All Questions
2
votes
1answer
35 views
Limit of sequence $u_1,u_3,u_5,\dots$ with $u_{n+1}=1+\frac{1}{u_n}$
We have a sequence of numbers defined recursively by $$u_{n+1}=1+\frac{1}{u_n},$$for $n\geqslant 1$. It is also given that $u_1=1$. Find the limit $l$ of the sequence $u_1,u_3,u_5,\dots$.
So I ...
2
votes
0answers
156 views
Books to understand the construction of all groups of a specific order
The algorithms introduced by Besche–Eick (1999) were used to construct (or count) the groups of order up to 2000 in Besche–Eick–O'Brien (2002), yet I find the algorithms somewhat inaccessible.
How ...
1
vote
1answer
35 views
Number of solutions for $\sum_{i=1}^{4} x_i < 22$ with condition.
I'm looking for the number of solutions to $\displaystyle\sum_{i=1}^{4} x_i < 22$ where $x_i > i$
Any help is appreciated.
I tried solving it using combinations to do $C(11,4)$ but that ...
0
votes
1answer
28 views
Is the Uniform family of Distributions dominated by the Lebesgue Measure?
The answer to this question should be fairly easy, but I can not just see it.
I want to say something like: let us consider a measure $P_{\theta}\in\mathcal{P}$ where ...
0
votes
0answers
36 views
$|z|$ is no where differentiable in complex plane?
Could any one tell me how do I show that $|z|$ is no where differentiable in complex plane?
$f(z)=\sqrt{x^2+y^2}$ if I apply CR equ I get $u_x={x\over\sqrt{x^2+y^2}}=v_y=0$
...
0
votes
2answers
30 views
What is the equation representing a constant elasticity of 1?
I'm reading the chapter in my textbook about the price elasticity of demand, and it was pointed out that most demand curves do not represent a constant elasticity of demand - even linear curves like ...
1
vote
1answer
36 views
A problem about holomorphic functions not continuous to the boundary.
I'm stuck in the the following argument. I believe the following to be true but not able to prove.
Let $\varphi_i\in\mathcal{O}(\mathbb{D},\mathbb{D}),\,i=1,\,2.$ Fix a point ...
1
vote
1answer
28 views
Finding the expectation and characteristic function of a mixed distribution.
I'm having difficulty with a practice exam question. Here's a modified version.
First, some notation. Let $E$ denote the exponential distribution, and $B$ the Bernoulli distribution. Also, given a ...
1
vote
1answer
160 views
Change of variables for triple integrals
Question: Consider the one-to-one transformation $(u; v;w) \to (x; y; z)$ defined by the equations $u = x + y + z; uv = y + z; uvw = z;$ which maps the unit cube $U$ defined by $0 \lt u \lt 1, 0 < ...
1
vote
1answer
51 views
A bounded sequence in $L^\infty$ has a weak-$^*$ convergent subsequence
Suppose $u_n$ is bounded in $L^\infty(\Omega)$. $\|u_n\|_{L^\infty(\Omega)}<M$.
Then $u_{n_k}\to u$ weak star in $L^\infty(\Omega)$ for some $n_k\uparrow \infty$ and $u\in L^\infty$.
I want to ...
-4
votes
0answers
45 views
Do carm exe 7 . section 1. [closed]
Let w be a differential n-form on G invariant on the left that is, L*x(w)=w for all x belong to G . prove that w is right invariant
G is a compact connected lie group . and dimG =n
2
votes
1answer
21 views
For given $p$, a regular mapping $F:M \to N$ of surfaces can be diffeomorphism
Want to show :
For given point $p$ of M, a regular mapping $F:M \to N$ of surfaces has a neighborhood $U$ such that $F|_{U}$ is a diffeomorphism of $U$ onto a neighborhood of $F(p)$ in N.
I learned ...
8
votes
1answer
126 views
Show that $2^n>1+n\sqrt{2^{n-1}}$
If $n$ be a positive integer greater than $1$, then prove that $$2^n>1+n\sqrt{2^{n-1}}$$
I found this problem under $AM \ge GM$ chapter. Help me to solve this problem using $AM \ge GM$. ...
0
votes
0answers
27 views
Task sort optimization
I am trying to come up with a solution to the following problem that will always give me the same answer if I run it again with the same parameters.
Let us say I need to schedule tasks (in sequence). ...
2
votes
0answers
28 views
Exercise in Mechanism Design
I found an exercise with solution in the field of Mechanism Design. The problem is I don't understand the solution.
Exercise. Use the characterization of incentive compatible direct-revelation ...
3
votes
0answers
72 views
Understanding Newman's proof of the prime number theorem
I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand.
In (III), ...
0
votes
1answer
28 views
How does this expression arise: $\pi(10.5) = \phi (-z_{1-\alpha} + \sqrt{n} \frac{\mu_0-\mu}{2})$?
$X_i$ is $N(\mu,\sigma^2)$ distributed and the following is given $H_0: \mu \geq 12, H_a: \mu < 12$, and $\alpha=0.01$. I'm asked to calculate $\beta=P[TII]$ if in fact $\mu=10.5$
Now this is the ...
0
votes
0answers
29 views
When is the image of a closed point closed under a morphism between schemes?
Let $f: X \rightarrow Y$ be a morphism between schemes. When is the image of a closed point closed? In another question , some remarks were already made. For example if $X$ and $Y$ are of finite type ...
9
votes
3answers
180 views
Why the sum of $n$ seems equals the period of the binary expansion of $1/n$?
The sum of $n$(suppose $n$ is positive odd,using $n=23$ as an example):
Step 1 : Get the odd part of $23 + ~~1 $, which is $~~3$,$~~3\times2^3=23 + ~~ 1$,get $s_1 = 3$
Step 2 : Get the odd part of ...
0
votes
1answer
40 views
Getting a values from nodes
The goal: get horizontal values of vertical level N where level 1 is pinacle node (1).
Example: level 4 as input should produce: | 1 | 3 | 3 | 1 |
Note: the sum ...
0
votes
2answers
52 views
What “whisker” means in box-and-whisker plot?
This is a bit off-topic but I can't help thinking about the reason behind naming box-and-whisker plot.
"Whisker" according to dictionary is "any of the long stiff hairs that grow near the mouth of a ...
3
votes
1answer
39 views
Monotonicity of $\ell_p$ norm
Consider a $n$ dimensional space, it is known (Wikipedia) that for $p>r>0$, we have
$$
\|x\|_p\leq\|x\|_r\leq n^{(1/r-1/p)}\|x\|_p.
$$
I have two questions about the above inequality.
$(\bf ...
0
votes
1answer
149 views
Finding set of values that satisfy constraint equation
We want to find set of all values that satisfy the following equation: $(a+ky)(a-ky)=gx$ All values are assumed to be nonzero integers. How does one set $x$ so that $a$ is not multiples of $y$ while ...
1
vote
0answers
27 views
Relation between semidirect product group and kernels of homomorphism
I want to ask a question related to semidirect product. If we have 2 groups $G$ and $H$ and 2 homomorphisms $\varphi_{1}, \varphi_{2}: H \to \operatorname{Aut(G)}$ and we know that ...
0
votes
0answers
18 views
Calculate price score
I want to create an algorithm/calculation that helps me figure out if the price on a used vehicle is high or low.
My thoughts are that to calculate this i need a critical mass of previous vehicle ...
0
votes
0answers
33 views
Integral transform of gaussian function
I am looking for an integral transform of $f(x)=\exp(-\frac{x^2}{2\sigma^2})$ such that:
$$(Tf)(u)=\int_{x_1}^{x_2}\, K(x,u)f(x)\,dx\,\stackrel{??}{=}\,\frac{a}{g(u)}$$
with $g(u)$ having zeros ...
5
votes
3answers
124 views
Arnold Trivium 49 : $ \oint_{|z|=2}\frac{dz}{\sqrt{1+z^{10}}}$.
Calculate : $$ \oint_{|z|=2}\frac{dz}{\sqrt{1+z^{10}}}.$$
If you find it too easy, then just post hints.
1
vote
1answer
43 views
The example related to separate sets in $\Bbb R^n$
I have another question. I am studying separate sets ın $\Bbb R^n$ and I have understand some theorems related to separate sets. And then, I see an example related to it. But I dont understand. I ...
1
vote
3answers
45 views
Linear Algebra: Orthogonal Theorem presented in two different ways. Why and how is it possible?
Book says:
According to the Pythagorean Theorem, two vectors are perpendicular if and only if:
Then, later is says: Let $u$ and $v$ be vectors in $\Bbb R^n$. Then u and v are orthogonal if and only ...
5
votes
2answers
85 views
Show that $\frac 1 2 <\frac{ab+bc+ca}{a^2+b^2+c^2} \le 1$
If $a,b,c$ are sides of a triangle, then show that $$\dfrac 1 2 <\dfrac{ab+bc+ca}{a^2+b^2+c^2} \le 1$$
Trial: $$(a-b)^2+(b-c)^2+(c-a)^2 \ge 0\\\implies a^2+b^2+c^2 \ge ab+bc+ca $$ But how I ...
0
votes
1answer
28 views
exterior covariant derivative of $ End(E)$-valued $p$-form
How can I calculate exterior covariant derivate of $ End(E)$-valued $p$-form i.e., $d_D(\eta)$?
5
votes
5answers
102 views
How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$
I don't know if I apply for this case sin (a-b), or if it is the case of another type of resolution, someone with some idea without using derivation or L'Hôpital's rule? Thank you.
...
0
votes
1answer
24 views
Sequences of i.i.d. subgaussian RVs and uniform integrability
Consider a sequence of i.i.d. subgaussian RVs $\{a_{j}\}^{n-1}_{j = 0}$; is $\{a^2_{j}\}^{n-1}_{j = 0}$ uniformly integrable (UI)?
Intuitively it appears to be so; if we take for example $\{a_j\}$ ...
3
votes
2answers
61 views
Sequence convergence and parentheses insertion
find an example for a series $a_{n}$
that satisfies the following:
$a_{n}\xrightarrow[n\to\infty]{}0$
${\displaystyle \sum_{n=1}^{\infty}a_{n}}$
does not converges
There is a way to insert ...
1
vote
1answer
28 views
Finding % of remaining students in the class would be girls?
In class of $120$ students, boys constitute $40$% of total. If $\dfrac 13^{rd}$ of boys and $4$ girls drop out of class to join a camp , what % of remaining students in the class would be girls ?
1
vote
2answers
23 views
what % of remaining of remaining coats are full length
A garment supplier stores $800$ coats in a warehouse of which $15$% are full length coats.
If $500$ of shorter length coats are removed from warehouse what % of remaining of remaining coats are full ...
2
votes
4answers
73 views
Using residue theory, show that $\oint_C\frac{e^{z/2}}{1+e^z}dz = 2\pi$
Using residue theory, show that $$\oint_C\frac{e^{z/2}}{1+e^z}dz = 2\pi$$
I've been attempted this problem using residue theory and the Cauchy integral formula that over a closed contour ...
2
votes
3answers
41 views
What is the mean score for the $20$ rolls?
A fair die is rolled twenty times. The results are shown in the bar graph. What is the mean score for the $20$ rolls?
2
votes
1answer
186 views
Period of linear congruential generator
How can you calculate the probability distribution of the period length of a linear congruential generator? That is $X_{n+1} = (aX_n + c) \bmod m$ where $a$ is chosen uniformly at random from ...
2
votes
1answer
37 views
Maximum cycle in a graph with a walk of length $k$
I don't understand why this stands:
Let $G$ be a graph containing a cycle $C$, and assume that $G$ contains a path of length at least $k$ between two vertices of $C$.
Then $G$ contains a cycle ...
0
votes
1answer
259 views
Volume of a cylinder: liquid vs solid
Apologies if this question is too much high school maths for the site, I'm a refugee from the webmaster stack site, and thought I might find some help here.
The question concerns candle making (in ...
4
votes
3answers
75 views
Abstract Algebra and Parallel Computing
I've recently been learning a bit about parallel computation. Two things I recently learned about are Reduce and Scan.
Where Reduce is defined as 2-Tuple of a Set of elements and a binary operation ...
2
votes
2answers
38 views
Laplace transform converging to zero
I have a sequence $X_n$ of random variables, whose Laplace transform $F_n(\lambda)=\mathbb{E}(e^{-\lambda X_n})$ satisfy for every $\lambda\geq0$
$$F_n(\lambda)\to1\qquad (n\to\infty).$$
Is it enough ...
-2
votes
0answers
42 views
A question about semigroup theory [closed]
How can I prove this lemma:
$$[P]={P} \cup P.X \cup X.P\cup X.P.X,\quad \forall P\in X$$
?
0
votes
1answer
116 views
Showing that $e^{ix}$ covers the whole unit circle
In my book analysis they argue that $e^{ix}$ covers the whole unit circle as follows.
Suppose that $w = u +iv$, such that $|w| = 1$.
Then if $v \geq 0$ we pick $x \in [0,\pi]$ with $\cos x = u$.
...
3
votes
0answers
103 views
+50
nonegative inverse eigenvalue problem
I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form:
$$
\mathbf{M} = \begin{vmatrix}
\mathbf{A} & \mathbf{b} \\
...
0
votes
1answer
29 views
Surface integrals of second kind
In the formula for calculating surface integrals of second kind, we have:
But, this integral is denoted by $\int \int _S \vec{F}\cdot \hat{n}dS $ . So, should we always normalize the expression $ ...
1
vote
2answers
33 views
Perpendicular distance question?
I've been having trouble with the following question:
"The point $P(x,y)$ is equidistant from the lines $2x+y-3=0$ and $x-2y+1=0$, which intersect at $A$. Use the distance formula to show that ...
3
votes
0answers
33 views
Lower bound on binomial coefficient
I encountered the following claim
$$\frac{1}{n+1}2^{nH_2(k/n)} \le \binom{n}{k} \le 2^{nH_2(k/n)}$$
where $H_2$ is the binary entropy function. The upper bound is rather well known but how does one ...
1
vote
0answers
33 views
$\epsilon$ - Nash Equilibrium exceeds Nash Equilibrium
Recently I found a pretty interesting exercise in the field of game theory.
Exercise. Show, for every $\epsilon$ > 0, a two player game where there is an $\epsilon$-Nash equilibrium in which both ...
