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bdd7d7bc7bab8ada63f9cb75856f7905ea51a1eb | where | Λ | {\displaystyle |\lambda |} represents | where | Λ | {\displaystyle |\lambda |} represents the volume of the parallelepiped formed by the vectors { v,..., v }. this periodic function is often referred to as the sampled spectrum and can be interpreted as the analogue of the discrete-time fourier transform (dtft) in higher dimensions. if the original wavenumber... | wikipedia |
ea91dc6ecfa9c9be8014490e90159356758884f7 | the generalization of the poisson summation formula to | the generalization of the poisson summation formula to higher dimensions can be used to show that the samples, { f (x): x ∈ Λ } {\displaystyle \{f(x):x\in \lambda \}}, of the function f (⋅) {\displaystyle f(\cdot)} on the lattice Λ {\displaystyle \lambda } are sufficient to create a periodic summation of the function f... | wikipedia |
b4f39e8713d2098b2e0ff2bd89f291220f08fcc9 | let Λ {\displaystyle \lambda } denote a lattice | let Λ {\displaystyle \lambda } denote a lattice in ℜ n {\displaystyle \re ^{n}} and Γ {\displaystyle \gamma } the corresponding reciprocal lattice. the theorem of petersen and middleton states that a function f (⋅) {\displaystyle f(\cdot)} that is wavenumber-limited to a set Ω ⊂ ℜ n {\displaystyle \omega \subset \re ^{... | wikipedia |
becb5e7ea3bf6038d535ab0677feecc57c1c5e01 | where the vectors u i {\displaystyle u_{i}} are | where the vectors u i {\displaystyle u_{i}} are chosen to satisfy ⟨ u i, v j ⟩ = δ i j {\displaystyle \langle u_{i},v_{j}\rangle =\delta _{ij}}. that is, if the vectors u i {\displaystyle u_{i}} form columns of a matrix a {\displaystyle a} and v i {\displaystyle v_{i}} the columns of a matrix b {\displaystyle b}, then ... | wikipedia |
5cd0071e1aad0f6738a71f8a82d767e5cfa9c853 | similarly, the configuration of uniformly spaced sampling points | similarly, the configuration of uniformly spaced sampling points in one-dimension can be generalized to a lattice in higher dimensions. a lattice is a collection of points Λ ⊂ ℜ n {\displaystyle \lambda \subset \re ^{n}} of the form Λ = { ∑ i = 1 n a i v i | a i ∈ z } {\displaystyle \lambda =\left\{\sum _{i=1}^{n}a_{i}... | wikipedia |
aa63fc7f300d9ffa7d341c4ab06154d5c6e39e6a | where x and ξ are n -dimensional vectors, | where x and ξ are n -dimensional vectors, and ⟨ x, ξ ⟩ {\displaystyle \langle x,\xi \rangle } is the inner product of the vectors. the function f (⋅) {\displaystyle f(\cdot)} is said to be wavenumber-limited to a set Ω {\displaystyle \omega } if the fourier transform satisfies f ^ (ξ) = 0 {\displaystyle {\hat {f}}(\xi)... | wikipedia |
cc74a055f6aa27e0829dc066f560a364f0719e73 | the concept of a bandlimited function in one | the concept of a bandlimited function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions. recall that the fourier transform of an integrable function f (⋅) {\displaystyle f(\cdot)} on n -dimensional euclidean space is defined as: | wikipedia |
7cee571675c3b829060f154cc31f4ef049982dd4 | as with the nyquist–shannon sampling theorem, this theorem | as with the nyquist–shannon sampling theorem, this theorem also assumes an idealization of any real-world situation, as it only applies to functions that are sampled over an infinitude of points. perfect reconstruction is mathematically possible for the idealized model but only an approximation for real-world functions... | wikipedia |
099de64d3db91f10f047f08e43c81269c66c57f2 | in essence, the petersen–middleton theorem shows that a | in essence, the petersen–middleton theorem shows that a wavenumber-limited function can be perfectly reconstructed from its values on an infinite lattice of points, provided the lattice is fine enough. the theorem provides conditions on the lattice under which perfect reconstruction is possible. | wikipedia |
99f4ca1b71728a7cb8e76a4fc080c7b71bc4ac14 | in digital signal processing, multidimensional sampling is the | in digital signal processing, multidimensional sampling is the process of converting a function of a multidimensional variable into a discrete collection of values of the function measured on a discrete set of points. this article presents the basic result due to petersen and middleton on conditions for perfectly recon... | wikipedia |
847184193f950133f67b31d470685d491365e2ef | kompressor is a marketing name for forced induction | kompressor is a marketing name for forced induction (supercharged) mercedes-benz engines. the term is not widely used by other motor manufacturers. the first mercedes supercharger was developed in 1921 by a daimler-benz team with assistance from ferdinand porsche. mercedes became the first manufacturer to install super... | wikipedia |
6bc53141f32d4af69d48314ff0ad68d70106c0e0 | amg phased out the supercharged 5.4 l engines | amg phased out the supercharged 5.4 l engines in favor of the new m156 v8 beginning in 2006, which was paired with 7g-tronic. however, some enthusiasts were disappointed because the m156 produces less torque than the supercharged m155. | wikipedia |
a4d4dc313d509c5ac67d728232e81b92ba748e07 | the v8 s55 amg had a comparable output | the v8 s55 amg had a comparable output to the v12-powered s600 throughout their production. the s55 amg (2001–02) was outfitted with a 5.4l 354 hp (264 kw) v8 motor while the later versions (2003–06) sported the same motor, but supercharged to a rated 493 hp (386 kw). the s600 (2001–02) was outfitted with a 5.8l 362 hp... | wikipedia |
8d4d1cbd242497ea305d354356c961578a93570a | the m113k was mated to the speedshift 5-speed | the m113k was mated to the speedshift 5-speed automatic transmission, which has a torque capacity of 796 lb⋅ft (1,079 n⋅m), as the newer 7g-tronic introduced in 2003 is limited to 542 lb⋅ft (735 n⋅m), not enough to handle the torque from the supercharged v8. | wikipedia |
dac42354eaa2715ae23867118c920e7d439a1d28 | the m113 kompressor, also nicknamed m113k, is supercharged | the m113 kompressor, also nicknamed m113k, is supercharged variant of the m113 v8. it is a 5.4 l v8 engine with a lysholm type twin screw supercharger and water-to-air intercooler. the supercharger was developed in conjunction with ihi and features teflon -coated rotors, is found in the rest of the amg 55 models which ... | wikipedia |
5667d2c28435b42fe9dc71a2103a71dd76b4ede5 | the c32 had a smaller engine than its | the c32 had a smaller engine than its predecessors, the c36 amg with the m104 3.6l i6 engine, and the c43 amg powered by the m113 4.3l v8 engine. nonetheless, the c32 amg was capable of runs to 60 in 4.5 seconds, when reviewed by autocar on june 27, 2001, beating the bmw m3 and the porsche 911. | wikipedia |
1cbab2a5c1292ad027bd050fb89bb1dd7986df97 | while rival bmw m developed the smg-ii automated | while rival bmw m developed the smg-ii automated manual for the bmw m3, the c32 and slk32 have a 5-speed automatic transmission's "speedshift" system which now has quicker response (up to 35 percent) to accelerator and shift selector movements. | wikipedia |
02effe38ff7e12894af354c961bb3c6b689cb51e | powertrain consists of amg speedshift black series 5-speed | powertrain consists of amg speedshift black series 5-speed automatic transmission mated to an amg 3.2 liter v6 kompressor engine with an output of 260 kw/349 hp and 332 lb⋅ft (450 n⋅m) @ 4,400 rpm. the engine is a special version of the 3.2 l (3199 cc) m112 e32, fitted with a helical twin-screw supercharger and water-t... | wikipedia |
e7b7100e6eb161db23d6625b3455b5f345c83a25 | initial engine options comprised the c 180 (139 | initial engine options comprised the c 180 (139 ps), c220 (143 ps), c 200 kompressor, and c 230 kompressor. in 2003, mercedes-benz added the c 180 kompressor, followed by the c 200 compressor in 2003-2007 (187 hp), and finally the c 160 kompressor in 2005. the c 230 sportcoupé was powered by a 2.3-litre supercharged, f... | wikipedia |
6893483f4493753f63305dd062ad8d2729657dd6 | in 2002 for the 2003 model year, a | in 2002 for the 2003 model year, a new family of supercharged four cylinder engines, dubbed m271, debuted for the entire c-class range. all of them used the same 1.8-litre engine, with different designations according to horsepower levels, including a version powered by natural gas. the c 230 kompressor variant sported... | wikipedia |
a514d940cc2b9ef235ca572e14473a80419dcfbf | the first mercedes supercharger was developed in 1921 | the first mercedes supercharger was developed in 1921 by a daimler-benz team with assistance from ferdinand porsche. mercedes became the first manufacturer to install superchargers on some production models. the designation "k" on mercedes usually means "kurz", or short, but can mean "kompressor". | wikipedia |
840c90d8bbd38a6b0f8d1248a3357d5029869b73 | lund was president of the british phycological society | lund was president of the british phycological society in 1957. he was elected a fellow of the royal society (frs) in 1963 and made commander of the most excellent order of the british empire (cbe) in 1975. in 1975 he and hilda mabel canter-lund received the prescott award from the phycological society of america "for ... | wikipedia |
3d724fbd3e069c44c1ac7f81b91bdfb7e0203bbd | in 1944 he joined the freshwater biological association | in 1944 he joined the freshwater biological association (fba) as scientific officer for algology, moving to windermere with the association's fritsch collection of freshwater algae in 1954. he retired in 1978, was appointed deputy chief scientific officer and then honorary advisor to the fba. in 1991 he was elected an ... | wikipedia |
dabdc6b800cb18328d99cb0708a1a909b80ad292 | he worked for a time as a lecturer | he worked for a time as a lecturer at the university of sheffield, and then joined the west midlands forensic science laboratory as staff biologist. while there he worked on the case of the unsolved murder victim who has become known as wych elm bella. | wikipedia |
245c01835ab7cc3fd235ceff5fdd19f2c635bf8b | lund was born in 1912 and was educated | lund was born in 1912 and was educated at sedbergh school. he studied for his bachelor of science and master of science degrees at the university of manchester, before moving to the university of london in 1935. he was awarded his phd in 1939 and his dsc in 1951. in 1949 he married hilda mabel canter and they had two c... | wikipedia |
f0bf4f4695da34e67a498d6c60b13bc6aadea050 | nicolas fuss was married to albertine benedikte philippine | nicolas fuss was married to albertine benedikte philippine luise euler (1766-1822). albertine euler was the daughter of leonhard euler's eldest son johann albrecht euler (1734-1800) and his wife anna sophie charlotte hagemeister. pauline fuss, a daughter of nicolas and albertine, married russian chemist genrikh struve.... | wikipedia |
7d1c1194591ecf33f80c155c44e3212ccd87dccb | in 1797, he was elected a foreign member | in 1797, he was elected a foreign member of the royal swedish academy of sciences. from 1800 – 1826, fuss served as the permanent secretary to the imperial academy of sciences in saint petersburg. he was elected a foreign honorary member of the american academy of arts and sciences in 1812. he died in saint petersburg. | wikipedia |
a046880b557db4bb51cc0d3fed66f742c415d636 | fuss was born in basel, switzerland. he moved | fuss was born in basel, switzerland. he moved to saint petersburg to serve as a mathematical assistant to leonhard euler from 1773 – 1783, and remained there until his death. he contributed to spherical trigonometry, differential equations, the optics of microscopes and telescopes, differential geometry, and actuarial ... | wikipedia |
ca8dd0c08aed173fce150b593624bfdd7d8e9849 | jutta maria jokiranta is a finnish theologian and, | jutta maria jokiranta is a finnish theologian and, since 2018, a professor in hebrew bible and cognate studies at the university of helsinki. she is a former university lecturer of the hebrew bible at the university of helsinki and an academy research fellow of the academy of finland. her area of specialization is qumr... | wikipedia |
8e113f87178195f6d62003dff0a447fd8cfe2bae | jutta maria jokiranta (née miettinen; born 14 april | jutta maria jokiranta (née miettinen; born 14 april 1971 in salla) is a finnish theologian and, since 2018, a professor in hebrew bible and cognate studies at the university of helsinki. she is a former university lecturer of the hebrew bible at the university of helsinki and an academy research fellow of the academy o... | wikipedia |
9676aa2c21b29d4ac51df2e6607b3994b88c8e1f | a separator is a permeable membrane placed between | a separator is a permeable membrane placed between a battery's anode and cathode. the main function of a separator is to keep the two electrodes apart to prevent electrical short circuits while also allowing the transport of ionic charge carriers that are needed to close the circuit during the passage of current in an ... | wikipedia |
a80a52752be505bb54b1d034bdf279f0505b0d90 | inorganic polymer separators have also been of interest | inorganic polymer separators have also been of interest as use in lithium-ion batteries. inorganic particulate film/ poly(methyl methacrylate) (pmma) /inorganic particulate film trilayer separators are prepared by dip-coating inorganic particle layers on both sides of pmma thin films. this inorganic trilayer membrane i... | wikipedia |
2d1a23289322caeae300e1d224eba6135c1bd4b5 | ni/mh batteries of bipolar design (bipolar batteries) are | ni/mh batteries of bipolar design (bipolar batteries) are being developed because they offer some advantages for applications as storage systems for electric vehicles. this solid polymer membrane gel separator could be useful for such applications in bipolar design. in other words, this design can help to avoid short-c... | wikipedia |
9e17bd16fcd63150b3ac3d44d1c0dffc26026789 | the sealed rechargeable nickel-metal hydride battery offers significant | the sealed rechargeable nickel-metal hydride battery offers significant performance and environmental friendliness above alkaline rechargeable batteries. ni/mh, like the lithium-ion battery, provides high energy and power density with long cycle lives. this technology's greatest problem is its inherent high corrosion r... | wikipedia |
e749038f91d888a142d1ee6b7ef6abcb1233c17e | plasma technology was used to modify a polyethylene | plasma technology was used to modify a polyethylene membrane for enhanced adhesion, wettability and printability. these are usually performed by modifying the membrane on only its outermost several molecular levels. this allows the surface to behave differently without modifying the properties of the remainder. the sur... | wikipedia |
cfa76b0fd464d46bd1ced887d3fba30ede5a9cd5 | in addition to polymer separators, there are several | in addition to polymer separators, there are several other types of separators. there are nonwovens, which consist of a manufactured sheet, web, or mat of directionally or randomly oriented fibers. supported liquid membranes, which consist of a solid and liquid phase contained within a microporous separator. additional... | wikipedia |
9172a6c4c4dcfeed6f5a6b275c7fbfbd3b0e7ac3 | separators are also subjected to numerous stresses during | separators are also subjected to numerous stresses during battery assembly and battery usage. common stresses include tensile stresses from dry/wet processes and compressive stresses from the volumetric expansion of electrodes and required forces to ensure sufficient contact between components. dendritic lithium growth... | wikipedia |
dd83401a578dea51c3aabfbbc2b0320bff91fe14 | polymer separators, similar to battery separators in general, | polymer separators, similar to battery separators in general, act as a separator of the anode and cathode in the li-ion battery while also enabling the movement of ions through the cell. additionally, many of the polymer separators, typically multilayer polymer separators, can act as “shutdown separators”, which are ab... | wikipedia |
ef0ad3e5f591cdd00556763ffc1089b1578f8f31 | many structural defects can form in polymer separators | many structural defects can form in polymer separators due to temperature changes. these structural defects can result in a thicker separators. furthermore, there can be intrinsic defects in the polymers themselves, such as polyethylene often begins to deteriorate during the stages of polymerization, transportation, an... | wikipedia |
806dfbdfc7dc4e09a9f296b647fa2d13aef7b4eb | specific types of polymers are ideal for the | specific types of polymers are ideal for the different types of synthesis. most polymers currently used in battery separators are polyolefin based materials with semi-crystalline structure. among them, polyethylene, polypropylene, pvc, and their blends such as polyethylene-polypropylene are widely used. recently, graft... | wikipedia |
2a329ebe82bf5a37b5d2bcbcd0693806415d599e | the wet process is suitable for both crystalline | the wet process is suitable for both crystalline and amorphous polymers. wet process separators often use ultrahigh-molecular-weight polyethylene. the use of these polymers enables the batteries with favorable mechanical properties, while shutting it down when it becomes too hot. | wikipedia |
24081257b7c33331f29f205e22ab950a29515a2f | the wet process consists of mixing, heating, extruding, | the wet process consists of mixing, heating, extruding, stretching and additive removal steps. the polymer resins are first mixed with, paraffin oil, antioxidant and other additives. the mixture is heated to produce a homogenous solution. the heated solution is pushed through a sheet die to make a gel-like film. the ad... | wikipedia |
4bbf10e2e61716171656660b29270f1869dfbbfa | after processing, separators formed from the dry process | after processing, separators formed from the dry process possess a porous microstructure. while specific processing parameters (such as temperature and rolling speed) influence the final microstructure, generally, these separators have elongated, slit-like pores and thin fibrils that run parallel to the machine directi... | wikipedia |
01c981c95da9959af9e9ab2474bef02885c2572e | the dry process is only suitable for polymers | the dry process is only suitable for polymers with high crystallinity. these include but are not limited to: semi-crystalline polyolefins, polyoxymethylene, and isotactic poly (4-methyl-1-pentene). one can also use blends of immiscible polymers, in which at least one polymer has a crystalline structure, such as polyeth... | wikipedia |
db29968b8de932d8915f01de4f21cec17f899651 | the dry process involves extruding, annealing and stretching | the dry process involves extruding, annealing and stretching steps. the final porosity depends on the morphology of the precursor film and the specifics of each step. the extruding step is generally carried out at a temperature higher than the melting point of the polymer resin. this is because the resins are melted to... | wikipedia |
8e597f6453bfeb916036362b347c28b29e24bb86 | membranes synthesized by dry processes are more suitable | membranes synthesized by dry processes are more suitable for higher power density, given their open and uniform pore structure, while those made by wet processes are offer more charge/discharge cycles because of their tortuous and interconnected pore structure. this helps to suppress the conversion of charge carriers i... | wikipedia |
a1336d1719d299340f52c5ef6c303ef19955955f | materials include nonwoven fibers (cotton, nylon, polyesters, glass), | materials include nonwoven fibers (cotton, nylon, polyesters, glass), polymer films (polyethylene, polypropylene, poly (tetrafluoroethylene), polyvinyl chloride), ceramic and naturally occurring substances (rubber, asbestos, wood). some separators employ polymeric materials with pores of less than 20 Å, generally too s... | wikipedia |
bff573b6de94fbefe47a49687038cf529a8dd58a | initially, lithium cobalt oxide was used as the | initially, lithium cobalt oxide was used as the cathode and polyacetylene as the anode. later in 1985, it was found that using lithium cobalt oxide as the cathode and graphite as the anode produced an excellent secondary battery with enhanced stability, employing the frontier electron theory of kenichi fukui. this enab... | wikipedia |
a84a079986cfd13af78a06cc7b49858fc6d5e0a2 | unlike many forms of technology, polymer separators were | unlike many forms of technology, polymer separators were not developed specifically for batteries. they were instead spin-offs of existing technologies, which is why most are not optimized for the systems they are used in. even though this may seem unfavorable, most polymer separators can be mass-produced at a low cost... | wikipedia |
5f85290d1bea373a5f1e5dd4e18e624143ff10e6 | separators are critical components in liquid electrolyte batteries. | separators are critical components in liquid electrolyte batteries. a separator generally consists of a polymeric membrane forming a microporous layer. it must be chemically and electrochemically stable with regard to the electrolyte and electrode materials and mechanically strong enough to withstand the high tension d... | wikipedia |
f2eb025107ce6a8a2be07484f6b9c871383b225c | a separator is a permeable membrane placed between | a separator is a permeable membrane placed between a battery's anode and cathode. the main function of a separator is to keep the two electrodes apart to prevent electrical short circuits while also allowing the transport of ionic charge carriers that are needed to close the circuit during the passage of current in an ... | wikipedia |
671599a28d39cde13732f3dd1a10e69524e8e612 | the all-ireland senior hurling championship of 1997 was | the all-ireland senior hurling championship of 1997 was the 111th staging of ireland's premier hurling competition. clare won the championship, beating tipperary 0-20 to 2-13 in the final at croke park, dublin. | wikipedia |
fb723dd393dbfbf72632df5bd1beadd681d4ce3d | quarter-finals: (2 matches) these are two lone matches | quarter-finals: (2 matches) these are two lone matches between the connacht champions and the leinster runners-up, and the ulster champions and the munster runners-up. two teams are eliminated at this stage, while the winners advance to the all-ireland semi-finals. | wikipedia |
0e00068c4ce4b2a07d1e3b63fcf689ab417c9d5b | final: (1 match) the winner of the semi-final | final: (1 match) the winner of the semi-final joins another ulster team to contest this game. one team is eliminated at this stage, while the winners advance to the all-ireland quarter-final where that play the munster runners-up. | wikipedia |
b8e0763e8b9400eba4fcf1216ac9fcb6f4ba05cd | quarter-finals: (2 matches) the winner of the preliminary | quarter-finals: (2 matches) the winner of the preliminary round game joins three other leinster teams to make up the two quarter-final pairings. two teams are eliminated at this stage, while two teams advance to the leinster semi-finals. | wikipedia |
0ea76275e1221cf9032d33693da1761390e9dcda | prior to the opening of the championship former | prior to the opening of the championship former all-ireland-winning manager Éamonn cregan gave his predictions for the upcoming championship in the irish times. as with the previous three championships the field was wide open. although they were the reigning champions, wexford were given little chance of making it two-... | wikipedia |
f7535262a183e62fc13052d1e803bedbc3708de8 | at the start of 1996 these proposals looked | at the start of 1996 these proposals looked unlikely of being introduced, however, a whistle-stop tour undertaken by the committee's secretary frank murphy and pat daly, the gaa's games development officer, had changed the position. in april 1996 the committee's proposals were accepted at the gaa's annual congress. mos... | wikipedia |
89df259f4b2f32a18717ef69fb963a446d9831d1 | in 1995 the hurling development committee began investigating | in 1995 the hurling development committee began investigating a way of improving hurling in general and revamping the championship. their proposals involved allowing the defeated munster and leinster finalists to re-enter the all-ireland championship. although the two provincial final winners would automatically qualif... | wikipedia |
a2623b34d08d1f1de9e2f646ae31a18d829e0e91 | since its inception in 1887 the championship had | since its inception in 1887 the championship had been played on a straight knock-out basis. if any team was defeated at any stage of the provincial or all-ireland competitions it meant automatic elimination. this system was deemed the fairest as the all-ireland champions would always be the team who won all of their ga... | wikipedia |
6067cbc98def749fb7e5c2eabfb229c8d5b393ec | the all-ireland senior hurling championship of 1997 (known | the all-ireland senior hurling championship of 1997 (known for sponsorship reasons as the guinness hurling championship 1997) was the 111th staging of ireland 's premier hurling competition. clare won the championship, beating tipperary 0-20 to 2-13 in the final at croke park, dublin. | wikipedia |
4328be4e4bb12c639615a71c8af2188faa32f0c9 | in mathematics, the set of positive real numbers, | in mathematics, the set of positive real numbers, r > 0 = { x ∈ r ∣ x > 0 } , is the subset of those real numbers that are greater than zero. the non-negative real numbers, r ≥ 0 = { x ∈ r ∣ x ≥ 0 } , also include zero. although the symbols r + and r + are ambiguously used for either of these, the notation r + or r + f... | wikipedia |
8658dd31416aeeafe91d33a0b507db5cde88f43a | the realms of business and science abound in | the realms of business and science abound in ratios, and any change in ratios draws attention. the study refers to hyperbolic coordinates in q. motion against the l axis indicates a change in the geometric mean x y, {\displaystyle {\sqrt {xy}},} while a change along h indicates a new hyperbolic angle. | wikipedia |
146dde875e470d0ccbf7c5a63654c1967073d462 | since r > 0 {\displaystyle \mathbb {r} _{>0}} | since r > 0 {\displaystyle \mathbb {r} _{>0}} is a group, q {\displaystyle q} is a direct product of groups. the one-parameter subgroups l and h in q profile the activity in the product, and l × h {\displaystyle l\times h} is a resolution of the types of group action. | wikipedia |
a5c29a458b00a6323a1cebd502b69d37f72bdb01 | the l ∪ h {\displaystyle l\cup h} forms | the l ∪ h {\displaystyle l\cup h} forms a trident while l ∩ h = (1, 1) {\displaystyle l\cap h=(1,1)} is the central point. it is the identity element of two one-parameter groups that intersect there: { { (e a, e a): a ∈ r }, × } on l and { { (e a, e − a): a ∈ r }, × } on h. {\displaystyle \{\left\{\left(e^{a},\ e^{a}\r... | wikipedia |
5083325eee027f5675d342c25c6ce3a505141b1a | let q = r > 0 × r | let q = r > 0 × r > 0, {\displaystyle q=\mathbb {r} _{>0}\times \mathbb {r} _{>0},} the first quadrant of the cartesian plane. the quadrant itself is divided into four parts by the line l = { (x, y): x = y } {\displaystyle l=\{(x,y):x=y\}} and the standard hyperbola h = { (x, y): x y = 1 }. {\displaystyle h=\{(x,y):xy=... | wikipedia |
ca41c7198ffdfd09bbd0962f9a9e880d77f3a35a | including 0, the set r ≥ 0 {\displaystyle | including 0, the set r ≥ 0 {\displaystyle \mathbb {r} _{\geq 0}} has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice of base giving a logarithmic unit) gives an isomorphism with the log semiring (with 0 corresponding to − ∞ {\displaystyle -\infty... | wikipedia |
f4aefa263477a85fdd963cf44ceae2aed3f42e03 | the utility of this measure is shown in | the utility of this measure is shown in its use for describing stellar magnitudes and noise levels in decibels, among other applications of the logarithmic scale. for purposes of international standards iso 80000-3, the dimensionless quantities are referred to as levels. | wikipedia |
b5279219a2d59e11e6b21ba9a6bc75f16d077bbf | if ⊆ r > 0 {\displaystyle \subseteq \mathbb | if ⊆ r > 0 {\displaystyle \subseteq \mathbb {r} _{>0}} is an interval, then μ () = log (b / a) = log b − log a {\displaystyle \mu ()=\log(b/a)=\log b-\log a} determines a measure on certain subsets of r > 0, {\displaystyle \mathbb {r} _{>0},} corresponding to the pullback of the usual lebesgue measure on the real numbe... | wikipedia |
8d69e2d79064b76fa175252f709881bcc7d40330 | among the levels of measurement the ratio scale | among the levels of measurement the ratio scale provides the finest detail. the division function takes a value of one when numerator and denominator are equal. other ratios are compared to one by logarithms, often common logarithm using base 10. the ratio scale then segments by orders of magnitude used in science and ... | wikipedia |
a7a23e9a81e918b9e67e1bb17c3ee472cbf371fb | in the study of classical groups, for every | in the study of classical groups, for every n ∈ n, {\displaystyle n\in \mathbb {n},} the determinant gives a map from n × n {\displaystyle n\times n} matrices over the reals to the real numbers: m (n, r) → r. {\displaystyle \mathrm {m} (n,\mathbb {r})\to \mathbb {r}.} restricting to invertible matrices gives a map from... | wikipedia |
7cf47adc8901be770baaaff1f95a558e9d0a97da | the ordered set (r > 0, >) {\displaystyle | the ordered set (r > 0, >) {\displaystyle \left(\mathbb {r} _{>0},>\right)} forms a total order but is not a well-ordered set. the doubly infinite geometric progression 10 n, {\displaystyle 10^{n},} where n {\displaystyle n} is an integer, lies entirely in (r > 0, >) {\displaystyle \left(\mathbb {r} _{>0},>\right)} and... | wikipedia |
8a5f741b7c2c89d5db82f6d1fbe6bcffa7485eba | r > 0 = (0, 1) ∪ { | r > 0 = (0, 1) ∪ { 1 } ∪ (1, ∞) {\displaystyle \mathbb {r} _{>0}=(0,1)\cup \{1\}\cup (1,\infty)} and the multiplicative inverse function exchanges the intervals. the functions floor, floor:, {\displaystyle \left,} which is a sequence of integers obtained from the floor function after the excess has been reciprocated. f... | wikipedia |
bb684b41638df273594c59c5a77c7fbf5bfb12be | for a given positive real number x, {\displaystyle | for a given positive real number x, {\displaystyle x,} the sequence { x n } {\displaystyle \left\{x^{n}\right\}} of its integral powers has three different fates: when x ∈ (0, 1), {\displaystyle x\in (0,1),} the limit is zero; when x = 1, {\displaystyle x=1,} the sequence is constant; and when x > 1, {\displaystyle x>1... | wikipedia |
cefc84290fb7a042fe8838ea1d976df529437bcb | the set r > 0 {\displaystyle \mathbb {r} | the set r > 0 {\displaystyle \mathbb {r} _{>0}} is closed under addition, multiplication, and division. it inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or of an additive topological semigroup. | wikipedia |
14006abbbeb539fb745ed22679c9d060dd58e83f | in a complex plane, r > 0 {\displaystyle | in a complex plane, r > 0 {\displaystyle \mathbb {r} _{>0}} is identified with the positive real axis, and is usually drawn as a horizontal ray. this ray is used as reference in the polar form of a complex number. the real positive axis corresponds to complex numbers z = | z | e i φ, {\displaystyle z=|z|\mathrm {e} ^{\... | wikipedia |
93ca67c7ee89153ad5a1934344d42e3d83161c51 | in mathematics, the set of positive real numbers, | in mathematics, the set of positive real numbers, r > 0 = { x ∈ r ∣ x > 0 }, {\displaystyle \mathbb {r} _{>0}=\left\{x\in \mathbb {r} \mid x>0\right\},} is the subset of those real numbers that are greater than zero. the non-negative real numbers, r ≥ 0 = { x ∈ r ∣ x ≥ 0 }, {\displaystyle \mathbb {r} _{\geq 0}=\left\{x... | wikipedia |
2bd6218f50b4adec85d693e8d78fd8eb8438afb7 | a rangefinder is a device used to measure | a rangefinder is a device used to measure distances to remote objects. originally optical devices used in surveying, they soon found applications in other fields, such as photography, the military, and space travel. they were especially useful for finding the range of a target, such as in naval gunnery and anti-aircraf... | wikipedia |
f8ffcb4700a1768758c27d0af4f5117efb3d6ed8 | rangefinders may be used by users of firearms | rangefinders may be used by users of firearms over long distances, to measure the distance to a target to allow for projectile drop. until the development of electronic means of measuring range during the second world war, warships used very large optical rangefinders—with a baseline of many meters—to measure range for... | wikipedia |
3280d613b10b17413d5615d281ec400c561e1def | laser rangefinders are used in golf to measure | laser rangefinders are used in golf to measure the yardage of a particular shot but also to gauge slope and wind as well. there has been debate over whether they should be allowed in tournaments. while their use is banned on the professional level, they are becoming widely used on the amateur level. | wikipedia |
44f23cb45db34169f39615149ce2be4c3046c765 | world war ii -era rangefinders worked optically with | world war ii -era rangefinders worked optically with two telescopes focused on the same target but a distance apart along a baseline. the range to the target is found by measuring the difference in bearing of the two telescopes and solving the skinny triangle. solutions can be obtained automatically, using tables or, r... | wikipedia |
38853b81909ed6d6c578f30ea1601c1a18e1c996 | in 1881, the british royal artillery adopted the | in 1881, the british royal artillery adopted the depression range finder, which had been developed by captain h.s.s. watkin for use by coastal artillery. it used the measurement of the angle of depression from the observer, sited on a high vantage point, to the waterline of the target vessel. | wikipedia |
f5e0462fb30e6978c2b1a4d6b7acb9f421a50a3a | in 1778, georg friedrich brander invented the coincidence | in 1778, georg friedrich brander invented the coincidence rangefinder. two mirrors set a distance apart horizontally in a long slim box, similar to a subtense bar but located at the measuring station, and forming two images. this rangefinder does not require a measuring rod at the target and could perhaps be considered... | wikipedia |
154b81da81db8e4342786df5bf6078374973b15b | several others have been credited with the invention | several others have been credited with the invention of the rangefinder telemeter at one time or another. the royal society of arts gave an award to w. green for its invention in 1778, even though they were made aware of watt's priority. | wikipedia |
b911ecc0947dd294b984ddcdf62b33ffbcd07f9c | the first rangefinder telemeter was invented by james | the first rangefinder telemeter was invented by james watt in 1769 and put to use in 1771 in surveying canals. watt called his instrument a micrometer, a term now used with a different meaning in engineering (the micrometer screw gauge). it consisted of two parallel hairs in the focal plane of a telescope eyepiece cros... | wikipedia |
29c4da2659e9292128ad4c0394671d9d69445a65 | a rangefinder (also rangefinding telemeter, depending on the | a rangefinder (also rangefinding telemeter, depending on the context) is a device used to measure distances to remote objects. originally optical devices used in surveying, they soon found applications in other fields, such as photography, the military, and space travel. they were especially useful for finding the rang... | wikipedia |
214a3daed65c8b321d051e08ad3a03fa6c3edc4c | representation theory is a branch of mathematics that | representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. in essence, a representation makes an abstract algebraic object more concrete by describing... | wikipedia |
3f47d37f17a5a67079850d137ad441188b03962c | has an impact in number theory via automorphic | has an impact in number theory via automorphic forms and the langlands program. there are diverse approaches to representation theory. the same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topolog... | wikipedia |
354a03fac6253fbe39571afb135aa9c6a52ebdf3 | one special case has had a significant impact | one special case has had a significant impact on representation theory, namely the representation theory of quivers. a quiver is simply a directed graph (with loops and multiple arrows allowed), but it can be made into a category (and also an algebra) by considering paths in the graph. representations of such categorie... | wikipedia |
1f7fa76f60300d1b634e3c119495682f7321af43 | since groups are categories, one can also consider | since groups are categories, one can also consider representation of other categories. the simplest generalization is to monoids, which are categories with one object. groups are monoids for which every morphism is invertible. general monoids have representations in any category. in the category of sets, these are mono... | wikipedia |
de35f535cfb8bb594303b8183bb1b84546afd99c | every group g can be viewed as a | every group g can be viewed as a category with a single object; morphisms in this category are just the elements of g. given an arbitrary category c, a representation of g in c is a functor from g to c. such a functor selects an object x in c and a group homomorphism from g to aut(x), the automorphism group of x. | wikipedia |
b2b3587b9b6b154c7df969e162e65c4f2936cf36 | this condition and the axioms for a group | this condition and the axioms for a group imply that ρ (g) is a bijection (or permutation) for all g in g. thus we may equivalently define a permutation representation to be a group homomorphism from g to the symmetric group s of x. | wikipedia |
ab04ae3102eb9c68cc5c49b8def46fb7ec8f048f | a set-theoretic representation (also known as a group | a set-theoretic representation (also known as a group action or permutation representation) of a group g on a set x is given by a function ρ from g to x, the set of functions from x to x, such that for all g, g in g and all x in x: | wikipedia |
dd28db81c1878511b7e8a0f8182be058632ca1c8 | the hopf algebras associated to groups have a | the hopf algebras associated to groups have a commutative algebra structure, and so general hopf algebras are known as quantum groups, although this term is often restricted to certain hopf algebras arising as deformations of groups or their universal enveloping algebras. the representation theory of quantum groups has... | wikipedia |
2efb0bb9aff8d0cafaa058f86f050cfe3a541cbe | hopf algebras provide a way to improve the | hopf algebras provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and lie algebras as special cases. in particular, the tensor product of two representations is a representation, as is the dual vector space. | wikipedia |
d822043fed562c8c9558f8e2aa749f2d82152257 | when considering representations of an associative algebra, one | when considering representations of an associative algebra, one can forget the underlying field, and simply regard the associative algebra as a ring, and its representations as modules. this approach is surprisingly fruitful: many results in representation theory can be interpreted as special cases of results about mod... | wikipedia |
bab5e89648fb37ad89042f32adeb211c1a1c9903 | in one sense, associative algebra representations generalize both | in one sense, associative algebra representations generalize both representations of groups and lie algebras. a representation of a group induces a representation of a corresponding group ring or group algebra, while representations of a lie algebra correspond bijectively to representations of its universal enveloping ... | wikipedia |
4cc248f02717e1589d7aabe5485477449b1699f6 | before the development of the general theory, many | before the development of the general theory, many important special cases were worked out in detail, including the hilbert modular forms and siegel modular forms. important results in the theory include the selberg trace formula and the realization by robert langlands that the riemann–roch theorem could be applied to ... | wikipedia |
7cb267b7f26d072ef9c394cb7008fd0a6c39afd5 | automorphic forms are a generalization of modular forms | automorphic forms are a generalization of modular forms to more general analytic functions, perhaps of several complex variables, with similar transformation properties. the generalization involves replacing the modular group psl (r) and a chosen congruence subgroup by a semisimple lie group g and a discrete subgroup Γ... | wikipedia |
0251f82eca9261c3f15a0f065c6d7feefd44d4ff | the representation theory of semisimple lie groups has | the representation theory of semisimple lie groups has its roots in invariant theory and the strong links between representation theory and algebraic geometry have many parallels in differential geometry, beginning with felix klein 's erlangen program and Élie cartan 's connections, which place groups and symmetry at t... | wikipedia |
868de828062cec844a53ff2bf73b71a8d4526389 | invariant theory of infinite groups is inextricably linked | invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. another subject with strong mutual influence is projective geometry, where invariant theory can be used to organize the subject, and during the 1960s, new life... | wikipedia |
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