Squares Definition and 383 Threads

Let A and B be two coprime integers. Find X = zero mod A such that Y = 2*X +1 = 0 mod B. Then 8*(Y +2*N*A*B)*(X + N*A*B) + 1 is a square for all integer N. If A = 5,B = 7, X = 10, Z = 21 then the sequence of square roots of the Squares for N = -3 to 3 is -379, -249, -99, 41, 181, and 321. Of...

Number theory--product of squares Homework Statement Suppose a and b are two integers, which can each be written as the sum of two squares. Prove that ab can be expressed as the sum of two squares. Homework Equations The Attempt at a Solution a=c^{2}+d^{2} b=x^{2}+y^{2} I'm...

This is a problem that I just thought of the other day and am intrigued by, let me set it up. You have an n x n grid of dots, so that each dot is 1 away from the four dots closest in each direction. My question is; for a grid of n2 dots, what is the total number of possible combinations of...

Hi guys, Thanks for taking the time to read the post. I have a question related to curve fitting and polynomials that i was hoping someone might be able to help me with. I have a set of x and y data points, all on a graph. I have then calculated the 4th order least squares polynomial...

Homework Statement is an expression that is a difference of two squares considered to be a quadratic. For example, would x2 - 4 be a quadratic? What about x4 - 4? Homework Equations Ax2 + Bx + C The Attempt at a Solution I know we can factor a DOTS into two binomials like a...

I found a claim in a paper (BSSA, Vol 81, No. 6: "A Waveform Correlation Method for Identifying Quarry Explosions", By D.B. Harris) concerning finding filter coefficients. The statement is given without proof. I cannot locate a reference or theorem for the following, and have not been able thus...

Homework Statement y''+(1/y)*(y')2=0 Homework Equations The Attempt at a Solution This is another problem I am having trouble with. I have done searches around the internet, but seen that all "non linear" ODE of second order involves a non linear form in a non differential term...

Hey, I have a graph for which I am supposed to fit two linear least squares line and minimize the combined residuals (the lines intersect)... I would really appreciate some info about how to do this or what this type of data analysis is called so i can google the step-by-step method. Thanks!

Hey, Not sure if this is the right section to post this but ... I have a graph for which I am supposed to fit two linear least squares line and minimize the combined residuals (the lines intersect)... I would really appreciate some info about how to do this or what this type of data analysis...

Hi, as no unique unique analytical solution exists to my problem (as another poster pointed out), I have taken to solving it through a least squares method. My equation is as follows: (s1x1 + s2x2 - I).^2 = min. Here s1 and s2 are shift matrices (I know them), and I is a matrix of size...

Homework Statement Solve the following linear equations simultaneously by the Least Squares solution and calculate the residuals. Homework Equations 3x + 2y + z = 5 x + 6y - z = -7 x - y + 2z = 3 5x - 2y = 1 The Attempt at a Solution This is a question from this guy at...

Hi, I have a problem that I'm a bit stuck on, and need some direction: I need to find \forall_n within a certain domain that can satisfy this equation: \left( 3n-1 \right) \left( n+1 \right) = m^{2} where m,n \in \mathbb{Z} Or, to put it in a different context, I'm looking for...

Imagine a cube wrapped in Latin squares and try to solve the following puzzles. Please be aware the symbols at the borders are shared between neighboring cube's sides. Let me know if you like it or not. This is rather straightforward: This is a bit harder

1. Find the number of pairs (m, n) such that 2m - 2n = 63 in which m and n are nonnegative integers. 2. What is the units digit of 625 - 324 ? 3. How many perfect squares divide the number 4!*5!*6! ? 4. What is the last digit of the sum 1! + 2! + 3! + ... + 2010! + 2011! ? Thanks!

Hello y'all, If I have n data points (xi, yi) each with error bars in both x and y (xi_err, yi_err), should I use 1/(xi_err^2+yi_err^2) as the weight in a weighted least squares linear fit, or should the weight be a different value that has nothing to do with error bars? I've never used WLS...

Homework Statement Let G be a finite abelian group, and assume that |G| is odd. Show that every element of G is a square. The Attempt at a Solution So we want to show that \forall g \in G, \exists h \in G, g = h^2 . Let g \in G be arbitary, and consider the subgroup generated by g, denoted...

I recently got interested in number theory and have been fiddling around with Scilab trying to find interesting things. I came across the following mildly interesting property, which I couldn't find much about on Google. Form the difference d_n between the sum of the squares of the prime...

I know that any prime p = 1 mod 4 can be expressed as sum of 2 squares. But how many different pairs of integers a,b such that p = a^2+b^2? (with a>b!) It seems there is only one pair. How to prove it? I try in this way: assume p = a^2+b^2 = c^2+d^2 (with a>c>d>b) and try to show it has...

Hi, I am trying to do a best least-squares fit to a set of data which is described by the following equation: y=a\exp(-b\ln^2(c/x)) Where a,b,c are constant parameters I am trying to find values for. Any advice on how to proceed?

Homework Statement Test these two equations, using least-squares fitting of the data (ti, bi), i = 1, 2, . . . , 100:1. b(t) = d_{1} + d_{2}te^{-t} + d_{3}t^{2}e^{-2t}2. b(t) = d_{1} + d_{2}\sqrt{t}e^{-\sqrt{t}} + d_{3}te^{-2\sqrt{t}} where d1, d2, d3 in R are unknown. For both theories...

Homework Statement I must find the best fitting function of the form ax²+bx+c using least squares. The points are (-1,6.1), (0,2.8), (1,2.2), (3,6) and (6,26.9). 2. Homework Equations + attempt at a solution A\vec x= \vec b, I'm looking for \vec x =\begin {pmatrix} a \\ b \\ c \end {pmatrix}...

With MATLAB or something. Basically, I just have a bunch of data points where I should do a linear least squares fit, but each of the points have error bars around them.

Hi everyone, I have to prove that every element z of a finite field F is a sum of 2 squares. Really not sure how to go about proving this, though I've done some research and it is suggested to start with a function that maps F* to itself, defined by f(x) = x^{2} . I guess if I could show...

Homework Statement Three small squares, S1, S2, and S3, each with side 0.1 and centered at the point (4,5,7), like parallel to the xy, yz, and xz planes respectively. The squares are oriented counterclockwise when viewed from the positive z, x, y axes respectively. A vector field G has...

Hi, All: Given a normed vector space (X,||.||), and an inconsistent system Ax=b, the generalized least squares solution x^ to Ax=b is the point in the span of Ax that is closest to b, i.e., given a fixed matrix A, we define AX={Ax: x in X}, and then: x^:={ x in AX...

Homework Statement Let A= |2 -1 -1| |-1 2 -1| |-1 -1 2| and B= |1| |2| |3| Homework Equations Find the x in which minimizes ||Ax-b||2 The Attempt at a Solution I tried to solve it by using this formula (A**A)-1A**b=x but i get the inverse of A*A equal 0

How should I go about solving this problem? This is only to get a better understanding of how NLLS works. F(x;a) = (1+a1*x)/(a2+a3*x) (so n = 3) I am choosing a1,a2,a3 to be 2,3,5 respectively. I am also picking 6 data points (so m = 6): (0, 0), (-1/4, 1/4), (-1/2, 1/10), (1/4, 1/4)...

sum of two squares? If an Even number could be expressed in the form a2 + b2 . And if there exits two other numbers m,n such that a2 + b2 = m2 + n2 then , my question is is there any relation between (a,b) and (m,n) apart from a2 + b2 = m2 + n2 ??

Homework Statement Magic squares. An n × n matrix that is filled with the numbers 1, 2, 3, ..., n2 is a magic square if the sum of the elements in each row, in each column, and in the two diagonals is the same value. Write a program that reads in n2 values from the keyboard and tests...

Now that this has happened a second time, I'm really puzzled over what I am experiencing, just wondering if any of you have heard of this. The other day I was lying in bed and I noticed that there was a large circle about 1 inch in diameter made of bright hot pink filaments in my vision. It...

I'm going through some methods to solve the LLS method of minimization and have come upon 3 general methods to solve the problem. The 3 methods I am looking at are normal equations, QR factorization, and SVD. I've come upon a fact that I can't find an explanation for: Can anyone explain why...

I know this may sound dumb, but is zero and one perfect squares? I have no idea.

Homework Statement For the system Ax = b, find least squares solution. A = (1 2) ; b = (3, 2, 1)T ----(2 4) ----(1 -2) Homework Equations I know if A is an m x n matrix of rank n, the normal equations ATAx = ATb have a unique solution x =...

Homework Statement Suppose the columns of A are not independent. How could you find a matrix B so that P=B(BTB)^-1BT does give the projection onto the column space of A? (The usual formula will fail when AT A is not invertible). T is transpose. Homework Equations The Attempt at a Solution I...

Hello, Sorry if this is in the wrong forum, I wasn't sure so I just picked General. Is there a formula to determine the squares that will produce a given difference. For example x^2 - y^2 = 21. From a little experimentation it seems that for odd numbers the problem can be solved with this...

Homework Statement I have to prove or disprove the following: Part a) If p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 - c2) Part b) f p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 + c2) Homework Equations The Attempt at a Solution Part a)...

Homework Statement Roll a fair die n times. Let Sn denote the sum of squares of the rolls. Thus, Sn is the sum of Xi^2, where Xi represents one roll. What are the mean and variance of sqrt(n) * (Sn/n - u), where u is the mean of Yn/n Homework Equations The Attempt at a Solution No...

Homework Statement Suppose a set of N data points {(xk,yk)}Nk=1 appears to satisfy the relationship for some constants a and b. Find the least squares approximations for a and b. Homework Equations The Attempt at a Solution I really have no idea about this problem.

Problem: A = n by m matrix x = m by 1 vector y = n by 1 vector C = c by m matrix E = e by m matrix Alpha, gamma and theta are constants. norm(Ax-y) = min subject to: norm(Cx) = alpha norm(Ex) = gamma transpose(Cx)*Ex = (alpha^2)*(gamma^2)*cos(theta) I read a paper on how to do this with 1...

I have a simple problem. I have a set of 3D data points and I want to fit a line through them using linear least squares. I understand the basic approach required: set up two matrices such that Ax = b, then make it a square matrix A^t*Ax = A^t*b, then solve for x using a Cholesky decomposition...

Homework Statement The distance traveled by a comet is described with the following equation: r = B + re cos (θ) B (Beta) and e are constants θ 0.88 1.10 1.42 1.77 2.14 2.91 3.85 r 8.27 5.49 3.53 2.50 1.95 1.52 5.21 These were some of the measurements, which can be written in...

I'm studying for my numerical analysis final on tuesday, and I know this is going to be one of the problems, so any help is greatly appreciated. Homework Statement State and prove existence and uniqueness for the solution of the linear least squares problem. Homework Equations y \approx...

I'm doing a replication paper and having a bit of issues. The author uses 2 instruments and has 4 endogenous variables. Since he is under-identified, he runs 4 separate regressions, so that each equation has 1 overidentifying restrictions. I want to extend this and run the four equations as a...

I have the following problem: I have a set of m measurements $\mathbf{\phi}$ and I estimate a set of 3 variables $\mathbf{x}$ The estimated value for $\mathbf{\phi}$ depends linearly on $\mathbf{x}$ : Hx=\Tilde{\phi} The solution through weighted linear least squares is: $\mathbf{x}$ =...

Hi All, So I was just wondering if there is a formula for the number of ways a number can be written as a sum of squares?(Without including negatives, zero or repeats). For example 5=2^2+1^2. (There is only one way for 5). Second question along this line is: In how many ways can a number...

Homework Statement The diagram below shows four congruent circles whose centres are the vertices of the square DEFG and whose circumference touch the sides of an isosceles triangle. Area of triangle ABC is 10000 units square. What is the radius of the circles, to the nearest unit...

a1=22+32+62 a2=32+42+122 a3=52+52+202 Using these, gIve a general term for an such that an is always a perfect square

Homework Statement For all natural numbers, a and b, if a and b are both even, then (a^2+b^2) is not a perfect square. (prove this) Homework Equations The Attempt at a Solution I tried proving by contradiction and got (2s)^2 +(2t)^2 =k^2. which translates to 4s^2 +4t^2=k^2. I...

Homework Statement For some natural N the no of positive integers x satisfying the equation 1! + 2! + 3! + ... + x! = (N)2 is: A)one B)two C)infinite D)none The Attempt at a Solution I have no idea of how to start.. never came across such problems. By trial i have got two values...

Hallo I'm new to this (wonderful) forum, and to SR too... I've a general question about the space time interval invariance. Say we have two points A and B, at rest each other, at distance AB. Now A and B simultaneously in their reference frame emit a flash of light. The space time interval...