2
votes
1answer
78 views

Identity between roots of polynomials

Let $A\in{\mathbb C}[X]$ be a monic polynomial of degree $n\geq 2$, with roots $\alpha_1,\alpha_2,\alpha_3, \ldots ,\alpha_n$. Let $B$ be the polynomial $$ B=\prod_{k=1}^{n} ...
0
votes
1answer
101 views

power series for $\tanh^{-1}(z)$

I'm trying to derive the power series for $\tanh^{-1}(z)$, the inverse $\tanh$ function, and I've found that we can write $\tanh^{-1}(z)$ as $1/2\log((1+z)/(1-z))$ but I don't know how to finish off ...
0
votes
1answer
53 views

What is the difference between these two given sums?

What is the difference between this: $\sum_{i=1}^n(x^i+9x*i)$ and $\sum_{i=0}^n x[x^i+9i+9]$ ? So far I know that the first terms are not different. The first term of the $\sum_{i=1}^n(x^i+9x*i)$ ...
0
votes
1answer
99 views

Using Mollifiers

If we take $f$ to be a smooth function, then how does it follow that we can write $f^{\epsilon}(x)-f(x) = \int_{B(0,1)}\eta(y)(f(x-\epsilon y)-f(x))dy$ where $f^{\epsilon} := \eta_{\epsilon}\ast f$ ...
1
vote
0answers
180 views

Every polynomial bounded on [a,b] is of bounded variation.

Question: Prove that every polynomial bounded on the compact interval [a,b] is of bounded variationMy proof: Let $f(x)$ be a bounded polynomial of n degree. Then $f '(x)$ is also a polynomial bounded ...
4
votes
2answers
94 views

Reflexive, separable containing all finite dimensional spaces almost isometrically

Is there a separable, reflexive Banach space $Z$ such that for every finite dimensional space $X$ and every $a>0$, there is a $1+a$-embedding of $X$ into $Z$? I can do the question without the ...
0
votes
2answers
84 views

Show that this function is measurable

Let $f$ be a measurable function, and define $g(x)=0$ if $f(x)$ is rational and $g(x)=1$ if $f(x)$ is irrational. Prove the $g$ is measurable. So what has to be proved is that for any open set $I ...
0
votes
4answers
89 views

Show that A = B

I need to show that A = B. Do I have to only work with one side of the equation and show that it equals the other side without using the other side? Or can I assume that they equal each other and ...
4
votes
3answers
117 views

limit of a sequence. might be related to Cesaro theorem

this it the limit to evaluate: $$\mathop {\lim }\limits_{n \to \infty } {1 \over {{n^{k + 1}}}}(k! + {{(k + 1)!} \over {1!}} + ... + {{(k + n)!} \over {n!}})$$ I've given an hint which is: $$(1 - ...
0
votes
1answer
33 views

Solving a probability equation.

One group of 30 people won a contest and as a reward they got a free vacation to Hawaii. The hotel they will be in has 10 rooms with 3-bed (bed for 3 people). The question: How many ways can we ...
0
votes
1answer
30 views

How to construct an $n$ equations in $n$ unknowns so that there is only a unique solution and all are equal

Suppose we are required to create $n$ equations in $n$ unknowns such that there is only 1 unique nontrivial solution and the solutions are equal to the others i.e. $x_1 = x_2 = \dots x_n$ What are ...
7
votes
0answers
216 views

Homomorphic Compression

Can there be an algorithm such that, given plaintext data P,Q, and compression function e, Such that if we treat P and Q as a number (a series of bits): $$\begin{eqnarray*}e(P + Q)& =& e(P) ...
1
vote
3answers
284 views

A Proof with Intersecting Lines

My geometry teacher has given me a question to try to solve which is: Prove that there exists lines a and b, such that a is not equal to b and a intersects b. I am not sure how to prove this or ...
0
votes
2answers
212 views

A (too?) simple argument for the undefinability of definable sets

Preliminaries (see e.g. Jech, Set Theory, p. 5): To every formula $\varphi(x)$ of ZF set theory corresponds a class $C = \lbrace x : \varphi(x)\rbrace$, but only to some formulas corresponds a set. ...
0
votes
1answer
36 views

Proof that $f[x_{i0},…,x_{ik}] = f[x_0,…,x_k]$

i try to show, that $f[x_0,...x_k]$ is a symmetric function of $x_i$. What means, that for a permutation $x_{i0},...,x_{ik}$ of numbers $x_0, ...,x_k$ applies: $$f[x_{i0},...,x_{ik}] = ...
1
vote
2answers
167 views

Integral of a complex function over a circle

I've got the problem. I was to compute: $\int_0^{2 \pi} \frac{1- cos(n\phi)}{1 - cos(\phi)} d\phi$ using analytic functions methods. My attempt: I came to this: On a unit circle, we have $z= ...
1
vote
2answers
147 views

Eigenvalues of matrix and its transpose.

Prove that if A is a square matrix then A and A transpose have same Eigen values. Kindly help me how to prove this by generally not considering matrix by ourselves.
1
vote
1answer
82 views

Approximating continuous functions with growth condition by lipschitz functions

I wonder about the following. Is it possible to approximate $x^p$ for $x\in\mathbb{R}_+$ with lipschitz continuous functions? If so, is it possible to approximate it in a dominating way, i.e. $f_n\to ...
0
votes
1answer
40 views

integrate the following expression again

Question: $x - x^2 + 1$ My answer: $\frac{x^2}{2} - \frac{x^3}{3} + {x} + C$ Correct answer: $\frac{x^2}{2} - \frac{x^2}{3} - \frac{x^3}{3} + x + C$ What am I doing wrong? thanks
3
votes
3answers
127 views

Calculate $i ^ {i+1}$ and also $i^{i^{i^{\dots}}}$

I just wanted to ask the following questions please. The first I have is calculate $i^{(i+1)}$ and also $i^i$ I was just wondering if anyone can nudge me in the right direction to solve these ...
1
vote
1answer
31 views

Limit of maximizer not equal to maximizer of limit

I am looking for functions $f_n,f$ defined on a subset of $\mathbb{R}$ with unique maximizers $\alpha_n, \alpha$, such that $f_n$ converges to $f$ pointwise, but the $\alpha_n$ do not converge to ...
1
vote
1answer
46 views

Theorem : If the moment of order t exists for an RV X, moments of order 0 < s < t exist.

An Introduction to probability and statistics - Rohatgi Pg. No. 74 Theorem 2 Theorem : If the moment of order $t$ exists for an RV $X$, moments of order $0 < s < t$ exist. The proof is given ...
4
votes
2answers
247 views

Is the empty function always a bijection?

Let $f_A:\emptyset\to A$ be the empty function with range $A$. The definition of a bijection as applied to this function is: $$\forall x,y \in \emptyset (x=y \implies f_A(x)=f_A(y))$$ negating you ...
3
votes
1answer
59 views

Family of sets which intersect isn't connected

Find any family of sets $A_n$ such that $A_n$ are connected sets and $A_{n+1} \subset A_n$ and $$\bigcap_{n=1}^{\infty}A_n$$ is not connected. I tried find family of sets such that ...
0
votes
3answers
57 views

Fit a function for these values.

This was asked to me in a interview its not a homework problem. I really wish my school would have given me such quality homework ;) Find out f(x) such that f(1) = 3 f(2) = 6 f(3) = 10
1
vote
1answer
85 views

find nested closed balls of polynomials s.t. intersection is empty (in a metric space)

the space is the set of all polynomials on $[0,1]$ with metric sup. the space is not complete. we need to find explicitly a nested family of closed balls with radius (of each closed ball) goes to ...
1
vote
1answer
156 views

How to apply perspective transform to Bezier curve?

I found that both Bezier curves and B-splines are described with a formula $p(t)=\sum\limits_{i=0}^d B^i_m p_i$ but in the case of B-splines $B^i_m$ are B-spline blending functions, while for Bezier ...
0
votes
3answers
102 views

integrate the following expression

Question: integrate $\sqrt{t}$ My answer: $t^{\frac{1}{2}} = \dfrac{t^{\frac{3}{2}}}{\frac{3}{2}} +c$ correct answer: $\frac{2}{3}.t^{\frac{3}{2}} +c$ what am I doing wrong? thank's
2
votes
2answers
104 views

An event that is very possible to happen, hasn't happened. Why? [closed]

I showed a friend an example of conditional probability, using the formula P(A|B) = P(A intersection B)/P(B). She asked an interesting question (a random question that didn't really have to do with My ...
1
vote
3answers
48 views

How to proof that more than half binary algebraic operations on a finite set are non-commutative?

We know that if $S$ is a set and $|S|=n$, then there are $n^{n^{2}}$ binary algebraic operations, right? The cardinality of $|S|=n$ and cardinality of $|S\times S|=n^{2}$. Also, the number of all ...
7
votes
2answers
1k views

Explaining Hypercomplex numbers to Children.

Imagine a highschool freshman walks up to you and asks you what hypercomplex numbers are. Explain to her, in a fair amount of detail, the different types of hypercomplex numbers in a way that any ...
1
vote
1answer
88 views

An integral relating to Bernoulli polynomials

Show that $$\int_{0}^{1}B_{2n+1}(x)(\cot({\pi}x)-2\sin(2{\pi}x))dx{\sim}0$$ where $B_{2n+1}(x)$ is the Bernoulli polynomials.
0
votes
4answers
207 views

Correct formal interval notation

I can't find any definitive answer on this topic, maybe that's because there isn't one, but I figured if there was a place to ask then SE was it! To describe a set in which $x$ and $y$ are in the ...
6
votes
2answers
239 views

$\sum\limits_{k=1}^{n-1} \frac{1}{1-cos(\frac{2\cdot \pi\cdot k}{n})} = \frac{1}{6} \cdot (n^2-1)$

I want to show that $\sum\limits_{k=1}^{n-1} \frac{1}{1-\cos(\frac{2\cdot \pi\cdot k}{n})} = \frac{1}{6} \cdot (n^2-1)$ is true. I don't know how I could prove that, with induction I don't know how ...
1
vote
1answer
344 views

Properties of the co-countable topology on $[0,1]$

I am trying to learn topology by myself. I have the following questions and my attempts of their proofs. Since I have no one to consult with, I am posting here. If anyone check my proofs that would be ...
0
votes
1answer
53 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
0
votes
1answer
57 views

Do pushouts exist in a cartesian closed category?

If $C$ is a cartesian closed category, then is it necessary that all pushouts exist?
1
vote
0answers
49 views

See elements of $H^1(\mathcal{O}_X)$ as repartitions

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$. Define the ring of repartitions of $X$ (or adeles) as the set whose elements are collections $\{ r_P \}_{P\in ...
0
votes
1answer
67 views

Independence (probability theory)

Having hard time with this. Hope someone can help me! We throw a dice 20 times. $A(i,j)$ is the event in which in $i$-th and $j$-th throw we get same number. Show that $\{ A(i,j) :1\leqslant i\lt j ...
0
votes
2answers
37 views

using $\sin x$ to get a function with range $[a,b]$

$\sin x$ is a nice function on $\mathbb{R}$ whose range is $[-1,1]$. can we 'adjust' it so that its range will be $[a,b]$. By 'adjusting' I mean changing the argument $x$ to some other argument which ...
0
votes
2answers
63 views

Drawing balls from a bin with a color-specific probabilistic discard step

(I migrated this question here myself from MathOverflow since it might be too low level there.) I have a bin with $N = (k_a + b_b)$ total balls, $k_r$ of which are red and $k_b$ of which are blue. I ...
1
vote
0answers
172 views

Rotating an $n$-dimensional hyperplane

Let $\mathcal{H}: \mathbf{x}^T\mathbf{w}+b=0$ be a hyperplane in the $n$-dimensional Euclidean space of column vectors. Is there a way of "rotating" the above hyperplane such that it coincides with ...
2
votes
6answers
189 views

Prove that $[a,b]$ is connected space.

Prove that $[a,b]$ is connected space. I know that $\mathbb{R}$ with euclidean metric is connected space. I would like find surjective function $f: \mathbb{R} \rightarrow [a,b]$. Because $\mathbb{R}$ ...
2
votes
1answer
142 views

a corollary of Cauchy's integral formula in several complex variables

I'm learning several complex variables. There is a corollary of Cauchy's integral formula that I don't know how to prove. Let $X\subset\mathbb{C}^n$ be a domain. For each multi-index $\nu$, for each ...
6
votes
6answers
300 views

which one is larger $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$?

Which is larger? $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$? Algebraic justilation does not help.
1
vote
1answer
41 views

Solving a first order non-linear ODE

I encountered a problem solving the following equation: $$t^2dy/dt+2ty-y^3=0$$ I have already tried the following steps, but to no avail: 1)Separation of variables 2.1)Integration Factor of 1 ...
5
votes
1answer
228 views

Probability: the expected value in a dice game. [duplicate]

If a dice is thrown till the sum of the numbers appearing on the top face of dice exceeds or equal to 100, what is the most likely sum?
5
votes
1answer
61 views

local gauge invariance of field's homotopy class? Every map $S^2\rightarrow \mathrm{group } G$ is homotopic to a constant map?

In a discussion of a gauge field theory with gauge group $G$, someone says we can use a celebrated result of E. Cartan to show the gauge invariance of matter field's homotopy class. And Cartan's ...
1
vote
1answer
32 views

Let $\, g \,$ be a function defined on $(a,b)$ such that $\, a<g(x)<x$

I am stuck on the following problem that says : What I guess option (A) is not possible. But, I am not sure about the other options. Can someone explain? Thanks in advance for your time.
2
votes
0answers
219 views

How find this sum $\sum_{k=1}^{n}[\arctan{f(k)}\arctan{g(k)}]$

Let $n$ be a positive integer. Compute ...

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