1
vote
2answers
28 views

Why if $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic? [duplicate]

In the notes I'm studying from ( again =) ) I read: If $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic Could someone give me a justification for ...
2
votes
2answers
45 views

Dual result of Fatou lemma

If $\{f_n\}\subset L^+$, $f\in L^+$ such that $\{f_n\}$ is dominated by $f$ where $\int f < \infty$ then $$\limsup\int f_n\leq \int \limsup f_n$$ Attempted proof - Consider the sequence $\{f - ...
0
votes
0answers
20 views

Finding expected value of $E[x^{2}y]$ [on hold]

X,Y are random Variables. X = 2,4 ; Y = 1,3,5; I have to find $E[x^{2}y]$. I know that - $E[g(x)] = \sum_{r} g(x=r)P(x=r)$ But i dont know what is $E[x^{2}y]$, Somebody can help me ? Thanks.
2
votes
1answer
32 views

Is $(X,\mathcal T)$ a $T_0$-space?

Let $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ be topological spaces. Now define $\mathcal T=\mathcal T_1 \cap \mathcal T_2$. If $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ are $T_1$-spaces, is $(...
7
votes
4answers
139 views

Confused about notation “:=” versus plain old “=” [duplicate]

Relating to sets, I find the following in a text book: "...the set S := {1, 2, 3}". The book has an extensive notation appendix, but the":=" notation is not included. What exactly does ":=" mean, and ...
1
vote
0answers
10 views

Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
2
votes
1answer
38 views

Prove that Standard Deviation is always $\geq$ Mean Absolute Deviation

Where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ and $$ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$ I came up with a sketchy proof for the case of $2$ values, but I would like ...
0
votes
3answers
24 views

Simple Contour Integral

I have forgotten much of the complex analysis I once knew. How do I go about using the Cauchy Integral Formula / Residue Theorem to solve this contour integral? The region is the unit circle. $$\...
-1
votes
2answers
46 views

Help with limit proofs [on hold]

I'm having trouble finding the following limit. Any help for a different approach would be great. Thank you!
1
vote
0answers
53 views

I hope you resolve the question with surrounding solution method [on hold]

That we know that: $$(i-\sqrt 3)^x-(i+\sqrt 3)^y=2^{xy}$$ Find the value of: $x+y$ .
6
votes
3answers
280 views

How can I rewrite recursive function as single formula?

There is following recursive function $$ \begin{equation} a_n= \begin{cases} -1, & \text{if}\ n = 0 \\ 1, & \text{if}\ n = 1\\ 10a_{n-1}-21a_{n-2}, & \text{if}\ ...
2
votes
3answers
47 views

Proving a function to be a difference

I am concerned with a function $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(f(x_1,x_2),f(x_3,x_2))=f(x_1,x_3)$$ for any $x_1,x_2,x_3\in \mathbb{R}$, and $$f(x,0)=x$$ $$f(x,x)=0$$ ...
0
votes
0answers
23 views

Linear transformation $T$ such that for every extension $\overline{T}$, $\|\overline{T}\|>\|T\|$.

Let $E$ and $F$ be normed spaces such that $\dim F < \infty$, $G$ a subspace of $E$ and $T:G\rightarrow F$ a continuous linear map. I know that there exists a continuous linear extension $\overline{...
0
votes
0answers
48 views

How mathematics would be different if the first derivations, conjectures and theorems would be others?

I've realised that mathematics is nothing else that an implication of some assumptions (plus the assumptions themselves, of course). We have axioms and we derive new "things", new rules, ideas, ...
1
vote
2answers
26 views

Describe all solutions of Ax = 0 (2)

Let $A = \begin{bmatrix}1&-5&-3&2\\4&-20&-12&8\end{bmatrix}$ Describe all solutions of $Ax = 0$ $x = x_2 \begin{bmatrix}\\\\\end{bmatrix} + x_3 \begin{bmatrix}\\\\\...
1
vote
1answer
37 views

How can I find the inverse Fourier transform of $e^{x^2}$ and similar functions? [on hold]

I'm honestly clueless; I tried writing it as ${\lmoustache}_{-\infty}^{+\infty} {e^{x^2}} {e^{ix}} {dx}$ , but I don't know where to go from there.
0
votes
0answers
18 views

How am I to understand this notation with regards to bdS and the int S? $S=\bigcap_{n=1}^{\infty}\left(-\infty,7+\frac{1}{n}\right]$

I'm trying to find the largest $\epsilon$ such that the neighborhood centered at $x$ of radius $\epsilon$ is contained in $S$. That part I think I can do, but I just don't know understand the below ...
0
votes
2answers
26 views

Proving this quotient space is a Hausdorff space

Define $S^1 = \{x \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$. Define the equivalence relation $\sim$ as follows: $(x,y) \sim (x',y')$ if and only if $y = y'$. Now prove that the quotientspace $X/\sim$ with ...
1
vote
0answers
25 views

Parallelizable open dense subset and integration

In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral ...
1
vote
2answers
22 views

Modules as morphisms to endomorphism rings

An $A$-module $M$ may be thought of as a (surjective) ring homomorphism $f: A \to E(M)$, where $E(M)$ is a ring of group endomorphisms of $M$. Then $am = f(a)(m)$. Is there any more to this ...
0
votes
1answer
19 views

Weak convergence implies convergence on continuous functions

Let $X$ be a metric space, and let $\mu_n$ be a sequence of measures on $X$ converging weakly to a measure $\mu$, meaning for all bounded continuous functions $f$, we have $\int_{X}fd\mu_n \rightarrow ...
1
vote
0answers
14 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
1
vote
1answer
25 views

A question about the definition of adjoint functors

Aluffi's "Chapter 0", on pg. 492, says the following: Let $C$ and $D$ be categories, and let $\mathcal{F}:C\to D, \mathcal{G}:D\to C$ be functors. We say that $\mathcal{F}$ and $\mathcal{G}$ are ...
8
votes
0answers
60 views

Is it possible to divide an equilateral triangle into N equal parts?

It’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an ...
0
votes
6answers
50 views

Arrange black and white balls so that each pair of white balls is separated by at least two black balls

I am trying to solve the following question: How many linear arrangements of $m$ white balls and $(n-m)$ black balls are possible such that each pair of white balls is separated by at least two ...
2
votes
0answers
35 views

Is incorrect the Landau's demonstration?

Leon Henkin says, at the end of his On Mathematical Induction text, Edmund Landau (Foundations of Analysis) failed to demonstrate the existence and uniqueness of adding natural numbers, because ...
-2
votes
0answers
40 views

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them. [on hold]

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them.
0
votes
0answers
21 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
5
votes
4answers
119 views

Why does $(128)!$ equal the product of these binomial coefficients $128! = \binom{128}{64}\binom{64}{32}^2 \dots \binom21^{64}$?

I'm working through some combinatorics practice sets and found the following problem that I can't make heads or tails of. It asks to prove the following: $$128! = \binom{128}{64}\binom{64}{32}^2\...
2
votes
1answer
51 views

Olympic Problem about Theory of numbers.

Let $Y=\{1,2,\ldots, 2014\} \subset \mathbb{N}$. Find the maximal subset $A\subset Y$ such that, $$\forall x\in A,\quad x\not\mid\sum_{y\in A\setminus\{x\}} y.$$ Example, $A'=\{2,4,6,\ldots,2014\}\...
0
votes
3answers
49 views

How to find a parametrization for a torus?

I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$ where $0<r<R$. I know I need to compute the metric tensor and ...
4
votes
3answers
29 views

Any undirected graph on 9 vertices with minimum degree at least 5 contains a subgraph $K_4$?

Let $G$ be simple undirected graph with degree of every vertices is at least 5. Prove or disprove that $G$ contains subgraph $K_4$. I came up with this question when I were trying to find Ramsey ...
1
vote
2answers
19 views

Logic of Elementary Row Operations to Create Equivalent Systems

Can anyone explain why the 3rd operation applied on a system creates an equivalent system with the same solution. Elementary Row Operations. 1. Interchange two rows. 2. Multiply a row with a ...
0
votes
2answers
19 views

Group order 168 having a normal subgroup of order 4, then G has a normal subgroup of order 28.

Prove that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. resolution: P4 is the ordeme subgroup 4 and P7 the order subgroup 7. As P4 is ...
1
vote
2answers
47 views

Describe all solutions of Ax = 0

Let $A = \begin{bmatrix}1&-5&3&-3&-4&-2\\0&0&1&1&0&-5\\0&0&0&0&1&-3\\0&0&0&0&0&0\end{bmatrix}$ Describe all ...
1
vote
1answer
35 views

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$?

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$? It is well known that most (in some suitable sense) polynomials $f \in \mathbb{Q}[x]$ of degree $d$ and coefficients $|...
0
votes
2answers
40 views

Find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$

I need to find $\frac{dy}{dx}$ if $x^3 + x^2y + xy^2 + y^3 = 81$ I am trying to first get y in terms of x, but that is quite lengthy and feels like I am doing something wrong. How do I go about this ...
3
votes
0answers
45 views

More convenient form of derivative of $\mathrm{sinc}(x)$

$\mathrm{sinc}(x)$ is defined as $\frac{\sin(x)}{x}$ except continuous at $x=0$ (insert the removable singularity). The derivative of $\mathrm{sinc}(x)$ is usually given as the derivative of $\frac{\...
0
votes
2answers
17 views

Independence of Random Variables From Expectation Counter Example

I know that if $X$ and $Y$ are independent random variables, then $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$. I also know that the converse is not true, although I cannot seem to find an easy ...
2
votes
1answer
17 views

X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list.

I'm teaching my self topology with the aid of a book. I'm trying to do the following problem: Let X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to ...
0
votes
0answers
17 views

Elliptic curve, different forms of.

y^2 = x^3 + mx + c An elliptic curve in the form defined in Wikipedia y^2 = x(x-A)(x+B) = x^3 +(B-A)x^2 + ABx Frey's curve has no term in x^2, but 2. does because from Fermat, A=a^n not equal ...
0
votes
1answer
12 views

Given a “polyline”, find a value x for the polyline that passes through the points (a, b)

Here's the description of the problem: There is a polyline going through points (0, 0) – (x, x) – (2x, 0) – (3x, x) – (4x, 0) – ... - (2kx, 0) – (2kx + x, x) – .... We know that the polyline passes ...
0
votes
1answer
29 views

Extending dimension of matrix to get it determinant. What I'm doing wrong? Or am I right?

Let the matrix of dimension 4 be: $$A=\begin{bmatrix} a11 & a12 & a13 & a14\\ a21 & a22 & a23 & a24\\ a31 & a32 & a33 & a34\\ a41 & a42 & a43 & a44 \...
1
vote
3answers
75 views

Find the values of $x$ such that $2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ is independent of $x$.

Find the values of $x$ such that $$2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$$ is independent of $x$. Checking for $x\in [-1,1]$ In the taken domain $\sin^{-1}\left(\frac{2x}{1+x^2}\...
2
votes
3answers
71 views

Question involving limit

How to find the following limit: $$\lim_\limits{n\to\infty} (ne\sqrt[n]{\ln{(1+e^n)}-n}-n)$$ Thanks in advance!
2
votes
2answers
17 views

Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
-1
votes
1answer
19 views

Expected value conditioned [on hold]

Given $X_1, \ldots, X_n$ r.s.s. from a random variable with probability function $$f_{\theta}(x)=\frac{1}{\theta}\text{ for }x=1, \ldots, \theta$$ Let $T_1=2X_1-1$ and $T_2=X_{(n)}$ (maximum of $X_1, \...
0
votes
1answer
17 views

Advice about formula for exact differential equation

When I realize that I have exact differential equation I know that is good to use specific formula. But this formula has two forms. Can you tell me when I must use first and when second form? ...
1
vote
2answers
22 views

Expressing the orthogonal projections on a linear operator $T$'s eigenspaces as polynomials in $T$

In the inner product space $\mathbb{C}^{2}$ with its standard inner product, let $$ T\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3x+4y\\-4x+3y \end{pmatrix} $$ a linear operator. Express the ...

15 30 50 per page