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| {{Infobox scientist
| | Whenever we were younger it didnt appear to matter what you ate you usually had a lot of vitality and for many folks that meant our fat would be kept down additionally. But because we get older our bodies changes plus the processes that we took for granted whenever we were younger dont function because well. Such is the case for metabolism. Unfortunately metabolism has become an easy target over the years because something to blame fat gain on. But that really isnt the case; you require metabolism incredibly when you are elder. One method to grow it really is from strength training.<br><br>On the other hand, overweight people who can [http://safedietplans.com/bmr-calculator bmr calculator] try to consume only 500 calories a day may virtually likely be starving themselves. Since the body is more utilized to taking in more than 2000 calories a day or even more, then the abrupt drop of calorie intake will signal the body into starvation mode. In this way, the body's metabolism will slow down to conserve vitality.<br><br>The basal metabolic rate delivers a good baseline for minimum calories. Obviously, the proper foods plus exercise are important to the success. A diet of sugary foods and/or an exercise program consisting of endless walking on a treadmill might create weight reduction difficult. But in the event you use a BMR because a starting point, you'll understand to not go below that level plus add food plus exercise accordingly to create a calorie deficit.<br><br>Our bodies need 1,800 to 2,000 calories a day in order to function properly. Some individuals want less or even more, nevertheless this is the average. So don't try to drop a noticeable amount of calories at once. There is no have to put the body through starvation. If you do, the metabolic rate really slows down, that makes the whole process harder.<br><br>How do you understand how several calories we need? There is no magic number which fits everybody. Many factors have a role in determining what the right number is for you. Some of these factors include age, present weight, activity level, height, plus bmr.<br><br>The 500 calories a day diet might be right for certain persons, nevertheless it really depends on the following factors: gender, activity level, height, body sort and total health plus health.<br><br>I hope you're inside the regular range, however in the event you are overweight you are able to plan certain weight loss system considering the BMR and current activity level plus hopefully boost the wellness. |
| |name = Ferdinand Georg Frobenius
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| |image = GeorgFrobenius.jpg
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| |image_size = 150px
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| |caption = Ferdinand Georg Frobenius
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| |birth_date = {{birth date|df=y|1849|10|26}}
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| |birth_place = [[Charlottenburg]]
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| |death_date = {{death date and age|df=y|1917|08|3|1849|10|26}}
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| |death_place = [[Berlin]]
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| |nationality = [[Germans|German]]
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| |field = [[Mathematics]]
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| |work_institutions = [[Humboldt University of Berlin|University of Berlin]]<br>[[ETH Zurich]]
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| |alma_mater = [[University of Göttingen]]<br>University of Berlin
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| |doctoral_advisor = [[Karl Weierstrass]]<br>[[Ernst Kummer]]
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| |doctoral_students = [[Richard Fuchs]]<br>[[Edmund Landau]]<br>[[Issai Schur]]<br>[[Konrad Knopp]]<br>[[Walter Schnee]]
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| |known_for = [[Differential equations]]<br>[[Group theory]]<br>[[Cayley–Hamilton theorem]]<br>[[Frobenius method]]
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| |influences =
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| |influenced =
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| |awards =
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| }}
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| '''Ferdinand Georg Frobenius''' (26 October 1849 – 3 August 1917) was a [[Germans|German]] [[mathematician]], best known for his contributions to the theory of [[elliptic functions]], [[differential equations]] and to [[group theory]]. He is known for the famous determinantal identities, known as Frobenius-Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as [[Padé approximants]]), and gave the first full proof for the [[Cayley–Hamilton theorem]]. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as [[Frobenius manifolds]].
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| ==Biography==
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| Ferdinand Georg Frobenius was born on 26 October 1849 in [[Charlottenburg]], a suburb of [[Berlin]]<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/BirthplaceMaps/Berlin.html|date=October 26, 2010|title=Born in Berlin}}</ref> from parents Christian Ferdinand Frobenius, a [[Protestant]] parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven.<ref name="Bio">{{cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Frobenius.html|title=Biography|date=26 October 2010}}</ref> In 1867, after graduating, he went to the [[University of Göttingen]] where he began his university studies but he only studied there for one semester before returning to Berlin, where he attended lectures by [[Kronecker]], [[Ernst Eduard Kummer|Kummer]] and [[Karl Weierstrass]]. He received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. His thesis, supervised by [[Karl Weierstrass|Weierstrass]], was on the solution of differential equations. In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics.<ref name="Bio" /> Frobenius was only in Berlin a year before he went to [[Zürich]] to take up an appointment as an ordinary professor at the [[ETH Zurich|Eidgenössische Polytechnikum]]. For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. It was there that he married, brought up his family, and did much important work in widely differing areas of mathematics. In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed. In 1893 he returned to Berlin, where he was elected to the [[Prussian Academy of Sciences]].
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| ==Contributions to group theory==
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| [[Group theory]] was one of Frobenius' principal interests in the second half of his career. One of his first contributions was the proof of the [[Sylow theorems]] for abstract groups. Earlier proofs had been for [[permutation group]]s. His proof of the first Sylow theorem (on the existence of Sylow groups) is one of those frequently used today.
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| * Frobenius also has proved the following fundamental theorem: If a positive integer ''n'' divides the order |''G''| of a [[finite group]] ''G'', then the number of solutions of the equation ''x''<sup>''n''</sup> = 1 in ''G'' is equal to ''kn'' for some positive integer ''k''. He also posed the following problem: If, in the above theorem, ''k'' = 1, then the solutions of the equation ''x''<sup>''n''</sup> = 1 in ''G'' form a subgroup. Many years ago this problem was solved for [[solvable group]]s.<ref>Marshall Hall, Jr., ''The Theory of Groups'', 2nd ed. (Providence, Rhode Island : AMS Chelsea Publishing, 1999), pages 145-146, [http://books.google.com/books?id=oyxnWF9ssI8C&pg=PA145#v=onepage&q&f=false Theorem 9.4.1.]</ref> Only in 1991, after the [[classification of finite simple groups]], was this problem solved in general.
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| More important was his creation of the theory of [[Character theory|group characters]] and [[group representation]]s, which are fundamental tools for studying the structure of groups. This work led to the notion of [[Character theory|Frobenius reciprocity]] and the definition of what are now called [[Frobenius group]]s. A group ''G'' is said to be a Frobenius group if there is a subgroup ''H'' < ''G'' such that
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| :<math>H\cap H^x=\{1\}</math> for all <math>x\in G-H</math>.
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| In that case, the set
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| :<math>N=G-\bigcup_{x\in G-H}H^x</math>
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| together with the identity element of ''G'' forms a subgroup which is [[nilpotent group|nilpotent]] as Thompson showed in his PhD thesis. All known proofs of that theorem make use of characters. In his first paper about characters (1896), Frobenius constructed the character table of the group <math>PSL(2,p)</math> of order (1/2)(''p''<sup>3</sup> − p) for all odd primes ''p'' (this group is simple provided ''p'' > 3). He also
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| made fundamental contributions to the [[representation theory of the symmetric and alternating groups]].
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| ==Contributions to number theory==
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| Frobenius introduced a canonical way of turning primes into [[conjugacy classes]] in [[Galois group]]s over '''Q'''. Specifically, if ''K''/'''Q''' is a finite Galois extension then to each (positive) prime ''p'' which does not [[ramification|ramify]] in ''K'' and to each prime ideal ''P'' lying over ''p'' in ''K'' there is a unique element ''g'' of Gal(''K''/'''Q''') satisfying the condition ''g''(''x'') = ''x''<sup>''p''</sup> (mod ''P'') for all integers ''x'' of ''K''. Varying ''P'' over ''p'' changes ''g'' into a conjugate (and every conjugate of ''g'' occurs in this way), so the conjugacy class of ''g'' in the Galois group is canonically associated to ''p''. This is called the Frobenius conjugacy class of ''p'' and any element of
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| the conjugacy class is called a Frobenius element of ''p''. If we take for ''K'' the ''m''th [[cyclotomic field]], whose Galois group over '''Q''' is the units modulo ''m'' (and thus
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| is abelian, so conjugacy classes become elements), then for ''p'' not dividing ''m'' the Frobenius class in the Galois group is ''p'' mod ''m''. From this point of view,
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| the distribution of Frobenius conjugacy classes in Galois groups over '''Q''' (or, more generally, Galois groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree extensions of '''Q''' depends crucially on this construction of Frobenius elements, which provides in a sense a dense subset of elements which are accessible to detailed study.
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| ==See also==
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| *[[List of things named after Ferdinand Georg Frobenius]]
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| ==Publications==
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| *{{Citation | last1=Frobenius | first1=Ferdinand Georg | author1-link=Ferdinand Georg Frobenius | title=Gesammelte Abhandlungen. Bände I, II, III | publisher=[[Springer-Verlag]] | location=Berlin, New York | editor-first= J.-P.|editor-last= Serre | isbn=978-3-540-04120-7 | mr=0235974 | year=1968}}
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| ==References==
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| {{Reflist}}
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| *{{Citation | last1=Curtis | first1=Charles W. | authorlink = Charles W. Curtis | title=Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer | url=http://books.google.com/books?isbn=0821826778 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-2677-5 | mr=1715145 | year=2003}} [http://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00867-3/ Review]
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| ==External links==
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| *{{MacTutor Biography|id=Frobenius}}
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| * {{MathGenealogy|id=4642}}
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| *[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/frobenius_-_hypercomplex_i.pdf G. Frobenius, "Theory of hypercomplex quantities"] (English translation)
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| {{Authority control|PND=119045605|LCCN=n/84/801120|VIAF=56675988}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Frobenius, Ferdinand Georg
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = German mathematician
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| | DATE OF BIRTH = 26 October 1849
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| | PLACE OF BIRTH = [[Charlottenburg]]
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| | DATE OF DEATH = 31 August 1917
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| | PLACE OF DEATH = [[Berlin]]
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| }}
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| {{DEFAULTSORT:Frobenius, Ferdinand Georg}}
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| [[Category:1849 births]]
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| [[Category:1917 deaths]]
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| [[Category:19th-century German mathematicians]]
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| [[Category:20th-century mathematicians]]
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| [[Category:German mathematicians]]
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| [[Category:Group theorists]]
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| [[Category:Members of the Prussian Academy of Sciences]]
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| [[Category:People from Berlin]]
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| [[Category:People from the Province of Brandenburg]]
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| [[Category:University of Göttingen alumni]]
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| [[Category:Humboldt University of Berlin alumni]]
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| [[Category:Humboldt University of Berlin faculty]]
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| [[Category:ETH Zurich faculty]]
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Whenever we were younger it didnt appear to matter what you ate you usually had a lot of vitality and for many folks that meant our fat would be kept down additionally. But because we get older our bodies changes plus the processes that we took for granted whenever we were younger dont function because well. Such is the case for metabolism. Unfortunately metabolism has become an easy target over the years because something to blame fat gain on. But that really isnt the case; you require metabolism incredibly when you are elder. One method to grow it really is from strength training.
On the other hand, overweight people who can bmr calculator try to consume only 500 calories a day may virtually likely be starving themselves. Since the body is more utilized to taking in more than 2000 calories a day or even more, then the abrupt drop of calorie intake will signal the body into starvation mode. In this way, the body's metabolism will slow down to conserve vitality.
The basal metabolic rate delivers a good baseline for minimum calories. Obviously, the proper foods plus exercise are important to the success. A diet of sugary foods and/or an exercise program consisting of endless walking on a treadmill might create weight reduction difficult. But in the event you use a BMR because a starting point, you'll understand to not go below that level plus add food plus exercise accordingly to create a calorie deficit.
Our bodies need 1,800 to 2,000 calories a day in order to function properly. Some individuals want less or even more, nevertheless this is the average. So don't try to drop a noticeable amount of calories at once. There is no have to put the body through starvation. If you do, the metabolic rate really slows down, that makes the whole process harder.
How do you understand how several calories we need? There is no magic number which fits everybody. Many factors have a role in determining what the right number is for you. Some of these factors include age, present weight, activity level, height, plus bmr.
The 500 calories a day diet might be right for certain persons, nevertheless it really depends on the following factors: gender, activity level, height, body sort and total health plus health.
I hope you're inside the regular range, however in the event you are overweight you are able to plan certain weight loss system considering the BMR and current activity level plus hopefully boost the wellness.