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Definiteness of a matrix

In linear algebra, a symmetric n × n real matrix M is said to be positive definite if the scalar z T M z is positive for every non-zero column vector z of n real numbers. Here z T denotes the transpose of z; when interpreting M z as the output of an operator, M, acting on an input, z, the property of positive definiteness implies that the output always has a positive inner product with the input, as observed in physical processes. More a complex n × n Hermitian matrix M is said to be positive definite if the scalar z ∗ M z is positive for every non-zero column vector z of n complex numbers. Here z ∗ denotes the conjugate transpose of z. Note that z ∗ M z is automatically real since M is Hermitian. Positive semi-definite matrices are defined except that the above scalars z T M z or z ∗ M z must be positive or zero. Negative definite and negative semi-definite matrices are defined analogously. A matrix, not positive semi-definite and not negative semi-definite is called indefinite; the matrix M is positive definite if and only if the bilinear form ⟨ z, w ⟩ = z T M w is positive definite.

This is a coordinate realization of an inner product on a vector space. Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones. In the following definitions, x T is the transpose of x, x ∗ is the conjugate transpose of x and 0 denotes the n-dimensional zero-vector. A n × n symmetric real matrix M is said to be positive definite if x T M x > 0 for all non-zero x in R n. Formally, A n × n symmetric real matrix M is said to be positive semidefinite or non-negative definite if x T M x ≥ 0 for all x in R n. Formally, A n × n symmetric real matrix M is said to be negative definite if x T M x < 0 for all non-zero x in R n. Formally, A n × n symmetric real matrix M is said to be negative semidefinite or non-positive definite if x T M x ≤ 0 for all x in R n. Formally, A n × n symmetric real matrix, neither positive semidefinite nor negative semidefinite is called indefinite; the following definitions all involve the term x ∗ M x.

Notice that this is always a real number for any Hermitian square matrix M. A n × n Hermitian complex matrix M is said to be positive definite if x ∗ M x > 0 for all non-zero x in C n. Formally, A n × n Hermitian complex matrix M is said to be positive semi-definite or non-negative definite if x ∗ M x ≥ 0 for all x in C n. Formally, A n × n {\display

2021 Australian census

The 2021 Australian census will be the eighteenth national population and housing census in Australia. The census will be conducted with effect in August 2021; the Australian Bureau of Statistics is legislated to collect and disseminate census data under the Australian Bureau of Statistics Act 1975, the Census and Statistics Act 1905. The 2016 Australian census was the previous census completed. Participation in the census is compulsory which empowers the Statistician to request persons to complete forms. Like all censuses since the 2001 Australian census, personally-identified information kept and securely held by the National Archives of Australia for the 99 years starting from the 2001 census will be made public in 2100; the final 2021 census questions are anticipated to be published in 2020. The 2021 census topics have been developed by Australian Bureau of Statistics. ABS begins consultation Publishes details of submission process Public submissions open ABS publishes consultation summary and refined recommendations Approvals by Australian Government ABS publishes final 2021 census questions Due to public expectations of interacting with government through digital services.

People will be able to complete their Census using the paper form. From the 2006 to 2016 censuses, completing it online has been an optional method with sixty-three percent in 2016. People will again be able to complete their Census online in 2021. 1911 Australian census Census in Australia

Cristo Rey Brooklyn High School

Cristo Rey Brooklyn High School is a private, Roman Catholic high school in Brooklyn, New York. The school opened in August 2008 with an initial freshman class of 43 students, operates within the Roman Catholic Diocese of Brooklyn. In 2013 it moved to more spacious quarters in the East Flatbush neighborhood of Brooklyn; the school opened on the site of the former Our Lady of Lourdes Elementary School, which closed in June 2005. Lourdes Academy opened in August 2008 and graduated its first class in 2012. In July 2011, Lourdes Academy was renamed Cristo Rey Brooklyn High School, it is part of the Cristo Rey Network of high schools. Its goal is to give children from low-income families a better chance for a college education, assistance is available if they cannot pay the $2000 annual tuition. In 2013 Cristo Rey Brooklyn moved to the premises of the former Catherine McAuley High School in East Flatbush, Brooklyn. In partnered with Boys Hope Girls Hope New York to provide boarding facilities for 50 girls in the former convent on the premises, from Sunday through Friday each week.

Sports sponsored by the school: Activities sponsored by the school include: In 2016, twelve students participated in the third annual service trip, this year to Nicaragua. Raising funds for the trip through various fundraisers and through the organization Courts for Kids. Kearney, G. R. More Than a Dream: The Cristo Rey Story: How One School's Vision Is Changing the World. Chicago, Ill: Loyola Press, 2008. ISBN 978-0-82-942576-5 School website Cristo Rey Network Fr. John P. Foley honored with Presidential Citizen's Medal 60 minutes Cristo Rey Featured in WashPost column by George Will Boston Globe - With sense of purpose, students cut class for a day Bill & Melinda Gates Foundation - Success of Innovative Urban Catholic School Sparks Major Investment

Necrotizing vasculitis

Necrotizing vasculitis called Systemic necrotizing vasculitus is a category of vasculitis, comprising vasculitides that present with necrosis. Examples include giant cell arteritis, microscopic polyangiitis, granulomatosis with polyangiitis. ICD-10 uses the variant "necrotizing vasculopathy". ICD-9, while classifying these conditions together, doesn't use a dedicated phrase, instead calling them "polyarteritis nodosa and allied conditions"; when using the influential classification known as the "Chapel Hill Consensus Conference", the terms "systemic vasculitis" or "primary systemic vasculitides" are used. Although the word "necrotizing" is omitted, the conditions described are the same. Giant-cell arteritis and Takayasu's arteritis have much in common, but affect patients of different ages, with Takayasu's arteritis affecting younger people, giant-cell arteritis having a age of onset. Aortitis can be considered a large-vessel disease. Takayasu arteritis. Affects the aorta and its main branches.

At least 3 out of 6 criteria yields sensitivity and specificity of 90.5 and 97.8%: onset < 40 years affects young and middle -aged women claudication of extremities decreased pulsation of one or both brachial arteries at least 10 mmHg systolic difference in both arms bruit over one or both carotid arteries or abdominal aorta arteriographic narrowing of aorta, its primary branches, or large arteries in upper or lower extremities Ocular manifestation visual loss or field defects Retinal hemorrhages Neurological abnormalities Treatment: steroidsGiant cell arteritis. Chronic vasculitis of both large and medium vessels affecting cranial branches of the arteries arising from the aortic arch. At least 3 out of 5 criteria yields sensitivity and specificity of 95 and 91%: Age at onset ≥ 50 years New onset headache with localized tenderness Temporal artery tenderness or decreased pulsation Elevated ESR ≥ 50 mm/hour Westergren Temporal artery biopsy showing vasculitis with mononuclear cell infiltrate or granulomatous inflammation with multinucleated giant cells These conditions are sometimes considered together with the small vessel vasculitides.

Polyarteritis nodosa. Systemic necrotizing vasculitis and aneurysm formation affecting both medium and small arteries. If only small vessels are affected, it is called microscopic polyangiitis, although it is more associated with granulomatosis with polyangiitis than to classic PAN. At least 3 out of 10 criteria yields sensitivity and specificity of 82 and 87%: unexplained weight loss > 4 kg livedo reticularis testicular pain myalgias, weakness Abdominal pain, GI bleeding mononeuropathy or polyneuropathy new onset diastolic blood pressure > 90 mmHg elevated serum BUN or serum creatinine hepatitis B infection arteriographic abnormalities arterial biopsy showing polymorphonuclear cellsKawasaki disease. In children, it affects large and small vessels, prominently the coronary arteries. Associated with a mucocutaneous lymph node syndrome. Diagnosis requires fever lasting five days or more with at least 4 out of 5 criteria: bilateral conjunctival injection injected or fissured lips, injected pharynx, or strawberry tongue erythema of palms/soles, edema of hands/feet, periungual desquamation polymorphous rash cervical lymphadenopathy Isolated cerebral vasculitis.

Affects medium and small arteries over a diffuse CNS area, without symptomatic extracranial vessel involvement. Patients have CNS symptoms as well as cerebral vasculitis by leptomeningeal biopsy. There are several vasculitides. Granulomatosis with polyangiitis. Systemic vasculitis of medium and small arteries, including venules and arterioles. Produces granulomatous inflammation of the respiratory tracts and necrotizing, pauci-immune glomerulonephritis. Most common cause of saddle nose deformity in USA. All patients with GPA have c-ANCA, but not vice versa. Current treatment of choice is cyclophosphamide. At least 2 out of 4 criteria yields sensitivity and specificity of 88 and 92%. Nasal or oral inflammation abnormal CXR showing nodules, cavities microscopic hematuria or RBC casts vessel biopsy shows granulomatous inflammation Peak incidence: ages 40–60, males > femalesEosinophilic granulomatosis with polyangiitis. Affects medium and small vessels with vascular and extravascular granulomatosis. Classically may be generalized.

At least 4 criteria yields sensitivity and specificity of 85 and 99.7%. Asthma eosinophilia > 10% on CBC mononeuropathy or polyneuropathy migratory or transient pulmonary opacities on chest x-ray paranasal sinus abnormalities vessel biopsy showing eosinophils in extravascular areasMicroscopic polyarteritis/polyangiitis. Affects venules, or arterioles. Thought to be part of a group that includes granulomatosis with polyangiitis since both are associated with ANCA and similar extrapulmonary manifestations. Patients do not have symptomatic or histologic respiratory involvement. Hypersensitivity vasculitis. Due to a hypersensitivity reaction to a known drug. Drugs most implicated are penicillin and thiazide diuretics. There is presence of skin vaculitis with palpable purpura. Biopsy of these lesions reveal inflammation of the small vessels, termed leukocytoclastic vasculitis, most prominent in postcapillary venules. At least 3 out of 5 criteria yie

Noel Blanc

Noel Barton Blanc is a retired American voice actor. Blanc is the son of voice actor Mel Blanc, throughout Noel's childhood and early adulthood, he worked with his father on the Looney Tunes voices so that when the time came, he could take over for his father. Following his father's death, Noel voiced Elmer Fudd, The Tasmanian Devil, Porky Pig and other characters in Tiny Toon Adventures and Stewie Griffin: The Untold Story, but before it was revealed years - by Mel Blanc himself - that Noel had ghosted for him for several cartoons during his time of recovery from the 1961 car accident. Blanc appeared in the television booth during the 2002 Chevrolet Monte Carlo 400 due to a promotion with Looney Tunes at Richmond International Raceway in 2001 and 2002. Noel did his Bugs Bunny voice, he retired from voice acting in 2006. He spends much of his time residing in his father's waterside cabin at California. In 2017, Noel provided extensive, exclusive commentary concerning his famous father's career and private life for the book, Mel Blanc, the Voice of Bugs Bunny...and Me: Inside the Studio with Hollywood's Man of 1,000 Voices, published on

The writer, Chuck McKibben, was employed in 1972 by both men as Mel's personal recording engineer and studio operations manager of Mel Blanc Audiomedia, located in Beverly Hills, California. Mel was the CEO and Noel the President of this "audio boutique" that specialized in developing uniquely creative radio commercials and syndicated radio programs, he won many industry awards as a producer and director, although except for McKibben's book, little has been published about this significant phase of his career. Blanc has been married three times. Blanc married actress Martha Smith in 1977, the marriage lasted for nine years until they divorced in 1986. In 1998, Blanc married his third wife Katherine Hushaw, they were married at the Warner Bros. Studios on June 3, 1998. General Electric Carousel of Progress - Radio Personalities This Is Your Life – Himself Roger Rabbit and the Secrets of Toontown – Himself 50 Years of Bugs Bunny in 31/2 Minutes – Himself Happy Birthday, Bugs!: 50 Looney Years – Himself What's Up Doc?

A Salute to Bugs Bunny – Himself Entertaining the Troops – Himself Behind the Tunes – Himself 100 Greatest Cartoons – Himself The Chuck Woolery Show – Himself Vicki! – Himself Friz on Film – Himself Mel Blanc: The Man of a Thousand Voices – Himself King-Size Comedy: Tex Avery and the Looney Tunes Revolution – Himself I Know That Voice – Himself Tomorrowland – Radio Personalities Carousel of Progress – Radio Personalities Noel Blanc on IMDb

WÄ…chock Abbey

Wąchock Abbey is a Cistercian abbey in Wąchock, Poland. Located near the larger town of Starachowice in the Świętokrzyskie Mountains of south-eastern Poland, Wąchock is best known for the architecture of this Roman Catholic site; the abbey was founded by Cistercian monks. The Cistercians had a reputation for administering the building sites for abbeys and cathedrals, "made it a point of honour to recruit the best stonecutters." Today, the interiors of the abbey remain well-preserved, the buildings themselves "are recognized as the most precious monuments of Romanesque architecture in Poland." The abbey itself is believed to have been founded in 1179, based on the few extant records from the era. It is further presumed, on the evidence of a stone escutcheon of Gadka, Bishop of Kraków, by the entrance to the later-built church on the grounds, that he was the principal founder; the church, dedicated to the Blessed Virgin Mary and Saint Florian, was completed prior to the Tatar Invasion of 1241. This incursion, the Mongol invasions to follow, destroyed most of the monastery, the bulk of what Romanesque work stands today was rebuilt in the late 13th century.

The abbey, like many Cistercian monasteries, prospered over the next several centuries and earning income from metal mining and manufacturing operations. Another series of invasions, culminating with that of George II Rákóczi of Transylvania, left the abbey plundered and burnt; the monastery was rebuilt in 1696. It was suppressed, the church converted to a parish church, in 1819 following the Congress of Vienna, which had created the "Kingdom of Poland" five years earlier as a de facto puppet state of the Russian Empire under the Romanov Tsars. In 1951, Cistercians from Mogila were able to return to the monastery at Wąchock, in 1964 the parish once again returned to its former status as an abbey. In 1991, the Fathers opened a museum on the abbey premises