id stringlengths 40 40 | title stringlengths 15 120 | text stringlengths 41 3.14k | source stringclasses 2
values |
|---|---|---|---|
ae91711d8b9e52bf7aad5dbdcc6cee0891945154 | gillian shirley wratt is a new zealand botanist | gillian shirley wratt is a new zealand botanist and science administrator. she was the first woman director of the new zealand antarctic programme (1992–1996) and made a member of the new zealand order of merit for services to antarctica. | wikipedia |
742dc1bac469cf86bdce5c73d29ace171b23360b | she was awarded the new zealand suffrage centennial | she was awarded the new zealand suffrage centennial medal in 1993, and was appointed a member of the new zealand order of merit, for services to the new zealand antarctic programme, in the 2004 queen's birthday honours. | wikipedia |
df4e9419de964f7dba33e4355100e49c44d6303e | she has worked in a range of scientific | she has worked in a range of scientific and environmental management roles in addition to her antarctic roles, including environment negotiator for new zealand free trade agreements, and establishment unit convenor for the crop and food research institute. notably, wratt was chief executive of the not-for-profit cawthr... | wikipedia |
8c43aee9e5f8cedb4519ba4809f5c2b8c7c9b2bd | wratt was director of the new zealand antarctic | wratt was director of the new zealand antarctic programme from 1992 to 1996 (being its first woman director) and then chief executive of the newly formed new zealand antarctic institute (antarctica new zealand) from 1996 to 2002. she has also held a number of advisory positions relating to antarctic science and operati... | wikipedia |
02add5b17a467ab2262f9dd5975b1ce37c5b0e65 | wratt was born in motueka and attended motueka | wratt was born in motueka and attended motueka high school. she gained a b.sc. degree with honours in botany (1972–1975) from the university of canterbury in christchurch, new zealand. she joined the plant physiology division of the department of scientific and industrial research (dsir) in palmerston north in 1977 and... | wikipedia |
0695b6eb4285ac6b0ae44e6d53b415ca3d0b75bd | gillian shirley wratt mnzm (born 1954) is a | gillian shirley wratt mnzm (born 1954) is a new zealand botanist and science administrator. she was the first woman director of the new zealand antarctic programme (1992–1996) and made a member of the new zealand order of merit for services to antarctica. | wikipedia |
f1d6f5e02b14f555f80563bb5e08df6c44f61a9c | in quantum chemistry, a configuration state function (csf), | in quantum chemistry, a configuration state function (csf), is a symmetry-adapted linear combination of slater determinants. a csf must not be confused with a configuration. in general, one configuration gives rise to several csfs; all have the same total quantum numbers for spin and spatial parts but differ in their i... | wikipedia |
b86f868a5f56a533074d21dea3a44be8cf46479b | similar tables can be constructed for atomic systems, | similar tables can be constructed for atomic systems, which transform according to the point group of the sphere, that is for s, p, d, f … {\displaystyle \ldots } orbitals. the number of term symbols and therefore possible couplings is significantly larger in the atomic case. | wikipedia |
c08aafd4b18206368c7a839c87145e4affe9df3a | the situation for orbitals in abelian point groups | the situation for orbitals in abelian point groups mirrors the above table. the next table shows the fifteen possible couplings for a π {\displaystyle \pi } orbital. the δ, ϕ, γ, … {\displaystyle \delta,\phi,\gamma,\ldots } orbitals also each generate fifteen possible couplings, all of which can be easily inferred from... | wikipedia |
88053a236e79d477572092c8ff063b23b49d8ea2 | basic quantum mechanics defines the possible single orbital | basic quantum mechanics defines the possible single orbital wavefunctions. in a software implementation, these can be provided either as a table or through a set of logic statements. alternatively group theory may be used to compute them. electrons in a single orbital are called equivalent electrons. they obey the same... | wikipedia |
3e64506ecdcab92a906fdee87fc8c0afa82342a4 | the lowdin projection operator formalism may be used | the lowdin projection operator formalism may be used to find the coefficients. for any given set of determinants d i {\displaystyle d_{i}} it may be possible to find several different sets of coefficients. each set corresponds to one csf. in fact this simply reflects the different internal couplings of total spin and s... | wikipedia |
bcdc10db0f44e30bda531e82bb7bf6bf6a15054a | if we then specify the overall coupling that | if we then specify the overall coupling that we wish to achieve for the configuration, we can now select only those slater determinants that have the required quantum numbers. in order to achieve the required total spin angular momentum (and in the case of atoms the total orbital angular momentum as well), each slater ... | wikipedia |
6401690a35f1458ee140b7b2c9c786543de58f20 | we can think of the set of spin | we can think of the set of spin orbitals as a set of boxes each of size one; let's call this m {\displaystyle m} boxes. we distribute the n {\displaystyle n} electrons among the m {\displaystyle m} boxes in all possible ways. each assignment corresponds to one slater determinant, d i {\displaystyle d_{i}}. there can be... | wikipedia |
7731e441856038ca3ac9eb39608bf30e22e06414 | are the one electron spin-eigenfunctions for spin-up and | are the one electron spin-eigenfunctions for spin-up and spin-down respectively. similarly, for the 1 π {\displaystyle 1\pi } orbital in a linear molecule (c ∞ v {\displaystyle c_{\infty v}} point group) we have four spin orbitals: | wikipedia |
2b82cb708e8fe628be0abe84524d56311b5b4124 | from any given configuration we can, in general, | from any given configuration we can, in general, create several csfs. csfs are therefore sometimes also called n-particle symmetry adapted basis functions. for a configuration the number of electrons is fixed; let's call this n {\displaystyle n}. when we are creating csfs from a configuration we have to work with the s... | wikipedia |
68a0a29d35ccb7f9a44c578fd2ac400059ea3cca | csfs are derived from configurations. a configuration is | csfs are derived from configurations. a configuration is just an assignment of electrons to orbitals. for example, 1 s 2 {\displaystyle 1s^{2}} and 1 π 2 {\displaystyle 1\pi ^{2}} are example of two configurations, one from atomic structure and one from molecular structure. | wikipedia |
ca5716aef4d63929cdccdecd2d3423563b8e267e | in linear molecules, l ^ 2 {\displaystyle {\hat | in linear molecules, l ^ 2 {\displaystyle {\hat {l}}^{2}} does not commute with the hamiltonian for the system and therefore csfs are not eigenstates of l ^ 2 {\displaystyle {\hat {l}}^{2}}. however, the z-projection of angular momentum is still a good quantum number and csfs are constructed to be eigenstates of l ^ z,... | wikipedia |
3e10d38c968ae08c4d5c22fc12e19d061a81b4e9 | where ψ k {\displaystyle \psi _{k}} denotes the | where ψ k {\displaystyle \psi _{k}} denotes the set of csfs. the coefficients, c k {\displaystyle c_{k}}, are found by using the expansion of Ψ {\displaystyle \psi } to compute a hamiltonian matrix. when this is diagonalized, the eigenvectors are chosen as the expansion coefficients. csfs rather than just slater determ... | wikipedia |
c702ffc8c1c97e13ac71ebc22b34ff3e09d30069 | a configuration state function (csf), is a symmetry-adapted | a configuration state function (csf), is a symmetry-adapted linear combination of slater determinants. it is constructed to have the same quantum numbers as the wavefunction, Ψ {\displaystyle \psi }, of the system being studied. in the method of configuration interaction, the wavefunction can be expressed as a linear c... | wikipedia |
f1d6f5e02b14f555f80563bb5e08df6c44f61a9c | in quantum chemistry, a configuration state function (csf), | in quantum chemistry, a configuration state function (csf), is a symmetry-adapted linear combination of slater determinants. a csf must not be confused with a configuration. in general, one configuration gives rise to several csfs; all have the same total quantum numbers for spin and spatial parts but differ in their i... | wikipedia |
b32a27ec7a72b3237bc74139552991fe8c8aa170 | ray l. birdwhistell was an american anthropologist who | ray l. birdwhistell was an american anthropologist who founded kinesics as a field of inquiry and research. birdwhistell coined the term kinesics, meaning "facial expression, gestures, posture and gait, and visible arm and body movements". he estimated that "no more than 30 to 35 percent of the social meaning of a conv... | wikipedia |
bc7b3b2b992db2ecdf6cd62286bd126cdd0315c1 | birdwhistell viewed communication as a continuous, multichannel (today, | birdwhistell viewed communication as a continuous, multichannel (today, the more common term is multimodal) process through which and in which social interaction occurs. although he is best known for inventing kinesics, his influence was much larger: he helped establish the logical underpinnings of language and social ... | wikipedia |
9321b62737c98eb111e3da5d46f17211502d82e7 | birdwhistell's first book introduction to kinesics, was published | birdwhistell's first book introduction to kinesics, was published in 1952, but as this was essentially an internal publication for the department of state, his second book, kinesics and context has been cited far more often, and, along with a brief encyclopedia article on kinesics, has had far greater influence on the ... | wikipedia |
7df1092327208f79f31a403d8f3809cfea7a37a3 | birdwhistell pointed out that "human gestures differ from | birdwhistell pointed out that "human gestures differ from those of other animals in that they are polysemic, that they can be interpreted to have many different meanings depending on the communicative context in which they are produced". and, he "resisted the idea that " body language " could be deciphered in some abso... | wikipedia |
a7eed20905956540c45e5503edd5777ba7877c52 | goffman became one of the best-known sociologists with | goffman became one of the best-known sociologists with an international reputation, and nearly all of his publications became best sellers. birdwhistell influenced lomax's development of cantometrics and choreometrics. byers was quite important in the study of visual communication. winkin went on to develop the anthrop... | wikipedia |
d33c840652fd2558c2da6866034985dfae84ed56 | through his involvement in the multidisciplinary projects at | through his involvement in the multidisciplinary projects at the foreign service institute, at the macy conferences, and most especially through the natural history of an interview project, birdwhistell helped to establish the study of nonverbal behavior as a central part of communication, as well as influencing critic... | wikipedia |
2cf8d0620001cca543c5da675b54bf42c0360669 | birdwhistell reputedly came to the attention of margaret | birdwhistell reputedly came to the attention of margaret mead and gregory bateson when he attended a showing of one of their ethnographic films (they were pioneers of the use of film as an ethnographic tool). "legend has it that birdwhistell was a younger anthropologist listening to mead and others comment on a balines... | wikipedia |
77fe8cee28107af0f0416f666ea6e40461a33cfe | from 1969 until he retired in 1988, birdwhistell | from 1969 until he retired in 1988, birdwhistell held the position of professor at the annenberg school for communication at the university of pennsylvania, where he worked closely with dell hymes and erving goffman, brought gregory bateson in as a guest speaker, and influenced a new generation of students. it was comm... | wikipedia |
f1e9e2ef27374c757dd64ab2210120c6a595f8d3 | much of the work at eppi was a | much of the work at eppi was a continuation of the natural history of an interview project, working mostly with scheflen, while brosin continued different parts of the same project from the western psychiatric institute & clinic in pennsylvania with adam kendon, william s. condon, kai erikson, harvey sarles, and occasi... | wikipedia |
a1325198a719c82c90381806146afb63ddd7127b | 3. lecture on kinesics by ray l. birdwhistell | 3. lecture on kinesics by ray l. birdwhistell at the second linguistic-kinesic conference nov. 4–7, 1964, is simply a documentary record of two lectures birdwhistell presented to a seminar group assembled for a few days to learn from his research team at eppi in 1964. seminar participants were primarily senior research... | wikipedia |
fd4e5587610a3012c4af14ebaa76718e88373f9a | 2. tdr- 009, an eighty-minute 16 mm black-and-white | 2. tdr- 009, an eighty-minute 16 mm black-and-white sound film of an english pub scene in a middle class london hotel. birdwhistell and van vlack observed behavior of listeners in relationship to speakers during the film. | wikipedia |
86dd90086de88629262c800181961866ee56731e | 1. microcultural incidents in ten zoos, an edited | 1. microcultural incidents in ten zoos, an edited version of a birdwhistell and van vlack presentation from an american anthropological association convention, compares family interactions while feeding elephants at 10 zoos based in 7 countries (england, france, italy, india, japan, hong kong, and the united states). f... | wikipedia |
d898086719c9f95fa265dc658b5c7d482a9b1c97 | birdwhistell argued strongly for the use of film | birdwhistell argued strongly for the use of film as an essential tool in the study of nonverbal behavior as a way to permit "observation and analysis of human social behavior which has hitherto been hidden from comparative analysis". together with jacques van vlack (the filmmaker), he prepared a series of films that we... | wikipedia |
97d3340ae17dd812b1bc40d09d53fca74c09ec69 | birdwhistell taught at the state university of new | birdwhistell taught at the state university of new york at buffalo from 1956 to 1959. in 1959 he was appointed senior research scientist at the eastern pennsylvania psychiatric institute, and simultaneously professor of research in anthropology at temple university in philadelphia. at eppi he managed a lab that include... | wikipedia |
d6e9dbb8b2745af5ef8a03d6b5ef75536f1e18d7 | through the 1950s he participated in multiple interdisciplinary | through the 1950s he participated in multiple interdisciplinary collaborations: at the foreign service institute of the united states department of state, where he first outlined his ideas about the study of nonverbal behavior, working with edward t. hall, henry lee smith, george l. trager, charles f. hockett; at the m... | wikipedia |
acf8e7d648e7fe1cab0f5c2d6ec0df1e198283b2 | in 1946 he took a position at the | in 1946 he took a position at the university of louisville, kentucky, where he taught for 10 years, and helped in racial integration of the university. while there he established the interdisciplinary committee on culture and communication, and organized a series of annual seminars on culture and communication, resulti... | wikipedia |
925e6da9135a21802f898c4bc7087d0a27e13228 | birdwhistell was born in cincinnati on september 29, | birdwhistell was born in cincinnati on september 29, 1918, and died october 19, 1994. he was raised and went to school in ohio. he graduated from fostoria high school in 1936, and was involved in the history club, debate team, journalism, and school plays. birdwhistell received his ba in sociology in 1940 from miami un... | wikipedia |
5735ed612841b13248d028cf29e62ee6e7fdc54b | ray l. birdwhistell (september 29, 1918 – october | ray l. birdwhistell (september 29, 1918 – october 19, 1994) was an american anthropologist who founded kinesics as a field of inquiry and research. birdwhistell coined the term kinesics, meaning "facial expression, gestures, posture and gait, and visible arm and body movements". he estimated that "no more than 30 to 35... | wikipedia |
a998e1687f2b9cdc567744722dcc4048a7563638 | mews was selected seventh overall in the 2022 | mews was selected seventh overall in the 2022 ontario hockey league (ohl) priority selection draft. during his first season in the ohl, the 2022–23 season, he scored 12 goals and 31 points in 55 games. | wikipedia |
f82f30516cec2e239195fb2744f250b86b2e2b3c | playing for the toronto jr. canadians at the | playing for the toronto jr. canadians at the u16 aaa level in the 2021–22 season, mews scored 14 goals and 64 assists for 78 points in 52 games. at the 2022 ohl cup, he recorded one goal and five points in six games. | wikipedia |
0ddc2c5b6dd46bf3d64ae707d550c59e03540c45 | switzerland has universal health care, regulated by the | switzerland has universal health care, regulated by the swiss federal law on health insurance. there are no free state-provided health services, but private health insurance is compulsory for all persons residing in switzerland. health insurance covers the costs of medical treatment and hospitalisation of the insured. ... | wikipedia |
64ef01e926bd725313c1403360bb227efbcfbe7e | healthcare costs in switzerland are 11.4% of gdp | healthcare costs in switzerland are 11.4% of gdp (2010), comparable to germany and france (11.6%) and other european countries, but significantly less than in the usa (17.6%). by 2015 the cost had risen to 11.7% of gdp -the second highest in europe. benefits paid out as a percentage of premiums were 90.4% in 2011. tota... | wikipedia |
2de0fb419b8e18158fc42bed48e1b290e4b68d04 | in 2007, the swiss federal government approved a | in 2007, the swiss federal government approved a national strategy for adoption of e-health. a central element of this strategy is a nationwide electronic health record. following the federal tradition of switzerland, it is planned that the infrastructure will be implemented in a decentralized way, i.e., using an acces... | wikipedia |
d547a19cb20b9cbb534af3bac8a951bcd4d36aa6 | the insured person has full freedom of choice | the insured person has full freedom of choice among the recognised healthcare providers competent to treat their condition (in their region) on the understanding that the costs are covered by the insurance up to the level of the official tariff. there is freedom of choice when selecting an insurance company (provided i... | wikipedia |
02fda74909aef90c47ba6c402ef72ed7bc00e9c4 | as far as the compulsory health insurance is | as far as the compulsory health insurance is concerned, the insurance companies cannot set any conditions relating to age, sex or state of health for coverage. although the level of premium can vary from one company to another, they must be identical within the same company for all insured persons of the same age group... | wikipedia |
ef0ada46da7ad61e2a8b25959f9fa2fe437f2d05 | the compulsory insurance can be supplemented by private | the compulsory insurance can be supplemented by private "complementary" insurance policies that allow for coverage of some of the treatment categories not covered by the basic insurance or to improve the standard of room and service in case of hospitalisation. this can include complementary medicine, routine dental tre... | wikipedia |
a902f747afbd4b381f42557057b70e1fa0dc5332 | insurance premiums vary from insurance company to company | insurance premiums vary from insurance company to company (health insurance funds; german: krankenkassen; french: caisses-maladie; italian: casse malati), the excess level chosen (franchise), the place of residence of the insured person and the degree of supplementary benefit coverage chosen (complementary medicine, ro... | wikipedia |
a9f332a06cf01d23bf5a53e8cfa0f6dc62468346 | since 1994, all swiss residents are required by | since 1994, all swiss residents are required by federal law to purchase basic health insurance, which covers a range of treatments detailed in the swiss federal law on health insurance (german: krankenversicherungsgesetz (kvg); french: la loi fédérale sur l’assurance-maladie (lamal); italian: legge federale sull’assicu... | wikipedia |
835dd77750e1d6139c3108b38c75f506b4d3132f | health insurance covers the costs of medical treatment | health insurance covers the costs of medical treatment and hospitalisation of the insured. however, the insured person pays part of the cost of treatment. this is done (a) by means of an annual deductible (called the franchise), which ranges from chf 300 (ppp-adjusted us$ 489) to a maximum of chf 2,500 (ppp-adjusted $4... | wikipedia |
e4025d559793f45ecb0b4eaf7065b85631bd769d | switzerland has universal health care, regulated by the | switzerland has universal health care, regulated by the swiss federal law on health insurance. there are no free state-provided health services, but private health insurance is compulsory for all persons residing in switzerland (within three months of taking up residence or being born in the country). | wikipedia |
6c8309384ce6116c569ad48e4b11a671b1bd05e0 | in sound processing, the mel-frequency cepstrum (mfc) is | in sound processing, the mel-frequency cepstrum (mfc) is a representation of the short-term power spectrum of a sound, based on a linear cosine transform of a log power spectrum on a nonlinear mel scale of frequency. mel-frequency cepstral coefficients (mfccs) are coefficients that collectively make up an mfc. they are... | wikipedia |
2de1519112664a100af28b1ebc6c6db935e4d9c7 | the mfccs are the amplitudes of the resulting | the mfccs are the amplitudes of the resulting spectrum. there can be variations on this process, for example: differences in the shape or spacing of the windows used to map the scale, or addition of dynamics features such as "delta" and "delta-delta" coefficients. the european telecommunications standards institute in ... | wikipedia |
0be11802be0722f718009acdc38effe46c86fd33 | many authors, including davis and mermelstein, have commented | many authors, including davis and mermelstein, have commented that the spectral basis functions of the cosine transform in the mfc are very similar to the principal components of the log spectra, which were applied to speech representation and recognition much earlier by pols and his colleagues. | wikipedia |
1b3786a7eacedf59e69b74803e529e114f51178e | bridle and brown used a set of 19 | bridle and brown used a set of 19 weighted spectrum-shape coefficients given by the cosine transform of the outputs of a set of nonuniformly spaced bandpass filters. the filter spacing is chosen to be logarithmic above 1 khz and the filter bandwidths are increased there as well. we will, therefore, call these the mel-b... | wikipedia |
3e0c1d9cbb3b92a48a9d92ddf515b09d4fa051a6 | mfcc values are not very robust in the | mfcc values are not very robust in the presence of additive noise, and so it is common to normalise their values in speech recognition systems to lessen the influence of noise. some researchers propose modifications to the basic mfcc algorithm to improve robustness, such as by raising the log-mel-amplitudes to a suitab... | wikipedia |
d11d5ea237f5e1ed555bf6d77f26ab37cb0d12b7 | an mfcc can be approximately inverted to audio | an mfcc can be approximately inverted to audio in four steps: (a1) inverse dct to obtain a mel log-power spectrogram, (a2) mapping to power to obtain a mel power spectrogram, (b1) rescaling to obtain short-time fourier transform magnitudes, and finally (b2) phase reconstruction and audio synthesis using griffin-lim. ea... | wikipedia |
1f6598bc638b90976d20ff2b0b4fdf487e925307 | where c i = c y (i) {\displaystyle | where c i = c y (i) {\displaystyle c_{i}=c_{y}(i)} corresponds to the i {\displaystyle i} -th mfcc coefficient, n f {\displaystyle n_{f}} is the number of triangular filters in the filter bank, s n {\displaystyle s_{n}} is the log energy output of n {\displaystyle n} -th filter coefficient, and l {\displaystyle l} is t... | wikipedia |
66178d1f3992a0cc94a144ebcd65e186be097add | c i = ∑ n = 1 n | c i = ∑ n = 1 n f s n cos (i (n − 0.5) (π n f)) {\displaystyle c_{i}=\sum _{n=1}^{n_{f}}s_{n}\cos \left(i(n-0.5)\left({\frac {\pi }{n_{f}}}\right)\right)}, i = 1, …, l {\displaystyle i=1,\dots,l}, | wikipedia |
efee68bc5be44f61e5ce5b3c3bff180394130dec | where xew(f) is the excitation function, x v | where xew(f) is the excitation function, x v (f) {\displaystyle x_{v}(f)} is the vocal tract transfer function for speech in the p t h {\displaystyle p^{th}} frame and h ′ (f) {\displaystyle h'(f)} is the equivalent transfer function that characterizes the cell phone. | wikipedia |
e93fcc920bc8cbdfd9823c70276e567ab7663913 | h (f) {\displaystyle h(f)} can be considered as | h (f) {\displaystyle h(f)} can be considered as a concatenated transfer function that produced input speech, and the recorded speech y p w (f) {\displaystyle y_{p}w(f)} can be perceived as original speech from cell phone. | wikipedia |
8ea595495da24ddfb7e64d3d0d693de11ab8e971 | this type of mobile device recognition is possible | this type of mobile device recognition is possible because the production of electronic components in a phone have tolerances, because different electronic circuit realizations do not have exact same transfer functions. the dissimilarities in the transfer function from one realization to another becomes more prominent ... | wikipedia |
ddde030255346eaf357d5af755e438bafa4991c3 | since mel-frequency bands are distributed evenly in mfcc | since mel-frequency bands are distributed evenly in mfcc and they are much similar to the voice system of a human, mfcc can efficiently be used to characterize speakers. for instance, it can be used to recognize the speaker's cell phone model details and further the details of the speaker. | wikipedia |
02a6f61dc3b5dbe3fd19455c7e127933f15067b7 | there can be variations on this process, for | there can be variations on this process, for example: differences in the shape or spacing of the windows used to map the scale, or addition of dynamics features such as "delta" and "delta-delta" (first- and second-order frame-to-frame difference) coefficients. | wikipedia |
832c8d567979e3be7326285e6a2a1dbf06710207 | mel-frequency cepstral coefficients (mfccs) are coefficients that collectively | mel-frequency cepstral coefficients (mfccs) are coefficients that collectively make up an mfc. they are derived from a type of cepstral representation of the audio clip (a nonlinear "spectrum-of-a-spectrum"). the difference between the cepstrum and the mel-frequency cepstrum is that in the mfc, the frequency bands are ... | wikipedia |
369dcc80dab6e060200293e69a71f91f3d6c3ebc | in additive number theory, fermat's theorem on sums | in additive number theory, fermat's theorem on sums of two squares states that an odd prime p can be expressed as: with x and y integers, if and only if the prime numbers for which this is true are called pythagorean primes. | wikipedia |
5bd1f4a73198e5d99dad3bc87a953018399db55d | for example, the primes 5, 13, 17, 29, | for example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: on the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. this is the easier... | wikipedia |
969647615c9005e64c0c4e360b2ff70c8505c1a7 | in 2016, a. david christopher gave a partition-theoretic | in 2016, a. david christopher gave a partition-theoretic proof by considering partitions of the odd prime n {\displaystyle n} having exactly two sizes a i (i = 1, 2) {\displaystyle a_{i}(i=1,2)}, each occurring exactly a i {\displaystyle a_{i}} times, and by showing that at least one such partition exists if n {\displa... | wikipedia |
6fa79cbda26925f8997fdba3a0e003eeb93b8874 | this proof is equivalent to a geometric or | this proof is equivalent to a geometric or "visual" proof using "windmill" figures, given by alexander spivak in 2006 and described in this mathoverflow post by moritz firsching and this youtube video by mathologer. | wikipedia |
6a6cc9bb78b1b20bc9203426bfb9a4ead359f9d1 | this proof, due to zagier, is a simplification | this proof, due to zagier, is a simplification of an earlier proof by heath-brown, which in turn was inspired by a proof of liouville. the technique of the proof is a combinatorial analogue of the topological principle that the euler characteristics of a topological space with an involution and of its fixed-point set h... | wikipedia |
8b74946fd6cb6a268934b7aeb05b0218424cf71a | which has exactly one fixed point (1, 1, | which has exactly one fixed point (1, 1, k) {\displaystyle (1,1,k)}. this proves that the cardinality of s {\displaystyle s} is odd. hence, s {\displaystyle s} has also a fixed point with respect to the obvious involution. | wikipedia |
982740d365d9f59b2700e04c11529a9672930d3e | let p = 4 k + 1 {\displaystyle | let p = 4 k + 1 {\displaystyle p=4k+1} be prime, let n {\displaystyle \mathbb {n} } denote the natural numbers (with or without zero), and consider the finite set s = { (x, y, z) ∈ n 3: x 2 + 4 y z = p } {\displaystyle s=\{(x,y,z)\in \mathbb {n} ^{3}:x^{2}+4yz=p\}} of triples of numbers.then s {\displaystyle s} has two... | wikipedia |
e4a42eedb9c09efaf14b968a2cb42024c17aaacd | the area of the fundamental parallelogram of the | the area of the fundamental parallelogram of the lattice is p {\displaystyle p}. the area of the open disk, d {\displaystyle d}, of radius 2 p {\displaystyle {\sqrt {2p}}} centered around the origin is 2 π p > 4 p {\displaystyle 2\pi p>4p}. furthermore, d {\displaystyle d} is convex and symmetrical about the origin. th... | wikipedia |
25ce7eb92b0071dd04f0d9b555e39afedae213e5 | for p {\displaystyle p} congruent to 1 {\displaystyle | for p {\displaystyle p} congruent to 1 {\displaystyle 1} mod 4 {\displaystyle 4} a prime, − 1 {\displaystyle -1} is a quadratic residue mod p {\displaystyle p} by euler's criterion. therefore, there exists an integer m {\displaystyle m} such that p {\displaystyle p} divides m 2 + 1 {\displaystyle m^{2}+1}. let i ^, j ^... | wikipedia |
f2a35858b10363f405f36bb93af1927fe1703a8f | for p {\displaystyle p} congruent to 1 {\displaystyle | ^{2}\equiv a^{2}+(am+bp)^{2}\equiv a^{2}(1+m^{2})\equiv 0{\pmod {p}}}. thus p {\displaystyle p} divides ‖ w → ‖ 2 {\displaystyle \|{\vec {w}}\|^{2}} for any w → ∈ s {\displaystyle {\vec {w}}\in s}. | wikipedia |
c7cc33bf0a6d0a8e91c12959f3765aa752f8b2ba | 2. second proof. this proof builds on lagrange's | 2. second proof. this proof builds on lagrange's result that if p = 4 n + 1 {\displaystyle p=4n+1} is a prime number, then there must be an integer m such that m 2 + 1 {\displaystyle m^{2}+1} is divisible by p (we can also see this by euler's criterion); it also uses the fact that the gaussian integers are a unique fac... | wikipedia |
437545ede396142ab7833ed0343f80f9996a6181 | kummer had already established that if f ∈ | kummer had already established that if f ∈ {1,2} is the order of the frobenius automorphism of z /(p), then the ideal (p) {\displaystyle (p)} in z would be a product of 2/ f distinct prime ideals. (in fact, kummer had established a much more general result for any extension of z obtained by adjoining a primitive m -th ... | wikipedia |
b34e51306a19b715ea68a836bd597777bfda62fe | since the automorphism fixes the elements of z | since the automorphism fixes the elements of z /(p). in the current case, p = 4 n + 1 {\displaystyle p=4n+1} for some integer n, and so in the above expression for ω, the exponent (p − 1) / 2 {\displaystyle (p-1)/2} of − 1 is even. hence the right hand side equals ω, so in this case the frobenius endomorphism of z /(p)... | wikipedia |
48c2dd8cefff95b4162dd42cd59af88f288de173 | 1. first proof. if p is an odd | 1. first proof. if p is an odd prime number, then we have i p − 1 = (− 1) p − 1 2 {\displaystyle i^{p-1}=(-1)^{\frac {p-1}{2}}} in the gaussian integers. consequently, writing a gaussian integer ω = x + iy with x,y ∈ z and applying the frobenius automorphism in z /(p), one finds | wikipedia |
1873f7a3d382c9ec60a74b4448a60acef264010c | richard dedekind gave at least two proofs of | richard dedekind gave at least two proofs of fermat's theorem on sums of two squares, both using the arithmetical properties of the gaussian integers, which are numbers of the form a + bi, where a and b are integers, and i is the square root of −1. one appears in section 27 of his exposition of ideals published in 1877... | wikipedia |
1df8a4b876cda6606664d7611541be89f8e898e1 | we claim such a square root of − | we claim such a square root of − 1 {\displaystyle -1} is given by k = ∏ k = 1 p − 1 2 k {\displaystyle k=\prod _{k=1}^{\frac {p-1}{2}}k}. firstly it follows from euclid's fundamental theorem of arithmetic that a b ≡ 0 (mod p) ⟺ a ≡ 0 (mod p) or b ≡ 0 (mod p) {\displaystyle ab\equiv 0{\pmod {p}}\iff a\equiv 0{\pmod {p}}... | wikipedia |
4d83ffcd7bf738259d58e85c5c0d440a03f45a17 | we claim such a square root of − | let l = ∏ l = p + 1 2 p − 1 l {\displaystyle l=\prod _{l={\frac {p+1}{2}}}^{p-1}l}: from what has just been observed, k l ≡ − 1 (mod p) {\displaystyle kl\equiv -1{\pmod {p}}}. but by definition, since each term in k {\displaystyle k} may be paired with its negative in l {\displaystyle l}, l = (− 1) p − 1 2 k {\display... | wikipedia |
444f57a7010243fc497750d63bf2cf91dd4a04b7 | where the first coefficient a = p {\displaystyle | where the first coefficient a = p {\displaystyle p} was chosen so that the form represents p {\displaystyle p} by setting x = 1, and y = 0, the coefficient b = 2 m is an arbitrary even number (as it must be, to get an even discriminant), and finally c = m 2 + 1 p {\displaystyle c={\frac {m^{2}+1}{p}}} is chosen so that... | wikipedia |
58022ce0c88e15be2f224b37a3a484dedb6c4a03 | lagrange proved that all positive definite forms of | lagrange proved that all positive definite forms of discriminant −4 are equivalent. thus, to prove fermat's theorem it is enough to find any positive definite form of discriminant −4 that represents p {\displaystyle p}. for example, one can use a form | wikipedia |
0d2f4440b4805cd33b0177bcdffb93b052ed76c8 | with α δ − β γ = ± | with α δ − β γ = ± 1 {\displaystyle \alpha \delta -\beta \gamma =\pm 1} such that, when substituted into the first form, yield the second. equivalent forms are readily seen to have the same discriminant, and hence also the same parity for the middle coefficient b {\displaystyle b}, which coincides with the parity of th... | wikipedia |
b22d0107c98f0c7f3978c52102c4e27f0a741a04 | two forms a x 2 + b x | two forms a x 2 + b x y + c y 2 {\displaystyle ax^{2}+bxy+cy^{2}} and a ′ x ′ 2 + b ′ x ′ y ′ + c ′ y ′ 2 {\displaystyle a'x'^{2}+b'x'y'+c'y'^{2}} are equivalent if and only if there exist substitutions with integer coefficients | wikipedia |
44e98c9a7a1091049217bc2bf9592008418052d8 | the discriminant of the quadratic form is defined | the discriminant of the quadratic form is defined to be b 2 − 4 a c {\displaystyle b^{2}-4ac}. the discriminant of x 2 + y 2 {\displaystyle x^{2}+y^{2}} is then equal to − 4 {\displaystyle -4}. | wikipedia |
54918f94443bb3effd51aa40e15316acffeabedd | an (integral binary) quadratic form is an expression | an (integral binary) quadratic form is an expression of the form a x 2 + b x y + c y 2 {\displaystyle ax^{2}+bxy+cy^{2}} with a, b, c {\displaystyle a,b,c} integers. a number n {\displaystyle n} is said to be represented by the form if there exist integers x, y {\displaystyle x,y} such that n = a x 2 + b x y + c y 2 {\... | wikipedia |
1603cc7674bebe3f67970bf6bbd3e3fcdf9a2495 | lagrange completed a proof in 1775 based on | lagrange completed a proof in 1775 based on his general theory of integral quadratic forms. the following presentation incorporates a slight simplification of his argument, due to gauss, which appears in article 182 of the disquisitiones arithmeticae. | wikipedia |
63fb3f4d214360cedb57ae6f7f76ce404140a880 | 4. if a {\displaystyle a} and b {\displaystyle | 4. if a {\displaystyle a} and b {\displaystyle b} are relatively prime positive integers then every factor of a 2 + b 2 {\displaystyle a^{2}+b^{2}} is a sum of two squares. (this is the step that uses step (3.) to produce an 'infinite descent' and was euler's proposition 4. the proof sketched below also includes the pr... | wikipedia |
f86504b820e571e3c59003adae22b0fa9b6341bb | 3. if a number which can be written | 3. if a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares. (this is euler's second proposition). | wikipedia |
f8d5b6835302c3d1a472f0fad35bebfce920260c | for the avoidance of ambiguity, zero will always | for the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. | wikipedia |
3ab5e09b34d8fba8561f8de76e4188061ebbea7e | euler succeeded in proving fermat's theorem on sums | euler succeeded in proving fermat's theorem on sums of two squares in 1749, when he was forty-two years old. he communicated this in a letter to goldbach dated 12 april 1749. the proof relies on infinite descent, and is only briefly sketched in the letter. the full proof consists in five steps and is published in two p... | wikipedia |
96ed9ca73d334ec4f1d374dc84d021ecd03f4bef | fermat usually did not write down proofs of | fermat usually did not write down proofs of his claims, and he did not provide a proof of this statement. the first proof was found by euler after much effort and is based on infinite descent. he announced it in two letters to goldbach, on may 6, 1747 and on april 12, 1749; he published the detailed proof in two articl... | wikipedia |
8d20c4821ce0a799f81ab7b20663fd2f36752437 | take p = 97 {\displaystyle p=97}. a possible | take p = 97 {\displaystyle p=97}. a possible quadratic non-residue for 97 is 13, since 13 97 − 1 2 ≡ − 1 (mod 97) {\displaystyle 13^{\frac {97-1}{2}}\equiv -1{\pmod {97}}}. so we let x = 13 97 − 1 4 = 22 (mod 97) {\displaystyle x=13^{\frac {97-1}{4}}=22{\pmod {97}}}.the euclidean algorithm applied to 97 and 22 yields: ... | wikipedia |
12a1f6db279ac658578f13a811c72b219f36663e | once x {\displaystyle x} is determined, one can | once x {\displaystyle x} is determined, one can apply the euclidean algorithm with p {\displaystyle p} and x {\displaystyle x}. denote the first two remainders that are less than the square root of p {\displaystyle p} as a {\displaystyle a} and b {\displaystyle b}. then it will be the case that a 2 + b 2 = p {\displays... | wikipedia |
6daf02cbabae8cf93e4d06c31df433fdc63f066e | such an x {\displaystyle x} will satisfy the | such an x {\displaystyle x} will satisfy the condition since quadratic non-residues satisfy q p − 1 2 ≡ − 1 (mod p) {\displaystyle q^{\frac {p-1}{2}}\equiv -1{\pmod {p}}}. | wikipedia |
97a8fe236ea964171b2d6ba660d007107d004473 | given an odd prime p {\displaystyle p} in | given an odd prime p {\displaystyle p} in the form 4 k + 1 {\displaystyle 4k+1}, first find x {\displaystyle x} such that x 2 ≡ − 1 (mod p) {\displaystyle x^{2}\equiv -1{\pmod {p}}}.this can be done by finding a quadratic non-residue modulo p {\displaystyle p}, say q {\displaystyle q}, and letting x = q p − 1 4 (mod p)... | wikipedia |
b2bce35bd68f6cea2b1c30f4892871b509bbb2f2 | a las vegas algorithm with a probabilistically polynomial | a las vegas algorithm with a probabilistically polynomial complexity has been described bystan wagon in 1990, based on work by serret and hermite (1848), and cornacchia (1908). the probabilistic part consists in finding a quadratic non-residue, which can be done with success probability ≈ 1 2 {\displaystyle \approx {\f... | wikipedia |
84df4b5c490971e6a03f4041dc560bb388ff57f7 | however the input size of the algorithm is | however the input size of the algorithm is log p, {\displaystyle \log p,} the number of digits of p (up to a constant factor that depends on the numeral base). the number of needed tests is of the order of p = exp (log p 2), {\displaystyle {\sqrt {p}}=\exp \left({\frac {\log p}{2}}\right),} and thus exponential in the ... | wikipedia |
581dd4ec2add76e67c64f1e3b59a8448a1a1ae4f | there is a trivial algorithm for decomposing a | there is a trivial algorithm for decomposing a prime of the form p = 4 k + 1 {\displaystyle p=4k+1} into a sum of two squares: for all n such 1 ≤ n < p {\displaystyle 1\leq n<{\sqrt {p}}}, test whether the square root of p − n 2 {\displaystyle p-n^{2}} is an integer. if this is the case, one has got the decomposition. | wikipedia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.