0
votes
0answers
4 views

To simplify $\sum\limits_{k|(2m, 2n), k\nmid m} \varphi(k)$

Consider the Euler phi-function $\varphi(n)$ of $n\in \mathbb N$ as the cardinality of $\{1\leq r\leq n: (r, n)=1\}$. I am willing to simplify $$S:=\sum\limits_{k|(2m, 2n), k\nmid m} \varphi(k).$$ ...
0
votes
0answers
6 views

Parametrization of a curve and by arclength

I am working on differential geometry and am having quite the blast, it's mathematics so what else can you have? Enough about that, I am working on the curves and we have various parametrizations and ...
0
votes
0answers
5 views

Does permuting indices of a cocycle leave the Cech cohomology class the same?

Consider a $X$ a space, $\{U_i\}_i$ an open covering, and $\mathcal{F}$ a sheaf on $X$. Consider $(c_{i_0,\ldots,i_n})$ a Cech $n$-cocycle. It is theorem that every Cech $n$-cocyle is cohomologous to ...
0
votes
0answers
9 views

How to prove that there is a bijection from the set Hom$(G_1\oplus G_2, G)$ to Hom$(G_1, G)\times $ Hom$(G_2, G)$?

Willing to solve the following problem but got stuck. Need your help. We denote the collection of all group homomorphisms from $G_1\oplus G_2$ to $G$ as $\text{Hom}(G_1\oplus G_2, G)$. Similarly ...
1
vote
1answer
14 views

Question in Probability Theory

Lets say that I have 3 sets $A,B,C\subseteq E$ and I know $|E|,|A|,|B|,|C|,|\overline{A}|,|A\cap B|,|B\cap C|,|C\cap A|,|\overline{A}\cap B|,|\overline{A}\cap C|$ and $|A\cap B\cap C|$. Is there any ...
2
votes
3answers
33 views

Show that $(2^n-1)^{1/n}$ is irrational

How to show that $(2^n-1)^{1/n}$ is irrational for all integer $n\ge 2$? If $(2^n-1)^{1/n}=q\in\Bbb Q$ then $q^n=2^n-1$ which doesn't seem right, but I don't get how to prove it.
0
votes
0answers
7 views

Integer solution for $x^ay^b-z^ct^d=1$.

P1 Find all nonnegative integer number $(a,b,c,d)$ such that $2^a5^b-3^c11^d=1$. P2 Find all nonnegative integer number $(a,b,c,d)$ such that $2^a3^b-5^c7^d=1$. I am looking for problems ...
0
votes
0answers
4 views

Showing $P=(o,m) \not\in 2E_m(\mathbb{Q})$ for an elliptic curve $E_m$

I have been having trouble with this question. Let $m\in \mathbb{Z}$ with $m > 0$ and define $E_m : y^2 = x^3 −x+m^2$ Then $E_m$ is an elliptic curve Determine the group sturcture of ...
1
vote
1answer
9 views

Universal property of kernel of a homomorphism

I am in the beginning of Category Theory. Let $\theta\colon G\rightarrow K$ be a group homomorphism, and $\ker\theta$ denote the kernel of this homomorphism. Let $\varepsilon\colon G\rightarrow K$ ...
0
votes
0answers
8 views

Radial & Cross-Radial Acceleration: A problem

A particle moves along $r=Ae^{\mu\theta}$ where $\theta=Bt$, prove that its acceleration is proportional to $r$ and makes a constant angle with the radius vector. Approach: $\dot{\theta}=B$ then ...
0
votes
0answers
21 views

What is the expansion in power series of ${x \over \sin x}$

How can I expand in power series the following function $$ {x \over \sin x} $$ I know $$ \sin x = x - {x^3 \over 3!} + {x^5 \over 5!} -+ \cdots $$ but substitution doesn't give me a hint to continue. ...
0
votes
2answers
15 views

Calculus: $f◦(g◦h)=f◦g+f◦h$ - Indirect reasoning

"Is it true that $f◦(g◦h)=f◦g+f◦h$?" I think I should use indirect reasoning to prove this but I don't know where to being. Any help?
0
votes
0answers
20 views

Multiplying $r$ times the function $f(x)=a\sqrt x+b$

I have the next function: $f(x)=a\sqrt{x}+b$ over the domain $\left[ 0, n-1 \right]$, where $n$ is a natural number. For any given $r$ inside the domain (also a natural number), I need to find a ...
-5
votes
0answers
23 views

Can someone solve this? [on hold]

If $$8432: 7 \\ 9213: 0 \\ 5144: 16 \\ 9064: 9 \\$$ Then $$7103: ?$$
0
votes
0answers
15 views

Permutation formula is wrong for this question.

The number of permutations of n distinct objects taken r at a time is nPr. You have four letters, a, b, c, d. Consider the number of permutations that are possible by taking two letters at a time ...
-1
votes
2answers
9 views

Total number of possible binary operations .

If there are n elements in a set the number of binary operations that can be defined are 2n, am I right or wrong ?
1
vote
2answers
21 views

Sum of (only certain) prime reciprocicals

It is well known that $$ \sum_{p\ is\ prime}\frac1{p}$$ diverges. Is there a simple proof that $$ \sum_{p\equiv 1\pmod 4}\frac1{p}$$ and $$ \sum_{p\equiv 3\pmod 4}\frac1{p}$$ also diverge? (p ...
0
votes
0answers
12 views

Probability of getting black or white

If I have a bag of black and white marbles, say 10 black and 10 white, I have a 50% chance of either getting black or white. What if the bag were infinite? Can one talk about probability when ...
0
votes
0answers
6 views

ZFC,unprovability of existence of a countable model,Skolem construction and paradox

The well-known Skolem construction yields a countable model of ZFC,elemetarily equivalent to the universe of sets $V$.Why this construction is not a proof of existence of models of ZFC,as such proofs ...
-2
votes
3answers
24 views

Trinomial Equation

$$x^n + x^m = 1$$Given $n,m$ are two positive distinct integers, prove that $x$ is irrational number which can be expressed only in terms of $n,m$ (Assume $n$ is greater than $m$)
0
votes
0answers
16 views

Is there a notation for saying that a function is defined on some subset of a set?

Let $X,Y$ be sets and let $f$ assign to every $x \in X$ some unique $y \in Y$. Then we may write $f: X \to Y$. This notation has the advantage that $f(X)$ need not be $Y$ but must be "within" $Y$. But ...
0
votes
0answers
9 views

Is the topological entropy of a continuous map $T\colon X\to X$ zero if $X$ is a finite compact topological space?

Let $X$ be a finite compact topological space and $T\colon X\to X$ continuous. As the title already suggests, I am wondering if the topological entropy of $T$, denoted by $h(X,T)$, then is $0$. As ...
0
votes
0answers
5 views

Which multivariate Gaussian has the highest expected norm?

Let $X$ be an $n$-dimensional Gaussian vector with zero mean and covariance matrix $K$ given by: $$K_{ij} = \begin{cases} p_i(1-p_i) & i=j \\ -p_ip_j & i\neq j\end{cases}~,$$ where ...
1
vote
2answers
31 views

Does anybody know a proof that the bases of two or more numbers with the same exponent can be multiplied together? (examples in the main text body).

I.e. how can we simply assume that $5^{1/2}\times 6^{1/2} = 30^{1/2}$ (i.e. all we have done is multiplied the bases and applied the exponent to the answer). In more general terms, how can we assume ...
0
votes
0answers
9 views

Is anyone familiar of a term called “Injected Matrix”?

Is anyone familiar of a term called "Injected Matrix"? I have been asked about it by one of my former students. He is taking a course in stochastic models and encountered the term there. Thank you!
1
vote
0answers
6 views

Big Oh symbol in Taylor expansions

Consider a remainder of some Taylor series: \begin{align}\frac{Mx^6}{C} + \frac{M'x^8}{C'} + \frac{M'' x^{10}}{C''} + ...\end{align} I want to replace this with $\mathcal{O}(x^\alpha)$ for the best ...
0
votes
1answer
10 views

Probability problem possibly based on principle of inclusion exclusion

The problem reads as follows: Probabilities that Rajesh passes in Physics, Math and Chemistry are $p$, $m$ and $c$ respectively. Of these subjects, Rajesh has $75%$ chance of passing in at least ...
0
votes
0answers
10 views

Finding the Number of Subfields of the Splitting Field of $x^{35}-1$ over $\mathbb{F}_8$

Let $E$ be the splitting field of $x^{35}-1$ over the field $\mathbb{F}_8$. Determine $|E|$ and the number of subfields of $E$. Attempt: I am confident that I computed $|E|$ correctly, but I am ...
1
vote
0answers
6 views

$X= \int_0^S e^{(S-s)A} B u(s) ds \Rightarrow X= \int_0^T e^{(T-s)A} B \bar{u}(s) ds$

Consider the ODE system $$X'(t) = AX(t)+Bu(t)$$ where $X(t) \in R^n, \; A \in R^{n \times n} \text{ and } B \in R^{n \times m}$. In control theory, we define the set of states reachable as $$A(0,T) = ...
0
votes
1answer
13 views

An inequality about primes

$p_1,\dots ,p_s$ are all given primes, prove the following inequality: $$\sum_{i_1,\dots ,i_s\geq 0}\frac{1}{p_1^{i_1}\cdots p_s^{i_s}}<(1-\frac{1}{p_1})^{-1}\cdots (1-\frac{1}{p_s})^{-1}$$
3
votes
1answer
13 views

Does a sequence bounded by a function of $L^{\infty}$, converges in $L^{\infty}$?

I want to know if this statement is right? Let $(g_n)$ be a measurable sequence from $[0,T]$ to $\mathbb{R}^n$ such that $\exists \beta \in L^{\infty}_{\mathbb{R}_+}([0,T])$ where $\|g_n(t)\| ...
-4
votes
0answers
28 views

How can I solve this logarithmic equation? [on hold]

$$\log_{0.7}\left(6+7^x\right) \geq x-1 $$ Please help.
0
votes
0answers
5 views

Minimal condition to restrict a curve on a surface

What's the necessary condition to make a curve stick on a parametrized surface?
0
votes
0answers
9 views

Attempted definition of a smooth manifold in $\mathbb{R}^{n}$ and an allied question

I have two questions; the former is about a definition of a smooth manifold in $\mathbb{R}^{n}$ and the latter is about the requirement of openness in the definition of a general manifold. I will ...
1
vote
1answer
8 views

Is it, in general, true that $\mu_f \leq \mu$, where $\mu_f, \mu$ are defined in the following way…

Let $(X, \mathcal{M}, \mu)$ be a measurable space and $f : X \to [0, \infty]$ a measurable function. Let $$\mu_f : \mathcal{M} \to [0, \infty], \mu_f(A) = \int_A f d\mu.$$ Is it, in general, true ...
0
votes
0answers
5 views

Steps in derivative of matrix expression

Can somebody tell me how to arrive at the following expressions \begin{align} \mathbf{D} &= (\mathbf{C}^{-1}\mathbf{t}\mathbf{t}^{T}\mathbf{C}^{-1}-\mathbf{C}^{-1}) \mathbf{\Phi} \mathbf{A}^{-1} ...
1
vote
1answer
18 views

Advanced introduction to riemmanian geometry

After some time of studying differential geometry/topology while avoiding riemmanian geometry I find mysef at a wierd position. I'd like to read a book that contains an introduction to riemannian ...
0
votes
0answers
13 views

Probability that One Can Find a Multiple of Uniformly Random Value

We define finite field $\mathbb{F}_q$, where $q=2p+1$ and $p$ is a large prime number (e.g. 256-bit). I pick uniformly at random value $\beta$ from the subfield such that $\beta>\frac{q}{2}$. ...
0
votes
0answers
14 views

Expected number of button clicks

Suppose we have $N$ buttons and each button can be clicked with probability $p_i$. The game stops when the player clicks the button with $i = 1$. What is the expected number of clicks? I am not able ...
-2
votes
4answers
37 views

Mathematical problem - Set theory

there is set A = {a,b,c,d,e,f}. Problem is: {d,b,f,a,e,c} ∈ A is it T or F and why? Thanks!
0
votes
1answer
16 views

Span and Matrices

I am currently working on two algebra questions that ask me to answer the following questions on matrix $A$, a $m \times n$ matrix given the following conditions: a) $n > m$ b) $m > n$ 1) ...
1
vote
2answers
28 views

Proof in square

there is a problem I'm trying to solve in the picture. So far I found x^2 and y^2 (that are |AP|^2 and |PD|^2). In order to prove, that AP=2PD, can I just divide $$ \sqrt{\frac{x^2}{y^2}} $$ and ...
0
votes
0answers
17 views

Attempt to apply Quotient Rule backwards

If Quotient Rule can be taken backwards I believe it introduces newer constants indefinitely depending on the order of integration. We have in case of differentiation w.r.t $t$ when $ \dfrac{y}{x}$ ...
1
vote
1answer
6 views

Suggestions for Computing $(\mathbb{Z}(i))^3/K$ where $K=((1,2,1), (0,0,5), (1,-i,6))$.

Let $D$ be the ring $\mathbb{Z}(i)$ and $M=D^3$ the free $D$-module of rank 3. Let $K$ be the submodule generated by $(1,2,1), (0,0,5), (1,-i,6)$. Prove that $M/K$ is finite, and determine its order. ...
-1
votes
1answer
45 views

Can anyone show me a solution where we solve a quintic and find its 5 roots properly using elliptic functions? [duplicate]

Kindly use this quintic equation:- $$ 5x^5 + 4x^4 + 3x^3 + 2x^2 + x^1 = 5 $$ Find roots of the above equation using elliptic function? I need this solution very badly?
3
votes
2answers
28 views

Uniform convergence of $\sum_{n=1}^{\infty}\left(\frac{nx}{1+n^{2}x^{2}}\right)^{n}$

Prove with Weierstrass M-Test, that given series $$\sum_{n=1}^{\infty}\left(\frac{nx}{1+n^{2}x^{2}}\right)^{n} ,\hspace{6mm} |x| < +\infty$$ is uniformly convergent First thing I tried to do, ...
0
votes
1answer
13 views

Prove convergence in distribution using the CLT

Problem. Let $X_1,X_2,...$ be independent and identically distributed random variables such that $EX_1^4<\infty$ and $0<V(X_1)=\sigma^2$. Put $$T_n = \frac{1}{n}\sum_{j=1}^n ...
2
votes
1answer
30 views

How to find $n$th term of two sequences both dependent on each other?

Given two sequences , $$\large a_{n+1}+b_{n+1}=2\sqrt{2}(a_{n})+2\sqrt{6}(b_{n})$$ $$\large a_{n+1}-b_{n+1}=2\sqrt{2}(b_{n})-2\sqrt{6}(a_{n})$$ Given $\large a_{0}$ and $\large b_{0}$ , How can i find ...
0
votes
1answer
8 views

Sine-Wave with increasing frequencies in Matlab

in my opinion, the following code should produce a sine-wave that has a frequency of 2Hz at t=20. but when i count the periods between t=19 and t=20, i count more than 3 periods. what am i doing ...
0
votes
0answers
11 views

Gaussian rationals with rational norm

Looking for information on Gaussian rationals with rational norm. A gaussian rational is a complex number of the form z = p + qi where p and q are rationals. Taking only those that have |z| = ...

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