# 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

{\displaystyle \square }
ABCD.

View More On Wikipedia.org
1. ### Inelastic collision of two squares

I know that the speed of the centre of mass is ##v_{cm}=(mv_0+mv_0)/(2m)=v_0##. But I don't know how to proceed from here with the angular speed around the centre of mass of the system. Any help will be appreciated.
2. ### I Exploring Nonlinear Least Squares for Regression Analysis

Hello, Regression analysis is about finding/estimating the coefficients for a particular function ##f## that would best fit the data. The function ##f## could be a straight line, an exponential, a power law, etc. The goal remains the same: finding the coefficients. If the data does not show a...
3. ### POTW Steenrod Squares over an Infinite Projective Space

Let ##u## be a generator of ##H^1(\mathbb{R} P^\infty; \mathbb{F}_2)##. Prove the relations $$\text{Sq}^i(u^n) =\binom{n}{i} u^{n+i}$$
4. ### B Combinatorics and Magic squares

Hi there. Happy new year. I am interested in magic squares. I am particularly interested in how to fill a square of order n in a symmetrical and logical way by analyzing the possible ways to achieve a given sum of numbers. My question is about combinatorics analyses. For example for a square...
5. ### A Convergence issue in this Least Squares calculation

I'm computing the trajectory of a moving body and my net is composed by 5 stations. My observations are DTOA: difference in time of Arrival (they have been linearized). I am trying to use Least Squares with a linear model: Y = Ax + b, where Y are the observed measurements (DTOA), A the design...
6. ### B Is ##\sum^n_{k=0} 2k+1 = n^2## useful? Has it been found already?

I was looking at the tiles of my home's kitchen when I realized that you can form squares by summing consecutive odd numbers. First, start with one tile, then add one tile to the right, bottom, and right hand corner (3), and so on. Can this be applied somewhere? And has someone found it already?
7. ### Proof: Twin Primes Always Result in Perfect Squares

Proof: Suppose ## p ## and ## p+2 ## are twin primes. Then we have ## p(p+2)+1=p^2+2p+1=(p+1)^2 ##. Thus, ## (p+1)^2 ## is a perfect square. Therefore, if ## 1 ## is added to a product of twin primes, then a perfect square is always obtained.
8. ### I Absolute difference between increasing sum of squares

Given are non-negative integer variables ##x##, ##y## and ##z##. I am trying to deduce the absolute difference between a certain value of ##C=x^2+y^2+z^2## and the very next smallest increase in ##C## possible. I'd like to do this so I can (dis)prove the following: Whether small absolute...
9. ### I Relationship between factorials and squares of natural numbers

Was fooling around and wrote down these two equations today that appear to work. I'm not all that bright and I'm positive these either have some proof or restate some conjecture--probably something in a textbook. Could somebody help me out? \forall n \in \mathbb{N}_0\smallsetminus\{0\} n^2 =...
10. ### MHB Least squares regression line (I'm very lost)

Hi! Basically this is the exercise: Given the covariance of x and y is -12 and the variance of x is 6,5, using the least squares line of best fit connecting x and y yo estimate the value of x when y=15 x 2 5 9 7 9 10 7 y 25 17 11 10 8 7 13 any help would mean everything, I'm desperate :(
11. ### MHB Why x2 +1 and x2 -1 are Not/Are Difference of Squares

Explain why x2 +1 is not a difference of squares and x2 -1 is
12. ### Applying least squares to measurement of nuclei masses and Q-value

Consider the problem in the attached image. The difference between A and B is 0.0020(20). How does one use the least squares method, particularly in matrix form, to find the best value of the masses of A and B respectively, as well as the Q-value? Aren't more measurements needed for the masses...
13. ### Using Least Squares to find Orthogonal Projection

I'm a little confused how to do this homework problem, I can't seem to obtain the correct answer. I took my vectors v1, v2, and v3 and set up a matrix. So I made my matrix: V = [ (6,0,0,1)T, (0,1,-1,0)T, (1,1,0,-6)T ] and then I had u = [ (0,5,4,0) T ]. I then went to solve using least...
14. ### Finding least squares solution of Ax=b?

Does anyone know the command or how to find the least squares solution of Ax=b on Ti-89 graphing calculator? I'm trying to check my answers on Ti-89 for those linear algebra problems.
15. ### I Can New Theorems Predict Incomplete 5x5 Magic Square Solutions?

A magic square is a NxN array of numbers from 1 to N2 such that the sum of elements of each row, column and diagonal adds up to the same number, 65 in the case of 5x5's. An example would be: 15 19 4 7 20 6 14 24 16 5 9 21 1 22 12 10 3 23 18 11 25 8...
16. E

### B Question about squares of operators

The magnitude of the momentum ##p## satisfies ##p^2 = p_x^2 + p_y^2 + p_z^2## and this implies the operator equation ##\hat{p}^2 = \hat{p}_x^2 + \hat{p}_y^2 + \hat{p}_z^2##, so we can say that ##\hat{p}^2 = -\hbar^2 (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} +...
17. ### MHB Dimension of the cut-out squares that result in largest possible side area

A topless square box is made by cutting little squares out of the four corners of a square sheet of metal 12 inches on a side, and then folding up the resulting flaps. What is the largest side area which can be made in this way? What information I have so far is that since the side of the...
18. ### I Least squares line - understanding formulas

Hello. I have listened to a great lecture, which gave helpful intuitive insight into correlation and regression (basic stuff). But there are formulas, which I cannot grasp intuitively and don't know their origin. To remember them I would like to understand what's happening in each part of the...
19. ### Zero Limit of Sum of Squares of Terms with Bounded Range

I don't know how to show that this limit is zero. It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one. Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.
20. ### I Linear least squares regression for model matrix identification

Summary: I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space. I need to do a linear model identification using least squared method. My model to identify is a...
21. ### I Newton-Raphson in Least Squares: How is it used? Cost Function?

I just went over analysis of a data set that was analzed using Linear Regression (OLS, I believe) and I saw Newton's method was used. Just curious, how is it used? I assume to minimize the cost function, but this function was not made explicit. Anyone know? Thanks.
22. ### MHB Least squares method : approximation of a cubic polynomial

Hey! :o I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method. So we are looking for a cubic polynomial $p(x)$...
23. ### MHB Finding Squares in a Square Box: n > 1

Determine all integers n> 1, for which in the square box of dimensions (n x n) you can enter different squares of integers, so that the sum of numbers in each row and in each column of the array is a square of an integer, and all the 2n sums are different.
24. ### MHB Optimizing Fuel Usage on a Circular Route with Squares: A Mathematical Proof

Hey! :o I am looking at an exercise that is formulated as follows: Finite number k of squares on a circular route. The whole fuel in all is enough for 1 circle. Show that there is a way to integrate the circle however the squares and the fuel are distributed. There is also the following...
25. ### I Other linear fitting than least squares

I'm analysing some data and my task is to get a line that best fits the data, using least square I'm getting these dashed curves (red and blue) with low correlation factors. Is there another method that takes into consideration the amount of data placed into the direction of a line?
26. ### I A formula for the number of structures composed of n squares

Hello, The problem I came up with deals with the structures that can be obtained by joining squares side to side or corner to corner. Specifically to this problem, structures, that are symmetrical to each other, are regarded the same. Ideally, I am looking for a formula that will tell how many...
27. ### MHB Ratio of areas of squares - Challenging problem

Hello all, I have encountered a very difficult question in geometry. The question has several parts. I really need your help. I have tried solving the first and second ones, not sure I did it correctly, and certainly don't know how to proceed and what the results means. I would really...
28. ### MHB How many regions can be obtained by drawing two squares?

by drawing two circles, Mike obtained a figure, which consists of three regions . at most how many regions could he obtain by drawing two squares? please can someone can explain...
29. ### I Can we construct a Lie algebra from the squares of SU(1,1)

I am trying to decompose some exponential operators in quantum optics. The interesting thing is that the operators includes operators from Su(1,1) algebra $$[K_+,K_-]=-2K_z \quad,\quad [K_z,K_\pm]=\pm K_\pm.$$ For example this one: $$(K_++K_-)^2.$$ But as you can see they are squares of it. I...
30. ### A Confused about Weighted Least Squares

I am trying to use Weighted Least Squares with a linear model: Y = Xβ + ε, where Y are some observed measurements and β is a vector of estimates. For example, in this case β has two terms: intercept and slope. The weighted least squares solution, as shown here, involves a weight matrix, W...
31. W

### Capacitor - Least squares fitting

Homework Statement We had a laboration for calculating ε_r in a parallel plate capacitor which we stuffed with plastic plates. All data we picked up was the area A, the distance d (and thus 1/d) and the capacitance C. We are now supposed to use the least squares-method to find ε_r, something we...
32. D

### Factoring Involving the Difference of Two Squares

Homework Statement Completely factor: $$5abc^4-80ab$$ Homework Equations N/A The Attempt at a Solution $$5abc^4-80ab$$ $$5ab\left(c^4-16\right)$$ $$5ab\left(c^2+4\right)\left(c^2-4\right)$$ [/B] The correct solution is ##5ab\left(c^2+4\right)\left(c+2\right)\left(c-2\right)## I can see...
33. ### MHB Counting Squares Challenge: Proving Formula and Evaluating Sum

We have an $n \times n$ square grid of dots ($n \ge 2$). Let $s_n$ denote the number of squares that can be constructed from the grid points. (a). Show, that $$s_n = \frac{n^4-n^2}{12}.$$ Note, that squares with "diagonal sides" also count. (b). Evaluate the sum: \[S = \sum_{k = 2}^{\infty...
34. ### Dyadic squares inside of a disc

Homework Statement Given ##\epsilon## > 0, show that the unit disc contains finitely many dyadic squares whose total area exceeds π − ##\epsilon##, and which intersect each other only along their boundaries. Homework EquationsThe Attempt at a Solution I don't really understand what the problem...
35. ### I Shaky model in least squares fit

I've come across a problem with my least squares fits and I think someone else must have analyzed this, but I don't know where to find it. I have a converged least squares fit of my spectroscopic data. Unfortunately, the physical model, on which the fit is based, is mediocre. The deviations...
36. ### MHB X^6 - y^6 As Difference of Squares

Factor x^6 - y^6 as a difference of squares. Solution: (x^3 - y^3)(x^3 + y^3) The problems states to use the difference of squares. I can apply the difference of cubes to the left factor and the sum of cubes to the right factor but how do I continue using the difference of squares?
37. ### Comp Sci C++: Method of Minimum Squares used for System Solving

Homework Statement Okay, this one is a bit big, and I'm attempting a translation, so I'm going to post a TL;DR version at the bottom just to be sure. Anyway, here goes: While doing an experiment, we write down the values of a physical/natural size y, for various values of a physical/natural...
38. ### MHB Rectangles & Squares - Finding a Numerical Measure

Hello I am looking for a mathematical measure, that will tell me, numerically, how far is any rectangle from a being a square. One obvious measure is the ratio between the sides of the rectangle. If the ratio is 1, it is a square. This measure is good, as it preserves a very important...
39. ### MHB What is the total area of the infinite number of inscribed squares?

Given a circle (radius $R$) with an inscribed square. Now inscribe a new circle in the square and then again a new square in the new circle etc. What is the total area of the infinite number of inscribed squares?
40. ### MHB Which method is easier and why?

I found the following problem in Section 1.3 of my Precalculus textbook by David Cohen. Factor x^6 - y^6 as a difference of squares and then as a difference of cubes. 1. Which method is easier and why? 2. Can someone get me started?
41. ### I Do perfect squares exist in physics?

Hello, I am a visual artist and currently working on a project about ‘being’. Under the assumption that ‘being’ or ‘to be’ is never square I was wondering if in physics (or astronomy/cosmology/biology etc) perfect squares or 90degrees corners exist. Isn't a square and perfect 90degrees corner...
42. ### I Matrix Mechanics and non-linear least squares analogy?

I have some experience with non-linear least squares curve fitting. For instance, if I want to fit a Gaussian curve to a set of data, I would use a non-linear least squares technique. A "model" matrix is implemented and combined with the observed data. The solution is found by applying well...
43. ### I Trying to understand least squares estimates

Hi, I'm trying to understand which mathematical actions I need to perform to be able to arrive at the solution shown in the uploaded picture. Thank you.
44. ### (Number theory) Sum of three squares solution proof

Homework Statement Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality." Homework Equations The Attempt at a Solution My informal proof attempt: Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4 Then x2, y2, y2 = (0 or 1) mod 4 So x2 +...
45. ### I How Does Summing Cubic Expansions Reveal the Formula for Sum of Squares?

I found a deduction to determinate de sum of the first n squares. However there is a part on it that i didn't understood. We use the next definition: (k+1)^3 - k^3 = 3k^2 + 3k +1, then we define k= 1, ... , n and then we sum... (n+1)^3 -1 = 3\sum_{k=0}^{n}k^{2} +3\sum_{k=0}^{n}k+ n The...
46. ### I Hessian of least squares estimate behaving strangely

I am doing a nonlinear least squares estimation on a function of 14 variables (meaning that, to estimate ##y=f(x)##, I minimize ##\Sigma_i(y_i-(\hat x_i))^2## ). I do this using the quasi-Newton algorithm in MATLAB. This also gives the Hessian (matrix of second derivatives) at the minimizing...
47. ### I Sum of squares of 2 non-commutating operators

Prof Adams does something rather strange, starting from 14:35 minutes in this lecture -- http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-videos/lecture-9/ He reminds us that for complex scalars, ##c^2+d^2=(c-id)(c+id)## and then proceeds to do the same with...
48. ### 24 divides m if n^2 -m and n^2 +m are perfect squares

Homework Statement Show that if n2 + m and n2 - m are perfect squares then m is divisible by 24. Homework Equations This problem comes from Larson's Problem-solving Through Problems in the section on modular arithmetic. It is the third part of a four-part problem, with the previous two parts...
49. ### I Two integers and thus their squares have no common factors

Integers ##p## and ##q## having no common factors implies ##p^2## and ##q^2## have no common factors. Could you prove this without using the fundamental theorem of arithmetic (every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique...
50. ### Least squares approximation of a function?

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...