Dimension Definition and 908 Threads

In general, how do you define the dimension of a singularity? E.g., we think of a Schwarzschild singularity as pointlike, so that its world-line is one-dimensional, and on a conformal diagram we represent it as a spacelike line, which seems to make sense. In point-set topology, we have...

Homework Statement find the dimension of the linear span of the given vectors v1 = ( 2, -3, 1) v2 = ( 5, -8, 3) v3 = (-5, 9, -4) Homework Equations The Attempt at a Solution so all i did was make it a matrix and put it in rref and i got [1 0 0] [0 1 0] [0 0 1] does this...

Hi, I was not entirely sure where to post this, but I think this will work. With the gravitational field we have that g^{\alpha\beta}g_{\alpha\beta}=4 which is the dimension of the manifold I believe. I have normally heard of g_{\alpha\beta} being interpreted as the gravitational field...

Consider the space of all polynomials in n variables of degree at most d. The dimension of that space is C(n+d,d). How do I calculate the dimension of that same space when I restrict the domain of the polynomials to the unit ball? In that case all the polynomials (sum(i=1..n) x_i^2)^p with p a...

I was hoping you guys could help me in understanding some vector spaces of infinite dimension. My professor briefloy touched n them (class on linear algebra), but moved on rather quickly since they are not our primary focus. He gave me the example of the closed unit interval where f(x) is...

Homework Statement A boy throws a ball with v velocity upwards as well as downwards . What will be the displacement covered by the two balls if air friction and external force or viscosity is neglected ? Homework Equations According to me the equations of motions are relevant here ...

My definition of diffeomorphism is a one-to-one mapping f:U->V, such that f and f^{-1} are both continuously differentiable. Now, how to prove that if f is a diffeomorphism between euclidean sets U and V, then U and V must be in spaces with equal dimension (using the implicit function theorem)?

With normal vectors i usually check there is the correct number of vectors i.e 3 for R3 2 for R2 etc and then just check for linear independence but reducing the matrix that results from c1v1+c2v2+..cnvn=0 and determining of unique solution or infinite solutions. There are the right number of...

I have a question regarding constructing subway platforms in curved line. As we know, in a curve, wagons get out of linear alignment and become closer in inner radius and separate from each other in outer radius. I want to know the relation between dimension of car, dimension of structure...

This question will probably make the most sense to those who have read Edwin Abbott Abbott's novel Flatland. But I'm sure many others know the answer. To explain, I'll have to use some dimensional analogy. Let's say you're a 2-dimensional being. You live in a two dimensional world and thus...

Homework Statement Having a symmetric tensor S^{a_1 ...a_n} forming a vector space V_n with indices taking values from 1 to 3; what is the dimension of such a vector space? Homework Equations The Attempt at a Solution essentially this reduces to picking a tensor of type S^{...

can a function in ONE dimension have NO inverse ?? i mean if given the inverse function f^{-1} (x) = g(x) + \sum_{k=-N}^{k=N}c_{k}exp(ixlogk) the first function g(x) is an smooth function , the last Fourier series is a 'noise correction' t o this function g , N is a big but finite...

In http://arxiv.org/abs/1010.1939 Rovelli describes his present model as a generalized TQFT. The current quantum groups stuff is also strongly TQFT inspired (as spin foams in general are). This makes me wonder whether LQG should not in fact be "higher dimensional". The reason is that I've...

Hi! It's been like two days since I have tried to make this work, still I got nothing. Searched Google etc. but no help there. I have a three dimensional matrix in form of {{a,b,c}, {d,e,f}, {g,h,i}, ... etc. } with a total of 51 elements, i.e. 51x3 matrix. What I want is to plot it as...

Box counting dimension! please help! Hi all, I am working on a problem from Chaos theory, I have to find the box counting dimension of the set {0}U{n^-p} where n is an integer and p>0. I started this problem by considering p=1. So, the set looks like {0,1,1/2,1/3,...}. If I take intervals...

Please help to find the dimension that will maximize...? The Park Service is building shelters for hikers along the Appalachian Trail. Each shelter has a back, a top, and two sides. Find the dimensions that will maximize the volume while using 384 square feet of wood. length (across the...

For SU(2) the three represented gauge fields are A_\mu^1, A_\mu^2 and A_\mu^3 and for U(1) the gauge field is B_\mu. The A_\mu^3 and B_\mu are electrically neutral. The photon \gamma and Z particle are combinations of these. My interest is the dimensions of the following parameters...

"Spacetime has No Time Dimension" http://www.dailygalaxy.com/my_weblog/2011/04/spacetime-has-no-time-dimension-new-theory-claims-that-time-is-not-the-4th-dimension.html#more

Homework Statement I'm currently trying to work my way through Frank Morgan's Geometric Measure Theory and I had a quick question regarding his definition of Hausdorff measure. Which is: Homework Equations $\mathscr{H}^m(A) = \lim_{\delta \rightarrow 0} \inf \left\{ \sum \alpha_m \left(...

Soft Article: http://www.sciencedaily.com/releases/2011/04/110420152059.htm Journal Article: http://prl.aps.org/abstract/PRL/v106/i10/e101101

I need some help with the equation of moments for this exercise: Each wheel of an automobile has a mass of 22 kg, a diameter of 575 mm, and a radius of gyration of 225 mm. The automobile travels around an unbanked curve of radius 150 m at a speed of 95 km/h. Knowing that the transverse distance...

I was given this question as a study question but no solution provided. I am unsure how to solve the question. Homework Statement A man is swimming 3m/s down a river. The man is 50m from the edge and 1000m from the top of the river. If I am in a boat at the top corner of the river...

Homework Statement If V and W are 2-dimensional subspaces of \mathbb{R}^{4}, what are the possible dimensions of the subspace V \cap W? (A). 1 only (B) 2 only (C) 0 and 1 only (D) 0, 1, 2 only (E) 0,1,2,3, and 4 Homework Equations dim(V + W) = dim V + dim W - dim(V \cap W) dim (V + W) \leq...

Homework Statement Two speakers are placed on the x axis, they produce monotonic sound waves of the same frequency. There is a microphone also on the x axis, but far away from the speakers. It detects a sound minimum when the second speaker is 20 cm behind the first. Then it detects a maximum...

Hi, I had a question about understanding some basic thing about the Hausdorff dimension. Specifically, I'm trying to understand why the two dimensional Hausdorff dimension of a 1-d line is zero. In terms of the two dimensional Lebesgue measure, I can see that I can cover the line by a...

Homework Statement x=ut- y^{2} * z^{2} / V x,y,z is length u is speed t is time V is volume The Attempt at a Solution m = m/s * s - (m^{2} * m^{2} / m^{3}) m=m- m^{4}/m^{3} m=m-m m=0 there is inconsistent? is this correct?

1.I am busy with an assignment based on a Vibration experiment in a Mechanical Engineering degree program The procedure is documented in the lab handout and one part is to compare the measured natural frequency to the calculated natural frequency The formula given in the handout for natural...

I have a similar question about rotation matrices. I'm trying to understand the dimension of the matrix given below which is a 3-D rotation. I think that its dimension is 3 but unsure. Any help appreciated. Thanks, John [(cosx sin x 0), (-sinx cosx 0), (0 0 1)] with ( ) = row,

Two bodies begin a free fall from rest from the same height. If one starts 1.0 s after the other, how long after the first body begins to fall will the two bodies be 10 m apart? x = x + vt +1/2at^2 I'm having trouble visualising a solution.

Is the 4th dimension time? 0D = dot 1D = line 2D = plane 3D = sphere 4D = time Am I correct?

Homework Statement Find a basis for each of the spaces and determine its dimension: The space of all matrices A=[a b, c d] (2x2 matrix) in R^(2x2) such that a+d=0 Homework Equations The Attempt at a Solution So I jumped at this question without knowing too much about spaces and...

Homework Statement 1. Consider three linearly independent vectors v1, v2, v3 in Rn. Are the vectors v1, v1+v2, v1+v2+v3 linearly independent as well? 2. Consider a subspace V of Rn. Is the orthogonal complement of V a subspace of Rn as well? 3. Consider the line L spanned by [1 2...

Homework Statement The mass of a van with a driver is 2000 kg . When the van accelerates, the velocity increases with a uniform acceleration of 3.0 m/s2. Homework Equations a) The van starts at rest. Find the velocity after 4.0 s. b) How far does the van travel in the first 4.0 s? c)...

Mathematicians are creating their own version of the periodic table that will provide a vast directory of all the possible shapes in the universe across three, four and five dimensions, linking shapes together in the same way as the periodic table links groups of chemical elements. The...

Homework Statement Prove that if A: V - >V is a linear map, dim V = n, and h1,...,hk (where 1,...,k are subscripts) are pairwise different eigenvalues of A such that their geometric multiplicities sum to n, then A does not have any other eigenvalues. Homework Equations Note sure if this is...

Homework Statement A rocket is launched from level ground with a speed of 30.0 m/s at an angle of 40 degrees above the horizontal. Find: a) Maximum height reached by the rocket. b) The time rocket is in the air. c) the horizontal distance where the rocket lands. Homework Equations...

Homework Statement Here is the problem verbatim (values have been slightly changed, also assume a frictionless environment): "A 4.3 kg block A and 6.0 kg block are connected by a string of negligable mass. Force FA = (15 N) acts on block A; force FB = (24 N) acts on block B. What is...

Homework Statement let V be a finite dimensional vector space of dimension n. For W \leq V define the codimension of W in V to be codim(W) = dim(V) - dim(W). Let W_i, 1 \leq i \leq r be subspaces of V and S = \cap_{i=1}^{r}W_i. Prove: codim(S) \leq \sum_{i=1}^{r} codim(W_i)Homework...

Homework Statement Two spacecraft are 13,500m apart and moving directly toward each other. The first spacecraft has in initial velocity of 525 m/s and accelerates at a constant -15.5 m/s^2. They want to dock, which means they have to arrive at the same position at the same time with...

Let A = k[x,y,z] and Y = \{(t,t^2,t^3)|t \in k\}, which is irreducible. It corresponds to the prime ideal p=(y-x^2,z-x^3). A(Y) is generated by x,y,z of degree 1 as a k-algebra in its graded ring structure. Each group corresponding to the degree d is spanned by the linearly independent...

Hi all The impulse response h(t) of an electric circuit (maybe in some special cases) is the derivative of the step response s(t) of the same circuit. right? So does it mean they have different dimension, namely if the dimension of s(t) is X, then the dimension of h(t)=ds/dt is x over...

Hi, I'm having trouble understanding a proof of the following theorem which allows it to be shown that all bases for a vector space have the same number of vectors, and that number is the dimension of the vector space (as you probably already know): Homework Statement Theorem: If S =...

First of all I would like to wish a happy new year to all of you, who have helped us understand college math and physics. I really appreciate it. Homework Statement Determine the dimension of the image of a linear transformations f^{\circ n}, where n\in\mathbb{N} and...

hi, what do you think that in general relativity, since space is curved, the curvature could be interpreted as a 4th space dimension "hyperspace"? in other words, the 4th space dimension would take the place of time, for a coordinate observer?

In three dimensions physical objects have an extent ( non-zero length ) along all three dimensions. If it is accepted that the four dimensions of general relativity correspond to physical reality, does that imply that physical objects also have a non-zero ( non-infinitesimal ) extent along...

1. Homework Statement Find the dimension of the set of 7x7 matrices with zero trace Relevant Equations The dimension of a standard basis matrix n x n is n^2 Zero trace = sum of diagonal elements = 0 Attempt at Solution I started with dim(M) = n^2 where M is an nxn matrix. Then I...

Homework Statement A football player kicks a field goal from a distance of 45m from the goalpost. The football is launched at 35 degrees above horizontal. What initial velocity is required so that the football just clears the goalpost crossbar that is 3.1m above the ground? ignore air...

How do we know that the (effective) dimensionality of "space" is three in a typical physical theory?

Homework Statement Let V\subset\,R^n be a subpace with dim(V) = D. Prove that any S vectors in V are linearly dependent if S > D. Homework Equations Rank-nullity theorem? The Attempt at a Solution dim(V) = D implies that there are D vectors in a basis for V. If S > D then there...

I just want to clear this up: why can't 5th till 11th dimension be represented in 3 dimensions no matter how small this extra dimensions are? One analogy for an extra dimension is a long rope that represents one dimension but this rope is composed of thinner twisting ropes which now represent...