Euclidean Definition and 204 Threads

gcd(a,b) unique in Euclidean domain?? Homework Statement In Hungerford's Algebra on page 142, the problem 13 describes Euclidean algorithm on a Euclidean domain R to find THE greatest common divisor of a,b in R. My question is that does this THE mean THE UNIUQE? I've heard from my lecturer in...

Homework Statement I have questions along the line of Use the Euclidean Algorithm to find d= gcd(a,b) and x, y \in Z with d= ax +by Homework Equations The Attempt at a Solution Ok so I use the euclidean algorithm as I know it on say gcd (83,36), by minusing of the the...

I am reading Kane - Reflection Groups and Invariant Theory and need help with two of the properties of reflections stated on page 7 (see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7) On page 6 Kane mentions he is working in \ell dimensional Euclidean space ie...

I would like to know the distribution of z as the euclidean distance between 2 points which are not centred in the origin. If I assume 2 points in the 2D plane A(Xa,Ya) and B(Xb,Yb), where the Xa~N(xa,s^2), Xb~N(xb,s^2), Ya~N(ya,s^2), Yb~N(yb,s^2), then the distance between A and B, would be...

Homework Statement Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space. Homework Equations A topological manifold is a Hausdorff space that are locally Euclidean, i.e., there's an n such that for each x, there's a neighborhood...

Can someone please describe to me how Euclidean Geometry is connected to the complex plane? Angles preservations, distance, Mobius Transformations, isometries, anything would be nice. Also, how can hyperbolic geometry be described with complex numbers?

Here's the definition I have: Extended Euclidean algorithm Takes a and b Computes r, s, t such that r=gcd(a, b) and, sa + tb = r (only the last two terms in each of these sequences at any point in the algorithm) Corollary. Suppose gcd(r0, r1)=1. Then r_1-1 mod r_0=t_m mod r_0. The...

Hello all, I was wondering if Hawking's approach is still relevant. Found a book on his compilation of papers an amazon and had heard a talk by him suggesting it as a view to continue research. With all the hoo ha on M-theory and etc, would it be possible to buy this collection of papers for...

Homework Statement I have this question about Euclidean vectors. in a coordinate system vector r and s and t are given . (there is an arrow on top of r, s and t but i can't put it in l r l is 3,48 and creates an angle of 44,3 degrees with x (x is a straight horizontal line) l s l is 4,16...

Hi, I'm trying to fix in my head a very precise definition of what to mean for an euclidean space, as we use it in multivariable calculus. The def. I had in my mind was that an ES is (1) a real vector space (2) of finite dimension (3) with the "standard" (dot) (4) inner product I'm...

I am interested to know how to realize this abstract surface as a subset of Euclidean space. The surface as a point set is the 2 dimenional Euclidean plane minus the origin. the metric is given by declaring the following 2 vector fields to be an orthonormal frame: e_{1} = x\partialx -...

Homework Statement Please see below... Homework Equations Please see below... The Attempt at a Solution Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution... Thank you...

Homework Statement There's a couple of questions that require the use of this, I'm having trouble with one of them, could anyone help? Homework Equations a) 520x - 1001y = 13 b) 520x - 1001y = -26 c) 520x - 1001y = 1The Attempt at a Solution The first two are easy to do, where you set 1001...

Hi forum, Here is a little challenge that I came up with that some of you may find interesting:- Using only a compass, ruler and pencil, can you draw a circle, of any diameter you wish, and divide its circumference exactly into 360 equal sections? It seems to me that one should be able to...

Homework Statement Let a,b\in\mathbb{Z}. Suppose r_{0}=a and r_{1}=b. By the algorithm, r_{i}=0 for some i\geq 2 is the first remainder that terminates. Show that r_{i-1}=\gcd(a,b). Homework Equations The Attempt at a Solution I've shown that c|r_{i-1}, and I know that I should...

"1-norm" is larger than the Euclidean norm Define, for each \vec{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n, the (usual) Euclidean norm \Vert{\vec{x}}\Vert = \sqrt{\sum_{j = 1}^n x_j^2} and the 1-norm \Vert{\vec{x}}\Vert_1 = {\sum_{j = 1}^n |x_j|}. How can we show that, for all \vec{x} \in...

All the perutations of elements in R(3)(three dimension euclidean space) form a permutation group. This group must be E(3)(euclidean group)?

Can anyone please help me with this because I'm really getting confused. Thanks! In R, we know that fine topology is equivalent to the Euclidean topology as convex functions are continuous. Now if instead of R we consider a subset of it say [0,1], the fine topology induced on [0,1] would...

What is a non-unitary CFT? Why are Euclidean CFTs allowed to be non-unitary? I assume the opposite of Euclidean is Lorentzian? Why are those not allowed to be unitary? These questions are from listening to Hartman's talk that mitchell porter recommended...

Well, I created this thread (under Geometry/Topology) about the Law of Sines, specifically for the three kinds of geometries. http://en.wikipedia.org/wiki/Law_of_sines http://mathworld.wolfram.com/LawofSines.html The Law of Sines states that, for a triangle ABC with angles A, B, C, and...

(This could maybe turn out to be a little longer post, so I'll bold my questions) Hi, I was reading a little about affine geometry, and something bothered me. Namely, in some books, there were some paragraphs that were written like "blabla, let's observe an affine plane for instance, and...

Homework Statement Let X=R^2 and the distance be the usual Euclidean distance. If C and D are non-empty sets of R^2 and we have: C+D := {y ϵ R^2 | there exists c ϵ C and dϵD s.t c+d = y} A) What is C+D if the open balls are C= ball((0.5,0.5);2) and D=ball((0.5,2.5);1) B) Same as A)...

Hi all, ok so below is an example of the Extended Euclidean Algorithm, and i understand the first part perfectly to find the g.c.d. 701 − 2 × 322 = 57, 322 − 5 × 57 = 37, 57 − 37 = 20, 37 − 20 = 17, 20 − 17 = 3, 17 − 5 × 3 = 2, 3 − 2 = 1, and 1 = 3 − 2 = 6 × 3 − 17 = 6 × 20 − 7...

While R^1, R^2, ... , R^n comes quite naturally, is it even conceivable to ponder the meaning of R^(-n)? Is this something that even can exist conceptually or is it just jibberish? This was just a random thought that rolled into my head earlier today, and it's something that I think COULD...

The word Euclidean space is applied to various distinct mathematical objects. One, kind of Euclidean space is the affine space (general sense of "affine space") defined by the Euclidean group of isometries, which don't including scaling. But wouldn't Euclid's axioms apply equally well if we...

Hi, I was wondering if someone can set this right , I'm discussing this with another person that says that If (working in spherical coordinates) we make r constant in a Euclidean 3d space, in the resulting slice (phi-theta plane) we can define 2-spheres. I say that in my opinion you can't have...

It seems that the tangent bundle of a hypersurface of Euclidean space is the bundle induced from the tangent bundle of the unit sphere under Gauss mapping. Is this true? The reason I think this is that tangent space at a point on the surface can be parallel translated to the tangent space on...

Homework Statement Euclidean space is the set of n-tuples with some operations and norm. I suddenly wonder if complex euclidean space can be defined. Is it also defined? Homework Equations The Attempt at a Solution

1. Question Let b = r_0, r_1, r_2,... be the successive remainders in the Euclidean algorithm applied to a and b. Show that every two steps reduces the remainder by at least one half. In other words, verify that r_(i+2) < (1/2)r_i for every i = 0,1,2,... 2. Attempt at a solution I take an...

Hey, This may seem like a simple question, but hopefully someone can answer it quickly. Consider the Euclidean 2-metric ds^2 = dx^2 + dy^2 . There are three killing fields, two translations K_1 = \frac{\partial}{\partial x}, \qquad K_2 = \frac{\partial}{\partial y} and a rotation. Now...

http://arxiv.org/abs/1102.0270 "The first hazard is well known in Euclidean quantum gravity. It is called “minbus” or “baby universe” [6]. ... The approach is called “causal dynamical triangulation” (CDT) and has been shown numerically to provide “birth control” [9]"

Hello, I am learning about manifolds but I am not understanding part of the definition. This is what I'm looking at for defining the n-manifold M. (i) M is Hausdorff (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets I have a problem with (ii)...

Homework Statement My problem is to calculate to calculate killing vectors in 3-D euclidean space(flat space). Homework Equations The relevante equations are killing equation : d_a*V_b+d_b*V_a=0 The Attempt at a Solution I found the solution in Ray D'Inverno and that is...

Deviations from the vacuum energy bring about deviations from a Euclidean spatial geometry. Fine; I am not questioning this principle. I am wondering why a Euclidean metric is the base from which everything deviates? An answer that it is the limit of more general metrics only begs the question...

As a newbie, I apologize if this topic has been discussed before. It seems to me that one result of quantum physics is that Euclidean geometry is artificial and cannot be represented in real space. For example, there can be no such thing as a straight line in granular quantum space. And...

1. Maths project to investigate compass and straightedge constructions 2. Most of the project is fine, but i need to find out the mimimum number of constructions to bisect an angle, a line segment, etc. 3. I can prove that you can bisect an angle, and it requires 4 steps to do it...

Homework Statement Suppose n \geq 3, x,y\in\mathbb{R}^n, ||x-y||=d>0 and r>0. Prove that if 2r>d, there are infinitely many z\in\mathbb{R}^n such that ||x-z||=||y-z||=r. Homework Equations N/A The Attempt at a Solution Well, I figure that no matter how large we choose n, it...

Classical Mechanics define our space is E^3 that's also assumed to be R^3x R_t I just wondering what if Q^3x Q_t? will it make any significant difference? will it cause any logical paradox? thanks for reading!

Hi, I have some doubts about the precise meaning of Euclidean space. I understand Euclidean space as the metric space (\mathbb{R}^n,d) where d is the usual distance d(x,y)=\sqrt{\sum_i(x_i-y_i)^2}. Now let's supose that we have our euclidean space (in 3D for simplicity) (\mathbb{R}^3,d)...

Hi, Suppose a stationary frame S' is observing frame S moving with velocity v=0.866c in the x-direction, and let points (4,0),(10,0) define the ends of a rod in S, so its distance is 6, but as measured from S' contracts to 3 because of the Lorentz factor gamma. I'm unable to determine...

proof the following using only euclidean geometry: Let S be any point inside a triangle ABC and let SP; SQ; SR be perpendicular to the sides BC;CA;AB respectively, then SA + SB + SC >= 2 (SP + SQ + SR) Hint: Set P1; P2 be the feet of the perpendiculars from R and Q upon BC. Prove first...

Hello my friends! My textbook has the following statement in one of its chapters: Chapter 8:Topology of R^n If you want a more abstract introduction to the topology of Euclidean spaces, skip the rest of this chapter and the next, and begin Chapter 10 now. Chapter 10 covers topological...

I would like to know the basic experimental observations or the logic which prove that the 3-d space which we inhabit is a close approximation of Euclidean Geometry. is it because parallel lines don't appear to converge or diverge? But how is this established, as we can't draw perfect straight...

Homework Statement Three-dimensional, euclidean space. We've got 2 non-parallel planes: \vec{OX} \cdot \vec{b_1}=\mu_1 and \vec{OX} \cdot \vec{b_2}=\mu_2. Find all the points Y such that Y lies on the first plane and Y+\vec{a} lies on the 2nd one. What did you get? Homework Equations Come in...

i really need help with this proof. suppose two circles intersect at points P and Q. Prove that the line containing the centers of the circles is perpendicular to line segment PQ

Ok so I was just thinking and realized that instead of a Cartesian plot, you can represent points in an n-dimensional space by drawing n parallel lines and marking a point on each line. Of course this is less appealing than the traditional plot because we perceive 3d space in a way more similar...

What is the famous theorem in Euclidean geometry that states that the sum of the interior angles of a triangle is 180 degrees?

In this thread, I plan to try to explain (with some apropos ctensor examples) in a simple and concrete context some basic techniques and notions about Riemannian two-manifolds which also apply to general Riemannian/Lorentzian manifolds. Suppose we have a euclidean surface given by a C^2...

I opted to not use the template because this is a pretty general question. I am not understanding how to find out if a set is an open subset of a Euclidean space. For example, {(x,y) belongs R2 | x squared + y squared < 1} The textbook is talking about open balls, greatly confusing me.

Homework Statement Find a positive integer solution to 1234x-4321y=1, both x and y will be positive. Homework Equations The Attempt at a Solution I created this array 4321 1234 619 615 4 3 1 3 1 1 153 1 1082 309 155 154 1 1...