0
votes
1answer
721 views

Every natural number greater than 1 is divisible by a prime number

Sorry if this question has been asked, but a couldn't find one using the method I need. I want to prove that every natural number greater than 1 is divisible by some prime number using the WOP. I ...
0
votes
1answer
64 views

Is there an Artinian ring with exactly two prime ideals which their product is non-zero?

Is there an Artinian ring with exactly two prime ideals which their product is non-zero? Clearly these prime ideals could not be zero on the other hand the summation of them is equal to R.
1
vote
0answers
41 views

Minimal immersion

Let $N$ be an $n$-dimensional manifold immersed in an $l$-dimensional manifold $L$ by immersion $\iota: N\to L$. And let $\iota$ be minimal immersion. On the other hand, let $\phi: M\to N$ be totally ...
5
votes
2answers
102 views

group of elements whose orders are a power of some integer

I am stuck with the following problem, from C. Pinter's "A Book of Abstract Algebra", p. 153 ex. C 6: Let $G$ be an abelian group, and $H_p$ the subset of $G$ such that the order of every $x \in H_p$ ...
2
votes
1answer
102 views

Colimits and epimorphisms.

I am working on a project, and I need to know the proof of this: Any functor which preserves all colimits preserves epimorphisms. So could you please tell me how or where I can find the proof ...
0
votes
0answers
60 views

Nondegenerate bilinear map

Let $V$ be the vector space consists of all $n\times n$ real matrices, and $f$ is a nonvanishing linear function on $V$ such that $$f(AB)=f(BA),\ \forall\ A,B\in V.$$ Show that $g(A,B)=f(AB)$ is a ...
0
votes
0answers
52 views

Tensor power of a quotient space

Let $E$ and $F$ be vector spaces, $F$ a subspace of $E$. Is there any canonical isomorphism between $E/F \otimes E/F$ and a quotient of the form $E \otimes E/G$, where $G$ is a subspace of $E \otimes ...
0
votes
1answer
80 views

Conservative Vector Fields

One of the theorems for a vector field to be conservative is that $$\frac{\partial N}{\partial x}=\frac{\partial M}{\partial y}$$ for $$F=\langle M,N\rangle.$$ To find the $$\int ...
4
votes
0answers
64 views

How prove this $f=C$ if $4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$

Question: if $f:\mathbb{Z}^2\to \mathbb{R}$ is bounded ,and for any $x,y\in \mathbb{Z}$,we have $$4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$$ show that $$f\equiv C$$ where $C$ is constant. My ...
1
vote
1answer
25 views

Fixing up a word with condition.

Please help me solve the next question. How many words can you fix from $\{0,1,2, ..., m-1\}$ in length $n$, when the number of zeros in the word is odd? Examples for valid words: $001, 00, ...
3
votes
1answer
35 views

Integral over two sets when the measure of their symetric difference equals zero

Let $A$ and $B$ be two measurable sets of $X$. Show that if $m(S(A,B))=0 $ then $$\int_{A} f\, dm =\int_{B} f\,dm,$$ where $S(A, B)$ is the symetric difference. my method was to prove that ...
1
vote
0answers
73 views

$q\sin q$ is small

I read from the book "Which Way did the Bicycle Go" that it is unknown whether for every $c>0$ there are infinitely many integers $n$ such that $|n\sin n|<c.$ Let $\mathbb{Q}_{m}$ be the set ...
0
votes
0answers
36 views

Problems with a basic proof in Aumann Structures

I am pretty sure this is more than trivial, but I have a problem with the proof of a basic results in Aumann structures (this is related to a more general problem I have with proofs that involve ...
1
vote
5answers
209 views

Prove an trigonometric identity. Can someone help me by solving it?

$$1-\frac{\sin^2 x}{1+\cot x}-\frac{\cos^2 x}{1+\tan x}=\sin x \cos x.$$ I cannot come to the result. I make that like $1-\dfrac{\sin^2 x}{1+\frac{\cos x}{\sin x}}$
0
votes
1answer
161 views

Functions $ \cos(2x)$, $\sin(2ax)$, $1$ independent and dependent

For which value(s) of $a$ are the functions $\cos(2x)$, $\sin(2ax)$, $1$ independent over the real numbers? For which $a$ are they dependent? I thought maybe to equate each (with the use of ...
2
votes
0answers
84 views

Derived functors and coboundary operator

I understand that one can define the cohomology of an object $A$ in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ...
0
votes
2answers
57 views

Summation and exponential problems

Solving $$\sum_{k=0}^\infty \frac{k(vt)^ke^{-vt}}{k!}$$ where v is a constant. How is the answer equals vt?
0
votes
0answers
39 views

Effective calculation of cummulative binomial distribution

Given $X \sim B(n,p)$, calculate $P(X \leq k)$. This probability is noted as $F(k;n,p)$. I search an effective method to calculate the cumulative binomial distribution, so it can be done with a ...
1
vote
1answer
67 views

Cardinality of the set

Is the set of all sequences of 0s and 1s, in which no more than two identical characters are nearby, finite, countable or continuum? My thoughts: Of course, it is not the finite. We can separate ...
1
vote
1answer
43 views

Problem about structure of Lie algebra

This is 2.13 exercise in Erdmann and Wildon's book. Define a center $$ Z(L) = \{ z\in L |\ [z,x]=0\ \forall \ x\in L \} $$ If $I$ is ideal of $L$ then let $$ B = C_L(I) = \{ z\in L|\ [z,x]=0\ ...
1
vote
0answers
35 views

Verify $lim\inf_{n\to\infty}[0,\frac{n}{1+n})=lim\sup_{n\to\infty}[0,\frac{n}{1+n})=[0,1)$

I want to verify the following. $lim\inf_{n\to\infty}[0,\frac{n}{1+n})=lim\sup_{n\to\infty}[0,\frac{n}{1+n})=[0,1)$ Using the definition ...
1
vote
0answers
51 views

inductive proof of two combinatorial identities

(please be tolerant if this question has already been answered on MSE - i am a newcomer here and it will take some time to adjust to the community ethos) set $$ a_{2m}= \sum_{j=0}^m ...
0
votes
2answers
135 views

Function to fit a curve

I am trying to fit a function to data that looks like the following. I have no idea what the function is, but by visual inspection it seems to be a parabola (or some other convex polynomial) + a ...
2
votes
1answer
97 views

Equivalence Relation, Quotient Space, Compactness

Define an equivalence relation on $\mathbb{R}^2$ by $a \times b$ ~ $c \times d$ iff $a-c$ and $b-d$ are both integers and let $T=(\mathbb{R}^2)^*$ denote the corresponding quotient space. (a) Show ...
1
vote
2answers
91 views

How find this $I=\iint_{\Sigma}(x^2+y^2+z^2)^{-\frac{3}{2}}(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4})^{-\frac{1}{2}}dS$

Find this Surface integral $$I=\iint_{\Sigma}(x^2+y^2+z^2)^{-\frac{3}{2}}\left(\dfrac{x^2}{a^4}+\dfrac{y^2}{b^4}+\dfrac{z^2}{c^4}\right)^{-\frac{1}{2}}dS$$ where ...
2
votes
2answers
89 views

What is the maximum number of distinct roots does the characteristic polynomial have?

Let $A$ be a $3\times 3$ matrix with real entries which commutes with all $3\times 3$ matrices with real entries. What is the maximum number of distinct roots that the characteristic polynomial of $A$ ...
0
votes
2answers
32 views

Does this modular arithmetic equation hold?

Does this modular arithmetic equation hold: $$a_1N_1+a_2N_2+a_3N_3+\cdots+a_mN_m \equiv a_1+a_2+a_3+\cdots+a_m \mod {N_1+N_2+N_3+\cdots +N_m+}$$
5
votes
1answer
95 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
0
votes
1answer
190 views

How can I prove that Sorgenfry line is Lindelöf space?

How can I prove that Sorgenfry line is Lindelöf space? Now, Sorgenfry line is $\mathbb{R}$ with the basis of $\{[a,b) \mid a,b\in\mathbb{R}, a<b\}$, and in general, topological space is called ...
6
votes
2answers
159 views

structure of the full symmetric group on a countably infinite set

trying to get a handle on the full symmetric group $S$ of permutations on a countable set $X$. i had never really thought much about this group, but now i look at it for the first time it appears a ...
1
vote
2answers
76 views

idea for the completion of a metric space

While doing the proof of the existence of completion of a metric space, usually books give an idea that the missing limit points are added into the space for obtaining the completion. But I do not ...
1
vote
1answer
54 views

Transforming a series to an integral with respect to counting measure

I'd really appreciate it if somebody could help me understand why we have this with a step-by-step explanation (i.e. in an argument complete way) : $$ \sum_{k=1}^{n} {\frac {n} {k^2+nk+1}} = \int ...
2
votes
1answer
230 views

invariant measure under irrational rotation on $S^1$

Prove that if $T:S^1 \to S^1$ is an irrational rotation, then the only probability measure on $S^1$ that is $T-$invariant is the lebesgue measure or a multiple or it. We are considering the ...
5
votes
2answers
190 views

surjective map of rings with same dimension

Let $A \to B$ be a surjective homomorphism between (unital) noetherian commutative rings with the same Krull dimension. Is the kernel of this map nilpotent ? Thanks to Makoto Kato and Martin ...
-3
votes
3answers
254 views

Norm of a fractional ideal of an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of rank $n$. The ideal theory of $R$ is useful at least when ...
0
votes
1answer
36 views

Language to describe a number smaller than, but related to Bell number

I understand that the Bell number $B_n$ is the number of partitions of a set of size $n$. Despite my incredible ineptitude at combinatorics, I also understand most of how the binomial coefficient ...
2
votes
2answers
84 views

Analysis question about projections

I just got back my exam and got $0$ marks in this question: consider a continuously differentiable function $f: [0,1] \to [0, \infty)$ with $|f'(x)| \leq M$ for every $x \in [0,1]$. For each $n \geq ...
0
votes
1answer
77 views

Find the volume of the region with triple integrals.

The volume of the region bounded by $y^2+z^2=1\ \text{and} \ z^2+x^2=1$ should be found. Cylindrical conditions are perhaps the most appropriate in this case but the limits of the integral are what ...
2
votes
1answer
115 views

Rotation/Inner Product between eigenspaces of Hermitian matrices

Given two Hermitian, positive definite matrices $\mathbf{T_1}$ and $\mathbf{T_2}$, is it generally true that the matrix $\mathbf{G}_{ij} = \lvert\langle \mathbf{u}_i \mathbf{v}_j \rangle\rvert^2$ of ...
8
votes
6answers
169 views

When $x$ goes to $0$ , what happens to $\sin\left(\frac{1}{x}\right) $ and $\cos\left(\frac{1}{x}\right)$?

When $x$ approaches $0$, do $\sin\frac{1}{x}$ and $\cos\frac{1}{x}$ converge or diverge? How do you show this?
0
votes
1answer
21 views

question about cauchy sequences [closed]

For which values of $\lambda$ is the sequence $f_n : [0,1] \to \mathbb{R}$ with $f_n(x) = 2^{- \lambda n x } $ when $x \in [0,2^{-n}]$ and $0$ otherwise, Cauchy sequence in $L^7([0,1])$??
1
vote
1answer
172 views

If $g$ is invariant under an ergodic map then it's almost everywhere constant

Let $(X,B(X),\mu)$ be a probability space, where $X$ is compact metrizable, and $B(X)$ are the Borel sets. Let $f:X\to X$ be a measurable function such that: i) $\forall A\in B(X)$ ...
3
votes
1answer
131 views

A visual proof of - Curved surface area of a hemisphere = 2(Area of circle)

Suppose we have a circle with radius $r$ . So its area is $\pi r^2$. Now suppose we have a hemisphere of the same radius ie. $r$. Then its curved surface area is $2 \pi r^2$. Which means it is equal ...
0
votes
1answer
23 views

Conditional expectation probability question

In my problem, $Y$~$Bin(4,p)$ and the conditional distribution of $X$ given that $Y=y$ Is $Poi(y)$. I have to work out $E(X|Y=y)$ and then use the formula $E(X)=E(E(X|Y))$ to find $E(X)$. For ...
2
votes
3answers
75 views

Limit of series $4\left( \frac {1}{8}+\frac {1}{12}\right) +6\left( \frac {1}{24}+\frac {1}{36}\right) +\ldots$

How to find this serie $4\left( \dfrac {1}{8}+\dfrac {1}{12}\right) +6\left( \dfrac {1}{24}+\dfrac {1}{36}\right) +8\left( \dfrac {1}{64}+\dfrac {1}{96}\right) +\ldots $ I think it's telescopic, ...
1
vote
1answer
64 views

What is the proof that $\sum \limits_{v \in V} deg(v) = 2|E|$?

My textbook gives $\sum \limits_{v \in V} deg(v) = 2|E|$ and has the proof If an edge is not a loop it gets counted twice b/c it's incident with 2 different vertices. If an edge is a loop, by ...
5
votes
3answers
808 views

Top homology of an oriented, compact, connected smooth manifold with boundary

Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Then, is it true that $n$-th singular homology of M, that is $H_n(M)$, is vanish? I can't make counterexamples ...
0
votes
1answer
47 views

Problem with cohomology (I)

I have some doubts regarding cohomology. As title suggests I will ask these one by one. Let $G$ be a group and $A$ be $G$-module. Let $C^n(G,A)$ denote the set of all maps from $G \times \cdots ...
0
votes
1answer
52 views

Isomorphism between Lie algebras

This is an exercise (cf. exe 2.11 in the Erdmann and Wildon's book) Define $$ gl_S(n, F) = \{ x\in gl(n,F)|\ x^t S = -Sx \} $$ Here $t$ is transpose. Then let $T=P^tSP$ and show that $$ ...
1
vote
5answers
103 views

Prove that if $7$ divides $6^n + 1$ then $n$ is odd

Prove that if $7$ divides $6^n + 1$ then $n$ is odd Attempt: We'll prove the contrapositive: $n$ is not odd if $7$ does not divide $6^n + 1$

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