# All Questions

35 views

30 views

### Question about proving a real number

If we know that for any, $\alpha \in \{0, 2\}^\mathbb{N}$ that $0 \le \sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \le 3$, then what property of real numbers do we have to use to prove that ...
25 views

35 views

### What do rank-2 tensor entries represent? What's their geometrical meaning?

I just followed a course on tensor analysis and I think I understood almost everything. Except the following: given a vector (with, say, 3 dimensions), if I multiply that with a second-rank tensor, I ...
27 views

I'm confused on the partial sums formula Why is $$\sum_{i=m+1}^\infty \frac{2}{3^i}=\frac{1}{3^m},$$ if $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$
45 views

### Class group of $\mathbb{Q}(\sqrt[4]{-2})$

I would like to show directly that $C(K)$ is trivial, where $K = \mathbb{Q}(\sqrt[4]{-2})$. Write $\delta = \sqrt[4]{-2}$. It is pretty easy to see that $\mathcal{O}_K = \mathbb{Z}[\delta] = R$. Then ...
52 views

### What does this paper mean by “$f(x)$ is practically a rational function”?

The paper "Infinite integrals involving Bessel functions by contour integration" by Qiong-Gui Lin gives a method to solve integrals of the form $\intop_{0}^{\infty}x^{v}f(x)J_{v}(qx)\, dx$. One of the ...
39 views

### If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N.$$ Suppose there exists a continuous function $g$ on ...
107 views

### Limit of factorial function: $\lim\limits_{n\to\infty}\frac{n^n}{n!}.$ [duplicate]

I am studying for a test and I am given this problem: $$\lim_{n\to\infty}\frac{n^n}{n!}.$$ How do I go about solving this limit? Intuitively I see how the numerator is growing much faster, but how ...
28 views

### Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
64 views

### Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4$. Somewhere along the proof we would ...
44 views

### What is the probability that Brian makes money on his first roll

Brian plays a game in which two fair, six-sided dice are rolled simultaneously. For each die, an even number means that he wins that amount of money and an odd number means that he loses that ...
20 views

Let X and Y be independent continuous random variables with marginal CDFs given by $F_X(x) =\begin{cases} 0 & \text{for }x < 0\\ x/3 & \text{for }0 \leq x \leq 3\\ 1 & \text{for }x ... 1answer 42 views ### Chain rule application in fundamental Theorem of Calculus I have attached a question that I came across in trying to understand the fundamental theorem of calculus. The solution (highlighted with an arrow). I have difficulty understanding why the chain rule ... 1answer 18 views ### Can I convert between a rotation about an axis and a rotation according to two angles (all in 3D) without solving a system of nonlinear equations? I am writing a program that needs to be able to switch between a rotation described by 2 angles to a rotation described an axis and one angle. I found one way to do this from this question, which ... 3answers 155 views ### Is there any theorem about figures of equal area and perimeter being congruent? I had an idea, that all geometric objects, that are different, as they're not a translation, rotation, and a reflection of one another cannot have the same area AND perimeter, as compared to ONE ... 3answers 53 views ### Transformation of two independent uniform random variables Suppose$X,Y \sim \text{Uniform} \left(0,1 \right)$are independent. Then I need to find the PDF for$W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ... 1answer 23 views ### How to get polarised electromagnetic TE wave differential equation from Maxwell's Equations? I wish to understand how the following equation: \frac{\partial^2 E_x}{\partial y^2} + \frac{\partial^2 E_x}{\partial z^2} + n^2 k_0 E_x = \frac{\text{d} (\ln \mu)}{\text{d}z}\frac{\partial ... 1answer 18 views ### Rounding Percentages Okay so I have been having a problem and would really appreciate any help. I have posted a table below with the aim of making this as simple/precise as possible. ... 0answers 30 views ### Discontinuous linear operator on \ell^{2} Let e_{n} = (0, 0, \ldots, 0, 1, 0, \ldots) where 1 is in the nth position. Then \{e_{n}\} is an orthonormal basis for the Hilbert space \ell^{2}(\mathbb{N}). Does there exists a linear ... 3answers 98 views ### Find the value of this infinitely nested radical (it appears to obtain multiple values) Find the value of$$\sqrt{1-\sqrt{\frac{17}{16}-\sqrt{1-\sqrt{\frac{17}{16}-\cdots}}}}$$This is not as simple as it looks for one reason - there are 2 real solutions to the equation ... 2answers 37 views ### The union of all the open sets in a family of topologies I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let \{\tau_\alpha\} be a family of topologies on X. [...] To say that ... 2answers 68 views ### Polynomial Division - “Define the largest natural number…” [on hold] Would someone mind helping me with this question? The more detailed possible so I can have 100% of understanding. Thanks. Question: Define the largest natural number m such that the polynomial ... 5answers 47 views ### How to show if A is denumerable and x\in A then A-\{x\} is denumerable My thoughts: If A is denumerable then it has a bijection with \mathbb{N} So therefore A\rightarrow \mathbb{N}. Then x is a single object in A and A is infinite. So if a single object is ... 2answers 32 views ### P/1 Actuary Exam Question I was doing problems and came across this one and was wondering why the P[1<=x<=2] is F(2) - lim (x->1-) F(x) rather than F(2)-F(1)? Could someone please explain this for me? 0answers 49 views ### What is the inverse Fourier transform of |k|^{-\alpha}? What is the inverse Fourier transform, \mathcal{F}^{-1}\{|k|^{-\alpha}\}? I am specifically interested in the case where 1<\alpha<2. To do this, I need to compute the integral ... 1answer 38 views ### Find the volume generated! [on hold] Find the volume generated by revolving the curve bound by y=4-x^2, the y-axis and x-axis about the x-axis using the disk method. 2answers 18 views ### 55/45% Partnership distribution [on hold] Partner 1 owns 55% of a business and Partner 2 owns 45% percent of the business. If they sell the business for \500,000, how much more will Partner 1 make than Partner 2 of the \500.000? What ... 0answers 4 views ### Legendrian isotopy I think the definition of Legendrian isotopy is that you can find an isotopy of the ambient manifold such that it takes a Legendrian knot to another through Legendrian knots. What I cannot figure out ... 1answer 43 views ### How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell Solving a diffusion-type ODE across a spherical shell, the equation is:$$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$with boundary conditions f(r_1)=f_1 and f(r_2)=f_2. The solution is: ... 1answer 24 views ### For what values of a > 0 does f(x,y)=(x^{2}+y^{a})^{-1} belong to L^{1}([0,1]^{2})? I am trying to understand for what values of a>0 the function$$f(x,y) = \frac{1}{x^2+y^a}$$belongs to L^1([0,1]^2). I think a \geq 2 should work. But how to show that it is not the case ... 1answer 34 views ### How to calculate the number of integer solution of a linear equation with constraints? If an equation is given like this ,$$x_1+x_2+...x_i+...x_n = S$$and for each x_i a constraint$$0\le x_i \le L_i$$How do we calculate the number of Integer solutions to this problem? 1answer 24 views ### Linear operator on a hilbert space Let \{e_{n}\}_{n = 1}^{\infty} be an orthonormal basis of a Hilbert space H. Suppose T: H \rightarrow H is a linear operator. For each x \in H, then x = \sum_{n}\langle x, e_{n}\rangle e_{n} ... 1answer 41 views ### How many ways it can be done? There are 5 chairs. Bob and Rachel want to sit such that Bob is always left to Rachel. How many ways it can be done ? I have solved this in the following way: seats: * * * * * ... 1answer 42 views ### Double integral proof, where is my mistake? The bounds are 0 < x < b , 0 < y < b.$$ \int_0^b \int_0^b e^{-(x^{2}+y^{2})} dxdy $$Since it is a square, x=y so we can write: =$$ (\int_0^b e^{-(x^{2}+x^{2})} )^{2} dxdy $$=$$ ... 1answer 21 views ### Marginal densities of a joint distribution Let$X$and$Y$be continuous random variables having joint density$f$given by$f(x,y) = \lambda^2 e^{-\lambda y}, 0 \leq x \leq y$and$f(x,y) = 0$elsewhere. Find the marginal densities of$X$... 1answer 23 views ### Is a ruled surface of degree>2 always singular? Let$X=\mathbb{C}\mathbb{P}^3$and let$V\subset X$be a closed algebraic sub variety. By V-is ruled, I mean that for every point in$V\$ there is a line passing through it which also lies in V. ...

15 30 50 per page