# trivial

1. ### the intersection number between a trivial loop and a meridian in the torus

Let A and B be two closed curves intersect on the torus transversally at a point, the intersection index of the crossing point is defined to be positive if the tangent vectors to A and B form an oriented basis for the tangent plane of the torus and negative otherwise. Then the intersection...
2. ### non trivial solution

I was thinking that non trivial solution occur in following case say whether i am right or wrong 1) when number of equation = number of variables 2)determinant of co-efficient is zero According to first 2x+3y=0 5x+5y=0 4x+5y+7z=0 It shoud have non trivial solution.am i right? Thank you
3. ### Trivial Solutions of the modified Wave Equation

I recently found the solution to a modified wave equation u_{tt}=c^2u_{xx}-ru_t where u=0 at the x boundaries, u(x,0)=\phi(x), u_t(x,0)=\psi(x). I used separation of variables, and found that I had to have a positive separation constant (eigenvalue?) in order to get non-trivial solutions, which...
4. ### If G has trivial center, what does Aut(G) ≈ Inn(G) imply?

If H has trivial center. I know that if Z(G) is trivial, then Aut(G) has trivial center, but I need help understanding what is happening when G has a trivial center and Aut(G) ≈ Inn(G). I know that when Z(G) is trivial, then G ≈ Inn(G) but when does Aut(G) ≈ Inn(G) imply?
5. ### Trivial Solution of a system of equations

What do we mean when we say that a system of equations has only a trivial solution i.e x=0,y=0 .... etc. We conclude that these vectors are linearly independent. But what does that visually mean ? Is it a set of diverging lines from the origin ? Thank you in advance ...
6. ### Direct sum of trivial subspaces

Hi guys. i need help to prove the following: "Let W be subspace of V, such that there exist one and only one subspace W' for wich W\oplus W'=V. Prove that W must be a trivial subspace." this is what i've tried so far: let's suppose there exist only one subspace W'. and let's suppose it is not...
7. ### Proof of the trivial subring

I have to prove that the ring (\{0\},+,\cdot) is a subring of any ring (R,+,\cdot) Let S=(\{0\},+,\cdot) and R=(R,+,\cdot) then S is a subring of R iff (R,+,\cdot) is a ring and S \subseteq R and S is a ring with the same operations. As we know S has an identity element of 0 --> 0+0=0 It has an...
8. ### Trivial covering map

Hello. I'm reading Munkres Topology, and I'm in chapter 9, section 53. Right on the first exercise I'm missing something important and I don't know what. The exercise says: Let Y have the discrete topology. Show that if p: XxY --> X is projection on the first coordinate, then p is a covering...
9. ### What is a trivial cocone?

Hello, The limit of a diagram of shape \bullet \rightarrow \bullet \leftarrow \bullet is known as a pullback. If we take the colimit of the diagram of shape \bullet \rightarrow \bullet \leftarrow \bullet, then we get a trivial cocone. I was just wondering if anyone knew what "trivial" means...
10. ### Need help with a Num. analysis question for unique, trivial and multiple solutions

Finding 2 unknown coefficents in a System of Linear Equations Hey so I have a question about a topic in my Numerical Analysis class (Civil engineering major). Here it is, 1.Give all possible values of A and B for the following system of equations to have the following types of solutions or no...
11. ### Define a meteric on a X so the associated topology is trivial

What is the topology determined by the meteric on X given by d(x,y)=1 for x doesnt equal y and d(x,y)=0 for x=y? for me it looks the discrete topology. Is it right? can we define a meteric on a given set X so that the associated meteric topology is trivial.!
12. ### Fundamental and trivial question on triangle inequality.

It's surprising(for me) that i will ask this but i have never met this. (Nerd) It's well known(e.g Recent Advances in Geometric Inequalities, Mitrinovic, et al) that the following is true: A,B,C are sides of a triangle if and only if A>0, B>0, C>0, A+B>C, A+C>B, C+B>A Of course the...
13. ### show columns are linearly independent if homogeneous system has only trivial solution

Let A be an m x n matrix. Show that the columns of A are linearly independent iff the homogeneous system Ax=0 has just the trivial solution. I think I need to use the fact that a homogeneous system of n linear equations in n unknowns has a nontrivial solution iff rank(A) < n
14. ### trivial solution for a homogeneous system

Show that if A = $\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$ then Ax=0 has only the trivial solution if and only if ad-bc ≠ 0
15. ### Tangent bundle of a Lie Group is trivial

Hello, Let G be a Lie Group and g his Lie Algebra, that is the tangent space at e \in G I have to show that the map \phi: G x g -> TG, \phi(h,X)=dL_h[X], (whereas dL_h[X](f)=X(f \circ L_h) with L_h left multiplication ) is a diffeomorphism and a linear isomorphism on each fibre. I'm...
16. ### Trivial Problem...

Apologies in advance, I'm rusty and having some trouble and really want to make more of an effort to keep up with my current year. Have a simple differential: dx/dt = x^3 Initial condition x(0) = x0 Apparently a simple integration by parts yields: x(t) = x0 / (1-2x0^2 t)^1/2 I know i'm...
17. ### trivial nullspace, basis.

If the nullspace of a transformation is trivial, then is a basis for the nullspace the emptyset?
18. ### Trouble with seemingly trivial integral

Hello everyone, I have been having a lot of trouble from a seemingly simple integral: \int arctan (arctan x)dx . I say that it's trivial because arctan (arctan x) is a function that can be plotted and does not go to infinity anywhere. In fact, it looks more or less like arctan, which does...
19. ### A trivial problem . . . and solution

Here's something that turned up from time to time. I type: . . \text{[m{a}th]}\theta\text{[/m{a}th]} is in a right triangle. . . And: \text{[m{a}th]}\backslash \text{tan} \backslash\text{theta} = \backslash \text{frac}\{\text{opp}\}\{\text{adj}\}\text{[/m{a}th]} And this is...
20. ### trivial zeros of the Riemann zeta function

I have read that the negative even integers are zeros for the Riemann zeta function because the function satisfies the equation and at s being a negative even integer, the sin(pi*s/2) part vanishes. This argument would mean that the positive even integers would also be zeros for the same...