text stringlengths 83 79.5k |
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H: If $V$ is right-orthogonal, does it hold $\langle AV,BV\rangle_F=\langle A,B\rangle_F$?
Let $A,B\in\mathbb R^{m\times n}$. It's easy to see that for the Frobenius inner product it holds $$\langle A,B\rangle_F=\operatorname{tr}B^\ast A=\operatorname{tr}A^\ast B.\tag1$$ So, if $U\in\mathbb R^{k\times m}$ is left-orth... |
H: Computing $\int_{0}^{1} x^2 \sin(2\pi nx)\sin(2\pi mx) \,dx$
I would like to know if there is any easy way or known formula to compute the following integral. For $n,m \in \mathbb{N}$, for $n \neq m$, $$ \int_{0}^{1} x^2 \sin(2\pi nx)\sin(2\pi mx) \,dx$$ I tried various graphs for $n\ne m$, and it seems the answer ... |
H: A question regarding partitions.
Number of partitions of n = p(n)
Number of partitions of n which has a part equal to 1 = p(n-1)
Number of partitions of n into k parts = p(n,k)
If for some k the following inequality holds
p(n,k) ≤ p(n-1)
Then does it necessarily imply that all the partitions of n into k parts ha... |
H: Definition of eigen space.
I am studying linear algebra and got confused in defining eigen space corresponding to eigen value.The thing wondering me is the same thing defined in two different books in different manners.let $\lambda$ be eigen value of matrix $A$ of order n over the field $\mathbb{F}$ then Hoffman & ... |
H: It is given that x, y, z are 3 real numbers such that $(x-y)/(2+xy)+(y-z)/(2+yz)+(z-x)/(2+zx)=0$
It is given that x, y, z are 3 real numbers such that $\frac{(x-y)}{(2+xy)}+\frac{(y-z)}{(2+yz)}+\frac{(z-x)}{(2+zx)}=0$. Is it true that at least two of three numbers must be equal? Justify your answer.
This is the giv... |
H: Partitioning a set of $11$ women and $7$ men - Combinatorics
Let $S (n, k)$ be the number of $k$-element partitions of an $n$-element set.
A set of eleven women and seven men is to be partitioned into four subsets.
None of the subsets should consist exclusively of women or men.
How many such partitions are t... |
H: Is it safe to say the following about the odd prime numbers other than 5.
I am studying basic number theory and have a habit of writing down interesting facts whenever I conclude something from the text or a problem itself. I was wondering whether I can write it down too:
All odd prime numbers other than 5, either... |
H: Some terminology: differences of term, formula, and expression in logic?
In the wikipedia article on logical terms it is written:
In analogy to natural language, where a noun phrase refers to an
object and a whole sentence refers to a fact, in mathematical logic, a
term denotes a mathematical object and a form... |
H: Cartan-Weyl basis for the complexified Lie algebra $L_{\mathbb{C}}(SU(N))$
I'm trying to construct the Cartan-Weyl basis for $L_{\mathbb{C}}(SU(N))$.
Looking at the basis for the complexified lie algebra for $SU(2)$ (consisting of the cartan element and the step operators), is there a straightforward generalisatio... |
H: Real valued continuous function is the unique difference of two positive functions
Let $X$ be a compact Hausdorff space and let $f: X \to \mathbb{R}$ be a continuous function. I want to prove that we have a unique decomposition
$$f= f_1 - f_2$$
where $f_1 f_2 = 0 = f_2 f_1$.
where $f_1, f_2: X \to \mathbb{R}$ are c... |
H: Paradox In the criteria for $a$ to be a removable singularity or a pole of $f$?
My complex analysis textbook stated the following proposition:
Let $a$ be an isolated singularity of $f$
If $\lim_{z\to a}(z-a)f(z)=0$, then $a$ is a removable singularity
If there exists a number $m \in \mathbb{N}$ such that: $\lim_{... |
H: How to prove that $\wp''$'s zeros are not at half-periods?
This is an exercise adapted from Apostol. The problem is stated as
Prove that
$$\wp''\left(\frac{\omega_1}{2}\right)=2(e_1-e_2)(e_1-e_3)$$
where $\omega_1,\omega_2$ generates the lattice for $\wp$.
I could see that by Weierstrass' differential equatio... |
H: First order linear non-homogenous ode
I'm learning how to solve ode and there's one thing in my lecture notes that I don't understand.
$y' +py = q \ $
$y(x_0) = y_0$
I understand that I can rewrite this to
$\phi = ce^{-P} + e^{-P} \int_{x_0}^{x}q(t)e^{P(t)}dt$
So I get:
$y_0 = \phi(x_0) = ce^{-P(x_0)} + e^{-P(x_0)... |
H: $f(0)=f(1)=0$, $f(x)=\frac{f(x+h)+f(x-h)}{2}$ implies $f(x)=0$ for $[0, 1]$
Question: Suppose $f$ is continuous on $[0, 1]$ with $f(0)=f(1)=0$. For $\forall x\in (0, 1)$, there $\exists h>0$ with $0\le x-h<x<x+h\le1$ such that $f(x)=\frac{f(x+h)+f(x-h)}{2}$. Show that $\forall x\in(0, 1), f(x)=0$.
I tried to prove... |
H: Frame vs vector basis in differential geometry
Let $E \to M$ a finite dimensional vector bundle. I faced a couple of times that a vector basis of the fiber $E_x$ over $x \in M$ was often called a 'frame'. Are there any differences between the the notation of a 'frame' and a 'vector basis'? Is a 'frame' just a termi... |
H: Finding a complement of $U=\{f\in C(\mathbb{R},\mathbb{R}) |f(0)=0\}$
Consider the vector space $V=C(\mathbb{R},\mathbb{R})$ and $V\ni U=\{f\in C(\mathbb{R},\mathbb{R}) |f(0)=0\}$. I want to find a complement of $U$, such that $V=U\oplus W$. This condition is the same as finding a set $W$ that satisfies $V=U+W$ and... |
H: Some doubts about Levy's Continuity Theorem proof - Convergence results
THEOREM (Levy's Continuity Theorem)
Let $(\mu_n)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$, and let $(\hat{\mu}_n)_{n\geq1}$ denote their characteristic functions (or Fourier transforms).
If $\hat{\mu}_n(u)$ converg... |
H: Weird Cauchy Problem
Can somebody help me in solving this weird Cauchy problem? I really don't know how to face it.
$$
\begin{cases}
y' = -\dfrac{(2x+y)\cos(x^2 + xy + 1) + y}{x\cos(x^2 + xy + 1) + x + 1}\\\\
y(0) = \sin(1)
\end{cases}
$$
I tried to perform $z = x^2 + xy+1$ and then $z' = \dfrac{y'}{x}$, yet this ... |
H: Integrate from $0$ to $2\pi$ with respect to $\theta$ the following $(\sin \theta +\cos \theta)^n$
$$\int_0^{2\pi} (\sin \theta +\cos\theta)^n d\theta$$
First I think about De Moivre's formula given by
$$(\cos x +i \sin x)^n=\cos (nx)+i\sin (nx)$$
I tried to apply it but I found myself lost !
Any tips or informati... |
H: Boundedness and openness of the set
Need to prove/disprove boundedness and openness of the set $S=\left\{f\in L_1[1,\infty):\;\displaystyle\int\limits_1^\infty x|f(x)|dx<1\right\}$.
There are no problems with boundedness. But I can’t check the openness. If $f_0\in S$ and $f\in B_r(f_0)$, then $\displaystyle\int\l... |
H: How to prove that $(\Bbb{Z}[t]+t\Bbb{R}[t])/t\Bbb{R}[t]\cong\Bbb{Z}\cong\Bbb{Z}[t]/t\Bbb{Z}[t]\cong\Bbb{Z}[t]/(\Bbb{Z}[t]\cap t\Bbb{R}[t])$?
I already proved that $(\mathbb{Z}[t]+t\mathbb{R}[t])/t\mathbb{R}[t]\cong\mathbb{Z}[t]/(\mathbb{Z}[t]\cap t\mathbb{R}[t])$ with the first isomorphism theorem but i do not know... |
H: Is $\mathbb{R}^2$ a Ring?
From what I know, $\mathbb{R}^2$ is a group under addition, defined as $(a, b) + (c, d) = (a+b,c+d)$. However, this answer on another question seems to suggest that $\mathbb{R}^2$ is actually a ring with multiplication defined as $(a, b)\cdot (c, d) = (ac, bd)$. I thought that we usually o... |
H: Solve the differential equation $y=2\sqrt{x}y^2y'+4xy'$
Solve the following differential equation:
$$
y=2\sqrt{x}y^2y'+4xy'
$$
The main problem for me is to understand what type of DE it is, since it is neither of these:
Separable
Homogeneous
Linear
Exact
Bernoulli
Riccati
Implicit
Lagrange
Perhaps I am mis... |
H: Calculating sum of series using derivative of a function
We're given the following problem:
"We know that $\frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k $ for $ -1 < x < 1 $. Using the derivative with respect to $x$, calculate the sum of the following power-series: $ \sum_{k=1}^{\infty} kx^k $ and $ \sum_{k=1}^{\infty}... |
H: why does $P(X) = X^3+X+1$ have at most 1 root in $F_p$?
why does $P(X) = X^3+X+1$ has at most 1 root in $F_p$ ?
I could fact check this on Sage for small values of $p$.
For example $p=5$ or $7$ or $19$; there is no root.
If $p = 11, 2$ is the only root.
If $p = 13, 7$ is the only root.
If $p = 17, 11$ is the only... |
H: Direct Product of Rings and isomorphism
Is there a way of finding all possible isomorphisms on the direct product of rings? I know that if the rings are of different size, direct product isomorphism induces automorphism on each rings, and if the rings are the same (isomorphically), then it also induces isomorphism ... |
H: Prove W is a subspace of V.
If W₁ ⊆ W₂ ⊆ W₃......, where Wᵢ are the subspaces of a vector space V, and W = W₁ ∪ W₂ ∪......
Prove that W ≤ V.
So I proved that:
If W₁ and W₂ are two subspaces of V and W₁ ∪ W₂ ≤ V then W₁ ⊆ W₂ or W₂ ⊆ W₁.
(I let u ∈ W₁ - W₂ and v ∈ W₂ - W₁ and it was trivial)
Now I don't know how to u... |
H: Dual representation of finite groups
If $V$ is a $\mathbb{C}G$ module, then $V^*$ is the dual module with the action
$$(gf)(v) = f(g^{-1}v) $$
for $g\in G,f\in V^*$ and $v\in V$. Where $V^* = \text{Hom}(V,\mathbb{C})$.
What I don't understand is why do we need the inverse of $g$ in $f(g^{-1}v)$, how does this agre... |
H: Convergence of series $\sum_{k=2}^{\infty} \frac{2^{k}}{\lfloor{\frac{k}{2}\rfloor}}$
I would like to inspect the convergence of the following series
$$\sum_{k=2}^{\infty} \frac{2^{k}}{\lfloor{\frac{k}{2}\rfloor}}$$.
Because I am new to the whole series part it would be very nice if someone could explain to me how... |
H: Inequality question.
Let $a,b,c>0$ with $a+b+c=1$. Show that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq 3 + 2\cdot\frac{\left(a^3 + b^3 + c^3\right)}{abc}$$
Ohhhkk. So first off,
\begin{align} a^3 + b^3+ c^3 & =a^3 + b^3+ c^3- 3abc +3abc\\
& =\ (a+b+c)(a^2+b^2+c^2-(ab+bc+ca))+3abc\\
& = \ (1-3(ab+bc+ca)) + 3abc... |
H: Equivalent definition of decomposable map
Let $A\subset B(H)$ be an operator system and $B$ be a $C^*$-algebra. (1)$u:A\rightarrow B$ is called a decomposable map if $u$ is in the linear span of $CP(A, B)$, where $CP(A,B)$ is the set of all completely positive maps from $A$ to $B$.
There is another equivalent defin... |
H: Show that there is a $\pi_i$-related smooth vector field for each smooth vector field $X_i \in \Gamma(M_i,TM_i)$
Assume $M_1, \dots,M_k$ are smooth manifolds and define $M:=M_1\times \dots \times M_k$. Denote the projections on the $i$-th factor with $\pi_i: M \rightarrow M_i$. I want to show that for each smooth v... |
H: For a given integer $k$, which of the following are false?
For a given integer $k$, which of the following are false?
$(1)$ If $k($mod $72)$ is a unit in $\mathbb{Z}_{72}$, then $k($ mod $9)$ is a unit in $\mathbb{Z}_9$
$(2)$ If $k($mod $72)$ is a unit in $\mathbb{Z}_{72}$, then $k($ mod $8)$ is a unit in $\mathbb{... |
H: Prove that (MB (φ))^k = MB (φ^k)
Let V be a K vector space with base B: = {b1,…, bn} and φ an endomorphism in V with a representation matrix MB(φ). Prove that (MB(φ))^k = MB(φ^k) for k = 1, ..., n applies.
AI: Because the diagram below is commutative and thus compatible with compositions of endomorphisms composing ... |
H: Let $a,b,c>0$ with$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$. Prove that $(a + 1)(b + 1)(c + 1) \geq 64$
Let $a,b,c>0$ with$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$. Prove that
$(a + 1)(b + 1)(c + 1) \geq 64$
Ohk so we are given that $abc=a+b+c$ with that now the inequality becomes $2abc+(a+b+c)+1 \geq 64$
How do ... |
H: How to prove the sequence $ \frac{c^n}{\sqrt{n}}, c \in (0, 1)$ is convergent
Given a sequence $a_n = \frac{c^n}{\sqrt{n}}$ where $c \in (0, 1), n = 1, 2, 3, \cdots$, how to prove that the sequence is convergent? What if $c \in (0, \infty)$?
AI: thats because if $c \in (0, 1]$ then $0 \le a_n \le 1/\sqrt{n}$ and $1... |
H: High school percentage question that I got wrong.
Calvin and Susie are running for class president. Of the first 80% of the ballots that are counted, Susie receives 53% of the votes and Calvin receives 47%. At least what percentage of the remaining votes must Calvin receive to catch up to Susie in the election?
Let... |
H: Finding area bounded by 3 function that aren't constant
Find an area that are bounded by $3$ functions:
\begin{align}
&= + 6
,\\
&= ^3
,\\
2 + &= 0
.
\end{align}
I only found the solution if one of the functions is constant, like $x=2$.
AI: \begin{align}
f_1(x)&=x+6
,\\
f_2(x)&=x^3
,\\
f_3(x)&=-x/2
.
\end{align... |
H: The sides of a pentagon are represented in centimeters by $x$, $10$, $2x$, $1$ and $3$. How many even values of $x$ satisfy this pentagon?
The sides of a pentagon are represented in centimeters by $x$, $10$, $2x$, $1$ and $3$. Determine how many even values of $x$ are there that satisfy this pentagon.
The answer... |
H: Help with differentiation chain rule with tensors in backpropagation
Say, we're given $N$ feature vectors $\mathbf{x}_i \in \mathbb{R}^{D \times 1}$ and assembled into a matrix $X \in \mathbb{R}^{D \times N}$. We also have a matrix $W \in \mathbb{R}^{D \times D}$, $W = XX^\top$ and a predictor matrix $Y \in \mathbb... |
H: Number of $3$-Letter words from the alphabet {A, B, C} that have no $2$ "A's" directly one after the other
What is the number of $3$-Letter words from the alphabet {A, B, C} that have no $2$ "A's" directly one after the other?
What am I doing wrong? I have the following calculation:
$(1 \cdot 2 \cdot 3) + (2 \cd... |
H: Prove that $(\mu_1 \otimes \mu_2)\circ {\Pi_1}^{-1}=\mu_1$
Suppose $\Pi_1 :(\Omega_1 \times \Omega_2, \mathcal{F_1} \otimes \mathcal{F_2} ,\mu_1 \otimes \mu_2) \rightarrow (\Omega_1, \mathcal{F_1},(\mu_1 \otimes \mu_2) \circ {\Pi_1}^{-1})$ is a projection map which basically maps $(x,y) \rightarrow x$.
We are to pr... |
H: Finding a closed form of an integral: $\int_0^k\ln(a\sin^2(x)+(a+b)\cos^2(x))dx$
I am trying to find a closed form for the following integral:
$$\int_0^k\ln(a\sin^2(x)+(a+b)\cos^2(x))dx$$
And I know that $a>0$, $b\ge0$ and $k=(\pi(1+n))/2$ where $n$ is a natural number.
How can I approach this problem?
AI: Assignme... |
H: For which ideal $I$ of $\Bbb Z[t]$ is $\mathbb{Z}[t]/I\cong\Bbb Z_{11}$?
Maybe for $I=(11,t-1)$ but i don't know how to prove it or if it is even right.
AI: We want to kill $11$ and $t$ and so the natural choice is $I=(11,t)$. This leads to the homomorphism $\phi: \mathbb{Z}[t] \to \mathbb{Z}_{11}$ given by $\phi(p... |
H: Is the degree m taylor polynomial, A polynomial field over R^n?
$x \in R[x]$, $x^2 \in R[x]$, since it is a field any operation * b/w these two's results must also be in $R[x]$, but $x^-1$ is not in $R[x]$
So clearly polynomials do not form a field.
However, Professor Ghrist says that it is a polynomial field at 1:... |
H: If $0 < \alpha < \alpha + \delta < \beta < \frac\pi2$ then $\tan\alpha + \tan\beta > \tan(\alpha + \delta) + \tan(\beta - \delta).$
For reasons that probably don't bear examination (I've rewritten my answer to this question, but I haven't posted the new improved version with added vitamins, because I wish to supple... |
H: Differences between polynomial quotient rings $\mathbb{Z}_m[x]/(x^n+1)$ and $\mathbb{Z}_m[x]/(x^n-1)$
As based on the definition of the polynomial quotient ring
$\mathbb{Z}_m[x]/(x^n+1) = \left\{a_{n-1}x^{n-1}+\cdots+a_1x+a_0:a_i\in\mathbb{Z}_m\right\}$,
does that imply that $\mathbb{Z}_m[x]/(x^n+1) = \mathbb{Z}_m... |
H: A closed discrete set
Let $V$ be a normed vector space.
Let $(b_n)\subseteq V, b_n \to b\in V.$ Show that $B := \{b,b_1,b_2\dots\}$ is closed.
I know that if $b_n\to b,$ then $b_n$ is Cauchy. That is, $\forall \epsilon > 0, \exists N\in\mathbb{N}, n,m\geq N\Rightarrow ||b_n-b_m|| < \epsilon.$ Also, if $(x_n)\sub... |
H: Olympiad Number theory Problem Solution Doubt
[Iberoamerican $1998]$ Let $\lambda$ be the positive root of the equation $t^{2}-1998 t-1=0 .$ Define the sequence $x_{0}, x_{1}, \ldots$ by setting
$
x_{0}=1, \quad x_{n+1}=\left\lfloor\lambda x_{n}\right\rfloor \quad(n \geq 0)
$
Find the remainder when $x_{1998}$... |
H: Dependence of two random variables
In a Bernoulli experiment of parameter $p$ let $T$ be the instant of first success and $U$ the instant of second success. Find the density of $U$ and tell if $T$ and $U$ are independent or not.
This is what I've done: since $U$ is the instant of second success we have that:
$$p_U(... |
H: Definition of derivative as a linear operator being applied to a vector
I have been told that, given a differentiable function $f: \mathbb{R}^n\longrightarrow \mathbb{R}$, we can view $f'(x)$ as a linear operator from $\mathbb{R}^n$ to $\mathbb{R}$ for any $x$, which makes sense because it is a vector, and thus a l... |
H: On the definition of compactness
Rudin, in Principles of Mathematical Analysis, defines compactness: A set in a metric space is compact if and only if for any open cover $\{G_\alpha\}$ of $E$ there exist a finite subcover $G_{\alpha_1},...,G_{\alpha_k}$ such that:
$E \subseteq G_{\alpha_1} \cup \cdots \cup G_{\a... |
H: The image of a locally constant function is coutable
Let $X\subset \mathbb{R}^m$. A function $f: X\to \mathbb{R}^n$ is said to be locally constant if, for every $x\in X$, there is $\epsilon_x>0$ such that, $f(y)=c_x$, for all $y\in B_{\epsilon_x}(x)\cap X$. Show that if $f:X\to \mathbb{R}^m$ is locally constant, th... |
H: Prove $\lim_{n \rightarrow \infty} f(x) f(2^2x) f(3^2x) \cdots f(n^2x) = 0$ for $f: \mathbb{R} \rightarrow \mathbb{R}$ in $L^1(\mathbb{R})$.
Here's another question that I'm stuck on from my studies for an upcoming exam. This one comes from another practice preliminary exam.
Problem
Let $f: \mathbb{R} \rightarr... |
H: How many of these unit squares contain a portion of the circumference of the circle?
Question: Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4n^2$ unit squares by drawing $2n−1$ horizontal and $2n−1$ vertical lines one unit apar... |
H: Prove that if $ \lim_{x\to\infty}f\left(x\right)=L $ then $ \lim_{n\to\infty}\intop_{0}^{1}f\left(n\cdot x\right)dx=L $.
let $ f $ be integrable function in any interval such [0,M].
assume $ \lim_{x\to\infty}f\left(x\right)=L $ for some $ L\in \mathbb{R} $
and prove that
$ \lim_{n\to\infty}\intop_{0}^{1}f\left(n\... |
H: Finding $\iint_W x^2y\,\mathrm{d}x\,\mathrm{d}y$ where $W$ is a rectangle
I'm learning double integrals and I'm trying to calculate the following integral:
$$\iint_{W} x^2y \,\mathrm{d}x\,\mathrm{d}y\,,$$ where $W$ is a rectangle given by points: $A=(0,1), B=(2,1), C=(2,2), D=(0,2)$.
Could you please help me calcul... |
H: If $\frac{z-\alpha}{z+\alpha},(\alpha \in R)$ is a purely imaginary number and $|z|=2$, can we find value of $\alpha$ geometrically?
If $\dfrac{z-\alpha}{z+\alpha},(\alpha \in R)$ is a purely imaginary number and $|z|=2$, then find value of $\alpha$.
Now I took $\dfrac{z-\alpha}{z+\alpha}=t$ and as t is purely imag... |
H: Conditions for a Euclidean domain to be a field or a polynomial ring over a field
I am having trouble proving the following.
Let $R$ be a Euclidean domain with degree function $\delta,$ i.e., $\delta(ab)=\delta(a)\delta(b)$ for all $a,b\in R-\{0\}$ and $\delta(a+b)\leq\textrm{max}(\delta(a),\delta(b))$. Show that ... |
H: Find the value of $n$ if $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\frac{a+b}{2}$
Now, this question looks simple, it did to me too, at first, but I got stuck at a point and can't get out.
This is how I did it, take a look :
$$\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}=\dfrac{a+b}{2}$$
By cross multiplication, we get :
$$2a^{n+1}+... |
H: How can I prove that a polynomial is irreducible over $\mathbb Z[x]$?
I am asked to prove that
$$P(x) = x^6 + x + 1$$ is irreducible over $\mathbb Z[x]$.
I tried using Eisenstein criteria by a doing a change of variable such as $x = y + a$ but I was unsuccessful.
AI: $x^6+x+1$ has no real roots, so it can't have ... |
H: Uniform Convergence of a series of functions using the Dirichlet's test
I have recently been trying out some questions on series of functions. I got stuck in one of those problems in which I am supposed to show that the below series of functions is uniformly convergent on any bounded interval.
The series is given b... |
H: Induced homomorphism example
Give an example of a commutative ring $R$, $R$ -modules $M,N,$ and$W$, and an injective $R$ module homomorphism $g:M \rightarrow N$ such that the induced homomorphism $Hom_{R}(N,W) \rightarrow Hom_{R}(M,W)$ is not surjective.
I'm just learning the basics of modules and I'm having troubl... |
H: How do I determine the area that is inside the circle r=3cos(θ) and outside the r=3sin(2θ) curve (For the first quadrant)
I have this graph here, but how do I find out the area that's inside the circle but outside the 3sin2θ rose shaped curve? I'm lost, please help me solve this
AI: The area of a region in polar co... |
H: Proving that a quantity is an integer
Suppose $gcd(a,b)=1$ and $a,b\geq 1$. How does one prove that $\dfrac{a+b\choose b}{a+b}$ is an integer?
I tried substituting for $a$ and $b$ using $ax+by=1$ for some $x,y$, also tried expanding but i cant figure it out. Any help is appreciated, thanks!
AI: HINT: $a$ and $a+b$... |
H: Derivative has Linear Growth Implies Lipshitz
Let $f\in C^\infty(\mathbb{R}^d)$. If $f$ has linear growth i.e
$$|\nabla f(x)|\leq C(|x|+1)$$
then is $f$ Lipshitz?
attempt at proof :
by Mean Value Theorem there exists $c\in (0,1)$ such that
\begin{align*}
|f(x)- f(y)| \leq |\nabla f((1-c)x+cy)||x-y|\leq & C(|x-y... |
H: GIF of the sum $\sum_{i=1}^{1000}\frac{1}{i^{2/3}}$
I am asked to find the GIF (greatest integer function) of the sum:$$\sum_{i=1}^{1000}\frac{1}{i^{2/3}}$$
I am able to find the lower limit of the sum by using the fact that
$$\sum_{i=1}^{1000}\frac{1}{i^{2/3}}$$ is greater than
$$\int_{1}^{1000}x^{-2/3}dx$$
But ... |
H: Showing two UFD elements are relatively prime without Bézout
I am struggling to prove a result about greatest common divisors that will lead eventually to Gauss's Lemma.
In particular, we let $R$ be a UFD with $\alpha,\beta \in R$. Define $\delta=\textrm{gcd}(\alpha,\beta)$, which exists because $R$ is a UFD. My b... |
H: Taylor series at $a=0$
I am trying to find Taylor series representation $\sum_{n=0}^\infty a_n x^n$ for the function
\begin{equation*}
f(x)=
\begin{cases}
\frac{x-\sin{x}}{x^2}, & \text{ when } x \neq 0\\
0, & \text{ when } x = 0,
\end{cases}
\end{equation*}
where $a_n = \frac{f^n(0)}{n!}$.
I am co... |
H: Show that $\sum_{1}^\infty\frac{\sin(nx)}{n^3}$ is differentiable everywhere
I have recently been trying out some questions on series of functions.In one of the questions, I was given a series $$\sum_{1}^\infty\frac{\sin(nx)}{n^3}$$
and now I am supposed to show that the above series is differentiable at every real... |
H: Proof verification and explanation in probability
Six regimental ties and nine dot ties are hung on a tie holder.
Sergio takes two simultaneously and randomly. What is the probability that both ties are regimental?
I have seen that the probability that, not counting the order, the two extracted are between 6 f... |
H: Classifying certain types of matrices
How many similarity classes of nilpotent $4 \times 4$ matrices over $\mathbb{C}$ are there?
I suspect the answer is connected to minimal polynomials, but I'm not sure. Any suggestions?
AI: An $n \times n$ matrix is nilpotent if and only if it has no eigenvalues except $0$, i.e.... |
H: Evaluate $\int \cos^2(x)\tan^3(x) dx$ using trigonometric substitution
How would I integrate to evaluate $\int \cos^2(x)\tan^3(x) dx$ using trigonometric substitution?
I made an attempt by making substitutions such as $$\cos^2(x)=1-\sin^2(x)$$
$$\tan(x) = \frac{\sin(x)}{\cos(x)}$$ and $$\tan^2(x)=1+\sec^2(x)$$
But ... |
H: Removing Gibbs Phenomenon
I am working with a sample of 20 points given from an unknown 1-periodic function that are plotted like this: Original sample
I am using Inverse Fast Fourier Transform (ifft) to recover the signal resampled in 1000 points at [0,1) that is plotted like this: Resampled
It is showing a Gibbs ... |
H: Uniform convergence and integrals.
I'm asked to tell if the following integral is finite:
$$\int_0^1 \left(\sum_{n=1}^{\infty}\sin\left(\frac{1}{n}\right)x^n \right)dx$$
I studied the series (which converges uniformly on $(-1,1)$ by d'Alembert's Criterion and in $-1$ by Leibniz's Criterion, so in general the conver... |
H: Clarify about local contractibility of quotient spaces
Consider these couple of spaces: the first is $A:=\{\frac 1 n :n \in \mathbb N\}\subset\mathbb R $; the other is $B:=[0,3)\subset \mathbb C$. I must describe the topology induced by the projections $h:\mathbb R\to \mathbb R/A$ and $k:\mathbb C\to \mathbb C/B$. ... |
H: Find $( \dotsb ((2017 \diamond 2016) \diamond 2015) \diamond \dotsb \diamond 2) \diamond 1$ given ...
For positive real numbers $a$ and $b,$ let
$$a \diamond b = \frac{\sqrt{a^2 + 4ab + b^2 - 2a - 2b + 9}}{ab + 6}.$$Find
$$( \dotsb ((2017 \diamond 2016) \diamond 2015) \diamond \dotsb \diamond 2) \diamond 1.$$
... |
H: The probability of getting $y$ new coupons from a batch of $k$
In the coupon collector process, the goal is to assemble a collection of $n$ distinct coupons, while we get a random coupon at each time.
I am looking at a generalization of this problem, where at each time we get a batch of $k$ random coupons (with rep... |
H: Show that the series $\sum_{n=1}^\infty \sin \left( \frac{x}{n^2} \right)$ does not converge uniformly
I asked this question about a week ago but I am little bit unsure about the way to solve it so I hope it is ok if I ask again about some things I do not fully understand.
I have to show that the series
$$
S = \sum... |
H: Moving a rocket between two points on a straight line, when to rotate from prograde to retrograde?
Imagine you have a rocket and you want to move it from point a to point b. The flight plan is as follows:
Fire the rocket engine for a constant acceleration of 1 m/s^2 until x meters is covered (prograde burn)
Turn t... |
H: Is $H_m - H_n$ a surjection onto $\mathbb{Q}^+$?
I was wondering whether, for each rational $q$, we may always write
$$q = \sum_{k=a}^b \frac 1k$$
For some positive integers $a \leq b$. I get the feeling that this is not true (although an immediate consequence of $\mathbb{R}^+$ being Archimedean is that the set of ... |
H: Is $1.r=r.1=r$ for a Non Commutative Ring
I am new to ring theory.
As per one of the axioms of a ring $R$ we have $\forall$ $r \in R$ ,$\:$$\exists$ $1 \in R$ such that $$1.r=r.1=r \tag{1}$$
So this is definitely commutative property for $1$ and $r$
Is this scenario true even for Non commutative Ring?
Does it mean... |
H: What power must we check to find the order of an element?
I know that $2^{100} \equiv 1 \pmod {125}$ because $\phi(125)=100$. $125=5^3$ is also the perfect power of an odd prime, so it has at least one primitive root. So, it is reasonable to check if $2$ is a primitive root mod $125$.
To check this, it would suffi... |
H: Is this a well known property of modular arithmetic
Let:
$p_1, p_2$ be primes
$x > 0$ be an integer where $p_1 \nmid x$ and $p_2 \nmid x$
I am interested in understanding the conditions where:
$x - p_1 \equiv 0 \pmod {p_2}$
$x - p_2 \equiv 0 \pmod {p_1}$
It seems to me that this is only true when:
$$x - p_1 - p... |
H: Finding all intersections of $f(x)= \sin(x)+1$ and $g(x)= \cos(x)$ on the interval $[0,4\pi]$
The question asks to find all the points where $f(x)= \sin(x)+1$ intersects with $g(x)= \cos(x)$ on the interval $[0,4\pi]$.
I started by setting both equations equal to each other resulting in the new equation:
$$\sin(x... |
H: Calculus proof of ln(ab)= lna + lnb
My calculus book states the following theorem of the properties of natural logarithms:
If a, b > 0 , then
ln(ab)= lna + lnb
The author goes on to prove this theorem as follows
I do not understand what property allowed the author to use the substitution U = t/a because the origin... |
H: $ \frac{1}{n}\sum_{i=1}^{n}{C_i}\text{ is weakly compact} $
Let $X$ be a separable Banach space. Let $C_1,...,C_n$ are nonempty weakly compact convex subsets of $X$. Why
$$
\frac{1}{n}\sum_{i=1}^{n}{C_i}\text{ is weakly compact}
$$
An idea please.
AI: In general for a topological vector space $X$ and compact subse... |
H: Minimal mutual information for the Binary Symmetric Channel
I am working on the following exercise:
Let $X, Y$ be RVs with values in $\mathcal{X} = \mathcal{Y} = \{0, 1\}$ and let $p_X(0) = p$ and $p_X(1) = 1−p$. Let $\mathcal{C} = (X , P, Y)$ be the channel with input RV $X$, output RV $Y$ and transition matrix
$... |
H: is this the correct way to solve this question?
when two fair dice are rolled,the odds of throwing a 'double'(two dice with the same number) is 1:5.
if two dice are rolled 400 times ,the best estimate of the number of times you would NOT get a double would be ___?
my work:
400/2=80/2=40
is this the correct answer ... |
H: How to compute the transition matrix of a channel composed of two channels?
I have a quick question on the following exercise:
Let $C = (X , P, Y)$ be a binary channel which is composed of two
binary channels in sequence, such that the output of the first channel $C1 = (X , P_1, Z)$ is the
input of the second ... |
H: Evaluation of a complex polynomial
As an intermediate step to a problem, I would like to know whether or not the following is true:
Let $0<r<1$, and let $\zeta_j$ denote the $n$th root of unity. Then define polynomial index by $j$ as
$$ f_j(z) = \frac{z\prod_{i} (z-r^{1/n}\zeta_i)}{z-r^{1/n}\zeta_j}.$$
Then $f... |
H: Finite dimension implies A[a]=A(a)
I was trying to prove following statement:
Let $A \subseteq F$ be field extension and $a \in F$. Then $A[a] = \{f(a)\,|\,f \in A[x]\}$. Prove that if $A[a]$ is finite dimensional as vector space over A, then $A[a]=A(a)$
All my attempts were unsuccessful, how do we prove such thing... |
H: Convert to a polynomial type integral
I came across a question where I had to convert $\int x^{32}\left(4+7x^3\right)^{2/9}\,dx$ to a "polynomial type integral". I have no idea where to start because I have never seen this type of question before. All I know is that this type of integral appears in engineering.
I c... |
H: Moment generating function for sum of independent random variables same as joint mgf
I'm seeing in general that for moment generating functions, the mgf of $X+Y$ where $X,Y$ are independent random variables is $M_{X+Y}(t) = M_X(t)M_Y(t)$. I'm also seeing that the joint mgf is given by $M_{(X,Y)}(t) = M_X(t_1)M_Y(t_... |
H: Show there is no ring map $R=\mathbb{Z}[\sqrt{-3}]\to\mathbb{Z}[i]=S$ such that $1_R\mapsto 1_S$
Let $R=\mathbb{Z}[\sqrt{-3}]$, and let $S=\mathbb{Z}[i]$. For sake of contradiction, assume $\varphi:R\to S$ is a ring map with $\varphi(1_R)=1_S$. Note that $\mathbb{Z}[\sqrt{-3}]$ is a free abelian group on the genera... |
H: Convergence $ \int_{-1}^1 \sqrt{1-\frac{x}{(1-x^2)^2}}$
I was trying to solve the following integral:
$$\int_{-1}^1 \sqrt{1-\frac{x}{(1-x^2)^2}}$$
But when I plugged it in to any online calculator, It said it couldn't find the integral and that it might not exist. Does this integral converge or is it just very hard... |
H: How to get the number of all possible combinations of k positive integers to reach a given product?
Let $n_1,\cdots,n_k$ be $k$ positive integers and $k>2$. Given that $\prod_{i=1}^kn_i=N$, how would you find all combinations of $(n_1,\cdots,n_k)$? Here the order does matter. For example， when $k=3$, $(2,3,2)$ and ... |
H: Metric properties.
i have the following problem:
Let $x=(x_{1},...,x_{n})$ and $y=(y_{1},...,y_{n})$ in $\mathbb{R^{n}}$.
let’s define $d(x,y)=|x_{i}-y_{i}|$ for some $i \leq n$ permanent.
What properties of metric does d have?
I have come to the conclusion that it is a metric, but according to the book I am follow... |
H: Linear Algebra Matrices to equations
Show that
$$\operatorname{det}\;\begin{bmatrix}
1&1&1\\x^2&y^2&z^2\\x^4&y^4&z^4
\end{bmatrix}=(y^2-x^2)(z^2-x^2)(z^2-y^2)$$
I am doing my homework,and this question came up.
I need to know about subject of this question so i can study and solve it.
I tried to look it up from... |
H: Correct notation to restrict parameter in equation
I'm trying to express the following equation using correct notation:
$\sin{\left(\frac{n\pi}{2}\right)},n\,\text{even} = 0$
I've already specified $n$ is a natural number, so presumably I don't need to respecify it? Would the following be better?
$\left\{\sin{\left... |
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