Squares Definition and 383 Threads

Homework Statement Find the least squares approximation of cos^3(x) by a combination of sin(x) and cos(x) over the interval (0, 2pi) Homework EquationsThe Attempt at a Solution I know how to find a least squares approximation with vectors, but I don't even know how to start with a function...

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...

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)...

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...

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##...

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 =...

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...

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...

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 -...

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 \\...

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...

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.

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...

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

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>...

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...

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?

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?

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!

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...

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 $r^m=1.$

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...

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...

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...

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...

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 $p=\dfrac{u^2+1}{2}$.

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

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...

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...

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...

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...

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...

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=...

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]...

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...

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

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...

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...

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...

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...

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##...

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

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...

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 −...

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) =...

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...

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

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...

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...

1 = 1^2 1 = 1^2 9 = 3^2 1 = 1^2 81 = 9^2 729 = 27^2 225 = 15^2 324 = 18^2 X 82944 = 288^2 176400 = 420^2 215296 = 464^2 3444736 = 1856^2 So, I am trying to find short method to find factorials. In order to achieve this, I imagined factorials as squares, one edge of which corresponding square...