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| In [[number theory]], the '''geometry of numbers''' studies [[convex body|convex bodies]] and <!-- [[lattice (group)|lattice]]s --> integer vectors in [[n-dimensional]] space. The geometry of numbers was initiated by {{harvs|txt|authorlink=Hermann Minkowski|first=Hermann|last= Minkowski|year=1910}}.
| | I gained a lot of weight when I was expecting with my son, plus my body happened to store only regarding every ounce left over for the next four years. It didn't feel wise understanding I hadn't reduction anything, and I'd had enough of feeling fat all the time. After seeing how heavy I really looked inside my brother's wedding images, I decided to lose fat by setting a New Year's resolution!<br><br>The difference in the overweight group is likely to be muscle. Folks with more muscle are more fit and healthy, but which muscle puts them inside the obese group for their height. Numbers which receive tossed about frequently are that 60% of Americans are obese plus half of those are fat. These numbers are based strictly off the BMI, and the group of overweight Americans is probably to be much lower.<br><br>The Basal Metabolic Rate is essentially the amount of calories the body must survive for 1 day while doing normal bodily functions like breathing plus pumping blood etc. Taking in less calories then this may force the body to burn fat because power. There's is a calculator found on the calculator page connected above.<br><br>In brief, for many adults, the [http://safedietplansforwomen.com/bmi-calculator bmi calculator men] for men is a wise method to receive an idea of healthy fat ranges. However it is not usually the final word in deciding if a person is obese or overweight. There are additional factors to think about whenever judging how much somebody should weigh. A individual with a high BMI ought to be evaluated by a health care provider, who may use alternative factors such as skin fold thickness (a measure of body fat), waist size, evaluations of diet plus family wellness issues, plus other tests to obtain out if a person's fat might pose a health risk. For instance, you are at a high risk if you carry nearly all of your weight around the abdomen. If you are obese, losing 5 to 10 per cent of the current weight at a rate of 2 to 4 lbs (1 to 2 kg) monthly is a healthy goal.<br><br>For female with low activity which has a weight at 149lbs and below, could try a 1,200 calorie diet to help their fat loss. Women 150lbs to 164 lbs must have 1,400 calories; 165 to 184lbs 1,600 calories; and finally females over 185lbs should have 1,800 calories.<br><br>In purchase to determine perfect weight for kids and teens (between age group 2-20 years), there is a specific reference tool called the BMI percentile chart. In case, when the percent value falls at 80, it means the kid is having more fat than 80 % of the kids of the same gender plus age group. According to the chart, a child's body mass index dropping inside 95 percentile or above indicates he is obese, while those with a BMI above 85 percentile have a risk of becoming overweight. On the alternative hand, when a kid's BMI percentile is 5 or lower, he/she is underweight.<br><br>In conclusion, DO NOT use BMI because an accurate gauge for the fat, you can end up inside tears like Sally, trust something more exact like body fat percentage, or conversely, lean body mass. |
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| The geometry of numbers has a close relationship with other fields of mathematics, especially [[functional analysis]] and [[Diophantine approximation]], the problem of finding [[rational number]]s <!-- or vectors with rational coordinates SIMPLIFY --> that <!-- accurately --> approximate an [[irrational number|irrational quantity]].<ref>Schmidt's books. Grötschel et alia, Lovász et alia, Lovász.</ref>
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| ==Minkowski's results==
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| {{main|Minkowski's theorem}}
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| Suppose that Γ is a [[Lattice (group)|lattice]] in ''n''-dimensional Euclidean space '''R'''<sup>''n''</sup> and ''K'' is a convex centrally symmetric body.
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| [[Minkowski's theorem]], sometimes called Minkowski's first theorem, states that if
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| <math>vol(K)>2^nvol(R^n/\Gamma)</math> | |
| then ''K'' contains a nonzero vector in Γ.
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| {{main|Minkowski's second theorem}}
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| The successive minimum λ<sub>''k''</sub> is defined to be the [[Infimum|inf]] of the numbers λ such that λ''K'' contains ''k'' linearly independent vectors of Γ.
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| Minkowski's theorem on [[successive minima]], sometimes called [[Minkowski's second theorem]], is a strengthening of his first theorem and states that<ref>Cassels (1971) p.203</ref>
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| :<math>\lambda_1\lambda_2\cdots\lambda_n vol(K)\le 2^n vol(R^n/\Gamma).</math>
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| ==Later research in the geometry of numbers==
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| In 1930-1960 research on the geometry of numbers was conducted by many [[number theorist]]s (including [[Louis Mordell]], [[Harold Davenport]] and [[Carl Ludwig Siegel]]). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.<ref>Grötschel et alia, Lovász et alia, Lovász, and Beck and Robins.</ref>
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| ===Subspace theorem of W. M. Schmidt===
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| {{main|Subspace theorem}}
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| {{see also|Siegel's lemma|volume (mathematics)|determinant|Parallelepiped}}
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| In the geometry of numbers, the [[subspace theorem]] was obtained by [[Wolfgang M. Schmidt]] in 1972.<ref>Schmidt, Wolfgang M. ''Norm form equations.'' Ann. Math. (2) '''96''' (1972), pp. 526-551.
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| See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.</ref> It states that if ''L''<sub>1</sub>,...,''L''<sub>''n''</sub> are [[linear independence|linearly independent]] [[linear]] [[algebraic form|forms]] in ''n'' variables with [[algebraic number|algebraic]] coefficients and if ε>0 is any given real number, then
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| the non-zero integer points ''x'' with | |
| :<math>|L_1(x)\cdots L_n(x)|<|x|^{-\varepsilon}</math>
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| lie in a finite number of [[linear subspace|proper subspaces]] of '''Q'''<sup>''n''</sup>.
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| ==Influence on functional analysis==
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| {{main|normed vector space}}
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| {{see also|Banach space|F-space}}
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| Minkowski's geometry of numbers had a profound influence on [[functional analysis]]. Minkowski proved that symmetric convex bodies induce [[normed space|norms]] in finite-dimensional vector spaces. Minkowski's theorem was generalized to [[topological vector space]]s by [[Kolmogorov]], whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a [[Banach space]].<ref>For Kolmogorov's normability theorem, see Walter Rudin's ''Functional Analysis''. For more results, see Schneider, and Thompson and see Kalton et alia.</ref>
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| Researchers continue to study generalizations to [[star-shaped set]]s and other [[convex set|non-convex set]]s.<ref>Kalton et alia. Gardner</ref>
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| == References ==
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| <references/>
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| ==Bibliography==
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| * Matthias Beck, Sinai Robins. ''Computing the continuous discretely: Integer-point enumeration in polyhedra'', Undergraduate texts in mathematics, Springer, 2007.
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| *{{cite journal|author=[[Enrico Bombieri]]|coauthors = Vaaler, J.|title = On Siegel's lemma|journal = Inventiones Mathematicae|volume = 73|issue = 1|date = Feb 1983|pages = 11–32|url = http://www.springerlink.com/content/k55042224131lp42|doi = 10.1007/BF01393823}}
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| *{{cite book|
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| author=[[Enrico Bombieri]] and Walter Gubler
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| |title=Heights in Diophantine Geometry
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| |publisher=Cambridge U. P.
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| |year=2006}}
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| * [[J. W. S. Cassels]]. ''An Introduction to the Geometry of Numbers''. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).
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| * [[John Horton Conway]] and [[N. J. A. Sloane]], ''Sphere Packings, Lattices and Groups'', Springer-Verlag, NY, 3rd ed., 1998.
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| *R. J. Gardner, ''Geometric tomography,'' Cambridge University Press, New York, 1995. Second edition: 2006.
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| *[[Peter M. Gruber|P. M. Gruber]], ''Convex and discrete geometry,'' Springer-Verlag, New York, 2007.
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| *P. M. Gruber, J. M. Wills (editors), ''Handbook of convex geometry. Vol. A. B,'' North-Holland, Amsterdam, 1993.
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| *M. Grötschel, [[Lovasz|L. Lovász]], [[Alexander Schrijver|A. Schrijver]]: ''Geometric Algorithms and Combinatorial Optimization'', Springer, 1988
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| *{{cite book
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| | author = Hancock, Harris
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| | title = Development of the Minkowski Geometry of Numbers
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| | year = 1939
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| | publisher = Macmillan}} (Republished in 1964 by Dover.)
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| * [[Edmund Hlawka]], Johannes Schoißengeier, Rudolf Taschner. ''Geometric and Analytic Number Theory''. Universitext. Springer-Verlag, 1991.
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| * {{citation
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| |last1=Kalton|first1=Nigel J.|author1-link=Nigel Kalton
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| |last2=Peck|first2=N. Tenney
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| |last3=Roberts|first3=James W.
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| | title = An F-space sampler
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| | series = London Mathematical Society Lecture Note Series, 89
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| | publisher = Cambridge University Press| publication-place = Cambridge
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| | year = 1984| pages = xii+240| isbn = 0-521-27585-7 | mr = 0808777}}
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| * C. G. Lekkerkererker. ''Geometry of Numbers''. Wolters-Noordhoff, North Holland, Wiley. 1969.
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| * {{cite journal | author = [[A. K. Lenstra|Lenstra, A. K.]]; [[H. W. Lenstra, Jr.|Lenstra, H. W., Jr.]]; [[Lovász|Lovász, L.]] | title = Factoring polynomials with rational coefficients | journal = [[Mathematische Annalen]] | volume = 261 | year = 1982 | issue = 4 | pages = 515–534 | id = {{hdl|1887/3810}} | doi = 10.1007/BF01457454 | mr = 0682664}}
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| *[[Lovasz|L. Lovász]]: ''An Algorithmic Theory of Numbers, Graphs, and Convexity'', CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
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| *{{Springer|id=G/g044350|title=Geometry of numbers|first=A.V. |last=Malyshev}}
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| *{{Citation | last1=Minkowski | first1=Hermann | author1-link=Hermann Minkowski | title=Geometrie der Zahlen | url=http://www.archive.org/details/geometriederzahl00minkrich | publisher=R. G. Teubner | location=Leipzig and Berlin | mr=0249269 | year=1910 | jfm=41.0239.03 }}
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| * [[Wolfgang M. Schmidt]]. ''Diophantine approximation''. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
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| * {{cite book | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=[[Springer-Verlag]] | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020 }}
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| *{{cite book | author = Siegel, Carl Ludwig | authorlink = Carl Ludwig Siegel | title = Lectures on the Geometry of Numbers | year = 1989 | publisher = [[Springer-Verlag]] }}
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| * Rolf Schneider, ''Convex bodies: the Brunn-Minkowski theory,'' Cambridge University Press, Cambridge, 1993.
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| * Anthony C. Thompson, ''Minkowski geometry,'' Cambridge University Press, Cambridge, 1996.
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| {{Number theory-footer}}
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| [[Category:Geometry of numbers| ]]
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I gained a lot of weight when I was expecting with my son, plus my body happened to store only regarding every ounce left over for the next four years. It didn't feel wise understanding I hadn't reduction anything, and I'd had enough of feeling fat all the time. After seeing how heavy I really looked inside my brother's wedding images, I decided to lose fat by setting a New Year's resolution!
The difference in the overweight group is likely to be muscle. Folks with more muscle are more fit and healthy, but which muscle puts them inside the obese group for their height. Numbers which receive tossed about frequently are that 60% of Americans are obese plus half of those are fat. These numbers are based strictly off the BMI, and the group of overweight Americans is probably to be much lower.
The Basal Metabolic Rate is essentially the amount of calories the body must survive for 1 day while doing normal bodily functions like breathing plus pumping blood etc. Taking in less calories then this may force the body to burn fat because power. There's is a calculator found on the calculator page connected above.
In brief, for many adults, the bmi calculator men for men is a wise method to receive an idea of healthy fat ranges. However it is not usually the final word in deciding if a person is obese or overweight. There are additional factors to think about whenever judging how much somebody should weigh. A individual with a high BMI ought to be evaluated by a health care provider, who may use alternative factors such as skin fold thickness (a measure of body fat), waist size, evaluations of diet plus family wellness issues, plus other tests to obtain out if a person's fat might pose a health risk. For instance, you are at a high risk if you carry nearly all of your weight around the abdomen. If you are obese, losing 5 to 10 per cent of the current weight at a rate of 2 to 4 lbs (1 to 2 kg) monthly is a healthy goal.
For female with low activity which has a weight at 149lbs and below, could try a 1,200 calorie diet to help their fat loss. Women 150lbs to 164 lbs must have 1,400 calories; 165 to 184lbs 1,600 calories; and finally females over 185lbs should have 1,800 calories.
In purchase to determine perfect weight for kids and teens (between age group 2-20 years), there is a specific reference tool called the BMI percentile chart. In case, when the percent value falls at 80, it means the kid is having more fat than 80 % of the kids of the same gender plus age group. According to the chart, a child's body mass index dropping inside 95 percentile or above indicates he is obese, while those with a BMI above 85 percentile have a risk of becoming overweight. On the alternative hand, when a kid's BMI percentile is 5 or lower, he/she is underweight.
In conclusion, DO NOT use BMI because an accurate gauge for the fat, you can end up inside tears like Sally, trust something more exact like body fat percentage, or conversely, lean body mass.