1
vote
2answers
84 views

Difficulty and established result in the classification of Galois groups of quintic polynomials.

This article provides a classification of Galois groups of cubic and quartic polynomials and criteria to determine it, then I try to find some paper about the same result in quintic equations. ...
1
vote
1answer
29 views

If $ a_1 a_2a_3 …a_{20} = 2^x * y! $ Then what is the value of (x+y)?

Let us consider a series with $ a_1 = 2012 $ and $ a_n = \frac{n}{a_{n-1}} $ . If $ a_1 a_2a_3 ...a_{20} = 2^x * y! $ Then what is the value of (x+y) ?
-1
votes
1answer
107 views

Expectation using a dirac measure as the probability measure

If $\varepsilon_a(dx)$ is the point mass at point $a\in\mathbb{R}$, I want to calculate $E_a[X]$ where $X$ is a real valued random variable and $E_a$ is expectation with respect to ...
14
votes
1answer
358 views

Searching for tighter bounds

I have to solve an equation $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where $$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$ The $z_i$ are all positive and add ...
1
vote
1answer
653 views

What is the transition matrix and stationary distribution of ${T}$

Let $T = (X_n:n \in \mathbb{N})$ denote a homogeneous Markov chain with state space $E=\lbrace 1, 2, 3\rbrace$ and $$\mathbb{P}(X_1=2\vert X_0=1) = \mathbb{P}(X_1=3\vert X_0=1)=\frac{1}{3}$$ as well ...
1
vote
1answer
33 views

LinAlg Vector Graphing

I'm looking for a graphing calculator to graph vectors - NOT vector fields. A Google search turned up many great calculators with vector-valued function capabilities. However, I'm looking for ...
0
votes
1answer
130 views

Question on singular homology

please where i can found the prove of this: If $X$ is a topological space and $(X_{\alpha})_{\alpha\in I}$ is the family of it's path connected components. Prove that for each $n\in \mathbb{N}$, ...
0
votes
1answer
61 views

Steady state and DE

Let $x_t = f (x_{t-1})$ a difference equation of order $1$, with $f(x) = ux(1-x)$; $u \in (0; 4)$. Show that $x^* = 0$ is always a steady state for all $u$ . Compute its positive steady ...
2
votes
0answers
101 views

The pushforward under the left action in the group of units of a Clifford algebra

The following I know to be true: let $A$ and $B$ be elements of $GL(m,\mathbb{R})$ and let $X \in T_BGl(m, \mathbb{R})$ and let $L_A:Gl(m, \mathbb{R}) \to GL(m, \mathbb{R})$ be the left multiplication ...
0
votes
1answer
52 views

3d Transformation

I am trying to understand 3d-transformation in html5, but can't understand how we get new (x1, y1) coordinates. For example, we have a plane on our screen with a point at coordinates (287, 431). We ...
0
votes
1answer
277 views

If $R$ is a UFD then $R[X,X^{-1}]$ is a UFD

Prove that the ring $R[X,X^{-1}]$ of Laurent polynomials over a UFD $R$ is a UFD. I'd like someone to give a full proof.
3
votes
1answer
70 views

Homology groups and inclusions

Let $X \subset \mathbb{R}^n$ be compact. Suppose that $\mathbb{R}^k \setminus (X \cap\mathbb{R}^k)$ is not connected for some $k < n$. Does it imply that $X$ has nontrivial homology groups? ...
2
votes
1answer
109 views

Sampling with replacement events vs. fraction coverage of a specified set

This question is related to a previous one of mine: Sampling with replacement events vs. probability of coverage Here, we are again provided a deck of $N$ cards, when $k \leq N$ of the cards bear a ...
1
vote
1answer
88 views

Is axiom of completeness an axiom?

The following statement is the axiom of completeness: Every non empty subset of $\mathbb R$ that is bounded above has a least upper bound. So I was wondering: is it an axiom or can it be proved? ...
12
votes
3answers
334 views

$x^2+xy+y^2$ and $x^2-xy+y^2$ are not both perfect squares

Prove that $x^2+xy+y^2$ and $x^2-xy+y^2$ cannot be both perfect squares. Surely $x$ and $y$ are natural numbers. If $x^2+xy+y^2 =a^2$ and $x^2-xy+y^2=b^2$ simultaneously then we have to show that ...
0
votes
1answer
54 views

Determining a fixed point

Let $G$ be a group of order $14$, and let $S$ be a set of order $5$ on which $G$ operates. Prove that there is a fixed point, an element $s$ of $S$ that is left fixed by every element of $G$.
0
votes
1answer
64 views

Conformal mappings of multiply-connected regions

I'm reading about conformal mappings in Ahlfors' text, and I got to this section: In the following $\Omega$ denotes a plane region of connectivity $n > 1$. The components of the complement are ...
1
vote
0answers
52 views

Multiple regression and hypothesis test $H_0$:$\beta_2=0$

Multiple regression model $H_0$:$\beta_2=0$, $H_1$:$\beta_2 \neq 0$ where $\beta_2$ is the vector of elements ($\beta_2, \beta_3, \dots, \beta_k$) and $\beta$ is slope of regression line. Why it is ...
0
votes
1answer
2k views

How to prove that Lebesgue outer measure is translation invariant?

I am trying to prove that lesbegue outer measure is translation invariant, i.e., $m^\ast (E+y)=m^\ast E$. I proceed as follows. Let $E$ be a set. Let $\{I_n\}$ be a collection of open intervals that ...
2
votes
1answer
28 views

If $\mathbb{Z}_m^*$ is cyclic, and $\mathbb{Z}_m^*=\langle\overline{g}\rangle$, is $\overline{g}$ a primitive root?

Based on the definition I have of a primitive root: If $(a,m)=1$ and $a$ has order $\varphi(m)$ modulo $m$, then $a$ is called a primitive root modulo $m$. it would appear that all the ...
2
votes
2answers
721 views

How many ten letter words are there with no repeated letters that contain neither the word ERGO nor the word LATER?

How many ten letter words are there with no repeated letters that contain neither the word ERGO nor the word LATER? I am thinking that there are 26^10 words with ten letters and 26P10 10 letter words ...
1
vote
0answers
79 views

Matrix Representation of Linear Transformation with Factor Modules

"Let $V = \mathbb{R}^4$ and $U =$ {${ \vec{v} = (v1, v2, v3, v4)^T \in \mathbb{R}^4: v1 = v2, v3 = v4}$} In Parts 1-3 of this question we show $U$ is a subspace of $V$, find a basis for $U$,which I ...
1
vote
1answer
110 views

Searching for random items in a set of lists

This is a programming question, but I'm interested in the math behind it so I think it's better asked here. Say I have a list of items and I'm looking for one item in the list that satisfies a ...
0
votes
1answer
318 views
0
votes
1answer
59 views

You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ , can the same be done with ${\left( {\frac{2}{3}} \right)^x}$?

You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ as: ${\left( {\frac{1}{2}} \right)^x} = {({2^{ - 1}})^x} = {2^{ - x}}$ But what about ${\left( {\frac{2}{3}} \right)^x}$? Can it be ...
0
votes
1answer
38 views

Finding a permutation

it's my first time here. I have an algorithmic problem. My friend said it's trivial, but I can't make it out. So, I'm given a multiset, like this one: $$\{1\rightarrow2; 2\rightarrow1; 3\rightarrow2; ...
2
votes
3answers
55 views

Intermediate Value Property

I am trying show that the function $f:[0,1]\to \mathbb{R}$ defined by $f(x)=\sin \dfrac{1}{x}$ if $x\neq 0$ and $f(0)=0$ possesses IVP. Though it looks easy, but I am not getting any clue how to start ...
0
votes
1answer
64 views

Entire function that decays faster than exponential on reciprocals of integers is $0$

If $f(z)$ is entire and $|f(1/n)|\le e^{-n}$ for all $n\in\mathbb{N}$ then $f=0$. My idea is to express $f$ as a power series centered at $0$ that converges on the entire complex plane, then look at ...
3
votes
1answer
270 views

Banach Algebra: $\sigma(xy)\cup\{0\} = \sigma(yx)\cup\{0\}$

It is Rudin excercise 10.4 where we aim to prove $\sigma(xy)\cup\{0\} = \sigma(yx)\cup \{0\}$ for elements $x,y\in A$ a Banach-algebra.( $\sigma$ being the spectrum) In (a) we prove that $e-yx$ ...
3
votes
1answer
45 views

Is this a valid re-write rule?

In my job (SQL developer) I frequently need to change search conditions (WHERE clauses, database constraints) from disjunctive normal form to conjunctive normal form (CNF). I confess I usually resort ...
1
vote
5answers
60 views

Finding the image of $f(x)=\frac{1}{1+x^{2}}$

$f(x)=\frac{1}{1+x^{2}}$ and $x\geq0$ To find the image: $y=f(x)$ $y=\frac{1}{1+x^{2}}$ $x=y^{-1}$ $x=\sqrt{y^{-1}-1}$ $y\geq1$ Then the image of $f(x)=\frac{1}{1+x^{2}}$ is $y\geq1$ Is this ...
3
votes
1answer
161 views

Integral Representation of Bessel Function (K)

There is an integral representation for the modified Bessel function of the second (or third depending on who you talk to) kind (denoted $K_\nu$) that says: $$K_\nu(z) = ...
3
votes
2answers
148 views

Approximation of pi

Given that $\frac{\pi^2}{6}=\sum_{n=1}^{\infty}\left(\frac{1}{n^2}\right)$, I have to write a program in C that finds an approximation of $\pi$ using the formula ...
2
votes
2answers
107 views

Closure boundary interior sets

If we denote for a set $A$: $A^{o}$ the interior points set; $\overline A$ the closure and $\delta A$ the boundary set and $A'$ the set of cluster points , do the following hold (give counter-examples ...
0
votes
1answer
37 views

Find the minimum value of $\operatorname{Im}(z^5)/(\operatorname{Im}(z))^5$ [closed]

If $z$ is a complex number, then find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5},$$ where $\operatorname{Im}(z)$ denotes the imaginary part of z.
0
votes
2answers
59 views

Are these ideals the same?

I have already proved that $(X^3-Y^3,X^2Y-X)\subseteq(X^2-Y,X-Y^2)$ since the elements $X^3-Y^3$ and $X^2Y-X $ can be written as a linear combination of $(X^2-Y,X-Y^2)$. However, I can't write ...
2
votes
0answers
134 views

Why to Use the Same Sign for Minus and Negative?

Using the same symbol for two different concepts may cause confusion. So if one decides to do so, they should justify this choice by showing its advantages over other choices. What about the minus ...
8
votes
2answers
165 views

Does there exist an elliptic curve $E$ such that $\#E(\Bbb{F}_{q^2})=(q+1)^2$ for all prime powers $q$?

The following (paraphrased) question is a homework exercise for a course on elliptic curves: Let $p\not\equiv1\pmod{12}$ be a prime number and let $q=p^k$. Show that there exists an elliptic curve ...
0
votes
1answer
46 views

Solution Verification - Combinatorial Card-Picking

I have a problem as such: How many ways are there to choose nine cards out of a standard deck of 52 cards in such a way that every suit is represented in the selection at least twice? Here's my ...
0
votes
1answer
35 views

Hanging Paintings in a Line

I am hanging ten paintings in a nice straight line. I don't want the Van Gogh to hang next to the DaVinci. I don't want the DaVinci to hang next to the Warhol. In how many ways can I hang my ...
0
votes
1answer
342 views

In compact-open topology, $C(X,Y)$ is Hausdorff if $Y$ is Hausdorff

Show that in the compact-open topology, $C(X,Y)$ is Hausdorff if $Y$ is Hausdorff and regular if $Y$ is regular In the first statement, let $f,g$ be 2 functions in $C(X,Y)$, we need to ...
1
vote
1answer
96 views

Can the characteristic function of a multivariate normal distribution be extended from a neighborhood of the origin?

Let $x$ be a scalar random variable. There is a theorem that states that if $E[\exp(ixs)]= \exp\Big( i{s}\mu - \tfrac{1}{2} {\sigma^2s^2} \Big)$ for some neighborhood around the origin (i.e. ...
0
votes
1answer
26 views

Convergence of Series over Multi-integers

Given a dimension $n\in\mathbb{N}$, is there an easy way to say why the following sum over the multi-integers should converge? (By $\|k\|$ I mean the Euclidian distance on $\mathbb{R}^n$.) ...
2
votes
1answer
91 views

What are the generators of $\mathbb{Z}_9^*$?

I am to understand that $\mathbb{Z}_9^*$ is cyclic because $9=3^2$, where $3^2$ is of the form $p^{\alpha}$, with $p$ an odd prime... but I can't find any generators for the set... ...
0
votes
1answer
41 views

Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}

Let $k$ be a field and let $A$ be a $k$-algebra. Denote by $End_A(A)$ the set of all $A$-homomorphisms of the regular $A$-module $A$ into itself. Fix $a ∈ A$, and define the $A$-module homomorphism $r_a ...
6
votes
2answers
117 views

Proving basic lemmas about categories with finite products and terminal/initial objects.

I would expect that in any category $\mathcal{C}$ with finite products and a terminal object $1$, the isomorphism $X \times 1 \cong X$ should hold, but I have a rather hard time finding the proof of ...
1
vote
1answer
49 views

How to calculate new position of a rectangle after translation and rotation?

I have a rectangle - lets say 100 long by 75 high. Origin been bottom left corner. I move the rectangle up and across by 10 and rotate by 3 degrees from centre of part. How do I calculate the new ...
0
votes
2answers
55 views

Complex value of a divergent series

Given the series: $$S=\sum_{k=1}^{\infty}\frac{2^k}{k^2}$$ the sum obviously doesn't converge. 'Maple' gives for the value of the series: $$S(a)=\sum_{k=1}^{\infty}\frac{a^k}{k^a}$$ $S(a)=Li_a(a)$ ...
2
votes
1answer
62 views

Minimization of norms

How do I minimize the following? $ min_{z>0} - zt + 1/2\ z\ ||\ Y + X_k\ /\ z\ ||_2^2 $ Also, $X_k^TX_k = 1 \ \ \forall k $ I am given that the answer should be : $ \sqrt{Y^T - 2t} + Y^TX$ ...
0
votes
1answer
132 views

A question about predictable stochastic process.

Let $X_t$ be predictable with respect to filtration $(\mathscr{F}_t:t\in[0,T] )$. If I observe the process over an interval $[0,s],0<s<T$, does that mean I can tell the value of $X_t$ over ...

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