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The axoplasm is the equivalent of cytoplasm in the cell. Microtubules form in the axoplasm at the axon hillock. They are arranged along the length of the axon, in overlapping sections, and all point in the same directiontowards the axon terminals. This is noted by the positive endings of the microtubules. This overlapp...
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Nodes of Ranvier (also known as myelin sheath gaps) are short unmyelinated segments of a myelinated axon, which are found periodically interspersed between segments of the myelin sheath. Therefore, at the point of the node of Ranvier, the axon is reduced in diameter. These nodes are areas where action potentials can be...
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Most axons carry signals in the form of action potentials, which are discrete electrochemical impulses that travel rapidly along an axon, starting at the cell body and terminating at points where the axon makes synaptic contact with target cells. The defining characteristic of an action potential is that it is "all-or-...
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Extracellular recordings of action potential propagation in axons has been demonstrated in freely moving animals. While extracellular somatic action potentials have been used to study cellular activity in freely moving animals such as place cells, axonal activity in both white and gray matter can also be recorded. Extr...
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Development The development of the axon to its target, is one of the six major stages in the overall development of the nervous system. Studies done on cultured hippocampal neurons suggest that neurons initially produce multiple neurites that are equivalent, yet only one of these neurites is destined to become the axon...
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Intracellular signaling During axonal development, the activity of PI3K is increased at the tip of destined axon. Disrupting the activity of PI3K inhibits axonal development. Activation of PI3K results in the production of phosphatidylinositol (3,4,5)-trisphosphate (PtdIns) which can cause significant elongation of a n...
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Cells called guidepost cells assist in the guidance of neuronal axon growth. These cells that help axon guidance, are typically other neurons that are sometimes immature. When the axon has completed its growth at its connection to the target, the diameter of the axon can increase by up to five times, depending on the s...
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The axons of neurons in the human peripheral nervous system can be classified based on their physical features and signal conduction properties. Axons were known to have different thicknesses (from 0.1 to 20 μm) and these differences were thought to relate to the speed at which an action potential could travel along th...
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In order of degree of severity, injury to a nerve in the peripheral nervous system can be described as neurapraxia, axonotmesis, or neurotmesis. Concussion is considered a mild form of diffuse axonal injury. Axonal injury can also cause central chromatolysis. The dysfunction of axons in the nervous system is one of the...
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History German anatomist Otto Friedrich Karl Deiters is generally credited with the discovery of the axon by distinguishing it from the dendrites. Swiss Rüdolf Albert von Kölliker and German Robert Remak were the first to identify and characterize the axon initial segment. Kölliker named the axon in 1896. Louis-Antoine...
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Ataxia (from Greek α- [a negative prefix] + -τάξις [order] = "lack of order") is a neurological sign consisting of lack of voluntary coordination of muscle movements that can include gait abnormality, speech changes, and abnormalities in eye movements, that indicates dysfunction of parts of the nervous system that coor...
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People with cerebellar ataxia may initially present with poor balance, which could be demonstrated as an inability to stand on one leg or perform tandem gait. As the condition progresses, walking is characterized by a widened base and high stepping, as well as staggering and lurching from side to side. Turning is also ...
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Dysmetria is thought to be caused by a deficit in the control of interaction torques in multijoint motion. Interaction torques are created at an associated joint when the primary joint is moved. For example, if a movement required reaching to touch a target in front of the body, flexion at the shoulder would create a t...
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Inability to perform rapidly alternating movements, known as dysdiadochokinesia, occurs, and could involve rapidly switching from pronation to supination of the forearm. Movements become more irregular with increases of speed. Inability to judge distances or ranges of movement happens. This dysmetria is often seen as ...
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Sensory The term sensory ataxia is used to indicate ataxia due to loss of proprioception, the loss of sensitivity to the positions of joint and body parts. This is generally caused by dysfunction of the dorsal columns of the spinal cord, because they carry proprioceptive information up to the brain. In some cases, the...
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Focal lesions Any type of focal lesion of the central nervous system (such as stroke, brain tumor, multiple sclerosis, inflammatory [such as sarcoidosis], and "chronic lymphocytyc inflammation with pontine perivascular enhancement responsive to steroids syndrome" [CLIPPERS]) will cause the type of ataxia corresponding...
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Vitamin B12 deficiency Vitamin B12 deficiency may cause, among several neurological abnormalities, overlapping cerebellar and sensory ataxia. Neuropsychological symptoms may include sense loss, difficulty in proprioception, poor balance, loss of sensation in the feet, changes in reflexes, dementia, and psychosis, whic...
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Hereditary ataxias Ataxia may depend on hereditary disorders consisting of degeneration of the cerebellum or of the spine; most cases feature both to some extent, and therefore present with overlapping cerebellar and sensory ataxia, even though one is often more evident than the other. Hereditary disorders causing ata...
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Gluten ataxia is an autoimmune disease derived from celiac disease, which is triggered by the ingestion of gluten. Early diagnosis and treatment with a gluten-free diet can improve ataxia and prevent its progression. The effectiveness of the treatment depends on the elapsed time from the onset of the ataxia until diagn...
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Diagnosis Imaging studies – A CT scan or MRI of the brain might help determine potential causes. An MRI can sometimes show shrinkage of the cerebellum and other brain structures in people with ataxia. It may also show other treatable findings, such as a blood clot or benign tumour, that could be pressing on the cereb...
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Physical therapy requires a focus on adapting activity and facilitating motor learning for retraining specific functional motor patterns. A recent systematic review suggested that physical therapy is effective, but there is only moderate evidence to support this conclusion. The most commonly used physical therapy inter...
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There are several assessment tools available to therapists and health care professionals working with patients with ataxia. The International Cooperative Ataxia Rating Scale (ICARS) is one of the most widely used and has been proven to have very high reliability and validity. Other tools that assess motor function, bal...
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In mathematics, the absolute value or modulus of a real number , is the non-negative value without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a ...
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From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers (their absolute difference) is the distance between them. The notion of an abstract distance function i...
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Since the product of any complex number and its with the same absolute value, is always the non-negative real number the absolute value of a complex number is the square root which is therefore called the absolute square or squared modulus This generalizes the alternative definition for reals: The complex abso...
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if another function is inside the absolute value. In the first case, the derivative is always discontinuous at in the first case and where in the second case. Distance The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from t...
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A real-valued function  on a field  is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms: {| cellpadding=10 |- | |Non-negativity |- | |Positive-definiteness |- | |Multiplicativity |- | |Subadditivity or the triangle inequality |} Where 0 denotes the ad...
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Every composition algebra A has an involution x → x* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x. The real numbers , complex numbers , and quaternions are all composition algebras with norms given by definite quadratic forms. The absolu...
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Arecales is an order of flowering plants. The order has been widely named as such only for the past few decades; until then, the accepted name for the order including these plants was Principes. The order includes palms and relatives. Taxonomy The APG IV system of 2016 places Dasypogonaceae in this order, after studie...
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In chemistry, an alcohol (), is a type of organic compound that carries at least one hydroxyl () functional group bound to a saturated carbon atom. Alcohols range from the simple, like methanol and ethanol, to complex, like sugars and cholesterol. The presence of an OH group strongly modifies the properties of hydrocar...
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The works of Taddeo Alderotti (1223–1296) describe a method for concentrating alcohol involving repeated fractional distillation through a water-cooled still, by which an alcohol purity of 90% could be obtained. The medicinal properties of ethanol were studied by Arnald of Villanova (1240–1311 CE) and John of Rupesciss...
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The 1657 Lexicon Chymicum, by William Johnson glosses the word as "antimonium sive stibium." By extension, the word came to refer to any fluid obtained by distillation, including "alcohol of wine," the distilled essence of wine. Libavius in Alchymia (1594) refers to "". Johnson (1657) glosses alcohol vini as "." The wo...
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Systematic names IUPAC nomenclature is used in scientific publications, and in writings where precise identification of the substance is important. In naming simple alcohols, the name of the alkane chain loses the terminal e and adds the suffix -ol, e.g., as in "ethanol" from the alkane chain name "ethane". When necess...
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Alcohols are then classified into primary, secondary (sec-, s-), and tertiary (tert-, t-), based upon the number of carbon atoms connected to the carbon atom that bears the hydroxyl functional group. The respective numeric shorthands 1°, 2°, and 3° are sometimes used in informal settings. The primary alcohols have gene...
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Methanol and ethylene glycol are more toxic than other simple alcohols. Their metabolism is affected by the presence of ethanol, which has a higher affinity for liver alcohol dehydrogenase. In this way, methanol will be excreted intact in urine. Physical properties In general, the hydroxyl group makes alcohols polar. ...
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Ziegler and oxo processes In the Ziegler process, linear alcohols are produced from ethylene and triethylaluminium followed by oxidation and hydrolysis. An idealized synthesis of 1-octanol is shown: Al(C2H5)3 + 9 C2H4 -> Al(C8H17)3 Al(C8H17)3 + 3O + 3 H2O -> 3 HOC8H17 + Al(OH)3 The process generates a range of alcoho...
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Substitution Primary alkyl halides react with aqueous NaOH or KOH to give alcohols in nucleophilic aliphatic substitution. Secondary and especially tertiary alkyl halides will give the elimination (alkene) product instead. Grignard reagents react with carbonyl groups to give secondary and tertiary alcohols. Related rea...
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Nucleophilic substitution Tertiary alcohols react with hydrochloric acid to produce tertiary alkyl chloride. Primary and secondary alcohols are converted to the corresponding chlorides using thionyl chloride and various phosphorus chloride reagents. Primary and secondary alcohols, likewise, convert to alkyl bromides u...
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Primary alcohols () can be oxidized either to aldehydes () or to carboxylic acids (). The oxidation of secondary alcohols () normally terminates at the ketone () stage. Tertiary alcohols () are resistant to oxidation. The direct oxidation of primary alcohols to carboxylic acids normally proceeds via the corresponding ...
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An abscess is a collection of pus that has built up within the tissue of the body, usually caused by bacterial infection. Signs and symptoms of abscesses include redness, pain, warmth, and swelling. The swelling may feel fluid-filled when pressed. The area of redness often extends beyond the swelling. Carbuncles and bo...
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The main symptoms and signs of a skin abscess are redness, heat, swelling, pain, and loss of function. There may also be high temperature (fever) and chills. If superficial, abscesses may be fluctuant when palpated; this wave-like motion is caused by movement of the pus inside the abscess. An internal abscess is more ...
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Pathophysiology An abscess is a defensive reaction of the tissue to prevent the spread of infectious materials to other parts of the body. Organisms or foreign materials destroy the local cells, which results in the release of cytokines. The cytokines trigger an inflammatory response, which draws large numbers of whit...
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Antibiotics Most people who have an uncomplicated skin abscess should not use antibiotics. Antibiotics in addition to standard incision and drainage is recommended in persons with severe abscesses, many sites of infection, rapid disease progression, the presence of cellulitis, symptoms indicating bacterial illness thr...
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Packing In North America, after drainage, an abscess cavity is usually packed, often with special iodoform-treated cloth. This is done to absorb and neutralize any remaining exudate as well as to promote draining and prevent premature closure. Prolonged draining is thought to promote healing. The hypothesis is that tho...
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Epidemiology Skin abscesses are common and have become more common in recent years. Risk factors include intravenous drug use, with rates reported as high as 65% among users. In 2005, in the United States 3.2 million people went to the emergency department for an abscess. In Australia around 13,000 people were hospital...
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The almond (Prunus amygdalus, syn. Prunus dulcis) is a species of tree from the genus Prunus. Along with the peach, it is classified in the subgenus Amygdalus, distinguished from the other subgenera by corrugations on the shell (endocarp) surrounding the seed. The fruit of the almond is a drupe, consisting of an outer...
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The fragrant flowers are white to pale pink, diameter with five petals, produced singly or in pairs and appearing before the leaves in early spring. Almond trees thrive in Mediterranean climates with warm, dry summers and mild, wet winters. The optimal temperature for their growth is between and the tree buds have a ...
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The bitter almond is slightly broader and shorter than the sweet almond and contains about 50% of the fixed oil that occurs in sweet almonds. It also contains the enzyme emulsin which, in the presence of water, acts on the two soluble glucosides amygdalin and prunasin yielding glucose, cyanide and the essential oil of ...
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Selection of the sweet type from the many bitter types in the wild marked the beginning of almond domestication. The wild ancestor of the almond used to breed the domesticated species is unknown. The species Prunus fenzliana may be the most likely wild ancestor of the almond, in part because it is native to Armenia and...
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Pollination The most widely planted varieties of almond are self-incompatible; hence these trees require pollen from a tree with different genetic characters to produce seeds. Almond orchards therefore must grow mixtures of almond varieties. In addition, the pollen is transferred from flower to flower by insects; ther...
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Sustainability Almond production in California is concentrated mainly in the Central Valley, where the mild climate, rich soil, abundant sunshine and water supply make for ideal growing conditions. Due to the persistent droughts in California in the early 21st century, it became more difficult to raise almonds in a su...
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The almond cultivar 'Marcona' is recognisably different from other almonds and is marketed by name. The kernel is short, round, relatively sweet, and delicate in texture. Its origin is unknown and has been grown in Spain for a long time; the tree is very productive, and the shell of the nut is hard. Toxicity Bitter ...
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Mandatory pasteurisation in California After tracing cases of salmonellosis to almonds, the USDA approved a proposal by the Almond Board of California to pasteurise almonds sold to the public. After publishing the rule in March 2007, the almond pasteurisation program became mandatory for California companies effectiv...
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Almonds are 4% water, 22% carbohydrates, 21% protein, and 50% fat. In a reference amount, almonds supply of food energy. The almond is a nutritionally dense food, providing a rich source (20% or more of the Daily Value, DV) of the B vitamins riboflavin and niacin, vitamin E, and the essential minerals calcium, copper...
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The young, developing fruit of the almond tree can be eaten whole (green almonds) when they are still green and fleshy on the outside and the inner shell has not yet hardened. The fruit is somewhat sour, but is a popular snack in parts of the Middle East, eaten dipped in salt to balance the sour taste. Also in the Midd...
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In French cuisine, alternating layers of almond and hazelnut meringue are used to make the dessert dacquoise. Pithivier is one of many almond cream-filled pastries. In Germany, Easter bread called Deutsches Osterbrot is baked with raisins and almonds. In Greece almond flour is used to make amygdalopita, a glyka tapsiou...
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In Indian cuisine, almonds are the base ingredients of pasanda-style and Mughlai curries. Badam halva is a sweet made from almonds with added colouring. Almond flakes are added to many sweets (such as sohan barfi), and are usually visible sticking to the outer surface. Almonds form the base of various drinks which are ...
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Milk Almonds can be processed into a milk substitute called almond milk; the nut's soft texture, mild flavour, and light colouring (when skinned) make for an efficient analog to dairy, and a soy-free choice for lactose intolerant people and vegans. Raw, blanched, and lightly toasted almonds work well for different pr...
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Almonds are a rich source of oil, with 50% of kernel dry mass as fat (whole almond nutrition table). In relation to total dry mass of the kernel, almond oil contains 32% monounsaturated oleic acid (an omega-9 fatty acid), 13% linoleic acid (a polyunsaturated omega-6 essential fatty acid), and 10% saturated fatty acid (...
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Similarly, Christian symbolism often uses almond branches as a symbol of the virgin birth of Jesus; paintings and icons often include almond-shaped haloes encircling the Christ Child and as a symbol of Mary. The word "luz", which appears in Genesis 30:37, sometimes translated as "hazel", may actually be derived from th...
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Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent developme...
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Computer science Requirements analysis – encompasses those tasks that go into determining the needs or conditions to meet for a new or altered product, taking account of the possibly conflicting requirements of the various stakeholders, such as beneficiaries or users. Competitive analysis (online algorithm) – shows h...
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The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule, added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery o...
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Statistics In statistics, the term analysis may refer to any method used for data analysis. Among the many such methods, some are: Analysis of variance (ANOVA) – a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts Boole...
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The field of intelligence employs analysts to break down and understand a wide array of questions. Intelligence agencies may use heuristics, inductive and deductive reasoning, social network analysis, dynamic network analysis, link analysis, and brainstorming to sort through problems they face. Military intelligence ma...
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Music Musical analysis – a process attempting to answer the question "How does this music work?" Musical Analysis is a study of how the composers use the notes together to compose music. Those studying music will find differences with each composer's musical analysis, which differs depending on the culture and history...
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Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting messages with a radio wave. In amplitude modulation, the amplitude (signal strength) of the wave is varied in proportion to that of the message signal, such as an audio signal. This technique contrasts ...
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In amplitude modulation, the angle term is held constant and the first term, A(t), of the equation has a functional relationship to the modulating message signal. The modulating message signal may be analog in nature, or it may be a digital signal, in which case the technique is generally called amplitude-shift keying...
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AM is also inefficient in power usage; at least two-thirds of the power is concentrated in the carrier signal. The carrier signal contains none of the original information being transmitted (voice, video, data, etc.). However its presence provides a simple means of demodulation using envelope detection, providing a fr...
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Analog telephony A simple form of amplitude modulation is the transmission of speech signals from a traditional analog telephone set using a common battery local loop. The direct current provided by the central office battery is a carrier with a frequency of 0 Hz. It is modulated by a microphone (transmitter) in the t...
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Continuous waves The first AM transmission was made by Canadian-born American researcher Reginald Fessenden on December 23, 1900 using a spark gap transmitter with a specially designed high frequency 10 kHz interrupter, over a distance of one mile (1.6 km) at Cobb Island, Maryland, US. His first transmitted words were,...
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Early technologies Early experiments in AM radio transmission, conducted by Fessenden, Valdemar Poulsen, Ernst Ruhmer, Quirino Majorana, Charles Herrold, and Lee de Forest, were hampered by the lack of a technology for amplification. The first practical continuous wave AM transmitters were based on either the huge, ex...
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Single-sideband In 1915, John Renshaw Carson formulated the first mathematical description of amplitude modulation, showing that a signal and carrier frequency combined in a nonlinear device creates a sideband on both sides of the carrier frequency. Passing the modulated signal through another nonlinear device can extr...
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A useful modulation signal m(t) is usually more complex than a single sine wave, as treated above. However, by the principle of Fourier decomposition, m(t) can be expressed as the sum of a set of sine waves of various frequencies, amplitudes, and phases. Carrying out the multiplication of 1 + m(t) with c(t) as above, ...
Amplitude modulation
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Another improvement over standard AM is obtained through reduction or suppression of the carrier component of the modulated spectrum. In figure 2 this is the spike in between the sidebands; even with full (100%) sine wave modulation, the power in the carrier component is twice that in the sidebands, yet it carries no u...
Amplitude modulation
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So if , carrier amplitude varies by 50% above (and below) its unmodulated level, as is shown in the first waveform, below. For , it varies by 100% as shown in the illustration below it. With 100% modulation the wave amplitude sometimes reaches zero, and this represents full modulation using standard AM and is often a t...
Amplitude modulation
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Modulation circuit designs may be classified as low- or high-level (depending on whether they modulate in a low-power domain—followed by amplification for transmission—or in the high-power domain of the transmitted signal). Low-level generation In modern radio systems, modulated signals are generated via digital signa...
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Plate modulation In plate modulation, the plate voltage of the RF amplifier is modulated with the audio signal. The audio power requirement is 50 percent of the RF-carrier power. Heising (constant-current) modulation RF amplifier plate voltage is fed through a choke (high-value inductor). The AM modulation tube plate ...
Amplitude modulation
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Ardipithecus is a genus of an extinct hominine that lived during the Late Miocene and Early Pliocene epochs in the Afar Depression, Ethiopia. Originally described as one of the earliest ancestors of humans after they diverged from the chimpanzees, the relation of this genus to human ancestors and whether it is a homini...
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In 1992–1993 a research team headed by Tim White discovered the first A. ramidus fossils—seventeen fragments including skull, mandible, teeth and arm bones—from the Afar Depression in the Middle Awash river valley of Ethiopia. More fragments were recovered in 1994, amounting to 45% of the total skeleton. This fossil wa...
Ardipithecus
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The less pronounced nature of the upper canine teeth in A. ramidus has been used to infer aspects of the social behavior of the species and more ancestral hominids. In particular, it has been used to suggest that the last common ancestor of hominids and African apes was characterized by relatively little aggression bet...
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On October 1, 2009, paleontologists formally announced the discovery of the relatively complete A. ramidus fossil skeleton first unearthed in 1994. The fossil is the remains of a small-brained female, nicknamed "Ardi", and includes most of the skull and teeth, as well as the pelvis, hands, and feet. It was discovered ...
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Ardipithecus kadabba is "known only from teeth and bits and pieces of skeletal bones", and is dated to approximately 5.6 million years ago. It has been described as a "probable chronospecies" (i.e. ancestor) of A. ramidus. Although originally considered a subspecies of A. ramidus, in 2004 anthropologists Yohannes Haile...
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However, some later studies still argue for its classification in the human lineage. In 2014, it was reported that the hand bones of Ardipithecus, Australopithecus sediba and A. afarensis have the third metacarpal styloid process, which is absent in other apes. Unique brain organisations (such as lateral shift of the c...
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In 2015, Australian anthropologists Gary Clark and Maciej Henneberg said that Ardipithecus adults have a facial anatomy more similar to chimpanzee subadults than adults, with a less-projecting face and smaller canines (large canines in primate males are used to compete within mating hierarchies), and attributed this to...
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According to Scott Simpson, the Gona Project's physical anthropologist, the fossil evidence from the Middle Awash indicates that both A. kadabba and A. ramidus lived in "a mosaic of woodland and grasslands with lakes, swamps and springs nearby," but further research is needed to determine which habitat Ardipithecus at ...
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An assembly line, often called progressive assembly, is a manufacturing process where the unfinished product moves in a direct line from workstation to workstation, with parts added in sequence until the final product is completed. By mechanically moving parts to workstations and transferring the unfinished product fro...
Assembly line
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In an assembly line, car assembly is split between several stations, all working simultaneously. When a station is finished with a car, it passes it on to the next. By having three stations, three cars can be operated on at the same time, each at a different stage of assembly. After finishing its work on the first car...
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The Industrial Revolution led to a proliferation of manufacturing and invention. Many industries, notably textiles, firearms, clocks and watches, horse-drawn vehicles, railway locomotives, sewing machines, and bicycles, saw expeditious improvement in materials handling, machining, and assembly during the 19th century, ...
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During the early 19th century, the development of machine tools such as the screw-cutting lathe, metal planer, and milling machine, and of toolpath control via jigs and fixtures, provided the prerequisites for the modern assembly line by making interchangeable parts a practical reality. Late 19th-century steam and ele...
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At Ford Motor Company, the assembly line was introduced by William "Pa" Klann upon his return from visiting Swift & Company's slaughterhouse in Chicago and viewing what was referred to as the "disassembly line", where carcasses were butchered as they moved along a conveyor. The efficiency of one person removing the sam...
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As a result of these developments in method, Ford's cars came off the line in three-minute intervals or six feet per minute. This was much faster than previous methods, increasing production by eight to one (requiring 12.5 man-hours before, 1 hour 33 minutes after), while using less manpower. It was so successful, pain...
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The massive demand for military hardware in World War II prompted assembly-line techniques in shipbuilding and aircraft production. Thousands of Liberty ships were built making extensive use of prefabrication, enabling ship assembly to be completed in weeks or even days. After having produced fewer than 3,000 planes fo...
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Since workers have to stand in the same place for hours and repeat the same motion hundreds of times per day, repetitive stress injuries are a possible pathology of occupational safety. Industrial noise also proved dangerous. When it was not too high, workers were often prohibited from talking. Charles Piaget, a skille...
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An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial . That is, it is a value for x for which the polynomial evaluates to zero. As ...
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Examples All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer and a (non-zero) natural number , satisfies the above definition, because is the root of a non-zero polynomial, namely . Quadratic irrational numbers, irrational solutions of a quadratic polynomial with intege...
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If a polynomial with rational coefficients is multiplied through by the least common denominator, the resulting polynomial with integer coefficients has the same roots. This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients. Given an algeb...
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For any two algebraic numbers , , this follows directly from the fact that the simple extension , for being either , , or (for ) , is a linear subspace of the finite-degree field extension , and therefore has a finite degree itself, from which it follows (as shown above) that is algebraic. An alternative way of sho...
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Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomia...
Algebraic number
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Artificial intelligence (AI), in its broadest sense, is intelligence exhibited by machines, particularly computer systems. It is a field of research in computer science that develops and studies methods and software that enable machines to perceive their environment and use learning and intelligence to take actions tha...
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