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Human blood is typical of that of mammals, although the precise details concerning cell numbers, size, protein structure, and so on, vary somewhat between species. In non-mammalian vertebrates, however, there are some key differences: Red blood cells of non-mammalian vertebrates are flattened and ovoid in form, and re...
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Cell production and degradation In vertebrates, the various cells of blood are made in the bone marrow in a process called hematopoiesis, which includes erythropoiesis, the production of red blood cells; and myelopoiesis, the production of white blood cells and platelets. During childhood, almost every human bone produ...
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Under normal conditions in adult humans at rest, hemoglobin in blood leaving the lungs is about 98–99% saturated with oxygen, achieving an oxygen delivery between 950 and 1150 ml/min to the body. In a healthy adult at rest, oxygen consumption is approximately 200–250 ml/min, and deoxygenated blood returning to the lung...
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Hemoglobin, the main oxygen-carrying molecule in red blood cells, carries both oxygen and carbon dioxide. However, the CO2 bound to hemoglobin does not bind to the same site as oxygen. Instead, it combines with the N-terminal groups on the four globin chains. However, because of allosteric effects on the hemoglobin mol...
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Hydraulic functions The restriction of blood flow can also be used in specialized tissues to cause engorgement, resulting in an erection of that tissue; examples are the erectile tissue in the penis and clitoris. Another example of a hydraulic function is the jumping spider, in which blood forced into the legs under p...
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General medical Disorders of volume Injury can cause blood loss through bleeding. A healthy adult can lose almost 20% of blood volume (1 L) before the first symptom, restlessness, begins, and 40% of volume (2 L) before shock sets in. Thrombocytes are important for blood coagulation and the formation of blood clots, ...
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Anemia Insufficient red cell mass (anemia) can be the result of bleeding, blood disorders like thalassemia, or nutritional deficiencies, and may require one or more blood transfusions. Anemia can also be due to a genetic disorder in which the red blood cells do not function effectively. Anemia can be confirmed by a bl...
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Substances other than oxygen can bind to hemoglobin; in some cases, this can cause irreversible damage to the body. Carbon monoxide, for example, is extremely dangerous when carried to the blood via the lungs by inhalation, because carbon monoxide irreversibly binds to hemoglobin to form carboxyhemoglobin, so that less...
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English blood (Old English blod) derives from Germanic and has cognates with a similar range of meanings in all other Germanic languages (e.g. German Blut, Swedish blod, Gothic blōþ). There is no accepted Indo-European etymology. History Classical Greek medicine Robin Fåhræus (a Swedish physician who devised the eryt...
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Blood is given particular emphasis in the Islamic, Jewish, and Christian religions, because Leviticus 17:11 says "the life of a creature is in the blood." This phrase is part of the Levitical law forbidding the drinking of blood or eating meat with the blood still intact instead of being poured off. Mythic references ...
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At the Council of Jerusalem, the apostles prohibited certain Christians from consuming blood – this is documented in Acts 15:20 and 29. This chapter specifies a reason (especially in verses 19–21): It was to avoid offending Jews who had become Christians, because the Mosaic Law Code prohibited the practice. Christ's b...
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Another ritual involving blood involves the covering of the blood of fowl and game after slaughtering (Leviticus 17:13); the reason given by the Torah is: "Because the life of the animal is [in] its blood" (ibid 17:14). In relation to human beings, Kabbalah expounds on this verse that the animal soul of a person is in ...
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Vampires are mythical creatures that drink blood directly for sustenance, usually with a preference for human blood. Cultures all over the world have myths of this kind; for example the 'Nosferatu' legend, a human who achieves damnation and immortality by drinking the blood of others, originates from Eastern European f...
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BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming languages designed for ease of use. The original version was created by John G. Kemeny and Thomas E. Kurtz at Dartmouth College in 1963. They wanted to enable students in non-scientific fields to use computer...
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Origin John G. Kemeny was the chairman of the Dartmouth College Mathematics Department. Based largely on his reputation as an innovator in math teaching, in 1959 the college won an Alfred P. Sloan Foundation award for $500,000 to build a new department building. Thomas E. Kurtz had joined the department in 1956, and f...
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Kemeny wrote the first version of BASIC. The acronym BASIC comes from the name of an unpublished paper by Thomas Kurtz. The new language was heavily patterned on FORTRAN II; statements were one-to-a-line, numbers were used to indicate the target of loops and branches, and many of the commands were similar or identical ...
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New Hampshire recognized the accomplishment in 2019 when it erected a highway historical marker in Hanover describing the creation of "the first user-friendly programming language". Spread on time-sharing services The emergence of BASIC took place as part of a wider movement toward time-sharing systems. First concept...
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A particularly important example was HP Time-Shared BASIC, which, like the original Dartmouth system, used two computers working together to implement a time-sharing system. The first, a low-end machine in the HP 2100 series, was used to control user input and save and load their programs to tape or disk. The other, a ...
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Explosive growth: the home computer era The introduction of the first microcomputers in the mid-1970s was the start of explosive growth for BASIC. It had the advantage that it was fairly well known to the young designers and computer hobbyists who took an interest in microcomputers, many of whom had seen BASIC on min...
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Micro-Soft, by this time Microsoft, ported their interpreter for the MOS 6502, which quickly become one of the most popular microprocessors of the 8-bit era. When new microcomputers began to appear, notably the "1977 trinity" of the TRS-80, Commodore PET and Apple II, they either included a version of the MS code, or q...
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As the popularity of BASIC grew in this period, computer magazines published complete source code in BASIC for video games, utilities, and other programs. Given BASIC's straightforward nature, it was a simple matter to type in the code from the magazine and execute the program. Different magazines were published featur...
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On Unix-like systems, specialized implementations were created such as XBasic and X11-Basic. XBasic was ported to Microsoft Windows as XBLite, and cross-platform variants such as SmallBasic, yabasic, Bywater BASIC, nuBasic, MyBasic, Logic Basic, Liberty BASIC, and wxBasic emerged. FutureBASIC and Chipmunk Basic meanwhi...
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In 2013, a game written in QBasic and compiled with QB64 for modern computers entitled Black Annex was released on Steam. Blitz Basic, Dark Basic, SdlBasic, Super Game System Basic, PlayBASIC, CoolBasic, AllegroBASIC, ethosBASIC, GLBasic and Basic4GL further filled this demand, right up to the modern RCBasic, NaaLaa, A...
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Mainstream support for the final version 6.0 of the original Visual Basic ended on March 31, 2005, followed by extended support in March 2008. Owing to its persistent remaining popularity, third-party attempts to further support it exist. On February 2, 2017, Microsoft announced that development on VB.NET would no long...
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Windows command-line QBasic, a version of Microsoft QuickBASIC without the linker to make EXE files, is present in the Windows NT and DOS-Windows 95 streams of operating systems and can be obtained for more recent releases like Windows 7 which do not have them. Prior to DOS 5, the Basic interpreter was GW-Basic. Quick...
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Legacy The ubiquity of BASIC interpreters on personal computers was such that textbooks once included simple "Try It In BASIC" exercises that encouraged students to experiment with mathematical and computational concepts on classroom or home computers. Popular computer magazines of the day typically included type-in p...
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Program flow control IF ... THEN ... {ELSE} used to perform comparisons or make decisions. Early dialects only allowed a line number after the THEN, but later versions allowed any valid statement to follow. ELSE was not widely supported, especially in earlier versions. FOR ... TO ... {STEP} ... NEXT repeat a secti...
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Input and output LIST displays the full source code of the current program. PRINT displays a message on the screen or other output device. INPUT asks the user to enter the value of a variable. The statement may include a prompt message. TAB used with PRINT to set the position where the next character will be s...
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Data types and variables Minimal versions of BASIC had only integer variables and one- or two-letter variable names, which minimized requirements of limited and expensive memory (RAM). More powerful versions had floating-point arithmetic, and variables could be labelled with names six or more characters long. There we...
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Most home computers BASIC versions, such as MSX BASIC and GW-BASIC, supported simple data types, loop cycles, and arrays. The following example is written for GW-BASIC, but will work in most versions of BASIC with minimal changes: 10 INPUT "What is your name: "; U$ 20 PRINT "Hello "; U$ 30 INPUT "How many stars do you...
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REM Main program follows INPUT "What is your name: ", UserName$ PRINT "Hello "; UserName$ DO INPUT "How many stars do you want: ", NumStars CALL PrintSomeStars(NumStars) DO INPUT "Do you want more stars? ", Answer$ LOOP UNTIL Answer$ <> "" Answer$ = LEFT$(Answer$, 1) LOOP WHILE UCASE$(Answer$) = "Y...
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Standards ANSI/ISO/IEC/ECMA Standard for Minimal BASIC: ANSI X3.60-1978 "For minimal BASIC" ISO/IEC 6373:1984 "Data Processing—Programming Languages—Minimal BASIC" ECMA-55 Minimal BASIC (withdrawn, similar to ANSI X3.60-1978) ANSI/ISO/IEC/ECMA Standard for Full BASIC: ANSI X3.113-1987 "Programming Languages Full...
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In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The term is closely associated with the work of the mathematician and meteorologist Edward Norton Lorenz. He...
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The idea that the death of one butterfly could eventually have a far-reaching ripple effect on subsequent historical events made its earliest known appearance in "A Sound of Thunder", a 1952 short story by Ray Bradbury. "A Sound of Thunder" features time travel. More precisely, though, almost the exact idea and the ex...
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The phrase refers to the effect of a butterfly's wings creating tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to i...
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While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied to work he published in 1969 which took the idea a step further. Lore...
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Illustrations {|class="wikitable" width=100% |- ! colspan=3|The butterfly effect in the Lorenz attractor |- | colspan="2" style="text-align:center;" | time 0 ≤ t ≤ 30 (larger) | style="text-align:center;" | z coordinate (larger) |- | colspan="2" style="text-align:center;"| | style="text-align:center;"| |- |colspan=3 | ...
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for some positive parameter a. The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems. The simplest mathematical framewor...
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The third kind of butterfly effect with finite predictability, as discussed in, was primarily proposed based on a convergent geometric series, known as Lorenz's and Lilly's formulas. Ongoing discussions are addressing the validity of these two formulas for estimating predictability limits in. A comparison of the two k...
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A recent study refers to the two-week predictability limit, initially calculated in the 1960s with the Mintz-Arakawa model's five-day doubling time, as the "Predictability Limit Hypothesis." Inspired by Moore's Law, this term acknowledges the collaborative contributions of Lorenz, Mintz, and Arakawa under Charney's lea...
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In quantum mechanics The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics, including atoms in strong fields and the anisotropic Kepler problem. Some authors have argued that extreme (exponential) dependence on init...
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Black is a color that results from the absence or complete absorption of visible light. It is an achromatic color, without hue, like white and grey. It is often used symbolically or figuratively to represent darkness. Black and white have often been used to describe opposites such as good and evil, the Dark Ages versus...
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Etymology The word black comes from Old English blæc ("black, dark", also, "ink"), from Proto-Germanic *blakkaz ("burned"), from Proto-Indo-European *bhleg- ("to burn, gleam, shine, flash"), from base *bhel- ("to shine"), related to Old Saxon blak ("ink"), Old High German blach ("black"), Old Norse blakkr ("dark"), Dut...
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For the ancient Egyptians, black had positive associations; being the color of fertility and the rich black soil flooded by the Nile. It was the color of Anubis, the god of the underworld, who took the form of a black jackal, and offered protection against evil to the dead. To ancient Greeks, black represented the unde...
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The German and Scandinavian peoples worshipped their own goddess of the night, Nótt, who crossed the sky in a chariot drawn by a black horse. They also feared Hel, the goddess of the kingdom of the dead, whose skin was black on one side and red on the other. They also held sacred the raven. They believed that Odin, the...
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14th and 15th centuries In the early Middle Ages, princes, nobles and the wealthy usually wore bright colors, particularly scarlet cloaks from Italy. Black was rarely part of the wardrobe of a noble family. The one exception was the fur of the sable. This glossy black fur, from an animal of the marten family, was the f...
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While black was the color worn by the Catholic rulers of Europe, it was also the emblematic color of the Protestant Reformation in Europe and the Puritans in England and America. John Calvin, Philip Melanchthon and other Protestant theologians denounced the richly colored and decorated interiors of Roman Catholic churc...
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18th and 19th centuries In the 18th century, during the European Age of Enlightenment, black receded as a fashion color. Paris became the fashion capital, and pastels, blues, greens, yellow and white became the colors of the nobility and upper classes. But after the French Revolution, black again became the dominant co...
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Some 19th-century French painters had a low opinion of black: "Reject black," Paul Gauguin said, "and that mix of black and white they call gray. Nothing is black, nothing is gray." But Édouard Manet used blacks for their strength and dramatic effect. Manet's portrait of painter Berthe Morisot was a study in black whic...
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In the 1950s, black came to be a symbol of individuality and intellectual and social rebellion, the color of those who did not accept established norms and values. In Paris, it was worn by Left-Bank intellectuals and performers such as Juliette Gréco, and by some members of the Beat Movement in New York and San Francis...
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In the visible spectrum, black is the result of the absorption of all light wavelengths. Black can be defined as the visual impression (or color) experienced when no visible light reaches the eye. Pigments or dyes that absorb light rather than reflect it back to the eye look black. A black pigment can, however, result ...
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Pigments The earliest pigments used by Neolithic man were charcoal, red ocher and yellow ocher. The black lines of cave art were drawn with the tips of burnt torches made of a wood with resin. Different charcoal pigments were made by burning different woods and animal products, each of which produced a different tone. ...
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Different civilizations burned different plants to produce their charcoal pigments. The Inuit of Alaska used wood charcoal mixed with the blood of seals to paint masks and wooden objects. The Polynesians burned coconuts to produce their pigment. Lamp black was used as a pigment for painting and frescoes, as a dye for ...
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A much richer and deeper black dye was eventually found made from the oak apple or "gall-nut". The gall-nut is a small round tumor which grows on oak and other varieties of trees. They range in size from 2–5 cm, and are caused by chemicals injected by the larva of certain kinds of gall wasp in the family Cynipidae. Th...
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India ink (or "Indian ink" in British English) is a black ink once widely used for writing and printing and now more commonly used for drawing, especially when inking comic books and comic strips. The technique of making it probably came from China. India ink has been in use in India since at least the 4th century BC, ...
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Astronomy A black hole is a region of spacetime where gravity prevents anything, including light, from escaping. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined boundary called an event horizon ...
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The current accepted answer is that, although the universe may be infinitely large, it is not infinitely old. It is thought to be about 13.8 billion years old, so we can only see objects as far away as the distance light can travel in 13.8 billion years. Light from stars farther away has not reached Earth, and cannot c...
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In Japan black can also symbolize experience, as opposed to white, which symbolizes naiveté. The black belt in martial arts symbolizes experience, while a white belt is worn by novices. Japanese men traditionally wear a black kimono with some white decoration on their wedding day. Black is associated with depth in Ind...
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Black was also adopted by Adolf Hitler and the Nazis in Germany. Red, white and black were the colors of the flag of the German Empire from 1870 to 1918. In Mein Kampf, Hitler explained that they were "revered colors expressive of our homage to the glorious past." Hitler also wrote that "the new flag ... should prove e...
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Patriotic resistance The Lützow Free Corps, composed of volunteer German students and academics fighting against Napoleon in 1813, could not afford to make special uniforms and therefore adopted black, as the only color that could be used to dye their civilian clothing without the original color showing. In 1815 the ...
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In Christianity, the devil is often called the "prince of darkness". The term was used in John Milton's poem Paradise Lost, published in 1667, referring to Satan, who is viewed as the embodiment of evil. It is an English translation of the Latin phrase princeps tenebrarum, which occurs in the Acts of Pilate, written in...
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Sports The national rugby union team of New Zealand is called the All Blacks, in reference to their black outfits, and the color is also shared by other New Zealand national teams such as the Black Caps (cricket) and the Kiwis (rugby league). Association football (soccer) referees traditionally wear all-black uniform...
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In general, the Negro race of African origin is called "Black", while the Caucasian race of European origin is called "White". In the United States, "Black Friday" (the day after Thanksgiving Day, the fourth Thursday in November) is traditionally the busiest shopping day of the year. Many Americans are on holiday beca...
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To blackball someone is to block their entry into a club or some such institution. In the traditional English gentlemen's club, members vote on the admission of a candidate by secretly placing a white or black ball in a hat. If upon the completion of voting, there was even one black ball amongst the white, the candidat...
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Associations and symbolism In the West, black is commonly associated with mourning and bereavement, and usually worn at funerals and memorial services. In some traditional societies, for example in Greece and Italy, some widows wear black for the rest of their lives. In contrast, across much of Africa and parts of Asi...
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Black formal attire is still worn at many solemn occasions or ceremonies, from graduations to formal balls. Graduation gowns are copied from the gowns worn by university professors in the Middle Ages, which in turn were copied from the robes worn by judges and priests, who often taught at the early universities. The m...
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Black and white have often been used to describe opposites, particularly light and darkness and good and evil. In Medieval literature, the white knight usually represented virtue, the black knight something mysterious and sinister. In American westerns, the hero often wore a white hat, the villain a black hat. In phil...
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Black is the color most commonly associated with elegance in Europe and the United States. Black first became a fashionable color for men in Europe in the 17th century, in the courts of Italy and Spain. In the 19th century, it was the fashion for men both in business and for evening wear. For women's fashion, the defin...
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A bone is a rigid organ that constitutes part of the skeleton in most vertebrate animals. Bones protect the various other organs of the body, produce red and white blood cells, store minerals, provide structure and support for the body, and enable mobility. Bones come in a variety of shapes and sizes and have complex i...
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Bone matrix is 90 to 95% composed of elastic collagen fibers, also known as ossein, and the remainder is ground substance. The elasticity of collagen improves fracture resistance. The matrix is hardened by the binding of inorganic mineral salt, calcium phosphate, in a chemical arrangement known as bone mineral, a form ...
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Cancellous bone or spongy bone, also known as trabecular bone, is the internal tissue of the skeletal bone and is an open cell porous network that follows the material properties of biofoams. Cancellous bone has a higher surface-area-to-volume ratio than cortical bone and it is less dense. This makes it weaker and more...
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Vascular supply Bone receives about 10% of cardiac output. Blood enters the endosteum, flows through the marrow, and exits through small vessels in the cortex. In humans, blood oxygen tension in bone marrow is about 6.6%, compared to about 12% in arterial blood, and 5% in venous and capillary blood. Cells Bone is met...
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Osteocyte Osteocytes are cells of mesenchymal origin and originate from osteoblasts that have migrated into and become trapped and surrounded by a bone matrix that they themselves produced. The spaces the cell body of osteocytes occupy within the mineralized collagen type I matrix are known as lacunae, while the osteoc...
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Bones consist of living cells (osteoblasts and osteocytes) embedded in a mineralized organic matrix. The primary inorganic component of human bone is hydroxyapatite, the dominant bone mineral, having the nominal composition of Ca10(PO4)6(OH)2. The organic components of this matrix consist mainly of type I collagen—"org...
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Woven bone is produced when osteoblasts produce osteoid rapidly, which occurs initially in all fetal bones, but is later replaced by more resilient lamellar bone. In adults, woven bone is created after fractures or in Paget's disease. Woven bone is weaker, with a smaller number of randomly oriented collagen fibers, but...
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In order to mineralise the bone, the osteoblasts secrete alkaline phosphatase, some of which is carried by vesicles. This cleaves the inhibitory pyrophosphate and simultaneously generates free phosphate ions for mineralization, acting as the foci for calcium and phosphate deposition. Vesicles may initiate some of the e...
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Five types of bones are found in the human body: long, short, flat, irregular, and sesamoid. Long bones are characterized by a shaft, the diaphysis, that is much longer than its width; and by an epiphysis, a rounded head at each end of the shaft. They are made up mostly of compact bone, with lesser amounts of marrow, ...
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In the study of anatomy, anatomists use a number of anatomical terms to describe the appearance, shape and function of bones. Other anatomical terms are also used to describe the location of bones. Like other anatomical terms, many of these derive from Latin and Greek. Some anatomists still use Latin to refer to bones....
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Endochondral ossification begins with points in the cartilage called "primary ossification centers". They mostly appear during fetal development, though a few short bones begin their primary ossification after birth. They are responsible for the formation of the diaphyses of long bones, short bones and certain parts of...
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The following steps are followed in the conversion of cartilage to bone: Zone of reserve cartilage. This region, farthest from the marrow cavity, consists of typical hyaline cartilage that as yet shows no sign of transforming into bone. Zone of cell proliferation. A little closer to the marrow cavity, chondrocytes mu...
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Bone development in youth is extremely important in preventing future complications of the skeletal system. Regular exercise during childhood and adolescence can help improve bone architecture, making bones more resilient and less prone to fractures in adulthood. Physical activity, specifically resistance training, sti...
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Functions Bones have a variety of functions: Mechanical Bones serve a variety of mechanical functions. Together the bones in the body form the skeleton. They provide a frame to keep the body supported, and an attachment point for skeletal muscles, tendons, ligaments and joints, which function together to generate an...
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Determined by the species, age, and the type of bone, bone cells make up to 15 percent of the bone. Growth factor storage—mineralized bone matrix stores important growth factors such as insulin-like growth factors, transforming growth factor, bone morphogenetic proteins and others. Fat storage – marrow adipose tissue ...
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The issue of having fragile bones during our childhood leads to an increase in certain disorders and conditions such as juvenile osteoporosis, though it is less common to see, the necessity for a healthy routine especially when it comes to bone development is essential in our youth. Children that naturally have lower b...
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Ultimately, the window that youth have for accruing and building resilient bones is very minimal. Being able to consistently meet calcium needs while also engaging in weight-bearing exercise is essential for building a strong initial bone foundation at which to build upon. Being able to reach our daily value of 1300mg ...
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Osteoblasts can also be stimulated to increase bone mass through increased secretion of osteoid and by inhibiting the ability of osteoclasts to break down osseous tissue. Increased secretion of osteoid is stimulated by the secretion of growth hormone by the pituitary, thyroid hormone and the sex hormones (estrogens and...
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A number of diseases can affect bone, including arthritis, fractures, infections, osteoporosis and tumors. Conditions relating to bone can be managed by a variety of doctors, including rheumatologists for joints, and orthopedic surgeons, who may conduct surgery to fix broken bones. Other doctors, such as rehabilitation...
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Tumor that can affect bone in several ways. Examples of benign bone tumors include osteoma, osteoid osteoma, osteochondroma, osteoblastoma, enchondroma, giant-cell tumor of bone, and aneurysmal bone cyst. Cancer Cancer can arise in bone tissue, and bones are also a common site for other cancers to spread (metastasise...
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Children makeup approximately 85% of Type 1 Diabetes cases and in America there was an average 22% rise in cases over the first 24 months of the Covid-19 Pandemic. With the increase of developing some form of diabetes across all ranges continually growing the health impacts on bone development and bone health in these ...
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Osteoporosis is a disease of bone where there is reduced bone mineral density, increasing the likelihood of fractures. Osteoporosis is defined in women by the World Health Organization as a bone mineral density of 2.5 standard deviations below peak bone mass, relative to the age and sex-matched average. This density is...
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Bone health is vastly important all throughout life due to a number of reasons, some of those being, without strong healthy bones we are more at risk for different chronic diseases, and fractures as well as day to day function being more difficult with poor bone health. Developing strong bones as a child is one of the ...
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A bird's beak is primarily made of bone as projections of the mandibles which are covered in keratin. Some bones, primarily formed separately in subcutaneous tissues, include headgears (such as bony core of horns, antlers, ossicones), osteoderm, and os penis/os clitoris. A deer's antlers are composed of bone which is ...
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Bone glue can be made by prolonged boiling of ground or cracked bones, followed by filtering and evaporation to thicken the resulting fluid. Historically once important, bone glue and other animal glues today have only a few specialized uses, such as in antiques restoration. Essentially the same process, with further r...
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Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a nonempty compact convex set to itself, there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interv...
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Convex compact setEvery continuous function from a nonempty convex compact subset K of a Euclidean space to K itself has a fixed point. An even more general form is better known under a different name: Schauder fixed point theoremEvery continuous function from a nonempty convex compact subset K of a Banach space to K...
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The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function , which is a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point. The analogous example works for the n-dimensional sphere (...
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Benzoic acid () is a white (or colorless) solid organic compound with the formula , whose structure consists of a benzene ring () with a carboxyl () substituent. The benzoyl group is often abbreviated "Bz" (not to be confused with "Bn," which is used for benzyl), thus benzoic acid is also denoted as BzOH, since the ben...
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Benzoic acid can be purified by recrystallization from water because of its high solubility in hot water and poor solubility in cold water. The avoidance of organic solvents for the recrystallization makes this experiment particularly safe. This process usually gives a yield of around 65%. By hydrolysis Like other ni...
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Precursor to sodium benzoate and related preservatives Benzoic acid and its salts are used as food preservatives, represented by the E numbers E210, E211, E212, and E213. Benzoic acid inhibits the growth of mold, yeast and some bacteria. It is either added directly or created from reactions with its sodium, potassium,...
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Biology and health effects Benzoic acid occurs naturally as do its esters in many plant and animal species. Appreciable amounts are found in most berries (around 0.05%). Ripe fruits of several Vaccinium species (e.g., cranberry, V. vitis macrocarpon; bilberry, V. myrtillus) contain as much as 0.03–0.13% free benzoic a...
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