What is Squares: Definition and 400 Discussions

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted



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{\displaystyle \square }
ABCD.

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  1. beyondlight

    Solve derivative of least squares matrix equation

    Homework Statement I am designing a MIMO communication system, with input signal s, channel H and transform matrix T. The received signal is corrupted by noise. Homework Equations [/B] The received signal is r = Hs+n And then it is transformed (compressed) by: y = Tr And then its...
  2. G

    Least squares problem: am I solving it correctly?

    Homework Statement In R^3 with inner product calculate all the least square solutions, and choose the one with shorter length, of the system: x + y + z = 1 x + z = 0 y = 0 2. The attempt at a solution So I applied the formula A^T A x = A^T b with A as being the matrix with row 1 (1,1,1)...
  3. benzun_1999

    Over constrained least squares analysis

    Hi, I have a over constrained least squares problem. I also have the correct solution to the problem. Now I need to determine which of the vectors are contributing how much information that is close to the correct solution. I am sure there should be some methodology for this analysis, but I...
  4. V

    Linearizing a nonlinear least squares model

    I have a nonlinear least squares problem with a set of parameters \bf{g}, where I need to minimize the function: \chi^2 = \sum_i \left( y_i - M(t_i ; {\bf g}) \right)^2 The t_i are some independent parameters associated with the observations y_i and the model function has the form M(t_i ...
  5. M

    MHB Seeking general solution for Graeco-Latin Squares of order 4N+2 with N>1

    Greetings from a newbie here to MHB. I am a director of a duplicate bridge club, and would like to be able to generate movements when the number of tables is 10, 14, 18 etc. I am aware that the most convenient movements are equivalent to Graeco-Latin Squares, and that these exist for all...
  6. RicardoMP

    Exponential Least Squares Method

    Homework Statement Hi! I've been interpolating a data set using Legendre polynomials and Newton's method and now I've been asked to, given a data set, approximate the function using the Least Squares Method with the following fitting function: ##g(r)=a+be^{cr}##, where ##a##, ##b## and ##c##...
  7. ognik

    MHB Matrix Sum of Squares: Rotate Coord System to Express as Diagonal

    Maybe I just need help understanding the question ... write $ x^2 + 2xy + 2yz + z^2 $ as a sum of squares $ (x')^2 -2(y')^2 + 2(z')^2 $ in a rotated coord system. The 1st expression $ = \left[ x, y, z \right]M \begin{bmatrix}x\\y\\z\end{bmatrix} $ and I get $ M =...
  8. B

    How to Find Least Square Estimates for Object Weights with Normal Error?

    Homework Statement Suppose that object 1 weighs \theta_1 and object two weights \theta_2. Then each object is weighed once and then together getting three observations y_1,y_2,y_3. The scale gives unbiased weights with normally distributed error (constant variance) Find the least square...
  9. S

    How Do You Calculate Predicted Values in Least Squares Regression?

    Homework Statement note a linear regression model with the response variable Y=(Y1..Yn) on a predictor variable X=(X1..Xn). the least squares estimates of the intercept and slope a(hat) and B(hat) are the values that minimize the function: (see attached image) and the problem reads on further...
  10. S

    Number Theory: Difference of Perfect Squares

    Homework Statement :[/B] Determine whether there exists an integer x such that x^2 + 10 is a perfect square. Homework Equations :[/B] N/A The Attempt at a Solution :[/B] Assume x^2 + 10 = k^2 (a perfect square). Solve for x in terms of k: x = ±sqrt(k^2 - 10) Since k is an integer and k^2 -...
  11. Samuel Williams

    What is the Geometric Approach to Proving Least Squares Approximation?

    My apologies for having to post in an image, my latex skills are not good enough for the question at hand :( a) There is no solution since the system has more unknowns than equations (the equations are equal giving 1=2 which does not make sense). b) I get a solution of \begin{bmatrix}1 \\1 \\...
  12. B

    How to handle squares of delta distributions

    Homework Statement I have to write equations of motion for a field, namely ## A ##. The full action has actually three terms, but my problem is with a part of the action reading: $$ S =\int d^{10}x \sqrt{-g} [ f(x_1, ... , x_{10}) + \delta (y) A ]^2 $$ In the 10 x's there is of course the...
  13. M

    MHB 25 x 25 Grid, find how many shaded squares

    pls help with the formula on solving these the easy way. It took time for me to find the answer by drawing it literally. my answer is 334 squares.
  14. J

    The linear in linear least squares regression

    It is my understanding that you can use linear least squares to fit a plethora of different functions (quadratic, cubic, quartic etc). The requirement of linearity applies to the coefficients (i.e B in (y-Bx)^2). It seems to me that I can find a solution such that a coefficient b_i^2=c_i, in...
  15. A

    MHB History of Sum of Squares: Pythagoras & Beyond

    I would like to know some history on the subject like who is the first to think about sum of squares of integers and what he/she was thinking about. I think maybe it is related to Pythagorean triples. Thanks
  16. Saitama

    MHB Writing a number as sum of squares

    Here's the problem statement from HackerRank: https://www.hackerrank.com/contests/programaniacs-june-15/challenges/sum-of-squares-1 Since the constraints are small, I tried a DP solution. Code I have written so far: #include <cmath> #include <cstdio> #include <vector> #include <iostream>...
  17. ORF

    Error in least squares fit: how to include error of points?

    Hello I have a doubt with the least squares fitting (linear fitting). The low-level statistics textbooks only take into account the statistical error of fitting, but not the error of the fitted points. How is the error of the fitted points taken into account, and included in the total error...
  18. matqkks

    Sum of Two Squares: Intro to Number Theory

    Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?
  19. matqkks

    MHB Sum of Two Squares: Applications & Motivation

    Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?
  20. patrickbotros

    Parameterization of Sum of Squares

    I've seen the parameterization of a^2+b^2=c^2 and also a^2+b^2=c^2+d^2, but I don't know how they arrived at those parameterizations. Would it be possible to parameterize something with two equalities like a^2+b^2=c^2+d^2=e^2+f^2? Any help is appreciated!
  21. V

    Least squares parameter correlation

    I am trying to solve a large least squares inversion (inverting data for the modeled sources), and find that my parameters describing 1 source are highly correlated with the parameters describing the second source. Can anyone recommend a technique or reference which discusses how to reduce the...
  22. P

    MHB Squares in a field with q^n elements

    Let $F$ be a field with $q^n$ elements, where $q$ is an odd prime. Write $q^n=2m +1$ with $m \in \mathbb{N}.$ If $r \in F^{\times},$ show that the equation $y^2= r$ has a solution iff $r^m=1.$
  23. V

    Large weighted least squares system

    I have a large weighted least squares system with millions of observations. I can't store the least squares matrix in memory, so I only store the normal equations matrix: A^T W A where A is dense and n-by-p and W = diag(w_1,...,w_n). It takes a very long time to build this normal equations...
  24. S

    Diophantine equation and squares

    The Diophantine equation below, $$ x_0^{2} - (x_1^{2}+x_2^{2}+x_3^{2}+x_4^{2}+x_5^{2}+x_6^{2}+x_7^{2}+x_8^{2})=1$$ 1. Does above equation have any specific name? 2. What are the solutions(a formula)?? 3. in the case,$$x_8^{2}=0$$ , does anything special happen?? 4. What is the general way...
  25. diracdelta

    Least squares method for cloud of atom

    Homework Statement After turnig of magnetic-optic pit, cold cloud of atom 87 Rb is expanding. Size of cloud after time t, is given with relation: where, k_B is Boltzman constant, m mass of 87 Rb. Draw a plot, then use least squares method to find temperature T, and initial size of cloud...
  26. END

    Least Squares Derivation—Simple Algebraic Simplification

    Hey, PF I'm reading the following derivation of least squares, and I'm trying to figure out how the author went from the last step at the bottom of pg. 7 to the final equation (11) at the top of pg. 8. [http://isites.harvard.edu/fs/docs/icb.topic515975.files/OLSDerivation.pdf] More...
  27. alexmahone

    MHB P is the sum of 2 consecutive squares

    Let $p$ be an odd prime. Prove that $p$ is the sum of 2 consecutive squares i.e. $p=a^2+(a+1)^2$ if and only if $p$ has the form $p=\dfrac{u^2+1}{2}$.
  28. anemone

    MHB Sum of Two Squares: Can $5^{64}-3^{64}$ Be Written?

    Is it possible to write $5^{64}-3^{64}$ as the sum of two squares?
  29. samgrace

    Weighted Least Squares Solution

    Homework Statement \begin{bmatrix} 3x_{1}& 7x_{2}& 4x_{3} \\ 3x_{1}& 4x_{2}& 5x_{3} \\ x_{1}& 10x_{2}& 8x_{3} \\ 8x_{1}& 8x_{2}& 6x_{3} \\ \end{bmatrix} = \begin{bmatrix} 26 \\ 16 \\ 33 \\ 46 \\ \end{bmatrix} the measurements represented by equations 1 and 3 above can be trusted more than those...
  30. D

    Least Squares Estimation for two parameters

    Homework Statement Hi guys, so the problem is as follows: A set of n independent measurements y_{i}, i=1...n are treated as Gaussian, each with standard deviations \sigma_{i}. Each measurement corresponds to a value of a control variable x_{i}. The expectation value of y is given by...
  31. M

    MHB Solve Towers of Hanoi: An Algorithm Example w/ Recursion

    Hey! :o The Towers of Hanoi problem consists of three pegs $A$, $B$ and $C$, and $n$ squares of varying size. Initially the squares are stacked on peg$A$ in order of decreasing size, the largest square on the bottom. The problem is to move the squares from peg$A$ to peg$B$ one at a time in such...
  32. Schnurmann

    Least Squares Estimation - Problem with Symbols

    Hi folks, 1. Homework Statement I don't fully understand the question statement, how is it supposed to be read? Question: Give a formula for the minimizer x* (to be read as x-star) of the function ƒ:ℝn → ℝ, x → ƒ(x) = ||Ax-b||22, where A∈ℝm×n and b∈ℝm are given. You can assume that A has rank...
  33. A

    How to do Linearization for Non-linear least squares?

    Hy I want to know how to make linearization for some function,...what should by in Non-linear least squares problems. In my book I have only this example how to do: http://i.imgur.com/MUFiHkr.pngSomeone could me help how to do, some receipt of method what I need to do? Non-linear least...
  34. R

    68068 as difference of two squares

    Homework Statement In how many ways can 68068 be written as the difference of two squares?Homework EquationsThe Attempt at a Solution Let (x+a) * (x+a) -x*x =68068=2*2*7*11*13*17 a (2x+a) =2*2*7*11*13*17 As 2x+a is odd ⇒ a is even ∴a=2b 2b (2x+2b) =2*2*7*11*13*17 b (x+b) =7*11*13*17 x=...
  35. 9

    Statistics: given total sum of squares, find R²

    Homework Statement Given: Σ(xi - x̄)² = 500 Σ(yi - ybar)² = 800 (total sum of squares, SST)) Σ(ŷ - ybar)² = 400 (total sum of estimators, SSE) Σ(xi - x̄)²(yi) = 200 Σ(xi - x̄)²(εi) = 0 n = 1000 s² = 4 Find (or explain why you cannot find): β1 β0 variance of β R² Homework Equations [/B]...
  36. L

    Statistics Sum of Squares x*y Proof

    Homework Statement Prove that \sum[(x_{i} - \overline{x})(y_{i} - \overline{y})] = \sum[(x_{i} - \overline{x})y_{i}] Homework Equations None. The Attempt at a Solution I tried using the fact that the sum of the mean values is just the mean value, because the sum of a constant...
  37. M

    MHB Finding Perfect Squares with $n^4 + 33$

    Find, with proof, all the positive integers $n$ such that $n^4 + 33$ is a perfect square.
  38. Medicol

    Counting squares of NxM lattice

    This is not a quiz but I am thinking how to write down a simple math formula to count the total number of squares present in a lattice of NxM points for my 12 year old nephew ? He'll sure be happy if I could turn this into, say, a common sense for pupils like him. :biggrin: For example, In a...
  39. T

    Algebraic manipulation of sum of squares.

    Hiya. I got to an interesting bit in a calculus book, but as usual I'm stumped by a (probably simple) algebraic step. The author goes from: (ds)^2=(dx)^2+(dy)^2 to: ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx I understand moving the square root across, but I don't understand how the...
  40. gfd43tg

    Least squares regression outputting function handle

    Homework Statement The number of twists ##y## required to break a certain rod is a function of the percentages ##u## and ##v## of each of two chemical components present in the rod. The following function is proposed ##y(u, v) = a_{1} + a_{2} exp(u^{2}) + a_{3}\sqrt{v} + a_{4}uv## (1) where the...
  41. D

    Weighted Least Squares for coefficients

    Hi, I have an ordinary least squares setup y = Ac where A is an NxM (N>>M) matrix, c the unknown coefficients and y the measurements. Now WEIGHTED least squares allows to weight the MEASUREMENTS if, for example, some measurements are more important or contain a lower variance. However...
  42. gfd43tg

    Linear regression with least squares for quadratic function

    Homework Statement We want to determine the coefficients of a polynomial of the form: ##p(x)=c_{1}x^2 +c_{2}x+c_{3}##The polynomial ##p(x)## must satisfy the constraint ##p(1)=1##. We would also like ##p(x)## to satisfy the following 4 constraints: ##p(−1)=5## ##p(0)=−1## ##p(2)=6##...
  43. anemone

    MHB Solving for Perfect Squares of $f(x)=x^2-19x+99$ for All Natural Numbers $x$

    Find all values of $x$ such that $f(x)=x^2-19x+99$ is a perfect square for all $x\in N$.
  44. C

    Does the Error Distribution Affect Least Squares Estimation in Simple Models?

    Suppose we have an observation y = c+ e where c is an unknown constant and e is error with the pdf = exp(-e-1) for e >-1 . We want to determine the least square estimator of c given by the c* which minimizes the error cost function E(c) = .5(y-c)^2 Minimizing the error cost is done by taking...
  45. P

    Completing the squares of a multivariable function

    Homework Statement completing the squares; ## x^2+y^2+2xy-2x-2y+43 = 0## The Attempt at a Solution I did it as follows, but i would like to know if there is a different 'nicer' method to complete it; ## x^2 + y^2 + 2xy − 2x − 2y + 43 ## ## = (x + y)^2 − 2x − 2y + 43 ## ## = (x + y)^2 −...
  46. B

    Problem with squares and roots

    Homework Statement Hi, I am currently studying for a exam and I have noticed I have difficulty with squares and roots. I decided to take a problem from an exam so that I can illustrate the problems I am having with it. Homework Equations If f(x) = √(x+1)2 - √(x-1)2 (a) f(x) = 2; (b) f(x) =...
  47. V

    Weighted least squares residuals

    Hello, When doing a weighted least squares fit of a model to data, I want to examine the residuals to see if their histogram matches the expected probability distribution. Since I am minimizing \chi^2 = \sum_i w_i \left[ y_i - Y(x_i) \right]^2 would I define my...
  48. B

    MHB Union of 2 Squares: How Many Regions Can Mike Get?

    By drawing two circles, Mike obtained a figure, which consists of three regions (see picture). At most how many regions could he obtain by drawing two squares? (A) 3 (B) 5 (C) 6 (D) 8 (E) 9
  49. V

    Uncertanty in a non-linear regression with least squares method

    Homework Statement Ok, so I'm trying to fit a set of data (21000 points to be exact) to a sine function. Homework Equations Y = A*sin(ωt) The Attempt at a Solution I used NumPy to get the parameters A and ω with the least squares method. So far, so good. However, i appear to...
  50. A

    Find Out How Many Combinations of n Squares Exist

    In the picture attached I have tried to list the different shapes you can get when you attach, 1, 2, 3, 4 squares, but, as you can imagine, when n gets bigger the number of combinations gets incredibly large. Is there are way to see how many possible configurations there is for n squares, and...
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