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. D

    Linear Least Squares Excel Template

    Homework Statement I had to make a Linear Least Squares Excel template for my first week's Physics lab to analyze linear data for the rest of the semester. All of my values come out correctly, except the standard deviation values. I was wondering if someone could take a look at my template and...
  2. G

    Proving a and b are squares of an integer

    Homework Statement If a, b and c are positive integers with (a,b) = 1 and c2 = ab, show that each of a and b is the square of an integer. I've been staring at this problem for a while now and I've got not clues. I don't see the relation between a and b being relatively prime and the rest...
  3. A

    MHB Toothpick Squares Sequence for nth # toothpicks

    Please help! I'm stuck on modeling the function for the # of toothpicks in the nth figure. Thank you so much. Renee
  4. O

    Confidence interval from least squares fitting?

    Hello, Let me get right to my problem. I have an experimental distribution and a single-parameter theoretical distribution. I want to find the value for the best fit theoretical distribution that agrees with my experimental data for the bulk of the distributions (the tails of my...
  5. Y

    Help with applying the least squares method for solving simultaneous equations

    Hi everyone given the system of equations A1Cx + B1Cy + C1Cz = D1 A2Cx + B2Cy + C2Cz = D2 A3Cx + B3Cy + C3Cz = D3 I need to solve for Cx, Cy, Cz All other variables are known and constants. However all other variables (A,b,c,d) come from experimentally measured data and thus I cannot...
  6. M

    Solving Integer Cubes & Squares: 7k & 3k?

    I am trying to solve how an integer is simultaneously that is simultaneously a square and a cube number must be either of the form 7k or 7k+1 and I am failing when i work (7k+2)^2, even (7k+3)^2....Can i interpret 7k+1 as 7k+(-1), i think i can't but then i fail in many steps! Also i know that...
  7. N

    Mathemaica: Nonlinear Least Squares

    Hi I like to fit in Mathematica using NonlinearModelFit. When I look at the fitted parameters, there is an entry called "P-value". Here is what it means: "The p-value is the probability of observing a t-statistic at least as far from 0 as the one obtained.". I'm not quite sure what this...
  8. P

    Linear Algebra problem (Least Squares?)

    Linear Algebra problem (Least Squares? - Distance between lines) Homework Statement We have two points R = (x,x,x) and S = (y,3y,-1). All we know is that they are on lines somewhere in 3-space and that they don't cross. Need to find an x and y that minimize || R - S ||2 Homework...
  9. K

    How can I improve regression results by adjusting the goodness of fit metric?

    I'm doing some line fitting on experimental data. Basically I have some array of pixels, and a value measured at each pixel, and I am fitting it with several constrained Gaussians. I'm using a Levenburg-Marquadt nonlinear least squares algorithm called mpfit to fit the parameters, but the...
  10. D

    Least squares approximation: Is smaller normal distance always better?

    I took a LA course in the spring, and was interested by the least squares method for building models. I decided to practice this concept by attempting to build a model that would predict ticket sales for the Mega Millions lottery given the jackpot amount. I have 249 data pairs of jackpot and...
  11. P

    Leas squares curve fit problem

    The main problem is that I have no idea what I am doing. But second to that I am attempting to fit a curve to some data using the equation y = ((1/A(x-B)^(C+1))+D)^-1 if that makes sense I need to get the values for A,B,C and D and so have been looking into lsqnonlin but I am having...
  12. M

    Approximate solution for square root of sum of squares

    Homework Statement If X^2=Sqrt(x1^2+x2^2+x3^2+...)=> X? and vice versa If X=x1+x2+x3+...=> X^2? Homework Equations The Attempt at a Solution
  13. H

    A very quick stupid question about square roots of squares

    If I see an expression like \sqrt{E^2c^2} I can just remove the square root sign right and replace it with Ec?
  14. C

    MHB Percent of perfect squares that have an odd digit for their hundreds place

    Suppose you consider the set of all perfect squares (the squares of all of the integers).What percent of them have an odd digit for their hundreds place?
  15. A

    Proof by induction of the sum of 2 squares

    Homework Statement (a^2+b^2)(c^2+d^2)=(ac-bd)^2 +(ad+bc)^2 prove by induction that r=a1,a2,a3...an where the a's represent the sums of 2 squares. it itself is a sum of two squares. Check it with: 2=1^2+1^2, 5=1^2+2^2, 8=2^2+2^2, for r=160, r=1600, r=1300, r=625 Homework Equations...
  16. R

    Line integral - confusion on squares and square root terms

    line integral -- confusion on squares and square root terms Homework Statement Do you see where they have sqrt(16 sin^2t etc = 5? How do they get that, the answer should be 7, the square root of 16 is 4, sin^2 + cos^2 is 1 and the square root of 9 is 3, 3 + 4 = 7. It's like they're...
  17. A

    Homogeneous least squares

    Given a homogeneous linear least squares problem: A^{T}y=0 What is the difference between minimizing y^{T}AA^{T}y (the least square error) and: y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y ? Thanks.
  18. P

    Determine optimal vectors for least squares

    Hello all, I have a set of measurements that I want to fit to a linear model with a bunch of parameters. I do this as \theta=(H^TH)^{-1}H^Tx, where θ are the parameters and x is the set of measurements. The problem is that I'd like to reduce the number of parameters in the fit. I'd like to...
  19. J

    MHB Number of Positive Integer Pairs for Perfect Squares

    the number of ordered pairs of positive integers $x,$y such that $x^2 +3y$ and $y^2 +3x$ are both perfect squares my solution...
  20. GreenGoblin

    MHB Regression least squares (boring question? or more to it)

    yi = a + bxi + ei is the simple liner regression model as per is usual "state the assumptions on the errors ei to justify a least squares fit" ? So is this just that E(ei)=0, i can't see what else is a 'must' for this? what about that they are normally distrubited? i know the properties of the...
  21. E

    Number Theory Problem (sums of consecutive squares)

    Homework Statement The sum of two consecutive squares can be a square: for instance, 32 + 42 = 52 (a) Prove that the sum of m consecutive squares cannot be a square for the cases m = 3; 4; 5; 6. (b) Find an example of eleven consecutive squares whose sum is a square. The Attempt at a...
  22. S

    Weird Sum of Squares as a Vector Norm and Gauss-Newton optimization

    Homework Statement A(\vec{x}) = (F + T * x )2 F is a constant, x is a 2×1 vector T is a (constant) 1×2 matrixB(\vec{x}) = || K.Z.x ||2 k:3\times3 matrix and Z:3\times2, x the same as aboveB(x) is also R2→RC(x) = A(x) + B(x) Homework Equations 1- I am confused...
  23. S

    Wierd sum of squares compilation to a single vector norm

    I've encountered a function like this: S(x) = [M(x) - F(x)] ^2 + || G(x) || ^ 2X being a 3*1 vector M and F: vector----->scalar G: vector------->vector and || G || meaning its norm To change S(x) into a single square, authors have described it like this: S(x) = || A + B || ^ 2 where A=(M(x) -...
  24. M

    The set of squares of rational numbers is inductive

    Homework Statement The set of squares of rational numbers is inductive Homework Equations definition of an inductive set The Attempt at a Solution sorry i know this is probably very easy to most but I am just learning analysis. Okay, so we can see that 1 is in the set because it is...
  25. ElijahRockers

    Why Do We Square the Deviation in Least Squares Regression?

    Homework Statement http://www.math.tamu.edu/~vargo/courses/251/HW7.pdf Given a set of points (xi,yi) and assuming f(xi) is linear, the deviation measured is F(m,b)=\sum_{i}^{n}(y_i - f(x_i))^2. There are a few different questions about this from the link above. The Attempt at a...
  26. J

    Matlab - Normal equation for linear least squares model fitting

    Homework Statement Im trying to construct a function in MATLAB that models linear least squares model fitting through the use of normal equations. Homework Equations Normal equation (A'*A*c)=(A'*y) A= [column vector of all x ; column vector of all 1] y= [column vector of all y] c = [b;a]...
  27. P

    Electron density in terms of squares of wavefunctions

    Dear all, I have a quantum system made up of two layers. An electric field is applied with two voltages applied at both layers. I want to calculate the electronic densities that reside at the two laters in terms of the wavefunctions obtained from solving the HamiltoniaN..Anyone knows of an...
  28. B

    Comp Sci Fortran Program for Least Squares Fit of Data

    (The usual format doesn't really fit this question, so I hope you won't mind if I set it up slightly differently) The task is to make a Fortran program that will read in a given set of data from a file, stopping when it reaches the end, and calculate the gradient and intercept of a best fit...
  29. C

    Least squares fitting by a constant

    say we have data set {y(t sub i), t sub i} Where i=1 2 3...m. I know how to fit these into a line of the form ax+b, but how about fitting into a constant??
  30. T

    Sum of Squares of Roots of quadratic equations.

    1. The problem: Find the value of \lambda for which the sum of squares of the roots of the equation: 2x^2 + (2\lambda +4)x^2-(1+ \lambda) = 0 has minimum value. 2. Homework Equations \alpha + \beta = -b/a \alpha\beta = c/a where a,b,c are coefficients of x^2 , x and constant...
  31. I

    Consecutive Numbers in the Fibbonacci Sequence and Sums of Two Squares

    I've noticed lots of interesting properties of the patterns of numbers in the Fibbonacci sequence that can be expressed as the sum of two squares. In fact, it's what got me into number theory in the first place. There seem to be no two adjacent entries that are not the sum of two squares- and it...
  32. C

    Extra Sum of Squares [Statistics]

    Homework Statement I'm trying to study off an old practice midterm for my upcoming statistics midterm. The midterm is attached. Part One is the following problem: The effect of a new antidepressant drug on reducing the severity of depression was studied in manic-depressive patients at...
  33. H

    Finding the Best Fit Line Using the Method of Least Squares

    Homework Statement number of years(x): [1,2,3,4,5,6,] number of accidents(y): [12,6,5,4,2,2] or (1,12),(2,6),(3,5),(4,4),(5,2),(6,2) Homework Equations the relation between x and y is on the form: y = a + b(1/x) by changing (1/x) to z, the model become linear instead of...
  34. S

    Integrating squares of sine & cosine

    Homework Statement find ∫4cosx*sin^2 x.dx Homework Equations The Attempt at a Solution ∫4cos x * 1/2 (1 - cos2x) ∫2cosx - 4cos^2 x. Then i don't know whereto go from here??
  35. X

    Calculating Least Squares: What is r and how do I use it?

    How do I calculate least squares? The way it's described is confusing to me. \hat{β}= r* sy/sx . sx and sy is the standard deviation of x and y, correct ? So let's say I have sets of numbers such as (16,19),(18,17) etc... 16 and 18 would be x and 19 and 17 would be y. So I have to calculate the...
  36. N

    Derivation of least squares containing vectors

    Hi, I'm trying to derive a result from this paper: http://www.research.yahoo.net/files/HuKorenVolinsky-ICDM08.pdf given cost function \min_{x_* , y_* } \sum_{u, i} c_{ui} (p_{ui} - x^T_u y_i)^2 + \lambda \left( \sum_u ||x_u||^2 + \sum_i ||y_||^2 \right) Where both xu and yi...
  37. M

    Least squares estimation with quadratic constraints (M*M = 0)

    Hello there, currently I am trying to solve a least squares problem of the following form: min_{M} ||Y - M*X||^2 where M is a 3x3 matrix and Y and X are 3xN matrices. However, the matrix M is of a special form. It is a rank 1 matrix which satisfies M*M = 0_{3x3} and the trace of M is zero...
  38. R

    Understanding Error Bars on Least Squares Fit Line

    Hello, What are the error bars on the least squares fit line? recall the variables are: x ; y ; x^2 ; xy ; y^2 ; yi=mx+c ; d =(y-yi) ; d^2 --------- is it d? Thanks
  39. O

    Uncertainty of a non linear least squares fit

    Hi, I have some experimental data as a function of time t and temperature T. I have done a least squares fit of the data with a function f=f(a1,a2,t,T) (the function is non linear in a1 and a2!). Optimization of e^2 = sum((yi-f(a1,a2,ti,Ti)^2) with Matlab's fminsearch gave me a1, a2 and the...
  40. V

    Product of a specific sequence of prime number squares

    Here is a good question for maths enthusiasts. I really find this sum very tough. Any help, advice or guidance for this question will be greatly appreciated. find the product upto n terms (1+(1/2^2)) . (1+(1/3^2)) . (1+(1/5^2)) . (1+(1/7^2)) . (1+(1/11^2))...upto n terms Where the nth...
  41. 6

    Cutting Up Squares: Problem Solving Method

    I don't really know if this is in the right place, or what the problem is called, if its in the wrong place I'm sorry. Anyway my question is apart from intuition is there any method to solve a problem like say we have an object like a square, and we are given for instance rectangles, that in one...
  42. B

    Sum of squares of differences of functions

    If you are looking for a minimum of the sum of squares of differences of functions should it be the same as the minimum for the product of squares of differences of functions? Also assume that no difference is equal to zero.
  43. N

    Proving the Parallelogram Theorem: Sum of Side Squares = Diagonal Squares

    Homework Statement Apply the formula for the distance between two points to prove the well-known theorem: In a parallelogram the sum of the squares of the sides is equal to the sum of the squares of the diagonals. Homework Equations It gave a hint saying to put one of the...
  44. T

    Probability of Sum of Squares of 2 Uniform RVs < 1

    If you were to pick two random numbers on the interval [0,1], what is the probability that the sum of their squares is less than 1? That is, if you let Y_1 ~ U(0,1) and Y_2 ~ U(0,1), find P(Y_1^2 + Y^2_2 \leq 1). There is also a hint: the substitution u = 1 - y_1 may be helpful - look for a beta...
  45. T

    Quadratic least squares equation

    f(x) = –3√x, 1 ≤ x ≤ 4 (a) Find the quadratic least squares approximating function g for the function f. g(x)=?
  46. A

    Exploring the Group Theory of Latin Squares

    As I'm studying permutation groups I remembered that when I was in elementary school my teacher introduced Latin squares to us and asked us to find all 4 by 4 Latin squares. I never succeeded in solving the problem and I found it so challenging at that time, even later in high school when I...
  47. W

    Least squares - why the square?

    I have a presentation to give on the least squares method this week, and am finding the least squares method very intuitive with the exception of one thing: why is the square of the offset used, as opposed to the absolute value? I did a bit of searching, but could not find a detailed answer...
  48. T

    Finding orthonormal set using Gram-Schmidt and least squares

    Homework Statement A) Find an orthonormal set q1, q2, q3 for which q1,q2 span the column space A [1 1; 2, -1; -2,4] (this is a 3x2 matrix). B) Which fundamental subspace contains q3? C) What is the least-squares solution of Ax=b if b=[1 2 7]T? Homework Equations Gram-Schmidt...
  49. T

    Fit with least squares, Levenberg–Marquardt algorithm

    Hello guys, I need your help with understanding a fitting Algorithm, so that I could make it in a C++ program. I have the following function: g(f; f0, phi0, a, b) = phi0 + a ArcTan((f-f0)/b) Which is a function of "f", frequency. I would like to fit this function with the...
  50. K

    Three colors to paint each side of a square, how many different squares?

    Homework Statement If you have a square and can paint each side with one of three different colours, how many completely different arrays can you get? (rotated squares don't count) Homework Equations The Attempt at a Solution I was thinking 3^4 / 4 but, that is not an integer, (3^4 - 1) / 4...
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