## Complex Eigenvalues and systems of 1st-order ODEs February 12    2

I am trying to solve \begin{aligned} \dfrac{dx}{dt} &= x+ y \\ \dfrac{dy}{dt} &= -10 x- y \end{aligned}.My strategy is to let $P = \left[\begin{array}{cc} 1 &1\cr -10 & -1 \end{array}\right]$ and find $\det(P - \lambda I) = 0$ and I get

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$−ψ''(x) − (ix)^ N ψ(x) = Eψ(x).$$ where $N$ can be any real number, the boundary condition $ψ( ... ## Determine all$k$such that$k^3+k+1$is divisible by 11 February 12 The task is the following:Determine all$\ k\in\mathbb Z$such that$k^3+k+1$is divisible by 11I assumed that "$k^3+k+1$is divisible by 11" is saying$11|k^3+k+1$. That means I can rewrite it as a linear combination $$k^3+k+1 = 11n\quad\forall ## Complex 3-D Euclidean spaceinner product February 12 1 1st question: Lets say we have a 3-D complex euclidean space. How do we geometrically draw this space? if 3-D real Euclidean space is represented by these base vectors:2nd question: Is there ... ## Fourier Series with Complex Exponentials February 12 3 In my Signals and Systems class, we learned that the Fourier Series of a signal x(t) is given by$$ x(t) = \sum_{k = -\infty}^{\infty} X_k e^{ik\omega_0t} $$where \omega_0 = 2\pi/p and$$ X_k = \frac{1}{p} \int_0^p x(t) e^{-ik\omega_0t} \, dt. $$I ## Calculate limit (\frac{2x+1}{x-1})^x as x goes to \infty February 12 2 I have to calculate the following limit:$$\lim_{x\rightarrow \infty} \left(\frac{2x+1}{x-1} \right)^x\lim_{x\rightarrow \infty} \left( \frac{2x+1}{x-1} \right)^x=\lim_{x\rightarrow \infty} \left(1+\frac{x+2}{x-1} \right)^x=\infty$$But is 2^\inft ## Complex euclidean tensor products February 12 Say you have Euclidean vectors \mathbf{a}=a_i \mathbf{p}_i and \mathbf{b}=b_j \mathbf{q}_j in \mathbb{R}^3, with bases \mathbf{p}_i and \mathbf{q}_j. Then you could use a typical inner product to find \mathbf{a}\cdot\mathbf{b}=a_i b_j (\ma ## prove that if a1 , then the limit of \frac{a^n}{n} is \infty as n goes to \infty February 12 1 prove that if a>1 , then the limit of \frac{a^n}{n} is \infty as n goes to \inftyI was trying to use the binomial theorem to replace a with 1+k, but then that n on the denominator became a problem, should I use a different term for the ## If the tensor product of algebras A \otimes B is unital, both A and B must be unital February 12 It is clear that if A and B are unital algebras (over a field), then the tensor product A \otimes B is also unital, with the unit being 1_A \otimes 1_B. I came across an exercise that questions about the converse statement. That is, if A \ot ## Measurablity of functions defined over sections of product measures February 12 I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it?Let (\Omega_1, \mathcal{F}_1) and (\Omega_2, \mathcal{F}_2) be measurable spaces. Take \mathcal{F}_1\otimes \mathcal{F}_2 ## Chain rule for the distributional derivative February 12 Do we have a chain rule for the distributional derivative? My guess is yes, but I do not know how to justify that.Can some one point out how to prove/disprove that?Thanks! ## assigning K_{m_i} (cmplte grf) to the i th vertex of a graph and 'join' if the corrs. vertices are adjac. is called February 12 1 Given a connected graph G with n vertices and given set of \{m_1,m_2,...,m_n\} n integers, we form a noe graph G^ by considering the complete graph K_{m_i} for each vertex i and 'join' (in the sense of graph theory) two of such complete g ## Hatcher deduce ring structure on \mathbb{R}P^\infty;\mathbb{Z} February 12 So in Hatcher they deduce the ring structure of H^\ast (\mathbb{R}P^\infty;\mathbb{Z}) by looking at the map \mathbb{Z}\rightarrow\mathbb{Z}_2, which induces maps on H^\ast(\mathbb{R}P^ ... ## Exponential of a complex variable February 12 1 Can someone please tell me if I am approaching this correctly? Given the following and asked to solve for the complex variable z:$$[e^z]^e^z=0$$My approach was purely algebraic and is why I have my doubts:$$[e^z]^3=5e^z[e^z]^2=5z=\frac ## Eigenvalues of matrix summation February 12 1 Let$A$be symmetric positive definite matrix with eigenvalues$\lambda_1,\lambda_2,\dots,\lambda_n$. Can we express the eigenvalues of$I-A$using eigenvalues of$A$? I can't find properties of eigenvalue related to this problem.Suppose we could dia ## Kernal of a map in Graph Theory (toric ideals) February 12 If we have an$n-$cycle with edges$e_1 =\{x_1,x_2 \}, e_2 = \{x_2, x_3 \},\dots, e_n = \{x_n,x_1\}$with a$K-$algebra homomorphism$\phi: k[e_1,\dots, e_n] \to k[x_1,\dots, x_n]$defined by$\phi(e_i) = x_1x_{i+1}$with$n \neq 3$. What is the kern ## How to compute the p value and the correct explanation of the overall experiment.(Is my answer correct) February 12 Hello community first of all thanks for helping me with my math problems. Here I'm again with Hypotesys test exercise. I want to know if I made some mistake in my answer and if someone can help me to calculate the value p for the problem, (I'm not su ## Show that$Y_1t- Y_2t \to 0$as long as$t \to \infty$Differential equations February 12 Let the differential equation$L[y] = a y'' + by' + cy = g(t)$, where$a$,$b$and$c$are strictly positive numbers. If$Y_1(t)$and$Y_2(t)$are solutions at the$L[y]$equation, show that$Y_1[t]- Y_2[t] \to 0$as long as$t \to \infty. I know th ## How to write absolute value as a true function February 12 Here's the basic absolute value ... what?\begin{align*} |x| = \left\{ \begin{array}{r@{\quad \mathrm{if} \quad}l} x & x > 0, \\ 0 & x = 0, \\ \!\! -x & x < 0. \end{array} \right. \end{align*}Is this a function, an "expression," You Might Also Like • I read that three vectors in\mathbb R^3$are coplanar if one is a linear combination of the other two. 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