# All Questions

85 views

### Find limit of the following sequence?

Find the limit of $\frac{\log(n+1)}{\log(n)}$ where $n\rightarrow\infty$. Here $n$ is a natural number so I guess we can't use L'Hopital
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### Direct Limit of finitely generated groups

Is every group the direct limit of its finitely generated subgroups? This is true for abelian groups, I have not seen this statement for nonabelian groups, so i am wondering if this is true. Seems ...
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I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
43 views

### Direct Sum Proof

I am reading Axler's Linear Algebra Done Right Book and I am on the part about the direct sum. He gives the following proposition: When he assumes that $a$ and $b$ hold to prove that the proof gives ...
24 views

### A question regarding harmonic function.

Can any one provide some hint on the following question? I have being thinking about this for a while but cannot figure out where to start. I have been thinking about Taylor expansion but it seems not ...
17 views

### Looking for a way to apply the Taylor Series expansion to find derivatives for a function.

This post references the Riemann-Siegel formula found at here and at here. I am writing a Java program which implements this formula. I am having trouble with the remainder terms. The Riemann-Siegel ...
30 views

### Integral upper bound

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...
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### Changing a heaviside function into a one line function

$$h(t) = \left\{\begin{array}{l}1,\, \pi\leq t<2\pi\\ 0,\, 0\leq t<\pi\text{ and }t\geq2\pi\end{array}\right.$$ I need to change $h(t)$ into a one line function. I believe it to be ...
58 views

### How to prove an identity (Trigonometry Angles--Pi/13)

In this page http://mathworld.wolfram.com/TrigonometryAnglesPi13.html I found equation (11) and (12). $$\cos^2\frac{\pi}{13}+\cos^2\frac{3\pi}{13}+\cos^2\frac{4\pi}{13}=\frac{11+\sqrt{13}}{8}$$ ...
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### Sizes of Quotient Rings of DVRs with Finite Residue Field

If $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{m}$ such that $R/\mathfrak{m}$ is finite, then all quotient rings of $R,$ namely $R/\mathfrak{m}^n$ for $n \in \mathbb{N},$ are ...
58 views

### What application/deeper meaning do countable and uncountable infinities have? [on hold]

Georg Cantor proved that there are two different infinities but what application does this proof have? Is this result used in some other more useful theorem?
24 views

### An inverse Laplace transform I

While viewing the problem "Find the inverse Laplace transform" the solution given by Amir Alizadeh can be reformulated into the form \begin{align} \mathcal{L}^{-1}\left\{ \frac{s \, (a - f(s))}{s-b} ...
29 views

### When is using the gradient to calculate distance not accurate?

I've read that if you have a function like $y=f(x)$ or $z=f(x,y)$, that you can get the distance from a point $P$ to the closest point on the function by using the gradient. Specifically, you plug ...
12 views

### What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
18 views

### A good, self-study statistical computing book

I'm looking for a book an introductory statistical computing that has proofs for the methods as well as examples. I'd like proofs that are about the same level as (or lower than) proofs in Statistical ...
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Doubt In third line of the proof, why is Q $\rightarrow$ R ? Thanks
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### Proving a sequence converges when combinations of consecutive terms converge

Problem: Let $\{x_n\}$ be a sequence of real numbers such that $$\lim_{n\to\infty} 2x_{n+1}-x_n=L \in \mathbf{R}.$$ Prove that $x_n \to L$ as $n\to\infty$. I can see that if $\{x_n\}$ converges to a ...
34 views

### calculate two-fold difference

These are a series of numbers that increase two folds: $$0.125, 0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024$$ If I pick up two numbers, say $0.5$ and $128$, I want to know know how may ...
40 views

### Trigonometric substitution and triangles

I'm learning trigonometric substitutions and am having a bit of trouble understanding the intuition behind the conversions (why do most use secant?). If you could explain the conversions geometrically ...
13 views

### Discrete-time Markov Chains

I am having trouble understanding this proof from Markov Chains by Norris (1997) How do we get the equality $P_j(X_n=j \text{ for infinitely many } n ) =P_j(X_n=j \text { for some } n \ge m+1)$ ?
21 views

### Translating a sentence to predicate logic 5 [on hold]

How to write "A and B but not C" in predicate logic?
11 views

### What is the derivative of the ReLu of a Matrix with respect to a matrix

I want to compute $\frac{\partial r(ZZ^tY)}{\partial Z}$ where the ReLu function is a nonlinear operator $r(x)=max(0,x)$ and $Z \in\mathbb{R}^{n\times m}$ is a matrix. I am wondering also if the ...