0
votes
4answers
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
2
votes
1answer
17 views

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 ...
0
votes
0answers
10 views

Time advance in Adaptive Mesh Refinement method

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 ...
0
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
1
vote
1answer
16 views

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 ...
3
votes
2answers
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). $$\cos^2\frac{\pi}{13}+\cos^2\frac{3\pi}{13}+\cos^2\frac{4\pi}{13}=\frac{11+\sqrt{13}}{8}$$ ...
1
vote
1answer
21 views

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 ...
-4
votes
2answers
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?
-1
votes
0answers
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} ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
4
votes
0answers
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 ...
0
votes
3answers
62 views

Please help me understand the proof

Doubt In third line of the proof, why is Q $\rightarrow$ R ? Thanks
7
votes
2answers
98 views

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 ...
2
votes
3answers
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 ...
1
vote
4answers
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 ...
1
vote
1answer
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 $P_j(X_n=j \text{ for infinitely many } n ) =P_j(X_n=j \text { for some } n \ge m+1)$ ?
-5
votes
0answers
21 views

Translating a sentence to predicate logic 5 [on hold]

How to write "A and B but not C" in predicate logic?
1
vote
0answers
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 ...
2
votes
2answers
36 views

The $\exp \circ \log$ function acts as the identity on unipotent matrices.

I am working through the exercises in "Lie Groups, Lie Algebras, and Representations" - Hall and can't complete exercise 9 of chapter 2 using the provided hint. In chapter 2, Hall defines the ...
-1
votes
0answers
63 views

Goldbach conjecture related thinking

Background The sequences of integers of the form $a_k = 2(p-1)k + p$ yield many prime numbers when $p$ is prime and $k$ is a natural number. (Test it) $2p \equiv 2 \bmod{2p-2} \equiv 2 \bmod ...
0
votes
1answer
81 views

Math replacing natural language [on hold]

Before reading any further. I ask yous to think creatively on this subject. I was in shower and was pondering over A.I. (Strong A.I. both at human level and beyond human level) as I do from time to ...
0
votes
0answers
24 views

Eigenvectors of the companion matrix

Suppose one has an Hermitian square matrix $A$ with $p$ is the characteristic polynomial $$ p(x)= a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ and define the companion matrix of $p$ as $$ ...
1
vote
0answers
19 views

Example of $S[1/a] \cong S[1/b]$ as rings via $\phi$, where $S$ is a UFD, $a, b \in S$, and $\phi(U(S)) \neq U(S)$.

Above, $U(S)$ refers to the units of $S$. This problem stems from reading a paper titled "Translates of Polynomials" from 2000, where a fact about a ring isomorphism between $S[1/a]$ and $S[1/b]$ is ...
0
votes
1answer
22 views

Is linear independence preserved through column transformations?

Let's say you have a $m\times n$ matrix $A$, and the $n$ column vectors are linearly independent. And let's say you have a transformation $T$. You perform the transformation on each column of $A$, ...
0
votes
1answer
25 views

Question about the definition of convergence in measure.

In my text book convergence in measure ${{f}_{k}}\mathop \to \limits^{m} {f}$ is defined as "$\forall \epsilon> 0$ we have $\mathop {\lim }\limits_{k \to \infty } |\{ x \in \Omega :\left| ...
2
votes
1answer
41 views

Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$

Problem : Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$ Then which of the following is always true $(a) f(2) = f(7)$ $(b) f(4) = f(10) $ $(c) f(2) =f(4) $ ...
0
votes
0answers
78 views

Prove $\{Y: Y \text{ is a subset of } X \}$ is a set. TAO Analysis 1 Ex 3.4.6

The definition of a set I am given is as follows: $\text{ A set is defined as an unordered collection of objects. If x is an object then we say that } $ x $ \text{is an element of A, otherwise x is ...
2
votes
1answer
18 views

Solution of Graph Isomorphism in current literature.

As of 2008, the best algorithm for graph isomorphism (Babai & Luks 1983) has run time $2^{O(\sqrt(n log n))}$ for graphs with n vertices. Does this algorithm gives a yes / no answer or provide ...
0
votes
0answers
9 views

Existence of affine parametrization

This is a question from General Relativity by Wald Chapter 3, problem 5. Given either pseudo-Riemannian or Riemannian metric $g_{ab}$ and manifold $M$. Assume the $\nabla$ is compatible with the ...
-7
votes
0answers
33 views

Show that $Nil(\mathbb{Z}_n) $ is a subgroup of $\mathbb{Z}_n$ [on hold]

Show that $\mathrm{Nil}(\mathbb{Z}_n) = \{\bar{x}\in \mathbb{Z}_n\mid \bar{x}\,^m=\bar{0}\text{ for some positive integer $m$}\}$ is a subgroup of $\mathbb{Z}_n$.
1
vote
1answer
29 views

Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
0
votes
0answers
19 views

Singularity type for ratio of functions

Let $f,g:\mathbb{C}\to\mathbb{C}$ be two functions with a pole of order $n$ in $z_0$. I need to classify the singularity type of $\frac{f(z)}{g(z)}$ in $z_0$. I would say (intuitively) that $f \over ...
9
votes
2answers
121 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
-4
votes
0answers
37 views

A number theory problem. [on hold]

If $\gcd(a,b) =1$, prove that $\gcd(a-b+bm, a-b+bn) = 1$ where $n= a + bm$.
0
votes
0answers
17 views

Smoothly interpolating between functions to create a bouncing wave

How can I create a function which allows me to control the roundness of a wave so I can transition between an Round Wave -> Linear Wave -> Inverted Round wave? I've made a function which creates a ...
-1
votes
0answers
10 views

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given? [on hold]

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given ?
0
votes
0answers
49 views

What kind of algorithm used by the following Chinese man to calculate $6^{13}$ so fast? [on hold]

Consider this video. What kind of algorithm used by the following Chinese man to calculate $6^{13}$ so fast? He can also solve more complicated expressions.
1
vote
2answers
13 views

Complete Toroidal Graphs

I've seen it referenced that $K_N$ is a toroidal graph for $N \leq 7$. Can anyone supply a proof (source link or outline) that $K_8$ is not a toroidal graph?
0
votes
2answers
27 views

why dividing a number by 1.25 gives back 20 percent less of original?

So i had to takeout the discount from price. price = 10 discount = 20% my default method has been: price - price*discount ...
1
vote
1answer
25 views

Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
0
votes
1answer
38 views

Projective bundles

I am studying about projective bundles now. And I have the following doubts. 1) If we have an exact sequence of vector bundles over a scheme $X$, $0\longrightarrow E'\longrightarrow E\longrightarrow ...
-1
votes
1answer
18 views

How many structurally different latin squares of order 5 do exist?

I know the number of latin squares order 5 which start with 1 2 3 4 5 in the 1st row or column, that is 1344, but the greater part of that number consists of structural duplicates of each other. So, I ...
5
votes
2answers
114 views

Why is the remainder uniformly distributed when 1,2,3,… are divided by an irrational number?

Let remainder $r$ be defined as $$ r = n - pq $$ where $n \in \mathbb{N}$ is the dividend , $q \in \mathbb{R}$ is the divisor, and $p = \mathrm{floor}(n/q)$. I calculated the remainders by dividing ...
1
vote
2answers
57 views

Find max of $S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$

Let $x$, $y$, $z$ are positive constants. $A$, $B$, $C$ are three angles of the triangle. Find max of $$S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$$
1
vote
1answer
15 views

Expected Shortfall alternative definition

Define: $$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid F_L(x)\geq \alpha\rbrace}=VaR_\alpha(L)$$ I want to prove that: $$ES_\alpha = ...
1
vote
2answers
59 views

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to ...

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