0
votes
0answers
49 views

Morphism of affine varieties.

I'm confused by the following thing I'm reading on the Wikipedia page Affine variety: Let $X = \operatorname{spec} A$, $Y = \operatorname{spec} B$ where $A$, $B$ are integral domains that are ...
0
votes
1answer
34 views

Biyective graph homomorphism implies isomorphism.

Given this definition: If $G_1=(V_1,E_1),G_2=(V_2,E_2)$ are graphs, then $\varphi:V_1\rightarrow V_2$ is a homomorphism iff $\{v_1,v_2\}\in E_1\Rightarrow \{\varphi(v_1),\varphi(v_2)\}\in E_2$ I ...
0
votes
3answers
98 views

How could I prove whether the infinite series $n!/n^n$ converges or diverges? [duplicate]

How would you go about proving the infinite series $\frac{n!}{n^n}$ converges or diverges?
1
vote
1answer
57 views

Derivative of a function on a manifold

I want to show that: Given $f,g \in C^\infty(M)$ defined in a differential manifold of dimension $n$ and $a \in M$, we have $$(dfg)_a=f(a)(dg)_a+g(a)(df)_a,$$ using the following proposition: ...
0
votes
1answer
62 views

how to calculate sum of a series? (me or Wangenmakers is wrong)

Wagenmakers in his critical article about p-values wrote that: $$\sum_{i=12}^{\infty} {{n-1} \choose {2}} \cdot \left(\frac{1}{2}\right)^n \approx .033$$ How could he do his calculations if the D'...
2
votes
2answers
76 views

Product of functions in $H^1(B)$ where $B \subset \mathbb{R}^2$

I'm rather new to Sobolev spaces and finding myself rather deficient of intuition. So when given a problem like the below where I need to "prove or disprove", I'm finding myself stuck. Suppose $B$ is ...
0
votes
1answer
11 views

Standardising Bivariate Normal

I don't understand why $\sigma_{Z_1}=\sigma_{Z_2}=1$ and $\mu_{Z_1}=\mu_{Z_2}=0$. I would understand if: $X_1$~$N(\mu_{x_1},\sigma_{x_1}^2)$ and $X_2$~$N(\mu_{x_2},\sigma_{x_2}^2)$ but this is not ...
0
votes
0answers
44 views

Describe the image of following map $w= \frac{1}{z}$ with $|z-\frac {1}{2}| \leq \frac {1}{2}$

Describe the image of following map $w= \frac{1}{z}$ with $|z-\frac {1}{2}| \leq \frac {1}{2}$ I know that the inverse function map the inside of the circl to the outside of the circle and vice versa....
2
votes
0answers
56 views

Infinitesimal Generator of A One Parameter Group

This is a small problem which drives me crazy. Let $\varphi(x,y,t)=(F_1(x,y,t),F_2(x,y,t))$ be a one paramter transformation group on $\mathbb{R}^2$. Let $F_3(x,y,z,t)=\frac{D_1F_2+zD_2F_2}{D_1F_1+...
0
votes
2answers
79 views

Is this transformation surjective?

Consider the transformation $T:C_{\mathbb R} [0,1] \to \mathbb R$ defined by $T(f(t)) = \int_0^1 f(t)dt$. Is this transformation surjective? It would be enough to show that $$\mathbb{R} \subseteq\...
3
votes
1answer
75 views

Some questions about $\omega$-consistency and arithmetic soundness

Here are two theorems about formal proof systems I have attempted to prove, and for which I would like to check my understanding and clarify any missing details or loose ends. Theorem 1. If a formal ...
0
votes
1answer
67 views

Calculating the information per symbol of a markov chain source

I have a 4-state 2nd order markov chain source with symbols 0 and 1. I have all the transition probabilities and have worked out the probabilities of each state. How do I go about finding the amount ...
1
vote
4answers
59 views

Does the inequality $ n! > A \cdot B^{2n+1}$ hold for sufficiently large $n$?

Suppose $A,B >0$ are given constants. Is it possible to find a large enough $n \in \mathbb{N}$ such that $$ n! > A \cdot B^{2n+1}?$$
2
votes
3answers
95 views

Proof of limit of $\left(\sqrt{n+1}-\sqrt{n}\right)\left(\sqrt{n+\frac{1}{2}}\right)=\frac{1}{2}$

I just started Courant's "Differential and Integral Calculus" to really get a grip on the math I should need in engineering. I'm currently stuck on this limit, which is Exercise 7 in Chapter I - $5$ (...
1
vote
0answers
105 views

sketch given region an its image under given mapping $2 \leq Im z \leq 5$ and $w=iz$

sketch given region an its image under given mapping $2 \leq Im z \leq 5$ and $w=iz$ Here is what I got so far $z=x+iy$ so $2\leq y\leq 5$ and $w=-y+ix$ sp $-5+ix \leq w \leq 2+ix$ this implies ...
0
votes
1answer
105 views

Mobius Transformations and cross ratio

Find the mobius transformations satisfying each of the following. Write your answers in standard form, as $\frac{az+b}{cz+d}$ (a)1$\rightarrow$0,2$\rightarrow$1,3$\rightarrow$$\infty$ (use the cross ...
2
votes
0answers
181 views

Eigenvalues for $y''+2y'=\lambda y$

I must find the eigenvalues and eigenfunction for $$y''+2y'=\lambda y$$ with initial conditions $y(0)=0$, $y'(1)=0$. I have found the non-trivial case, and made an attempt to solve for $\lambda$, but ...
0
votes
2answers
176 views

Set with infinitely many limit points not contained in S

I'm trying to find a set S with infinitely many limit points but none of the limit points themselves can be contained in S.
1
vote
1answer
21 views

Solve $\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$

I have to solve $$\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$$ For $a_1 \leq a_2 \leq ...\leq a_n$ My intuition says that this x is a point in the middle of the $a_i's$ but I am not sure that it is ...
1
vote
3answers
55 views

integration of $\frac{x}{(x^2-1)^\frac{1}{2}}$

When I use the substitution $u=\cosh x$ to integrate $\frac{x}{(x^2-1)^\frac{1}{2}}$, I get $\frac{1}{2}(x+\sqrt{x^2-1}-\frac{1}{x+\sqrt{x^2-1}})$ but when I check online the answer is $\sqrt{x^2-1}$ ...
2
votes
2answers
88 views

Odd number of students in odd number of classes

In a school there are an odd number of classes, and each class has an odd number of students. We want to choose a school council consisting of one student from each class. Prove that the following are ...
1
vote
0answers
54 views

How to manipulate this summation in the easiest way possible?

$$ D = \sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f \...
2
votes
2answers
224 views

How to calculate the sum of this Series?

How do you calculate $$\sum_{i=0}^\infty {{2i \choose i}\over 4^i\cdot (2i+1)}$$ Wolfram gives ${\pi \over 2}$, but I have no idea how it got this.
1
vote
3answers
85 views

Is $\exists x(P(x) \to Q(x)) \equiv (\exists xP(x) \to \exists xQ(x))$?

My intuition is that this statement is false and here is my proof. $\exists x(P(x) \to Q(x))$ $\exists x(\lnot P(x) \lor Q(x))$ using logical equivalence. $\exists x\lnot P(x) \lor \exists x Q(x)$ ...
1
vote
1answer
176 views

Expectation - Sample Covariance

I am trying to derive the expectation $\mathbb E$ of the sample covariance $$\overline{cov}_{X,Y} := \frac{1}{n-1}\cdot \sum_{i=1}^n (X_i-\overline X)(Y_i - \overline Y)$$ where $\overline X = \frac1n ...
-2
votes
1answer
57 views

Probability: busy street intersection problem

At a busy street intersection, it is estimated that a jaywalker will be hit by a car with probability $0.01$. Assuming the individual trips to be independent, find the probability that a jaywalker ...
0
votes
1answer
48 views

Where is a good starting place to do research on the algebra of quaternion numbers?

I'm doing a project for my intro to real analysis class and decided that the algebra of quaternion numbers would be interesting to do. I'm wondering what a good starting place would be.
1
vote
3answers
76 views

Proving $\sum_{n=1}^\infty \frac{\xi ^n}{n}$ is not uniformly convergent

Proving $\sum_{n=1}^\infty \frac{\xi ^n}{n}$ is not uniformly convergent for $\xi \in (0,1)$. I am trying to do the above. I have attempted to show it is not a cauchy sequence by considering $||\...
0
votes
1answer
113 views

Proving the transform of the Q-function

I have the Gaussian Q-function, given by: and I want to prove that it can be also expressed as: Can somebody help explaining how to obtain the second integral from the first?
2
votes
1answer
61 views

What is the stochastic integral of $\frac{dW_t}{W_t}$

Does anyone know the solution to the Ito integral with the scaling factor on $dW_t$ being $\frac{1}{w_t}$? In other words what is: $\int \frac{dW_t}{W_t}$ ? It looks dangerously close to what ...
6
votes
2answers
140 views

If $\frac{a+b}2$ is rational, can we say that $a,b$ are rational?

The question is if it's given that $$ {a+b\over 2} \in \Bbb Q $$ prove or disprove $a,b \in \Bbb Q$. Since it is to disprove, I tried the following method by using examples. Take $$a = 1 + \sqrt{2} ...
1
vote
1answer
40 views

If $p$ is prime and congruent to $1$, then show $((\frac{p-1}{2})!)^2 \equiv -1 \pmod p$ [closed]

I got another one. Quadratic residues are completely new to me... Thanks!
1
vote
0answers
60 views

Let $X$ and $Y$ be finite CW Complexes. Prove that $\Omega(X \times Y)=\Omega(X)\Omega(Y) $, where $\Omega$ is the euler characteristics.

Let $X$ and $Y$ be finite CW Complexes. Prove that $\Omega(X \times Y)=\Omega(X)\Omega(Y) $, where $\Omega$ is the euler characteristics. I have thought in this way that if $X= e_{1}^{0}\cup e_{1}^{...
0
votes
4answers
43 views

How to do a Proof by Induction

I need to prove the following statement below by using induction. The problem is I have no clue what induction is and how I can approach it. Thanks! Prove by Induction: $$3 + 7 + 11 + \cdots+ (...
0
votes
1answer
29 views

Distribution of student $t$ ratio under the wrong mean

Suppose that we have an i.i.d. sample of size $n$: $X_1,\ldots,X_n\sim N(\mu_0,\sigma_0^2)$. Define: $$ t_n(\mu)\equiv\frac{\sqrt{n}(\bar{X_n}-\mu)}{s_X}\quad\text{where}\quad\bar{X_n}=\frac{1}{n}\...
1
vote
0answers
70 views

Relation between Nuttal Q-function and Gaussian Q-function

I am trying to express the famous Nuttal Q-function, given as: $$\mathcal Q_{m,n}(p,q)=\int_q^\infty t^me^{-0,5\left[p^2+t^2\right]}I_n(pt)\;dt$$ where $m$, $n$, $p$, and $q$ are constants and $I_{n}...
2
votes
0answers
53 views

Does the sum $\sum_{n=1}^{\infty}{a_nb_n}$ converge(fourier series coefficients)?

Let $f\in H(0,2\pi)$, with inner product $<f,g>=\int_0^{2\pi}{f(t)g(t)dt}$ $S_f=a_0 + \sum_{n=1}^{\infty}{a_ncos(nx)}+\sum_{n=1}^{\infty}{b_nsin(nx)}$, is the fourier series for f. Where $a_0=...
1
vote
1answer
77 views

The Definition of Definition by Recursion

The following is presented as the Transfinite Recursion on well-founded relations in Kenneth Kunen's book. Assume that $R$ is set like and well founded on $A$ and $\forall{x,s}\exists{!y}\varphi{(x,s,...
0
votes
1answer
59 views

Gradient in a rotated reference frame

I'm sure this is an embarrassingly simple question, but what is the gradient in a different reference frame whose axes are at an angle $\theta$ to the original frame? i.e. what is $\nabla=\left(\...
4
votes
1answer
50 views

Show that there are no two positive integers verifying this equality

I'm trying to show that there is no pair of strictly positive integers $\alpha$ and $\beta$ such that $$180n^2+218n+66 = \alpha (30n+18) + \beta(36n+22)$$ but so far I haven't quite managed to do it. ...
0
votes
1answer
58 views

Understanding derivatives in simple terms

Im am trying to understand the idea of derivatives and how they relate to the real world. I understand if i have function, in pkysics first derivative is the velocity, and the second derivative is ...
1
vote
2answers
39 views

Number theory: Proof: Prove that $3 \in QR_p \iff p \equiv \pm1 \pmod{12}$

Prove $3 \in QR_p \iff p \equiv \pm1 \pmod{12}$. I'm not too sure where to go with this one. I was trying to use the fact that, for the case of $3$, $3 \in QR_p \iff p \in QR_3$... Thanks!
0
votes
1answer
27 views

Categories with basepoints from forgetful functor

The example of the forgetful functor: $$U: \text{Vect}_K \rightarrow Set$$ mapping the category of vector spaces over field $K$ to Set yields the category of elements consisting of based vector spaces ...
1
vote
0answers
39 views

Show that if $p$ is prime , then $a^p \equiv a \pmod p$ holds, whether $p$ divides $a$ or not

i don't know where to start. think I'm over thinking it. but here is what I have but can really connect the dot. Let $\{0, 1, 2, \ldots, p - 1\}$ be a complete residue system modulo $p$. We show that ...
3
votes
3answers
238 views

The limit $\lim_{x\to 0}\frac{e^{x\ln(x)}-1}{x}$

I would like to show that $$\lim_{x\to 0^{+}}\frac{e^{x\ln(x)}-1}{x}=-\infty$$ without using: L'Hôpital's rule expansion series. My thoughts $$ \lim_{x\to 0^{+}}\frac{e^{x\ln(x)}-1}{x}=\...
0
votes
2answers
229 views

Dividing the interval in subintervals of equal length

I am asked to describe the operation of the processure BUCKET SORT at the array $$A=\langle 0.75, 0.13, 0.16, 0.64, 0.39, 0.20, 0.89, 0.53, 0.71, 0.42, 0.19 \rangle $$ dividing the interval $[0, 1)$ ...
1
vote
0answers
50 views

preservation in unions of chains

Let $K=\{A_i:i\in\omega\}$ be a countable chain of infinite (not necessarily countable) N−substructures, where N is a binary relation and let A be the limit (union) of K. Let Ax be a $\Pi_2$ ...
-1
votes
1answer
58 views

Find the limit. (If an answer does not exist, enter DNE.) $lim_{x \to \infty} {\sqrt{9x^2 + x}− 3x}$ [duplicate]

Is my process is correct? Also, why is one to allowed to divide the variable in the radical by its highest power? $$ \lim_ {x-> \infty} \sqrt { 9 x^2 +x} - 3 x $$
1
vote
1answer
40 views

Generalisation of Schwarz lemma

So, the theorem states that if f is holomorphic in $U = B(0,1)$ (so, $\{z \in C : |z| < 1\}$) and if it holds that $|f(z)| \leq R + |a|$ for all $z \in U$ and $f(0) = 0$, then it holds that: 1)$|f(...
0
votes
1answer
89 views

If for any $\varepsilon$ exists $\delta$, does that mean that for every $\delta$ exists $\varepsilon$? [closed]

For any $\varepsilon \gt 0$ exists $\delta \gt 0$, does that mean that for any $\delta \gt 0$ exists $\varepsilon \gt 0$? If $\delta$ depends on $\varepsilon$ such as $\delta = \frac 1 \varepsilon$, ...

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