, a 1992 children's book about a musical elephant, illustrated by Ellen Weiss. Awards and honours In 2015 he was elected as a fellow of the American Mathematical Society "for contributions to combinatorics and probability, and for mentoring and exposition." Personal He is married to research psychologist Alexandra (Sandi) Gubin. They have a son Adam and a daughter Eliana. Notes External links Propp's website Year of birth missing (living people) Living people Harvard University alumni University of California, Berkeley alumni University of Wisconsin–Madison faculty Massachusetts Institute of Technology faculty Harvard University Department of Mathematics faculty Harvard University faculty Alumni of the University of Cambridge Recreational mathematicians Probability theorists 20th-century American mathematicians 21st-century American mathematicians Fellows
https://en.wikipedia.org/wiki/Ibrahim%20Al-Mukhaini%20%28footballer%2C%20born%201987%29
Ibrahim Sabir Marzouq Al-Mukhaini (; born 16 April 1987), commonly known as Ibrahim Al-Mukhaini, is an Omani who plays for Sur SC. Club career statistics International career Ibrahim was selected for the national team for the first time in 2008. He made his first appearance for Oman on 26 March 2008 in a 2010 FIFA World Cup qualification match against Thailand. References External links Ibrahim Al-Mukhaini at Goal.com 1987 births Living people Omani men's footballers Oman men's international footballers Men's association football defenders Sur SC players Al-Shabab SC (Seeb) players Oman Professional League players
https://en.wikipedia.org/wiki/Yaqoob%20Salem%20Al-Farsi
Yaqoob Salem Saleh Al Farsi (; born 18 April 1982) is an Omani footballer who plays for Sur SC. Club career statistics International career Yaqoob was selected for the Oman national football team for the first time in 2001. He earned his first international cap for Oman on 7 May 2001 against Philippines in a 2002 FIFA World Cup qualification match. He has represented the national team in the 2002 FIFA World Cup qualification and the 2010 FIFA World Cup qualification. References External links 1982 births Living people Omani men's footballers Oman men's international footballers Men's association football midfielders Sur SC players Oman Professional League players
https://en.wikipedia.org/wiki/Lacunarity
Lacunarity, from the Latin lacuna, meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to Benoit Mandelbrot, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis. Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular (see Applications). Measuring lacunarity In many patterns or data sets, lacunarity is not readily perceivable or quantifiable, so computer-aided methods have been developed to calculate it. As a measurable quantity, lacunarity is often denoted in scientific literature by the Greek letters or but it is important to note that there is no single standard and several different methods exist to assess and interpret lacunarity. Box counting lacunarity One well-known method of determining lacunarity for patterns extracted from digital images uses box counting, the same essential algorithm typically used for some types of fractal analysis. Similar to looking at a slide through a microscope with changing levels of magnification, box counting algorithms look at a digital image from many levels of resolution to examine how certain features change with the size of the element used to inspect the image. Basically, the arrangement of pixels is measured using traditionally square (i.e., box-shaped) elements from an arbitrary set of sizes, conventionally denoted s. For each , a box of size is placed successively on the image, in the end covering it completely, and each time it is laid down, the number of pixels that fall within the box is recorded. In standard box counting, the box for each in is placed as though it were part of a grid overlaid on the image so that the box does not overlap itself, but in sliding box algorithms the box is slid over the image so that it overlaps itself and the "Sliding Box Lacunarity" or SLac is calculated. Figure 2 illustrates both types of box counting. Calculations from box counting The data gathered for each are manipulated to calculate lacunarity. One measure, denoted here as , is found from the coefficient of variation (), calculated as the standard deviation () divided by the mean (), for pixels per box. Because the way an image is sampled will depend on the arbitrary starting location, for any image sampled at any there will be so
https://en.wikipedia.org/wiki/Enduleni
Enduleni is an administrative ward in the Ngorongoro District of the Arusha Region of Tanzania. The ward is home of the Laetoli prehistoric site. In 2016 the Tanzania National Bureau of Statistics report there were 13,537 people in the ward, from 13,537 in 2012. References Ngorongoro District Wards of Arusha Region
https://en.wikipedia.org/wiki/Kakesio
Kakesio is an administrative ward in the Ngorongoro District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,179 people in the ward, from 5,537 in 2012. References Ngorongoro District Wards of Arusha Region
https://en.wikipedia.org/wiki/Naiyobi
Naiyobi is an administrative ward in the Ngorongoro District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,191 people in the ward, from 9,133 in 2012. References Ngorongoro District Wards of Arusha Region
https://en.wikipedia.org/wiki/Oldonyo-Sambu
Oldonyosambu is an administrative ward in the Ngorongoro District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,657 people in the ward, from 5,233 in 2012. References Ngorongoro District Wards of Arusha Region
https://en.wikipedia.org/wiki/Orgosorok
Orgosorok is an administrative ward in the Ngorongoro District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,690 people in the ward, from 12,268 in 2012. References Ngorongoro District Wards of Arusha Region
https://en.wikipedia.org/wiki/Sale%20%28Tanzanian%20ward%29
Sale is an administrative ward in the Ngorongoro District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 4,892 people in the ward, from 4,384 in 2012. References Wards of Arusha Region
https://en.wikipedia.org/wiki/Songoro
Songoro is an administrative ward in the Chemba District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,685 people in the ward, from 10,751 in 2012. References Arumeru District Wards of Arusha Region
https://en.wikipedia.org/wiki/Makuyuni%2C%20Monduli
Makuyuni is an administrative ward in the Monduli district of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,529 people in the ward, from 11,228 in 2012. References Monduli District Wards of Arusha Region
https://en.wikipedia.org/wiki/Moita%20%28Tanzanian%20ward%29
Moita is an administrative ward in the Monduli District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,654 people in the ward, from 11,340 in 2012. References Monduli District Wards of Arusha Region
https://en.wikipedia.org/wiki/Monduli%20Juu%2C%20Monduli%20District
Monduli Juu is an administrative ward in the Monduli District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,457 people in the ward, from 15,914 in 2012. References Monduli District Wards of Arusha Region
https://en.wikipedia.org/wiki/Selela
Selela is an administrative ward in the Monduli District of the Arusha Region of Tanzania. The name Selela means Clean water in the Maasai language. In 2016 the Tanzania National Bureau of Statistics report there were 9,712 people in the ward, from 8,703 in 2012. Education In Selela there are a number of schools: - Selela Primary School is the largest and counts almost 1,200 children. . - In the subvillage of Mbaash (which means 'between the mountains in Maasai language), about 16 km from Selela, there's also primary school which counts more than 500 children. The head teacher is Mr. Paakwai Meitamei. - A third primary school is about 7 km from Selela village and is called Ndinyika Primary School, with more than children. Ndinyika means 'far from the boma' in Maasai language. . - Oltinga Secondary School is built on the escarpment and overviews Selela village. This school has almost 400 students from the region around Selela. Head teacher is Mr. Kitally. Health Selela has a dispensary and since 2016 a new dispensary at Mbaash has opened . Selela and Mbaash are supported by Tanzania Support Foundation. This organization especially helped the Selela schools with several supplies (for example school desks, computers, books, exercise books, storage cupboards, construction materials for classrooms and teachers houses et cetera) and the medical dispensary (for example several medical supplies, wheelchairs, walkers and a solar installation). Economy Every Wednesday there's a market day at Selela village. In the village there are some guesthouses and small shops. References Monduli District Wards of Arusha Region
https://en.wikipedia.org/wiki/Sepeko
Sepeko is an administrative ward in the Monduli District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,550 people in the ward, from 16,720 in 2012. References Monduli District Wards of Arusha Region
https://en.wikipedia.org/wiki/Endamarariek
Endamarariek is an administrative ward in the Karatu district of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 27,893 people in the ward, from 24,996 in 2012. The ward borders Lake Manyara National Park to the east. References Karatu District Wards of Arusha Region
https://en.wikipedia.org/wiki/Ambient%20calculus
In computer science, the ambient calculus is a process calculus devised by Luca Cardelli and Andrew D. Gordon in 1998, and used to describe and theorise about concurrent systems that include mobility. Here mobility means both computation carried out on mobile devices (i.e. networks that have a dynamic topology), and mobile computation (i.e. executable code that is able to move around the network). The ambient calculus provides a unified framework for modeling both kinds of mobility. It is used to model interactions in such concurrent systems as the Internet. Since its inception, the ambient calculus has grown into a family of closely related ambient calculi. Informal description Ambients The fundamental primitive of the ambient calculus is the ambient. An ambient is informally defined as a bounded place in which computation can occur. The notion of boundaries is considered key to representing mobility, since a boundary defines a contained computational agent that can be moved in its entirety. Examples of ambients include: a web page (bounded by a file) a virtual address space (bounded by an addressing range) a Unix file system (bounded within a physical volume) a single data object (bounded by “self”) a laptop (bounded by its case and data ports) The key properties of ambients within the Ambient calculus are: Ambients have names, which are used to control access to the ambient. Ambients can be nested inside other ambients (representing, for example, administrative domains) Ambients can be moved as a whole. Operations Computation is represented as the crossing of boundaries, i.e. the movement of ambients. There are four basic operations (or capabilities) on ambients: instructs the surrounding ambient to enter some sibling ambient , and then proceed as instructs the surrounding ambient to exit its parent ambient instructs the surrounding ambient to dissolve the boundary of an ambient located at the same level makes any number of copies of something The ambient calculus provides a reduction semantics that formally defines what the results of these operations are. Communication within (i.e. local to) an ambient is anonymous and asynchronous. Output actions release names or capabilities into the surrounding ambient. Input actions capture a value from the ambient, and bind it to a variable. Non-local I/O can be represented in terms of these local communications actions by a variety of means. One approach is to use mobile “messenger” agents that carry a message from one ambient to another (using the capabilities described above). Another approach is to emulate channel-based communications by modeling a channel in terms of ambients and operations on those ambients. The three basic ambient primitives, namely in, out, and open are expressive enough to simulate name-passing channels in the π-calculus. See also Lambda calculus Mobile membranes Type theory API-Calculus References External links Mobile Computational Ambie
https://en.wikipedia.org/wiki/Stable%20curve
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent to the condition that it is a complete connected curve whose only singularities are ordinary double points and whose automorphism group is finite. The condition that the automorphism group is finite can be replaced by the condition that it is not of arithmetic genus one and every non-singular rational component meets the other components in at least 3 points . A semi-stable curve is one satisfying similar conditions, except that the automorphism group is allowed to be reductive rather than finite (or equivalently its connected component may be a torus). Alternatively the condition that non-singular rational components meet the other components in at least three points is replaced by the condition that they meet in at least two points. Similarly a curve with a finite number of marked points is called stable if it is complete, connected, has only ordinary double points as singularities, and has finite automorphism group. For example, an elliptic curve (a non-singular genus 1 curve with 1 marked point) is stable. Over the complex numbers, a connected curve is stable if and only if, after removing all singular and marked points, the universal covers of all its components are isomorphic to the unit disk. Definition Given an arbitrary scheme and setting a stable genus g curve over is defined as a proper flat morphism such that the geometric fibers are reduced, connected 1-dimensional schemes such that has only ordinary double-point singularities Every rational component meets other components at more than points These technical conditions are necessary because (1) reduces the technical complexity (also Picard-Lefschetz theory can be used here), (2) rigidifies the curves so that there are no infinitesimal automorphisms of the moduli stack constructed later on, and (3) guarantees that the arithmetic genus of every fiber is the same. Note that for (1) the types of singularities found in Elliptic surfaces can be completely classified. Examples One classical example of a family of stable curves is given by the Weierstrass family of curves where the fibers over every point are smooth and the degenerate points only have one double-point singularity. This example can be generalized to the case of a one-parameter family of smooth hyperelliptic curves degenerating at finitely many points. Non-examples In the general case of more than one parameter care has to be taken to remove curves which have worse than double-point singularities. For example, consider the family over constructed from the polynomials since along the diagonal there are non-double-point singularities. Another non-example is the family over given by the polynomials which are a family of elliptic curves degenerating to a rational curve with a cusp. Properties One of the most important properties of stable curve
https://en.wikipedia.org/wiki/Kingman%27s%20formula
In queueing theory, a discipline within the mathematical theory of probability, Kingman's formula also known as the VUT equation, is an approximation for the mean waiting time in a G/G/1 queue. The formula is the product of three terms which depend on utilization (U), variability (V) and service time (T). It was first published by John Kingman in his 1961 paper The single server queue in heavy traffic. It is known to be generally very accurate, especially for a system operating close to saturation. Statement of formula Kingman's approximation states are equal to where τ is the mean service time (i.e. μ = 1/τ is the service rate), λ is the mean arrival rate, ρ = λ/μ is the utilization, ca is the coefficient of variation for arrivals (that is the standard deviation of arrival times divided by the mean arrival time) and cs is the coefficient of variation for service times. References Single queueing nodes
https://en.wikipedia.org/wiki/Edward%20Kasner
Edward Kasner (April 2, 1878 – January 7, 1955) was an American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department. Kasner was the first Jewish person appointed to a faculty position in the sciences at Columbia University. Subsequently, he became an adjunct professor in 1906, and a full professor in 1910, at the university. Differential geometry was his main field of study. In addition to introducing the term "googol", he is known also for the Kasner metric and the Kasner polygon. Education Kasner's 1899 PhD dissertation at Columbia University was titled The Invariant Theory of the Inversion Group: Geometry upon a Quadric Surface; it was published by the American Mathematical Society in 1900 in their Transactions. Googol and googolplex Kasner is perhaps best remembered today for introducing the term "googol." In order to pique the interest of children, Kasner sought a name for a very large number: one followed by 100 zeros. On a walk in the New Jersey Palisades with his nephews, Milton (1911–1981) and Edwin Sirotta, Kasner asked for their ideas. Nine-year-old Milton suggested "googol". In 1940, with James R. Newman, Kasner co-wrote a non-technical book surveying the field of mathematics, called Mathematics and the Imagination (). It was in this book that the term "googol" was first popularized: The Internet search engine "Google" originated from a misspelling of "googol", and the "Googleplex" (the Google company headquarters in Mountain View, California) is similarly derived from googolplex. Personal life Kasner was Jewish and was the son of Austrian immigrants. Works Edward Kasner and James R. Newman, Mathematics and the Imagination, Tempus Books of Microsoft Press, 1989. References External links History from the Google website 1878 births 1955 deaths Jewish American scientists Differential geometers 19th-century American mathematicians 20th-century American mathematicians City College of New York alumni Columbia University alumni Columbia University faculty
https://en.wikipedia.org/wiki/Alan%20Baker%20%28mathematician%29
Alan Baker (19 August 1939 – 4 February 2018) was an English mathematician, known for his work on effective methods in number theory, in particular those arising from transcendental number theory. Life Alan Baker was born in London on 19 August 1939. He attended Stratford Grammar School, East London, and his academic career started as a student of Harold Davenport, at University College London and later at Trinity College, Cambridge, where he received his PhD. He was a visiting scholar at the Institute for Advanced Study in 1970 when he was awarded the Fields Medal at the age of 31. In 1974 he was appointed Professor of Pure Mathematics at Cambridge University, a position he held until 2006 when he became an Emeritus. He was a fellow of Trinity College from 1964 until his death. His interests were in number theory, transcendence, linear forms in logarithms, effective methods, Diophantine geometry and Diophantine analysis. In 2012 he became a fellow of the American Mathematical Society. He has also been made a foreign fellow of the National Academy of Sciences, India. Research Baker generalised the Gelfond–Schneider theorem, itself a solution to Hilbert's seventh problem. Specifically, Baker showed that if are algebraic numbers (besides 0 or 1), and if are irrational algebraic numbers such that the set is linearly independent over the rational numbers, then the number is transcendental. Baker made significant contributions to several areas in number theory, such as the Gauss class number problem, diophantine approximation, and to Diophantine equations such as the Mordell curve. Selected publications ; Honours and awards 1970: Fields Medal 1972: Adams Prize 1973: Fellowship of the Royal Society References External links 1939 births 2018 deaths 20th-century English mathematicians 21st-century English mathematicians Fields Medalists Number theorists Alumni of University College London Alumni of Trinity College, Cambridge Fellows of Trinity College, Cambridge Fellows of the American Mathematical Society Fellows of the Royal Society Foreign Fellows of the Indian National Science Academy Institute for Advanced Study visiting scholars Cambridge mathematicians Mathematicians from London
https://en.wikipedia.org/wiki/Manin%20obstruction
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a variety X over a global field, which measures the failure of the Hasse principle for X. If the value of the obstruction is non-trivial, then X may have points over all local fields but not over the global field. The Manin obstruction is sometimes called the Brauer–Manin obstruction, as Manin used the Brauer group of X to define it. For abelian varieties the Manin obstruction is just the Tate–Shafarevich group and fully accounts for the failure of the local-to-global principle (under the assumption that the Tate–Shafarevich group is finite). There are however examples, due to Alexei Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points. References Diophantine geometry
https://en.wikipedia.org/wiki/Tensor%20rank%20decomposition
In multilinear algebra, the tensor rank decomposition or the decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum tensors. This is an open problem. Canonical polyadic decomposition (CPD) is a variant of the rank decomposition which computes the best fitting terms for a user specified . The CP decomposition has found some applications in linguistics and chemometrics. The CP rank was introduced by Frank Lauren Hitchcock in 1927 and later rediscovered several times, notably in psychometrics. The CP decomposition is referred to as CANDECOMP, PARAFAC, or CANDECOMP/PARAFAC (CP). PARAFAC2 rank decomposition is yet to explore. Another popular generalization of the matrix SVD known as the higher-order singular value decomposition computes orthonormal mode matrices and has found applications in econometrics, signal processing, computer vision, computer graphics, psychometrics. Notation A scalar variable is denoted by lower case italic letters, and an upper bound scalar is denoted by an upper case italic letter, . Indices are denoted by a combination of lowercase and upper case italic letters, . Multiple indices that one might encounter when referring to the multiple modes of a tensor are conveniently denoted by where . A vector is denoted by a lower case bold Times Roman, and a matrix is denoted by bold upper case letters . A higher order tensor is denoted by calligraphic letters,. An element of an -order tensor is denoted by or . Definition A data tensor is a collection of multivariate observations organized into a -way array where =+1. Every tensor may be represented with a suitably large as a linear combination of rank-1 tensors: where and where . When the number of terms is minimal in the above expression, then is called the rank of the tensor, and the decomposition is often referred to as a (tensor) rank decomposition, minimal CP decomposition, or Canonical Polyadic Decomposition (CPD). If the number of terms is not minimal, then the above decomposition is often referred to as CANDECOMP/PARAFAC, Polyadic decomposition'. Tensor rank Contrary to the case of matrices, computing the rank of a tensor is NP-hard. The only notable well-understood case consists of tensors in , whose rank can be obtained from the Kronecker–Weierstrass normal form of the linear matrix pencil that the tensor represents. A simple polynomial-time algorithm exists for certifying that a tensor is of rank 1, namely the higher-order singular value decomposition. The rank of the tensor of zeros is zero by convention. The rank of a tensor is one, provided that . Field dependence The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor. As an example, consider the following real tensor where . The rank of this tensor over the re
https://en.wikipedia.org/wiki/Tensor%20reshaping
In multilinear algebra, a reshaping of tensors is any bijection between the set of indices of an order- tensor and the set of indices of an order- tensor, where . The use of indices presupposes tensors in coordinate representation with respect to a basis. The coordinate representation of a tensor can be regarded as a multi-dimensional array, and a bijection from one set of indices to another therefore amounts to a rearrangement of the array elements into an array of a different shape. Such a rearrangement constitutes a particular kind of linear map between the vector space of order- tensors and the vector space of order- tensors. Definition Given a positive integer , the notation refers to the set of the first positive integers. For each integer where for a positive integer , let denote an -dimensional vector space over a field . Then there are vector space isomorphisms (linear maps) where is any permutation and is the symmetric group on elements. Via these (and other) vector space isomorphisms, a tensor can be interpreted in several ways as an order- tensor where . Coordinate representation The first vector space isomorphism on the list above, , gives the coordinate representation of an abstract tensor. Assume that each of the vector spaces has a basis . The expression of a tensor with respect to this basis has the form where the coefficients are elements of . The coordinate representation of is where is the standard basis vector of . This can be regarded as a M-way array whose elements are the coefficients . General flattenings For any permutation there is a canonical isomorphism between the two tensor products of vector spaces and . Parentheses are usually omitted from such products due to the natural isomorphism between and , but may, of course, be reintroduced to emphasize a particular grouping of factors. In the grouping, there are groups with factors in the group (where and ). Letting for each satisfying , an -flattening of a tensor , denoted , is obtained by applying the two processes above within each of the groups of factors. That is, the coordinate representation of the group of factors is obtained using the isomorphism , which requires specifying bases for all of the vector spaces . The result is then vectorized using a bijection to obtain an element of , where , the product of the dimensions of the vector spaces in the group of factors. The result of applying these isomorphisms within each group of factors is an element of , which is a tensor of order . Vectorization By means of a bijective map , a vector space isomorphism between and is constructed via the mapping where for every natural number such that , the vector denotes the ith standard basis vector of . In such a reshaping, the tensor is simply interpreted as a vector in . This is known as vectorization, and is analogous to vectorization of matrices. A standard choice of bijection is such that which is consisten
https://en.wikipedia.org/wiki/Regular%20conditional%20probability
In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution is a parametrized family of probability measures called a Markov kernel. Definition Conditional probability distribution Consider two random variables . The conditional probability distribution of Y given X is a two variable function If the random variable X is discrete If the random variables X, Y are continuous with density . A more general definition can be given in terms of conditional expectation. Consider a function satisfying for almost all . Then the conditional probability distribution is given by As with conditional expectation, this can be further generalized to conditioning on a sigma algebra . In that case the conditional distribution is a function : Regularity For working with , it is important that it be regular, that is: For almost all x, is a probability measure For all A, is a measurable function In other words is a Markov kernel. The second condition holds trivially, but the proof of the first is more involved. It can be shown that if Y is a random element in a Radon space S, there exists a that satisfies the first condition. It is possible to construct more general spaces where a regular conditional probability distribution does not exist. Relation to conditional expectation For discrete and continuous random variables, the conditional expectation can be expressed as where is the conditional density of given . This result can be extended to measure theoretical conditional expectation using the regular conditional probability distribution: . Formal definition Let be a probability space, and let be a random variable, defined as a Borel-measurable function from to its state space . One should think of as a way to "disintegrate" the sample space into . Using the disintegration theorem from the measure theory, it allows us to "disintegrate" the measure into a collection of measures, one for each . Formally, a regular conditional probability is defined as a function called a "transition probability", where: For every , is a probability measure on . Thus we provide one measure for each . For all , (a mapping ) is -measurable, and For all and all where is the pushforward measure of the distribution of the random element , i.e. the support of the . Specifically, if we take , then , and so , where can be denoted, using more familiar terms . Alternate definition Consider a Radon space (that is a probability measure defined on a Radon space endowed with the Borel sigma-algebra) and a real-valued random variable T. As discussed above, in this case there exists a regular conditional probability with respect to T. Moreover, we can alternatively define the regular conditional probability for an event A given a particular value t of the random variable T in the following manner: where the limit
https://en.wikipedia.org/wiki/Tucker%20decomposition
In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD). It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense. The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data. For a 3rd-order tensor , where is either or , Tucker Decomposition can be denoted as follows, where is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor respectively. are unitary matrices in respectively. The j-mode product (j = 1, 2, 3) of by is denoted as with entries as Taking for all is always sufficient to represent exactly, but often can be compressed or efficiently approximately by choosing . A common choice is , which can be effective when the difference in dimension sizes is large. There are two special cases of Tucker decomposition: Tucker1: if and are identity, then Tucker2: if is identity, then . RESCAL decomposition can be seen as a special case of Tucker where is identity and is equal to . See also Higher-order singular value decomposition Multilinear principal component analysis References Dimension reduction
https://en.wikipedia.org/wiki/Mathematical%20economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications. Broad applications include: optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy) is modeled as not changing comparative statics as to a change from one equilibrium to another induced by a change in one or more factors dynamic analysis, tracing changes in an economic system over time, for example from economic growth. Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximization, an early economic application of mathematical optimization. Economics became more mathematical as a discipline throughout the first half of the 20th century, but introduction of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics. This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics. History The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in German universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration. Gottfried Achenwall lectured in this fashion, coining the term statistics. At the same time, a small group of professors in England established a method of "reasoning by figures upon things relating to government" and referred to this practice as Political Arithmetick. Sir William Petty wrote at length on issues that would later concern economists, such as taxatio
https://en.wikipedia.org/wiki/Kolmogorov%27s%20criterion
In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem giving a necessary and sufficient condition for a Markov chain or continuous-time Markov chain to be stochastically identical to its time-reversed version. Discrete-time Markov chains The theorem states that an irreducible, positive recurrent, aperiodic Markov chain with transition matrix P is reversible if and only if its stationary Markov chain satisfies for all finite sequences of states Here pij are components of the transition matrix P, and S is the state space of the chain. Example Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round, Proof Let be the Markov chain and denote by its stationary distribution (such exists since the chain is positive recurrent). If the chain is reversible, the equality follows from the relation . Now assume that the equality is fulfilled. Fix states and . Then . Now sum both sides of the last equality for all possible ordered choices of states . Thus we obtain so . Send to on the left side of the last. From the properties of the chain follows that , hence which shows that the chain is reversible. Continuous-time Markov chains The theorem states that a continuous-time Markov chain with transition rate matrix Q is, under any invariant probability vector, reversible if and only if its transition probabilities satisfy for all finite sequences of states The proof for continuous-time Markov chains follows in the same way as the proof for discrete-time Markov chains. References Markov processes
https://en.wikipedia.org/wiki/Naylor%20Prize%20and%20Lectureship
The Naylor Prize and lectureship in Applied Mathematics is a prize of the London Mathematical Society awarded every two years in memory of Dr V.D. Naylor. Only those who reside in the United Kingdom are eligible for the prize. The "grounds for award can include work in, and influence on, and contributions to applied mathematics and/or the applications of mathematics, and lecturing gifts." Prize winners 1977 James Lighthill 1979 Basil John Mason 1981 H. Christopher Longuet-Higgins 1983 Michael J. D. Powell 1985 I C Percival 1987 Douglas Samuel Jones 1989 J D Murray 1991 Roger Penrose 1993 Michael Berry 1995 John Ball 1997 Frank Kelly 1999 Stephen Hawking 2000 Athanassios S. Fokas 2002 Mark H. A. Davis 2004 Richard Jozsa 2007 Michael Green 2009 Philip Maini 2011 John Bryce McLeod 2013 Nick Trefethen 2015 S. Jonathan Chapman 2017 John King 2019 Nicholas Higham 2021 Endre Süli 2023 Jens G. Eggers See also Forder Lectureship Whitehead Prize Senior Whitehead Prize Shephard Prize Fröhlich Prize Berwick Prize Pólya Prize (LMS) De Morgan Medal List of mathematics awards References Awards established in 1977 British science and technology awards Awards of the London Mathematical Society British lecture series 1977 establishments in the United Kingdom Recurring events established in 1977 Science lecture series Biennial events Mathematical events
https://en.wikipedia.org/wiki/Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined. History The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geomet
https://en.wikipedia.org/wiki/Mathematical%20beauty
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, (a position taken by G. H. Hardy) or, at a minimum, as a creative activity. Comparisons are made with music and poetry. In method Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean: A proof that uses a minimum of additional assumptions or previous results. A proof that is unusually succinct. A proof that derives a result in a surprising way (e.g., from an apparently unrelated theorem or a collection of theorems). A proof that is based on new and original insights. A method of proof that can be easily generalized to solve a family of similar problems. In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs being published up to date. Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity. In fact, Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published. Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, highly conventional approaches or a large number of powerful axioms or previous results are usually not considered to be elegant, and may be even referred to as ugly or clumsy. In results Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated. These results are often described as deep. While it is difficult to find universal agreement on whether a result is deep, some examples are more commonly cited than others. One such example is Euler's identity: This elegant expression ties together arguably the five most important mathematical constants (e, i, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of Euler's formula, which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics". Modern examples include the modularity theorem, which establishes an important connection between elliptic curves and modular forms (work on which led to the awarding of the Wolf Prize to Andrew Wiles and Robert Langlands), and "monstrous moonshine", which connects the Monster group to modular functions via string theory (for which Richard Borcherds was awarded the Fields Medal). Other examples of deep results include unexpected insights into mathematical structures. For example, Gauss's Theorema Egregium is a deep theorem
https://en.wikipedia.org/wiki/Chamwino
Chamwino is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 20,840 people in the ward, from 19,175 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Chihanga
Chihanga is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,959 people in the ward, from 11,004 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Dodoma%20Makulu
Dodoma Makulu is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 18,582 people in the ward, from 17,097 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Hombolo%20Makulu
Hombolo Makulu is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,659 people in the ward. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Ipala%2C%20Dodoma%20Municipal%20Council
Ipala is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,549 people in the ward, from 6,026 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Kikombo
Kikombo is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,067 people in the ward, from 8,343 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Kilimani%2C%20Dodoma%20Municipal%20Council
Kilimani is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,033 people in the ward, from 7,237 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Kiwanja%20cha%20Ndege
Kiwanja cha Ndege is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,009 people in the ward, from 10,129 in 2012. References Wards of Dodoma Region Constituencies of Tanzania
https://en.wikipedia.org/wiki/Kizota
Kizota is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 14,520 people in the ward, from 34,453 in 2012. References Wards of Dodoma Region Constituencies of Tanzania
https://en.wikipedia.org/wiki/Makutupora
Makutupora is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,683 people in the ward, from 14,430 in 2012. Transportation The railway will be replaced by the Standard Gauge Railway from Dar es Salaam and in early 2023 the construction of a freight station and a new Passenger station for Makutupora was near complete. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Mbabala
Mbabala is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,934 people in the ward, from 11,901 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Mbalawala
Mbalawala is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,597 people in the ward, from 8,830 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Miyuji
Miyuji is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 16,264 people in the ward, from 14,965 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Mkonze
Mkonze is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,602 people in the ward, from 12,515 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Mpunguzi
Mpunguzi is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,656 people in the ward, from 17,891 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Msalato
Msalato is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,301 people in the ward, from 6,718 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Nala%2C%20Dodoma%20Municipal%20Council
Nala is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,050 people in the ward, from 5,567 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/Nzuguni
Nzuguni is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 16,809 people in the ward, from 15,466 in 2012. References Wards of Dodoma Region
https://en.wikipedia.org/wiki/N%21%20conjecture
In mathematics, the n! conjecture is the conjecture that the dimension of a certain bi-graded module of diagonal harmonics is n!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's positivity conjecture about the Macdonald polynomials. Formulation and background The Macdonald polynomials are a two-parameter family of orthogonal polynomials indexed by a positive weight λ of a root system, introduced by Ian G. Macdonald (1987). They generalize several other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials. They are known to have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them. introduced a new basis for the space of symmetric functions, which specializes to many of the well-known bases for the symmetric functions, by suitable substitutions for the parameters q and t. In fact, we can obtain in this manner the Schur functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the zonal symmetric functions, the zonal spherical functions, and the elementary and monomial symmetric functions. The so-called q,t-Kostka polynomials are the coefficients of a resulting transition matrix. Macdonald conjectured that they are polynomials in q and t, with non-negative integer coefficients. It was Adriano Garsia's idea to construct an appropriate module in order to prove positivity (as was done in his previous joint work with Procesi on Schur positivity of Kostka–Foulkes polynomials). In an attempt to prove Macdonald's conjecture, introduced the bi-graded module of diagonal harmonics and conjectured that the (modified) Macdonald polynomials are the Frobenius image of the character generating function of Hμ, under the diagonal action of the symmetric group. The proof of Macdonald's conjecture was then reduced to the n! conjecture; i.e., to prove that the dimension of Hμ is n!. In 2001, Haiman proved that the dimension is indeed n! (see [4]). This breakthrough led to the discovery of many hidden connections and new aspects of symmetric group representation theory, as well as combinatorial objects (e.g., insertion tableaux, Haglund's inversion numbers, and the role of parking functions in representation theory). References to appear as part of the collection published by the Lab. de. Comb. et Informatique Mathématique, edited by S. Brlek, U. du Québec á Montréal. External links Bourbaki seminar (Procesi), PDF n! conjecture by François Bergeron n! homepage of Garsia http://www.maths.ed.ac.uk/~igordon/pubs/grenoble3.pdf http://mathworld.wolfram.com/n!Theorem.html Algebraic combinatorics Theorems in algebraic geometry Orthogonal polynomials Representation theory Conjectures that have been proved Factorial and binomial topics Theorems about polynomials Module theory Theorems in linear algebra
https://en.wikipedia.org/wiki/Tensor%20decomposition
In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generalize some matrix decompositions. Tensors are generalizations of matrices to higher dimensions (or rather to higher orders, i.e. the higher number of dimensions) and can consequently be treated as multidimensional fields. The main tensor decompositions are: Tensor rank decomposition; Higher-order singular value decomposition; Tucker decomposition; matrix product states, and operators or tensor trains; Online Tensor Decompositions hierarchical Tucker decomposition; block term decomposition Notation This section introduces basic notations and operations that are widely used in the field. Introduction A multi-way graph with K perspectives is a collection of K matrices with dimensions I × J (where I, J are the number of nodes). This collection of matrices is naturally represented as a tensor X of size I × J × K. In order to avoid overloading the term “dimension”, we call an I × J × K tensor a three “mode” tensor, where “modes” are the numbers of indices used to index the tensor. References Tensors
https://en.wikipedia.org/wiki/Burke%27s%20theorem
In queueing theory, a discipline within the mathematical theory of probability, Burke's theorem (sometimes the Burke's output theorem) is a theorem (stated and demonstrated by Paul J. Burke while working at Bell Telephone Laboratories) asserting that, for the M/M/1 queue, M/M/c queue or M/M/∞ queue in the steady state with arrivals is a Poisson process with rate parameter λ: The departure process is a Poisson process with rate parameter λ. At time t the number of customers in the queue is independent of the departure process prior to time t. Proof Burke first published this theorem along with a proof in 1956. The theorem was anticipated but not proved by O’Brien (1954) and Morse (1955). A second proof of the theorem follows from a more general result published by Reich. The proof offered by Burke shows that the time intervals between successive departures are independently and exponentially distributed with parameter equal to the arrival rate parameter, from which the result follows. An alternative proof is possible by considering the reversed process and noting that the M/M/1 queue is a reversible stochastic process. Consider the figure. By Kolmogorov's criterion for reversibility, any birth-death process is a reversible Markov chain. Note that the arrival instants in the forward Markov chain are the departure instants of the reversed Markov chain. Thus the departure process is a Poisson process of rate λ. Moreover, in the forward process the arrival at time t is independent of the number of customers after t. Thus in the reversed process, the number of customers in the queue is independent of the departure process prior to time t. This proof could be counter-intuitive, in the sense that the departure process of a birth-death process is independent of the service offered. Related results and extensions The theorem can be generalised for "only a few cases," but remains valid for M/M/c queues and Geom/Geom/1 queues. It is thought that Burke's theorem does not extend to queues fed by a Markovian arrival processes (MAP) and is conjectured that the output process of an MAP/M/1 queue is an MAP only if the queue is an M/M/1 queue. An analogous theorem for the Brownian queue was proven by J. Michael Harrison. References Single queueing nodes Probability theorems Queueing theory
https://en.wikipedia.org/wiki/1988%20Chinese%20Jia-A%20League
Statistics of Chinese Jia-A League for the 1988 season. Overview It was contested by 21 teams, and Liaoning F.C. won the championship. First round Second round Places 1–12 Group A Group B Final ranking Places 13–21 References China - List of final tables (RSSSF) Chinese Jia-A League seasons 1 China China 1988 establishments in China
https://en.wikipedia.org/wiki/1989%20Chinese%20Jia-A%20League
Statistics of the Chinese Jia-A League for the 1989 season. Overview It was contested by 8 teams, and China B won the championship. League standings References China - List of final tables (RSSSF) Chinese Jia-A League seasons 1 China China 1989 establishments in China
https://en.wikipedia.org/wiki/1990%20Chinese%20Jia-A%20League
Statistics of the Chinese Jia-A League for the 1990 season. Overview It was contested by 8 teams, and Liaoning F.C. won the championship. League standings References China - List of final tables (RSSSF) Chinese Jia-A League seasons 1 China China 1990 establishments in China
https://en.wikipedia.org/wiki/1991%20Chinese%20Jia-A%20League
Statistics of the Chinese Jia-A League for the 1991 season. Overview It was contested by 8 teams, and Liaoning F.C. won the championship. League standings References China - List of final tables (RSSSF) Chinese Jia-A League seasons 1 China China 1991 establishments in China
https://en.wikipedia.org/wiki/1992%20Chinese%20Jia-A%20League
Statistics of the Chinese Jia-A League for the 1992 season. Overview It was contested by 8 teams, and Liaoning F.C. won the championship. League standings References China - List of final tables (RSSSF) Chinese Jia-A League seasons 1 China China 1992 establishments in China
https://en.wikipedia.org/wiki/Volodymyr%20Chesnakov
Volodymyr Hennadiyovych Chesnakov (; born 12 February 1988) is a Ukrainian professional footballer who plays as a midfielder for Vorskla Poltava in the Ukrainian Premier League. Career statistics Club References External links Profile on Football Squads 1988 births Living people People from Hlobyne Ukrainian men's footballers FC Vorskla Poltava players Ukrainian Premier League players Ukraine men's under-21 international footballers Men's association football defenders Footballers from Poltava Oblast
https://en.wikipedia.org/wiki/Philip%20Saffman
Philip Geoffrey Saffman FRS (19 March 1931 – 17 August 2008) was a mathematician and the Theodore von Kármán Professor of Applied Mathematics and Aeronautics at the California Institute of Technology. Education and early life Saffman was born to a Jewish family in Leeds, England, and educated at Roundhay Grammar School and Trinity College, Cambridge which he entered aged 15. He received his Bachelor of Arts degree in 1953, studied for Part III of the Cambridge Mathematical Tripos in 1954 and was awarded his PhD in 1956 for research supervised by George Batchelor. Career and research Saffman started his academic career as a lecturer at the University of Cambridge, then joined King's College London as a Reader. Saffman joined the Caltech faculty in 1964 and was named the Theodore von Kármán Professor in 1995. According to Dan Meiron, Saffman "really was one of the leading figures in fluid mechanics," and he influenced almost every subfield of that discipline. He is known (with his co-author Geoffrey Ingram Taylor) for the Saffman–Taylor instability in viscous fingering of fluid boundaries, a phenomenon important for its applications in enhanced oil recovery, and for the Saffman–Delbrück model of protein diffusion in membranes which he published with his Caltech colleague and Pasadena neighbour Max Delbrück. He made important contributions to the theory of vorticity arising from the motion of ships and aircraft through water and air; his work on wake turbulence led the airlines to increase the minimum time between takeoffs of aircraft on the same runway. Saffman also studied the flow of spheroidal particles in a fluid, such as bubbles in a carbonated beverage or corpuscles in blood; his work overturned previous assumptions that inertia was an important factor in these particles' motion and showed instead that Non-Newtonian properties of fluids play a significant role. Along with his many research papers, Saffman wrote a book, Vortex Dynamics, surveying a field to which he had been a principal contributor. Russel E. Caflisch writes that "This book should be read by everyone interested in vortex dynamics or fluid dynamics in general." Awards and honours Saffman was elected a Fellow of the American Academy of Arts and Sciences and a Fellow of the Royal Society in 1986, and the recipient of the American Physical Society's Otto Laporte Award. His nomination for the Royal Society reads: Personal life Saffman was survived by his wife (Ruth Arion whom he married in 1954), three children (Mark, Louise, Emma), and eight grandchildren (Timothy, Gregory, Rae, Jenny, Nadine, Aaron, Miriam, Alexandra and Andrey. References P.A. Davidson, Y. Kaneda, K. Moffatt, and K.R. Sreenivasan (eds, 2011). A Voyage Through Turbulence, chapter 12, pp 393–425, Cambridge University Press 1931 births 2008 deaths Alumni of Trinity College, Cambridge 20th-century British mathematicians 21st-century British mathematicians Fluid dynamicists California Institute of Technol
https://en.wikipedia.org/wiki/1991%20Nepal%20census
The 1991 Nepal census was a widespread national census conducted by the Nepal Central Bureau of Statistics. Working with Nepal's Village Development Committees at a district level, they recorded data from all the main towns and villages of each district of the country. The data included statistics on population size, households, sex and age distribution, place of birth, residence characteristics, literacy, marital status, religion, language spoken, caste/ethnic group, economically active population, education, number of children, employment status, and occupation. This census was followed by the 2001 Nepal census. References See also List of village development committees of Nepal (Former) 2001 Nepal census 2011 Nepal census Censuses in Nepal Nepal 1991 in Nepal
https://en.wikipedia.org/wiki/Data
In common usage and statistics, data (; ) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data is usually organized into structures such as tables that provide additional context and meaning, and which may themselves be used as data in larger structures. Data may be used as variables in a computational process. Data may represent abstract ideas or concrete measurements. Data is commonly used in scientific research, economics, and in virtually every other form of human organizational activity. Examples of data sets include price indices (such as consumer price index), unemployment rates, literacy rates, and census data. In this context, data represents the raw facts and figures from which useful information can be extracted. Data is collected using techniques such as measurement, observation, query, or analysis, and is typically represented as numbers or characters which may be further processed. Field data is data that is collected in an uncontrolled in-situ environment. Experimental data is data that is generated in the course of a controlled scientific experiment. Data is analyzed using techniques such as calculation, reasoning, discussion, presentation, visualization, or other forms of post-analysis. Prior to analysis, raw data (or unprocessed data) is typically cleaned: Outliers are removed and obvious instrument or data entry errors are corrected. Data can be seen as the smallest units of factual information that can be used as a basis for calculation, reasoning, or discussion. Data can range from abstract ideas to concrete measurements, including, but not limited to, statistics. Thematically connected data presented in some relevant context can be viewed as information. Contextually connected pieces of information can then be described as data insights or intelligence. The stock of insights and intelligence that accumulates over time resulting from the synthesis of data into information, can then be described as knowledge. Data has been described as "the new oil of the digital economy". Data, as a general concept, refers to the fact that some existing information or knowledge is represented or coded in some form suitable for better usage or processing. Advances in computing technologies have led to the advent of big data, which usually refers to very large quantities of data, usually at the petabyte scale. Using traditional data analysis methods and computing, working with such large (and growing) datasets is difficult, even impossible. (Theoretically speaking, infinite data would yield infinite information, which would render extracting insights or intelligence impossible.) In response, the relatively new field of data science uses machine learning (and other artificial intelligence (AI)) methods t
https://en.wikipedia.org/wiki/William%20Floyd%20%28mathematician%29
William J. Floyd is an American mathematician specializing in topology. He is currently a professor at Virginia Polytechnic Institute and State University. Floyd received a PhD in mathematics from Princeton University 1978 under the direction of William Thurston. Mathematical contributions Most of Floyd's research is in the areas of geometric topology and geometric group theory. Floyd and Allen Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. In a 1980 paper Floyd introduced a way to compactify a finitely generated group by adding to it a boundary which came to be called the Floyd boundary. Floyd also wrote a number of joint papers with James W. Cannon and Walter R. Parry exploring a combinatorial approach to the Cannon conjecture using finite subdivision rules. This represents one of the few plausible lines of attack of the conjecture. References External links William Floyd's webpage, Department of Mathematics, Virginia Polytechnic Institute and State University Year of birth missing (living people) Living people 20th-century American mathematicians 21st-century American mathematicians Topologists Virginia Tech faculty Princeton University alumni
https://en.wikipedia.org/wiki/Kerry%20Bond
John Kerry Bond (born July 18, 1945) is a retired professional ice hockey forward, most notably for the Indianapolis Racers of the World Hockey Association. Career statistics References External links 1945 births Living people Canadian ice hockey left wingers Indianapolis Racers players Ice hockey people from Greater Sudbury
https://en.wikipedia.org/wiki/Dynamic%20equation
In mathematics, dynamic equation can refer to: difference equation in discrete time differential equation in continuous time time scale calculus in combined discrete and continuous time Dynamical systems
https://en.wikipedia.org/wiki/Bogomolov%20conjecture
In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometry. The conjecture was proven by Emmanuel Ullmo and Shou-Wu Zhang in 1998. A further generalization to general abelian varieties was also proved by Zhang in 1998. Statement Let C be an algebraic curve of genus g at least two defined over a number field K, let denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an such that the set   is finite. Since if and only if P is a torsion point, the Bogomolov conjecture generalises the Manin-Mumford conjecture. Proof The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998. Generalization In 1998, Zhang proved the following generalization: Let A be an abelian variety defined over K, and let be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety is called a torsion subvariety if it is the translate of an abelian subvariety of A by a torsion point. If X is not a torsion subvariety, then there is an such that the set   is not Zariski dense in X. References Other sources Further reading The Manin-Mumford conjecture: a brief survey, by Pavlos Tzermias Abelian varieties Diophantine geometry Conjectures that have been proved
https://en.wikipedia.org/wiki/Ko%20Chang-hyun
Ko Chang-Hyun (born September 15, 1983) is a South Korean football player who lastly played for Ulsan Hyundai FC. Club career statistics As of April 18, 2011. Honors Ulsan Hyundai AFC Champions League (1): 2012 References External links 1983 births Living people Men's association football midfielders South Korean men's footballers Suwon Samsung Bluewings players Busan IPark players Gimcheon Sangmu FC players Daejeon Hana Citizen players Ulsan Hyundai FC players K League 1 players
https://en.wikipedia.org/wiki/Walter%20Benz
Walter Benz (May 2, 1931 Lahnstein – January 13, 2017 Ratzeburg) was a German mathematician, an expert in geometry. Benz studied at the Johannes Gutenberg University of Mainz and received his doctoral degree in 1954, with Robert Furch as his advisor. After a position at the Johann Wolfgang Goethe University Frankfurt am Main, he served as a professor at Ruhr University Bochum, University of Waterloo, and University of Hamburg. Benz was honoured with the degree of a Dr. h.c. Based on his book Vorlesungen über Geometrie der Algebren (Springer 1973), certain geometric objects are called Benz planes. Inner product spaces over the real numbers provide the basis of a 2007 book by Benz: Classical Geometries in Modern Contexts. See also List of University of Waterloo people References Uta Hartmann (2009) "Walter Benz — 'Die Mathematik war mein Leben, ist mein Leben' ", Journal of Geometry 93:83–115. External links Personal web site Oberwolfach Photo Collection 1931 births 2017 deaths 20th-century German mathematicians Geometers Johannes Gutenberg University Mainz alumni Academic staff of the University of Waterloo Academic staff of the University of Hamburg Academic staff of Ruhr University Bochum People from Rhein-Lahn-Kreis
https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein%20theorem%20for%20measurable%20spaces
The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called Borel spaces. This theorem, whose proof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory of standard Borel spaces contains very strong results about isomorphic measurable spaces, see Kuratowski's theorem. However, (a) the latter theorem is very difficult to prove, (b) the former theorem is satisfactory in many important cases (see Examples), and (c) the former theorem is used in the proof of the latter theorem. The theorem Let and be measurable spaces. If there exist injective, bimeasurable maps then and are isomorphic (the Schröder–Bernstein property). Comments The phrase " is bimeasurable" means that, first, is measurable (that is, the preimage is measurable for every measurable ), and second, the image is measurable for every measurable . (Thus, must be a measurable subset of not necessarily the whole ) An isomorphism (between two measurable spaces) is, by definition, a bimeasurable bijection. If it exists, these measurable spaces are called isomorphic. Proof First, one constructs a bijection out of and exactly as in the proof of the Cantor–Bernstein–Schroeder theorem. Second, is measurable, since it coincides with on a measurable set and with on its complement. Similarly, is measurable. Examples Example 1 The open interval (0, 1) and the closed interval [0, 1] are evidently non-isomorphic as topological spaces (that is, not homeomorphic). However, they are isomorphic as measurable spaces. Indeed, the closed interval is evidently isomorphic to a shorter closed subinterval of the open interval. Also the open interval is evidently isomorphic to a part of the closed interval (just itself, for instance). Example 2 The real line and the plane are isomorphic as measurable spaces. It is immediate to embed into The converse, embedding of into (as measurable spaces, of course, not as topological spaces) can be made by a well-known trick with interspersed digits; for example, g(π,100e) = g(, ) = . …. The map is clearly injective. It is easy to check that it is bimeasurable. (However, it is not bijective; for example, the number is not of the form ). References S.M. Srivastava, A Course on Borel Sets, Springer, 1998. See Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94). Theorems in measure theory Descriptive set theory Theorems in the foundations of mathematics
https://en.wikipedia.org/wiki/Herbert%20Edelsbrunner
Herbert Edelsbrunner (born March 14, 1958) is a computer scientist working in the field of computational geometry, the Arts & Science Professor of Computer Science and Mathematics at Duke University, Professor at the Institute of Science and Technology Austria (ISTA), and the co-founder of Geomagic, Inc. He was the first of only three computer scientists to win the National Science Foundation's Alan T. Waterman Award. Academic biography Edelsbrunner was born in 1958 in Graz, Austria. He received his Diplom in 1980 and Ph.D. in 1982, both from Graz University of Technology. His Ph.D. thesis was entitled Intersection Problems in Computational Geometry obtained under the supervision of Hermann Maurer. After a brief assistant professorship at Graz, he joined the faculty of the University of Illinois at Urbana-Champaign in 1985, and moved to Duke University in 1999. In 1996, with Ping Fu (then director of visualization at the National Center for Supercomputing Applications and his wife), he co-founded Geomagic, a company that develops shape modeling software. Since August 2009 he is Professor at the Institute of Science and Technology Austria (ISTA) in Klosterneuburg. In 1991, Edelsbrunner received the Alan T. Waterman Award. He was elected to the American Academy of Arts and Sciences in 2005, and received an honorary doctorate from Graz University of Technology in 2006. In 2008 he was elected to the German Academy of Sciences Leopoldina. In 2014 he became one of ten inaugural fellows of the European Association for Theoretical Computer Science. He is also a member of the Academia Europaea. Publications Edelsbrunner has over 100 research publications and is an ISI highly cited researcher. He has also published four books on computational geometry: Algorithms in Combinatorial Geometry (Springer-Verlag, 1987, ), Geometry and Topology for Mesh Generation (Cambridge University Press, 2001, ), Computational Topology (American Mathematical Society, 2009, 978-0821849255) and A Short Course in Computational Geometry and Topology (Springer-Verlag, 2014, ). As Edelsbrunner's Waterman Award citation states, Research contributions Edelsbrunner's most heavily cited research contribution is his work with Ernst Mücke on alpha shapes, a technique for defining a sequence of multiscale approximations to the shape of a three-dimensional point cloud. In this technique, one varies a parameter alpha ranging from 0 to the diameter of the point cloud; for each value of the parameter, the shape is approximated as the union of line segments, triangles, and tetrahedra defined by 2, 3, or 4 of the points respectively such that there exists a sphere of radius at most alpha containing only the defining points. Another heavily cited paper, also with Mücke, concerns “simulation of simplicity.” This is a technique for automatically converting algorithms that work only when their inputs are in general position (for instance, algorithms that may misbehave when some three input
https://en.wikipedia.org/wiki/Kwon%20Hyuk
Kwon Hyuk (; born November 6, 1983 in Daegu, South Korea) is a baseball player from South Korea who won a gold medal at the 2008 Summer Olympics. References External links Career statistics and player information from the KBO League 1983 births Baseball players at the 2008 Summer Olympics KBO League pitchers Living people Medalists at the 2008 Summer Olympics Olympic baseball players for South Korea Olympic gold medalists for South Korea Olympic medalists in baseball Samsung Lions players South Korean baseball players Baseball players from Daegu South Korean Buddhists
https://en.wikipedia.org/wiki/Sigurd%20Zienau
Sigurd Zienau (1921–1976) was a physicist notable for the theory of the polaron. Education His undergraduate studies were in mathematics at Birkbeck College. His further studies in physics were very much in the 'old school' European style at the time and he variously studied under Walter Heitler, Wolfgang Pauli, and Herbert Fröhlich. Career In 1954, he became an ICI Fellow and lecturer at the University of Liverpool. Then in 1965, he became a Reader in Physics at University College London until his early death at the age of 55. As well as his work on polarons he is remembered for his insightful revisions of Walter Heitler's book Quantum Theory of Radiation and Nevill Francis Mott & Harrie Massey's book The Theory of Atomic Collisions. See also Polaron Edwin Power Walter Heitler References E. A. Power and F. F. Heymann, "Sigurd Zienau," (Obituary) Nature, Vol. 266, pp. 201–202, 1977. Notes External links Zienau letter to Needham History of Physics at UCL with reference to Zienau History of Physics at UCL with reference to Zienau, Osborn & Cordero 1976 deaths Alumni of Birkbeck, University of London Academics of University College London Academics of the University of Liverpool British physicists 1920s births
https://en.wikipedia.org/wiki/Mostafa%20Salehinejad
Mostafa Salehi Nejad is an Iranian football player who currently plays for Paykan in Iran's Premier Football League. Club career Club career statistics Last Update 19 October 2010 Assist Goals External links Persian League Profile Iranian men's footballers Men's association football midfielders Zob Ahan Esfahan F.C. players Persian Gulf Pro League players Footballers from Isfahan Living people 1981 births
https://en.wikipedia.org/wiki/1976%20Japan%20Soccer%20League
Statistics of Japan Soccer League for the 1976 season. League tables First Division Promotion/relegation Series Since Eidai dropped out of the league and folded in March 1977, Fujitsu was promoted, meaning no team was relegated. Second Division JSL promotion/relegation Series Nissan Motors, future Yokohama Marinos, currently Yokohama F. Marinos, joined the league for the first time. Nissan promoted. Furukawa Chiba was not relegated due to Eidai's withdrawal. References Japan - List of final tables (RSSSF) 1976 1 Jap Jap