0
votes
1answer
23 views

Matrix which has every vector in the space as an eigenvector is square, diagonal.

I've been trying to solve this linear algebra problem for some time and have gotten stuck. I've been asked to either prove or disprove the following statement: For $V$ an $\mathbb{ R } $ vector ...
1
vote
0answers
25 views

$f_x(y) = f(y-x)$, $L^p(\mathbb{R}^n)$ [on hold]

Let $x \in \mathbb{R}^n$ and $f \in L^p(\mathbb{R}^n)$, $f_x$ function on $\mathbb{R}^n$ defines $f_x(y) = f(y-x)$.Let fix $f$ and $1 \leq p < \infty$. Prove that is mapping $x \mapsto f_x$ ...
0
votes
0answers
28 views

On $O^{p'}(G)$ for a finite group $G$

Let $G$ be a finite group and let $N\unlhd G$. Consider $O^{p'}(G)$ which is the smallest normal subgroup of $G$ with factor group order coprime to $G$. Is $O^{p'}(G/N) = O^{p'}(G)/N$ if $N\le ...
5
votes
0answers
107 views
1
vote
1answer
74 views

If $v_1, \dots, v_m$ are linearly independent, then there is $w$ such that $\langle w, v_j \rangle > 0$ for all $j$

Suppose $v_1, \dots v_m$ is a linearly independent list in V. Show that there exists $w \in V$ such that $\langle w, v_j \rangle > 0$ for all $j \in {1, \dots ,m}$. I understand this question is ...
0
votes
0answers
11 views

Is there a bound on largest eigenvalue for covariance matrix of discrete random variable?

I have a random variable $Z=(Z_1,\ldots,Z_p)$. Each component can take values in {-1,0,1}. Is there a way to bound the largest eigenvalue of Cov(Z)? Actually, I have a latent multinormal variable ...
0
votes
0answers
10 views

ODE theorem solution existence

My memories from ODEs are a little vague. I need a theorem that would explain the following: If $\phi_t$ is a family of functions defined on $R^n$ with values in $R^n$ such that ...
2
votes
1answer
42 views

Intuition concerning Riemann Sums

I have just started learning integrals, and I want to know the following: In the definition of a riemann integral, it states that the interval that the integral is to be evaluated, is partitioned ...
0
votes
1answer
19 views

Is Champernowne's constant Liouville?

By looking at extreme spikes of Champernowne's constant and how well it's approximated by some rational numbers I think it's reasonable to think that this is a Liouville number. However, no source I ...
0
votes
1answer
40 views

Contrapositive Epsilon-Delta Limits?

Is the contrapositive form of the epsilon-delta limit definition valid, or am I missing something? -- If valid, are there any examples where it is easier in a proof? \begin{align*}\scriptsize ...
0
votes
0answers
16 views

maximum k-colorable subgraph

Problem: Given an arbitrary graph $G$ and $k$ colors, find a maximum $k$-colorable sub-graph of $G$. Question: This problem is known to be NP-complete. Is there any solution to this problem, whatever ...
-1
votes
1answer
49 views

Does anyone have any good resources for learning high level maths. [on hold]

Does anyone have any good resources for learning higher level maths? The topics I'm considering are algebra, triangles and anything you people think is cool and useful.
2
votes
3answers
45 views

Evaluate the limit of $\ln(\cos 2x)/\ln (\cos 3x)$ as $x\to 0$

Evaluate Limits $$\lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$$ Method 1 :Using L'Hopital's Rule to Evaluate Limits (indicated by $\stackrel{LHR}{=}$. LHR stands for L'Hôpital Rule) ...
3
votes
2answers
41 views

The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
3
votes
2answers
34 views

Largest Equilateral Triangle in a Polygon

Is there an algorithm to determine the largest equilateral triangle in a convex polygon?
1
vote
1answer
45 views

A different vector product

I am taking linear algebra, and have learned about the vector dot product and cross product. Is there a vector product defined by : $(a_1, a_2, \dots ,a_n)\times (b_1, b_2, \dots,b_n) = (a_1b_1, ...
0
votes
1answer
32 views

Finding the area of an equilateral triangle on an ellipse

The question is as follows: Let $E$ be an ellipse with major axis length $4$ and minor axis length $2$. Inscribed an equilateral triangle $ABC$ in $E$ such that $A$ lies on the minor axis and $BC$ is ...
3
votes
0answers
26 views

Best algebra text for Model Theory

I'm looking for an algebra book that is tailored towards some of the ideas in Model Theory, I'm currently slogging through Hodges' Model Theory. I'm a bit rusty with my algebra and was curious if ...
0
votes
0answers
10 views

Topologizing $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$

Given a separable Hilbert space $\mathcal{H}$ I would like to know how one could topologize the quotient algebra $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$? Here ...
1
vote
3answers
35 views

Proof for complex numbers and square root

Use the polar form of complex numbers to show that every complex number $z\neq0$ has two square roots. I know the polar form is $z=r(\cos(\alpha)+i \sin(\alpha))$. I'm just not sure how to do this ...
0
votes
1answer
41 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
1
vote
1answer
32 views

Showing $f_{n} \rightarrow f$ in $L^{1}$ given an integral condition

Let $f_{n}: [0, 1] \rightarrow [0, \infty)$ be a Borel measurable function such that $$\int_{0}^{1}f_{n}(x)\log(2 + f_{n}(x))\, dx < \infty.$$ If $f_{n} \rightarrow f$ Lebesgue almost everywhere. ...
1
vote
0answers
21 views

How to research a series, when only some elements are available

I want something like the On-Line Encyclopedia of Integer Sequences, but for series, not sequences. I'd like to know the name of a series, in what natural phenomenon it happens and so on.
0
votes
1answer
47 views

Mid level algebra headscratcher. [on hold]

I have 4 numbers. Base = 100 , Start = 0.4 , Count = 4 , Multiplier = 0.64448 I do 100 × 0.4 = 40 Then 4 times: 40 × 0.64448 = 25.7788368 25.7788368 × 0.64448 = 16.61371067 16.61371067 × ...
1
vote
0answers
19 views

Proof that Lipshitz function has a primitive

I was doing exercise 5 of this exercise sheet: http://didel.script.univ-paris-diderot.fr/claroline/backends/download.php?url=L1RENi5wZGY%3D&cidReset=true&cidReq=31UKMT42 And I don't know how ...
0
votes
1answer
20 views

Gradient of l2 norm squared

Could someone please provide a proof for the following rule: $$\nabla\|x\|_2^2 = 2x$$ I.E. why is the gradient of the $L_2$ norm square of $x$ equal to $2x$? Thanks
0
votes
0answers
19 views

A book on analytic geometry

It's easy to find good recommendation for books here for any subject other than analytic geometry ,therefore I'd like to ask for any suggestion of analytic geometry books ,the only charactrestic I'm ...
2
votes
2answers
19 views

how can I show measurable functions preserve mean independence?

Let $X$ and $Y$ be continuous R.Vs s.t. $\mathbb{E}(X\mid Y=y)=\mathbb{E}(X)$ for all $y\in \operatorname{Supp}(Y)$. How can I show that for any measurable function $f$ we have that $\mathbb{E}(X\mid ...
0
votes
1answer
17 views

What does “Choose N ~ Poisson(ξ), Choose θ ~ Dir ( α )” mean in the context of Latent Dirichlet Allocation

I'm reading http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ03.pdf and trying to understand the notation and concepts behind LDA, in order to implement it myself. I've followed some ...
1
vote
1answer
37 views

When is $R^n / ( I \cdot R^n) = (R/I)^n$, for ring $R$ and ideal $I$?

I am looking at a set of old lecture notes in which I scribbled: $R^n / ( I \cdot R^n) = (R/I)^n$. However, I cannot recall why this is true. The setting is that $R$ is commutative, has identity, and ...
3
votes
4answers
109 views

How would I go about evaluating $\int \frac{x}{(9-8x^2)^3}dx$?

So I have homework on webAssign (a site used by my college), and I am not understanding the logic as to why I am taking the steps into solving the integral it is telling me to take. So I'll list the ...
0
votes
2answers
16 views

Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
1
vote
0answers
8 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
4
votes
0answers
32 views

Differential forms and determinants

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$ But what if I have two concret ...
0
votes
1answer
36 views

Fundamental group computation

I am trying to compute the homology groups and the fundamental group of the space $X$ obtained as the disjoint union of a circle and a cylinder $S^1\times I$ by attaching the cylinder along its ...
2
votes
2answers
33 views

Is it still possible for their to exist a well-ordering on the reals such that there is always a least next element?

Let $a \leq b$ be a well-ordering on $\Bbb{R}$. And suppose for any $a_n \in \Bbb{R}$ there exists another element $a_{n+1} \in \Bbb{R}$ such that under the given ordering $(a_n, a_{n+1})$ is empty. ...
2
votes
1answer
7 views

Smallest triangle in a convex polygon triangulation

I have been working on this problem for quite a while and it seems necessary to prove or disprove this particular problem. Suppose $T$ is the set of all possible triangles made from the vertices of a ...
0
votes
3answers
106 views

How to show that the set of functions $1,x,x^2,x^3…$ is linearly independent? [on hold]

Show that the set of functions $1,x,x^2,x^3...x^n...$ is linearly independent on any interval $[a,b]$. If $$c_1+xc_2+x^2c_3+x^3c_4...=0$$ we should show $$c_i=0,\quad i=1,2, \ldots$$ how could I ...
1
vote
0answers
10 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
1
vote
1answer
34 views

Use polar complex numbers to find multiplicative inverse

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(cos(\alpha))+i(sin(\alpha))$. I can do ...
2
votes
1answer
82 views

Is $a^b+b^a$ unique for all integers a and b?

Is $a^b+b^a$ unique for all integers $a$ and $b$? Any proof?
1
vote
0answers
32 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
0
votes
1answer
34 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
3
votes
1answer
25 views

Why does $\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}$ converge to $\log\left[\frac{1}{\epsilon}\right]$?

while playing around with my equations, i found that the following has to hold for my universe to be consistent: $$\lim_{N \rightarrow ...
2
votes
2answers
28 views

Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
1
vote
0answers
19 views

$I \cdot R^n = I$ for ideal $I$ in commutative ring $R$ with identity?

Question: is $I \cdot R^n = I$, for a two sided ideal $I$, in a commutative ring $R$ with identity? Attempt: $I$ is closed under right multiplication with ring elements, so $I \cdot R^n \subseteq I$. ...
2
votes
2answers
235 views

Outcome of rolling a fair die 6 times

I'm failing to understand how to come to the answer to this question. If you roll a fair die six times, what is the probability that the numbers recorded are $1$, $2$, $3$, $4$, $5$, and $6$ in any ...
1
vote
0answers
28 views

Types of definitions

My question is about terminology. Consider the following two types of definitions of a circle: A circle is the collection of all points at the same distance from a given point. A circle is the ...
4
votes
0answers
59 views

$A\oplus C \cong B \oplus C$. Is $A \cong B$ when $C$ is finite, A and B infinite.

So my question is simply that for groups $A, B, C,$ if C is finite, A and B infinite and $A\oplus C \cong B \oplus C$, is $A \cong B$? My gut tells me this must be the case, and logically I can find ...
5
votes
1answer
69 views

What math have I missed as an Engineeering graduate? [on hold]

To explain, I have a Master's in Engineering from a well known university. We did a wide variety of mathematical topics, vector calc, perturbation methods, numerical methods, linear algebra, discrete ...

15 30 50 per page