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...hematics]], [[statistics]] and elsewhere, '''sums of squares''' occur in a number of contexts: * For partitioning of variance, see [[Partition of sums of squares]] ...

...17|issue=1|year=2014|pages=58–76|title=On a conjecture of Rudin on squares in arithmetic progressions|author=González-Jiménez, Enrique|author2=Xarles, Xa ...sserts (in [[big O notation]]) that <math>Q(N) = O(\sqrt { N })</math> and in its stronger form that, if <math>N > 6</math>, <math>Q(N) = Q(N; 24, 1)</ma ...

{{short description|Characterization by prime factors of sums of two squares}} ...g|thumb|upright=1.5|Integers satisfying the sum of two squares theorem are squares of possible distances between integer lattice points; values up to 100 are ...

{{AFC comment|1=User left "Here is your Wikipedia-style article in Wikipedia markup (wikitext) format:" from the ChatGPT response. Here is your Wikipedia-style article in Wikipedia markup (wikitext) format: ...

...proposed model fits well. The other component is the '''pure-error sum of squares'''. ...ue(s) of the independent variable(s). The remainder of the residual sum of squares is attributed to lack of fit of the model since it would be mathematically ...

In [[number theory]], '''Brahmagupta's function''' is a concept developed by the ancient India ...] and has connections to [[modular forms]].<ref>Cohen, H. (2007). ''Number Theory – Volume I: Tools and Diophantine Equations''. Springer.</ref>{{page needed ...

{{Short description|Puzzle published in 1921 by Percy MacMahon}} [[File:MacMahon Squares solution.png|alt=MacMahon Squares|thumb|A particular solution; this illustrates the B_20,0,0 constr ...

...d to the [[expected value]] of the distribution in a way analogous to that in which the [[quantile]]s of the distribution are related to the [[median]]. ...metric Least Squares Estimation and Testing," ''Econometrica'', volume 55, number 4, pp. 819–47. </ref> ...

...ion|How many ways a positive integer can be represented as the sum of four squares}} ...} can be represented as [[Sums of powers|the sum]] of four [[Square number|squares]] (of [[integer]]s). ...

...-09-01|title=Blockbusting and domineering|journal=Journal of Combinatorial Theory, Series A|language=en|volume=49|issue=1|pages=67–116|doi=10.1016/0097-3165( ...<math> n \times 1 </math> strip of squares called "parcels". Each player, in turn, claims and colors one previously unclaimed parcel until all parcels h ...

...legal move of the bishop; that is, there is an edge between two vertices (squares) if they occupy a common diagonal. When the chessboard has dimensions <math ...has two connected components, whose vertex sets are the red and the black squares, respectively. The reason is that the bishop's diagonal moves do not allow ...

==In a square== ...|first3=János |author3-link=János Pach |year=2005 |title=Research Problems in Discrete Geometry |publisher=Springer |location=New York |isbn=978-0387-238 ...

...for a least-squares approach the quality of the approximation is measured in terms of the squared differences between the two. ...e]] in the case of complex functions. The extension of Pythagoras' theorem in this manner leads to [[function space]]s and the notion of [[Lebesgue measu ...

...cription|Says when a natural number can be represented as the sum of three squares of integers}} ...' states that a [[natural number]] can be represented as the sum of three squares of integers ...

{{Short description|Unsolved problem in mathematics}} ...sp;=&nbsp;1,&nbsp;2,&nbsp;...,&nbsp;''M'',... for some integer&nbsp;''x''. In 1983, [[Douglas Hensley]] observed that Büchi's problem is equivalent to th ...

...cannot be written as a sum of squares, ''s''(''F'') = <math>\infty</math>. In this case, ''F'' is a [[formally real field]]. [[Albrecht Pfister (mathema If <math>s(F)\ne\infty</math> then <math>s(F)=2^k</math> for some [[natural number]] <math>k</math>.<ref name=R13>Rajwade (1993) p.13</ref><ref name=Lam379>La ...

...t Sussex, England-6April2011 (1).jpg|thumb|A square pyramid of cannonballs in a square frame]] ...can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. ...

...constraints. For problems with sum-of-squares cost functions, see [[Least squares]].}} ...optimization is also known as the '''Lasserre hierarchy''' of relaxations in [[semidefinite programming]]. ...

...where representations that differ only in the order of the [[summand]]s or in the signs of the numbers being squared are counted as different. It is den :<math>r_k(n) = |\{(a_1, a_2, \ldots, a_k) \in \mathbb{Z}^k \ : \ n = a_1^2 + a_2^2 + \cdots + a_k^2\}|</math> ...

...arning]] methods. Proto-value functions are generated by [[Spectral graph theory|spectral analysis]] of a graph, using the [[Laplacian matrix|graph Laplacia ...t introduced in the context of reinforcement learning by Sridhar Mahadevan in his paper, ''Proto-Value Functions: Developmental Reinforcement Learning'' ...