# All Questions

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 ...
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 ...
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 ...
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 ...
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 $u_x={x\over\sqrt{x^2+y^2}}=v_y=0$ ...
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 ...
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 ...
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 ...
160 views

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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 ...
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., $d_D(\eta)$?
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. ...
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 $\{a_j\}$ ...
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 ...
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 ?
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 ...
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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
42 views

### A question about semigroup theory [closed]

How can I prove this lemma: $$[P]={P} \cup P.X \cup X.P\cup X.P.X,\quad \forall P\in X$$ ?
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 $\cos x = u$. ...
103 views
+50

### $\epsilon$ - 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 ...