Squares Definition and 400 Threads

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. K

    MHB Is there any sense with these squares?

    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...
  2. S

    How to interpret this problem involving dyadic squares and unit disc?

    The following problem is from the textbook "Real Mathematical Analysis" by Charles Chapman Pugh. Given ##\epsilon > 0##, show that the unit disc contains finitely many dyadic squares whose total area exceeds ##\pi - \epsilon##, and which intersect each other only along their boundaries. I...
  3. S

    Least squares assumptions: finite and nonzero 4th moments

    This isn't a homework problem - I'm just confused by something in a textbook that I'm reading (not for a class, either). I'd appreciate an intuitive clarification, or a link to a good explanation (can't seem to find anything useful on Google or in my textbook). My book states that one of the...
  4. N

    Comp Sci Fortran77 woes- single parameter least squares minimisation stuck

    Hey guys! Having issues with a code I'm writing. It doesn't fit the typical format and I may have already accidentally posted it in the wrong forum, so here's a copy of what was there with the newest version of the code. It'll (eventually) be a four-parameter least squares minimisation, but for...
  5. N

    Fortran Fortran77 woes- single parameter least squares minimisation stuck

    Hey guys! Having issues with a code I'm writing. It'll (eventually) be a four-parameter least squares minimisation, but for the moment I'm just trying to get the single parameter version to work. I've got the first module done but, when I run it it gets caught up in a loop and doesn't actually...
  6. N

    Nonlinear Least Squares Fitting

    Hello, So I was hoping to get some help implementing a nonlinear least squares fitting algorithm. Technically this is an extension of my previous thread, however the problem I am having now is correctly computing the algorithm So the problem definition is this: Given two sets of n 3D points...
  7. B

    MHB What is the Difference of Two Squares in this Expression?

    is this a difference of two squares? $\displaystyle x^{\frac{1}{4}}-y^{\frac{1}{4}}$
  8. N

    Modified least squares fitting?

    Hi, I wonder if someone could give me some guidance on this problem please. I'm not a mathematician and I'm not even sure what the title of this problem should be - curve fitting, regression, function minimization? It started with something fairly simple, least squares fitting with 3...
  9. F

    Proving Finite Squares in n!+n^p-n+2 for p=3,5 (mod8)

    Homework Statement How to prove the following: Let p be a prime p=3,5 (mod8). Show that the sequence n!+n^p-n+2 contains at most finitely many squares. Should I build a contardiction or prove it directly? I really need some help 2. The attempt at a solution Use Fermats...
  10. M

    MHB How to Calculate Total Toothpicks and Squares in Any Figure of the Sequence?

    Please help! I need to be able to find the number of total toothpicks used in any given figure in this sequence, in addition to total squares in any figure, using the figure number. The only way I can figure this out is by listing them all out one by one, which is TERRIBLE :( Thank you so much...
  11. K

    MHB Induction for series of squares

    Prove that for all nEN 1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = (4n^3 - n) / 3My Solution) If n = 1, 4(1)^3 - 1 / 3 = 1 so base case holds. Assume 1^2 + 3^2 + ... + (2k-1)^2 = (4k^3 - k) / 3 What next?
  12. U

    Selecting squares from chessboard

    Homework Statement If three one by one squares are selected at random from the chessboard, then the probability that they form the letter 'L' is Homework Equations The Attempt at a Solution Total number of ways to choose 3 squares is 64C3. Starting from the 1st square at upperleft...
  13. hxthanh

    MHB [Unsolved] How many "magic squares" (combinatorial)?

    A "magic square" is an $n\times n$ table includes $n^2$ non-negative integers satisfying conditions sum on each row and each column are equal and equal to $r$. Define $S_3(r)$ is number of all $3\times 3$ magic square with row sum is $r$ Prove that: $\displaystyle S_3(r)={r+2\choose...
  14. Jameson

    MHB What is the proof for the residual sum of squares problem?

    Problem: Through transformation with orthogonal matrix $O$, the problem \hat{b}=\underset{b}{\operatorname{arg min}}||y-Xb||^2 is equivalent to \hat{b}=\underset{b}{\operatorname{arg min}}||y^{*}-X^{*}b||^2, where $y$ and $y^{*}$ are in $\mathbb{R}^m$, $X$ and $X^{*}$ are in $\mathbb{R}^{m...
  15. caffeinemachine

    MHB Sum of squares of average change is less than

    Hello MHB, I have no good ideas on how to go about solving the following: Let $f:[a,b]\to\mathbb R$ and $g:[a,b]\to\mathbb R$ be real values functions both of which are differentiable in $(a,b)$. Show that there is an $x\in(a,b)$ such that...
  16. S

    MHB How to Graphically Determine the Minimum of a Function Under a Constraint?

    Problem: The sum of two real numbers is 1. What is the minimum value of the sum of the squares of the two numbers? I have already managed to solve the problem algebraically (by substitution and completing-the-square we arrive at a minimum value of 0.5), but what I am interested in is a...
  17. K

    What is the pattern in factorials and squares?

    Hi there, I don't really have a question but I just thought I'd share something that I've found and see if anyone could make any sense of it, or find some sort of pattern in the results. I noticed that for some of the first few factorials (from 4! to 12!), (ceiling[(n!)0.5]2-n!)=a perfect...
  18. S

    Proving a^2 - b^2 = (a+b)(a-b) with Factorization

    Hello, how can I prove using factorization that a^2 - b^2 = (a+b)(a-b) ? a*a - b*b I should get something like a(a-b)+b(a-b) but how ? thanks
  19. J

    Mathematical Tripos, archived question about squares

    Homework Statement a, b, x, y are rational numbers (ay - bx)² + 4(a - x)(b - y) = 0 implies that x = a and y = b or 1 - ab and 1 - xy are squares of rational numbers 2. The attempt at a solution My attempts are hard to transcribe, is this mandatory for getting help? This is my first post, I...
  20. C

    Proof about difference of squares

    Homework Statement Show that an integer n can be represented as a difference of 2 squares if it is either odd or divisible by 4, otherwise not. The representation is unique if and only if n is a prime number. The Attempt at a Solution let x and y be integers so then we have...
  21. A

    MHB Arrange Numbers for Complete Squares Puzzle

    Arrange the numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 such that the summation of each two successive numbers is a complete square easy interesting question
  22. A

    Disjoint dyadic squares in unit disk

    Homework Statement Given ε > 0, show that there is a collection of disjoint dyadic squares in the unit disk that has a total area which exceeds π - ε. Homework Equations Define a dyadic interval as an interval of the form [a, b] such that a = p/2k and b = (p + 1)/2k, p and k are integers. A...
  23. J

    Least squares estimator and distribution (simple linear regression)

    Homework Statement Under the simple linear regression model Y= A + Bx + e, where A is the intercept (a known concept), B is the slope parameter (unknown) and e is a random error term satisfying the normality assumption. If (X1,Y1)...(Xn,Yn) are the n data points observed, find the least squares...
  24. MarkFL

    MHB Jacky L's question at Yahoo Answers regarding minimizing a sum of squares

    Here is the question: Here is a link to the question: (CALCULUS)The sum of two numbers is k. Show that the sum of their squares is AT LEAST (1/2)(k^2)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  25. Z

    Multiple least squares regression

    Homework Statement design a regression model that will use the dataset y trial x1 x2 x3 0.08536, 1, -1, -1, -1.00000 0.09026, 2, -1, -1, -1.00000 0.10188, 1, -1, -1, -0.33333 0.09301, 2, -1, -1, -0.33333 0.10362, 1, -1, -1, 0.33333 0.09920, 2, -1, -1...
  26. Z

    Does the through-origin least squares line pass through the point (ybar, xbar)?

    Homework Statement Does the "through-origin" least squares line yhat = b*X pass through the point (ybar, xbar)? The "through-origin" model is the least squares model without the intercept. Homework Equations b = sum[YX]/sum[X^2] yhat = b*X The Attempt at a Solution when I calculate a...
  27. twoski

    Finding the Curve with Least Squares Approximation: 15 hrs

    Homework Statement Given this data: hours / value ----------- 2 | 1.6 4 | 1.5 6 | 1.45 8 | 1.42 10 | 1.38 12 | 1.36 fit a curve of the form Y ≈ ae^{-bx} What value can you predict after 15 hours? The Attempt at a Solution So i can rewrite the equation as Y ≈ log(a)-bx...
  28. Z

    What is E[MSE] in Simple Linear Regression Analysis?

    Homework Statement This is with respect to simple linear regression.I do not have a book that goes into expected mean squares ( i have an elementary stats book ) but my professor wants us to list the expected mean squares in the general analysis of variance table. Along with SSE, SST...
  29. Sneakatone

    Where is the mass of the joined squares?

    I did Xcm=Lx/(xyz) Ycm=Ly/(xyz) Zcm=Lz/(xyz) I modeled it from the equation CM=(mx+...)/M
  30. K

    Does least squares solution to Ax=b depend on choice of norm

    To find the closest point to b in the space spanned by the columns of A we have: \mathbb{\hat{x}}=(A^TA)^{-1}A^T\mathbb{b} My question is, shouldn't this solution ##\hat{x}## depend on the choice of distance function over the vector space? Choosing two different distance functions might give...
  31. V

    Parameter Estimation, Non Linear Least Squares, Newton-Raphson

    Hello, Homework Statement I want to estimate a set of parameters by trying to minimize the sum of squares wrt to the parameter set given a simulation of a markov process and real data in Matlab. I have already implemented the Levenberg-Marquart (LM) Algorithm and it is converging to certain...
  32. Albert1

    MHB Big Square composed of Small Squares ?

  33. N

    Least Squares Method- What is the Measured mean value of y ?

    Homework Statement So I'm doing a Least Squares Analysis and I'm wondering about what the 'measured mean value of y for replicate measurements of the unknown' value is supposed to be. I have no idea in the world what it's asking for. The value it is speaking of is not the same as the average...
  34. L

    Find volume of circle. Cross-sections are squares. What am I doing wrong? :/

    Hi, everyone. I am just trying to do some practice problems on finding volume. So this is the one I'm working on: 1. "Find the volume of the solid described below: The solid lies between the planes perpendicular to the x-axis at x=6 and x=-6. The cross-sections perpendicular to the axis on...
  35. I

    Minimum table size to contain combinations 000 to 999 in adjacent squares

    Consider an m x n table containing single digits 0 to 9. So each cell contain the digits 0 to 9. My goal is to make sure that all combinations from 000 to 999 can be found in the table in adjacent cells. A cell is adjacent to the cells around it (i.e. above, below, to the left, to the right...
  36. binbagsss

    The weighted least squares formula - Quick question

    The weighted least squares formula - Quick question ! m=Ʃiwi Ʃiwixiyi - Ʃiwixi Ʃiw[SIZE="2"]iy[SIZE="2"]i - In this formula, does w[SIZE="2"]i mean to include the weighting of all points - i.e. both x and y points throughout. Thanks Alot !
  37. G

    Least squares adjustment/regression - two points known distance apart

    Hi All, I'm struggling with finding a solution to an adjustment I'm working on. Thought someone else may have some thoughts? I have a kinematic time series of X,Y positions for two points (X1,Y1,X2,Y2). I know that the two points were a distance D (e.g., 100 m) apart from each other (the...
  38. B

    Substitution to turn a non-linear least squares problem into a linear one

    Hi, I want to solve an overdetermined non-linear equation with the method of least squares. Assume it's f(x) = 1 + ax + a^2 + b, and I want to estimate a and b. This is non-linear, as I said, so the derivatives of the squared residuals involve a^3 terms and are difficult to solve. Now I thought...
  39. R

    Novel Multiplication and Division by odd squares

    let A_i be an odd integer, s_i be the square of a_i and t_i be the triangular number, (s_i -1)/8. Same for a_j , s_j, t_j, etc. Define Multiplication of n X A_i , etc to be n * s_i - t_j and division to be the reverse of this process. I found that n X A_i X A_j X A_k = n X A_k X A_j X...
  40. K

    Lagrange's Four-Square Theorem: 8n-1 Sum of 4 Squares?

    Lagrange's four-square theorem states that any natural number can be expressed as the sum of four integer squares. I've noticed that the first few values of 8n-1 can all only be expressed as a minimum of the sum of four squares. Is this true for all values of n? What's the proof behind it? Thanks.
  41. S

    Least Squares Solutions for Inconsistent Linear Systems

    Homework Statement Hi there! First time user, so I hope I do this right. The question is: Let A be an 8x5 matrix of rank 3, and let b be a nonzero vector in N(AT). First, prove that the system Ax=b must be inconsistent. Then, how many least squares solutions will the system have...
  42. V

    Sum of the squares of remainders

    Homework Statement The division of a polynomial f(x) by (x – 1)(x – 2) has remainder x + 1. If the remainder of the division of f(x) by (x – 1) and (x – 2) are, respectively, a and b. Then what is a^2 + b^2? Homework Equations I guess the remainder theorem could be useful here. The...
  43. D

    Fortran Creating a Loop for Sum of Digit Squares to Equal 1

    Hi, I'm writing a program in fortran that basically creates a loop from 1 until a certain number x (input by the user), and goes through each value between 1 and the certain number x in order to determine if each value meets certain criteria. The criteria is that the sum of the square of the...
  44. M

    A Theorem on Squares and Congruences

    Hey guys -- I was wondering whether the following theorem was already established, or if it was something novel. I couldn't find it in my abstract algebra textbook and a Google search was not productive ("a list of congruence theorems" just turns up elementary geometry theorems on triangle...
  45. P

    Comparing Least Squares and Maximum Likelihood?

    Hi, Below is my attempt at a comparison between the two above-mentioned methods of estimation. Does anything in the table lack in validity and/or accuracy? Should any properties, advantages/disadvantages be eked out? Any suggestions/comments would be most appreciated! MLE...
  46. L

    Uncertainty for least squares fit

    I am fitting data to a parabolic equation using the least squares fit method. Each data point that goes into the fit is the average of 5 data points at that x value, so that each point has error bars that come from the standard deviation of those 5 y values. I have a fitted equation, and I...
  47. M

    Number Theory (Modular Arithmetic and Perfect Squares)

    Homework Statement If k is an integer, explain why 5k +2 cannot be a perfect square. Homework Equations n/a The Attempt at a Solution I'm in way over my head and not really sure what type of proof I should be using. In my course, we just went over some number theory and modular algebra so...
  48. O

    Derivative of a function involving square root of sum of squares

    Provided is a function f(x)=\sum_{j=1}^n ||x-x_j||, for x being a two dimensional vector, where ||.|| denotes the Euclidean distance in 2D space. How could one obtain a derivative of such a function?
  49. T

    Understanding Least Squares Statistics in Homework Problems

    Homework Statement I am not sure whether the answer should be b or d. It says the average of the function and the vertical scale need to be determined. In other words I won't have to determine a_0 (the vertical shift) right? Because it did not mention that? Also what is the average...
  50. E

    Factoring x^6 - y^6 as a difference of squares vs cubes?

    Homework Statement Factor x6 - y6Homework Equations a3 - b3 = (a - b)(a2 + ab + b2) a2 - b2 = (a + b)(a - b) The Attempt at a Solution I'm confused. x6 - y6 = (x2)3 - (y2)3 = (x3)2 - (y3)2 So shouldn't they all have the same factors? When I factor (x2)3 - (y2)3 = (x3)2 - (y3)2 I get...
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