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H: Why $O(\lambda_\text{max}^{\tau/2})=O(1/n^2)$ for $\tau=\frac{4\ln n}{1-\lambda_\text{max}}$?
It appears in the proof of Lemma 9 in 'Bounds on the cover time' by A. Z. Broder and A. R. Karlin.
AI: I am assuming that $0 < \lambda_{max} < 1$.
Hint : Prove that $\frac{2 ln(\lambda)} {(1-\lambda)} < -2$. |
H: What is the difference between diagonalising a matrix and its eigendecomposition?
When we diagonalise a matrix, we write it in terms of a diagonal matrix $S$ which contains all of the eigenvalues of the matrix, a matrix $P$ which contains all of the corresponding eigenvectors of $A$, and $P^{-1}$.
$$A=P^{-1}SP$$
Ho... |
H: Positive integer solutions to $a^b-b!=a$
Conjecture: The only solutions to $a^b-b!=a$ such that $a,b\in\Bbb Z^+$ are $(a,b)=(2,2)$ and $(2,3)$.
Evidently, we have $a<b$, and extending the domain of $a,b$ to $\Bbb R^+$ reveals that asymptotically as $a\to+\infty$ either $b\sim1$ or $b\sim e(a-1)$ (using Stirling's... |
H: How to show that $I$ = {f $\in$ $F(X,R)$ : $f(a)=0$ $\forall a \in A$} is an ideal of $F(X,R)$.
I am studying ring theory and have come across a ring:
$F(X,R)$ ={all functions $X \to R$} with $(f+g)(x)=f(x)+g(x)$ and $(f \times g)(x) = f(x) \times g(x)$ with $X$ a nonempty set.
There is a question:
Show that, for $... |
H: If every subset of $X$ is saturated, then $(X,\tau)$ is a $T_1$ - space
In a general topology exercise the following question is asked:
Is it true that if the topological space $(X,\tau)$ is such that every subset is saturated, then $(X,\tau)$ is a $T_1$- space?
My approach to answering this:
If "every subset is... |
H: From the given definition of function $f(n)$, find $f^{-1}(-100)$
$f: \{1,2,3..\}\rightarrow \{\pm 1,\pm 2, \pm 3..\}$ is defined by $f(n)=\begin {matrix} \frac n2~~\text{if n is even} \\ -\frac{n-1}{2}~~\text{if n is odd}\end{matrix}$
$-100$ is even, so situation one would apply in which case
$$y=\frac n2$$
$$n=... |
H: Extending Taylor's theorem to differentiable, but not continuously differentiable functions
Taylor's theorem says that if $f:I\to\mathbb R$, $I\subseteq\mathbb R$ an open interval, is $n$ times continuously differentiable, then for $x_0\in I$ there exists a continuous function $R_{n,x_0}:I\to\mathbb R$ with $\frac{... |
H: Is there any matrix for which $A^{-1} = - A^t$?
where $A \in GL_{3}(\mathbb{R})$
My idea:
$A^T \cdot A = I$
and we know that
$A^{-1} \cdot A = I$ which implifies that,
$A^T \cdot A = I$
$(A^T⋅A)⋅A^{−1}=(I)⋅A^{−1}$
$A^{T}⋅(A⋅A^{−1})=A{−1}$
$A^T⋅(I)=A^{−1}$
$A^T=A^{−1}$
I'm not sure that my idea above is a correct ... |
H: Integral of $\int_{-\infty}^{\infty} e^{\alpha x}/({e^x+1})$
Show that, for $0<\alpha<1$:
$$\int_{-\infty}^{\infty} \frac{e^{\alpha x}}{{e^x+1}}\text{d}x=\frac{\pi}{\sin(\pi\alpha)}.$$
Hint: Use the recangular path $S_r=\left[-r, r, r+2\pi, -r+2\pi,-r \right]$ (see fig. 1).
My attempt: Denoting $f(z)=\frac{e^{\a... |
H: Proving a constructed approximating sequence in a Banach space converges
Suppose that $(V,\|\cdot\|)$ is a Banach space. Let $S_n$ be a sequence of closed sets such that $\bigcup_{n\in\mathbb{N}} S_n = V$ and $S_n \subseteq S_m$ for $n\leq m$.
Let $(x_n)_{n\in\mathbb{N}}$ be a converging sequence in $V$ and define ... |
H: Question about probability / mutually exclusive events
Decide whether this statement is true or false:
Let $(\Omega, \mathbb{F}, \mathbb{P})$ be a probability space, if for two events $A,B \in \mathbb{F}$ $\hspace{1cm}$ $\mathbb{P}(A \cup B)= \mathbb{P}(A) + \mathbb{P}(B)$ holds, then $ A \cap B= \emptyset$
In the ... |
H: Multidimensional Young diagrams
Consider a Young diagram defined as follows:
A Young diagram (also called a Ferrers diagram, particularly when
represented using dots) is a finite collection of boxes, or cells,
arranged in left-justified rows, with the row lengths in
non-increasing order. Listing the number of boxe... |
H: Conditional probability with inequality
It is known that: $P(X > a) = 1 - P(X \leq a)$
Is there a rule for $P(X > a | Y > b)$ ? Maybe something similar to:
$P(X > a | Y > b) = 1- P(X < a | Y < b)$
(I am not sure, just a guess)
Thank you!
AI: $P(X > a | Y > b) = 1- P(X \le a | Y >b)$ is true since
\begin{align*}
&P(... |
H: hereditary ring and Dedekind ring
On page 161 of Rotman's homological algebra book, it states that:
A Dedekind ring is a hereditary domain.
Following it is an example: If k is a field, R = k<x,y> is both left and right hereditary. This ring is neither left-noetherian or right-noetherian. However, Dedekind rings are... |
H: Is this function surjective? $F:(0,1) \times \mathbb{Z} \ni (a,b) \rightarrow a+b \in \mathbb{R}$
Consider the function
$$F:(0,1) \times \mathbb{Z} \ni (a,b) \rightarrow a+b \in \mathbb{R}$$
Is it surjective?
At first I thought that $F$ is not surjective, because we will not get integer numbers. But, what if $0,(... |
H: What Is Bigger $\frac{3}{e}$ or $\ln(3)$
Hello everyone what is bigger $\frac{3}{e}$ or $\ln(3)$?
I tried to square it at $e$ up and I got:
$e^{\frac{3}{e}} = \left(e^{e^{-1}}\right)^{3\:}$ and $3$ but I don't know how to continue I also tried to convert it to a function but I didn't find.
AI: Note that
$$\dfrac d... |
H: How can I solve $x^2.(y')^2-2.(x.y-3).y'+y^2=0$?
How can I solve the differential equation $x^2.(y')^2-2.(x.y-3).y'+y^2=0$ ? I don't have any idea of what method to use on this.
AI: Hint:
$$x^2.(y')^2-2.(x.y-3).y'+y^2=0$$
$$(xy'-y)^2=-6y'$$
This is Clairaut's differential equation:
$$y=xy'+f(y')$$
$$y=xy'\pm \sqrt ... |
H: Determining whether an angle is between two given angles on the unit circle
I am trying to find a way to determine whether an angle is between two given angles where all angles are provided as vectors on the unit circle i.e.: $\mathbf{a}=(\cos(\theta),\sin(\theta))$
Note that by inbetween I mean on the arc of the ... |
H: Given $f\in A(\{z\in\mathbb{C}:|z|<2\})$ and $f(1)=0,f'(1)\neq0$ calculate the angle
Given $f\in A(\{z\in\mathbb{C}:|z|<2\})$ and $f(1)=0,f'(1)\neq0$ setting $u=Re(f),v=Im(f)$ and assuming in a neighborhood oh1 the $u(z)=0$ define a smooth path $\gamma_0$ and the same for $v$ define $\gamma_1$.
Find the angle betwe... |
H: A simpler refutation of the General Comprehension Principle?
The famous Russell's Paradox in which $R = \{ x \; | \; x \notin x \}$ leads to a contradiction
$$
R \in R \Longleftrightarrow R \notin R,
$$
thereby showing that the General Comprehension Principle entails inconsistency. But I think I found an even simpl... |
H: Does the curve with the following equation holds a line segment?
For a smooth curve $\alpha:I\to \mathbb{R}^3$ with $[a,b]\subset I$,
if $$|\alpha(b)-\alpha(a)| = \int_a^b |\alpha'(t)|dt$$ holds
Does the curve with the equation hold a line segment?
We can show for any smooth curve with point $\alpha(a),\alpha(b)$ f... |
H: In a metric space a sequence with no converging subsequences is discrete (?)
I've been trying to prove that given a metric space $X$ (not necessarily complete) and a sequence $(x_n)_n \subseteq X$ which contains no convergent subsequences, there exists an open neighborhood $V_n$ of $x_n$ for each $n \in \mathbb{N}$... |
H: Using Euler's theorem and I still can't solve this question
let $n\in \Bbb Z$ suppose for every $q|n-1$ there is $a_q\in \Bbb Z_n^*$ that keeps equations $a_q^{n-1} \equiv 1 \pmod n$ and $a_q^{\frac{n-1}{q}}\neq1 \pmod n$. proof that n is prime number.
can you help me?
I've been told to prove that $\Bbb Z_n^*$ is c... |
H: If $A$ is an orthogonal matrix with $|A|=-1$, show that $|I-A|=0$
Let $A$ be an $n \times n$ orthogonal matrix where $A$ is of even order with $|A|=-1.$ Show that, $|I-A|=0,$ where $I$ denotes the $n \times n$ identity matrix.
My approach
$A \cdot A^{\top}=I$
$|A| \cdot\left|A^{\top}\right|=1 \quad$ or $\quad\left... |
H: $f(x) = \cos|x| - 2ax + b$ increases for all $x$. Find the maximum value of $2a + 1$
Here's how I approached the problem.
$f'(x) = -\sin x - 2a$
$f'(x) \geq 0$
$\Rightarrow -\sin x -2a \geq 0$
$\Rightarrow 2a \leq -\sin x$
$\Rightarrow 2a+1 \leq 1- \sin x \tag{*}$
Since range of $\sin x$ is $[-1,1]$,
$\Rightarrow ... |
H: Why is the finite-projective-plane minus a single edge r-partite?
Let $P_r$ be the finite projective plane in which each line contains $r$ points (when it exists). For example, $P_2$ is a triangle, $P_3$ is the Fano plane, and $P_r$ exists whenever $r-1$ is the power of a prime number.
Let $P_r'$ be $P_r$ with one ... |
H: In a nonabelian finite group $G$, if a prime $p$ divides order of $|G|$, $p$ divides order of centralizer of some element which is not in the center
Here's what I want to prove:
Suppose $G$ be nonabelian finite group and $p$ be a prime which divides the order of G. Then there is some element $b\in G$ such that $b \... |
H: Can every symmetric, unimodular and positive definite $G\in\mathbb{Z}^{n\times n}$ be written as $G=U^TU$?
Let $G\in\mathbb{Z}^{n\times n}$ be symmetric, unimodular and positive definite. Does there exist a unimodular matrix $U\in\mathbb{Z}^{n\times n}$ such that $G=U^TU$? I now that the result is true if $n=2$, bu... |
H: Reversing MODULO operation ? system of equations
i have 1000 prime numbers p1 ... p1000 .. which i use to encode a value
v % p1 = r1
v % p2 = r2
v % p3 = r3
....
v % p1000 = r1000
then I pick the 20 PRIMES which gives the SMALLEST reminders and store them.
Later I want to be able to recover back the VALUE (or... |
H: Getting a negative magnitude when solving for magnitude using the formula for angle between two vectors.
Problem:
$\vec{u},\vec{v}$ are two given vectors. It is known that
$\vec{u} \cdot \vec{v} = 5, \ \ \ ||\vec{v}|| = 2, \ \ \ \theta=2\pi/3$
Find $||\vec{u}||$
I am using the formula for the angle between two v... |
H: How to get $(ab)1 = (a1)(b1)$ in Galois field?
I'm reading Galois field from textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell.
Here $r,a,b \in \mathbb N$ and $1 \in \mathbb F$.
While I understand $(ab)1 = a(b1)$, I could not get $(ab)1 = (a1)(b1)$. Cl... |
H: If $\frac{a}{k} - \frac{a(c-1)}{kb} > 1$, is $\frac{a}{k+1} - \frac{a(c-1)}{(k+1)b} > 1$?
I'm trying an induction method, but not sure if I'm able to prove it or find a counterexample.
Suppose $3 \leq a < b$ and $1 \leq c \leq b$, and $k \geq 1$, where $a, b, c, k$ are integers.
Suppose $n = 1$ is proven. Then indu... |
H: Minimum value of the function $\sin5x/\sin^5x$
I recently came across a question as follows:
Find the minimum value of the function $\sin5x/\sin^5x$.
I tried differentiating the function but the calculation was messy. The resulting differentiated equation had many roots, difficult for me to identify which ones a... |
H: Inclusion map $i : C^{0,\beta}[0, 1] \rightarrow C^{0,\alpha}[0, 1] $ is linear, and therefore compact
Q) Given $0 < \alpha < \beta \leq 1$. Show that the inclusion map $i : C^{0,\beta}[0, 1] \rightarrow C^{0,\alpha}[0, 1] $ is compact.
Ans) Let {$u_n$}$_1^\infty$ ⊂ $C^{0,β}[0,1]$ such that $\|u_n\|_{C^{0,β}} \leq... |
H: The flow of a Killing vector field preserves submanifolds?
I am wondering if the flow of a Killing vector field preserves the submanifolds?
In fact, on a Riemannian manifold $(M,h)$ I have a Killing vector field $W$. Let $\varphi:I\times M\to M$ be its flow. We know that $\varphi_t:=\varphi(t,.):M\to M$ is an isome... |
H: $T$ is a normal operator, prove any eigenspace of $T+T^*$ is invariant under $T$
I've been asked to prove in a homework problem exactly what the title describes. This was the $3^{rd}$ part of a question whose first 2 parts were to prove that $\ker T=\ker TT^*$ and $\ker T=\ker T^n$, but I can't find any way to impl... |
H: Find coordinates of point in $\mathbb{R^4}$.
There is a problem here in Q. $5$ on the last page. It states to find coordinates of point $p$.
Taking point $a=(3,2,5,1), \ b=(3,4,7,1), \ c= (5,8,9,3)$.
Also, $b$ has two coordinates in common with $a$, and $p$ lies on the same line as $a,b$.
So, those two coordinate... |
H: Simplifying ${n\choose k} - {n-1 \choose k}$.
How would I simplify $${n\choose k} - {n-1 \choose k}$$
I've expanded them into the binomial coefficient form
$$\frac{n!}{k!(n-k)!} - \frac{(n-1)!}{k!(n-1-k)!}$$
but that's about all I've got.
I'm having trouble with factorials. Any suggestions?
AI: To explain why your ... |
H: Find the number of solutions for the equation $4\{x\}=x+[x]$
$$4\{x\}=[x]+\{x\}+[x]$$
$$3\{x\}=2[x]$$
$$\{x\}=\frac 23 [x]$$
$$0\le \frac 23 [x] <1$$
$$0\le [x]<1.5$$
So $[x]=0,1$
The solutions for $x$ should be infinite, but the given answer is 3.
Even if assume that the answer not being $\infty$ is upto interpret... |
H: Why is $\operatorname{res}(0, \cos(\frac{1}{z}))=0$?
The residue of a function $f$ represented by a Laurent Series in complex analysis is defined as the coefficient at $n=-1$ of the series ($a_{-1}$). Giving the definition of the cossine in complex analysis:
$$f(z)=\cos(\frac{1}{z})=\sum_{n=0}^\infty \frac{(-1)^n}{... |
H: Confusion regarding number of ordered pairs for symmetry/asymmetry
My Discrete Mathematics textbook says the following :
A relation is symmetric/antisymmetric/transitive even if there’s one pair/triplet that satisfies the condition.
This probably means that if I have a relation that consists of a number of ordere... |
H: Is the function convex?
Let's have the following function $f:\mathbb{R}^{2}\to\mathbb{R}$ defined by $|x+y|$, is it convex?
We have $\lambda\in (0,1),x,y\in$ dom$(f)$, so $|\lambda x+(1-\lambda)y|\le \lambda |x| + (1-\lambda)|y|$. It means the function is convex according to the definition of a convex function. Is ... |
H: Showing Lebesgue Measurable Set is Measure Zero
I'm trying to show that given $A \subseteq \mathbb{R}$ with $A$ Lebesgue measurable and given that $m(A\cap [a,b]) < \frac{b-a}{2}$ for every $a<b$, that $A$ must have measure zero. I've been trying to use continuity of measure in some way, but I've been unsuccessful ... |
H: Why can we expand an analytic function in such way?
In one of the answers to my questions on StackExchange (Open Mapping Theorem Serge Lang Proof) included the fact that a an analytic function can be expanded as $$f(z)=f(a)+C(z-a)^n + \ldots$$ but I am unsure about how the person got to this fact.
My best guess is ... |
H: Why can the $n_{\epsilon}$ of the definitions of convergence and Cauchy sequence be the same in the following proposition?
I have the following proposition proved in my lectures notes, but I think there are a couple of errors and there is one think I don't get:
If $p_n$ is a Cauchy sequence in a metric space $(X,d)... |
H: The countable product of Fréchet spaces is a Fréchet space
Let $\{E_n \; ; \; n \in \mathbb{N}\}$ be a family of Fréchet spaces. I want to prove that the product
$$E:= \prod_{n=1}^{\infty} E_n$$
is a Fréchet space, that is, $E$ is metrizable (Hausdorff space and admits a countable basis of neighborhoods of $0\in E$... |
H: Notation for matrix filled with zeros except for one row and one column
Is there existing succinct notation for a matrix $[A]_i$ whose elements are all $0$, except that the $i$th row and $i$th column is given by a particular vector, for example the vector
$$\mathbf{z} = \frac{\mathbf{v} - 2\mathbf{v}_i}{a}$$
I'm tr... |
H: Show that $A^{-1}+B^{-1}$ is also invertible
Let $A$ and $B$ be two invertible $n \times n$ real matrices. Assume that $A+B$ is invertible. Show that $A^{-1} + B^{-1}$ is also invertible.
My approach
\begin{aligned}
&|\mathrm{A}|\left|A^{-1}+B^{-1}\right||\mathrm{B}|=|\mathrm{B}+\mathrm{A}| \neq 0 \\
\Rightarrow... |
H: Determine the order of the subgroup H of $S_n$ for $n \geq 3$
From the chapter on Permutation Groups in Gallian's 'Abstract Algebra' 9th Ed., we are asked to prove that for $n \geq 3,$ $$H=\{\beta \in S_{n}|\beta (1) \in \{1,2\} \land \beta (2)\in\{1,2\}\}$$ is a subgroup of $S_{n}$ and then determine (not prove) w... |
H: Show that $\psi_{b}=\psi_{c}\Leftrightarrow b=c$.
Let $K$ finite field with $|K|=p^{n}$,where $p$ is prime number.Let the transformation (Trace) :
$$Tr:K\to K,\ Tr(a)=a+a^{p}+a^{p^{2}}+\cdots a^{p^{n-1}}.$$
For every $a\in K$ we khow it's true that $Tr(a)\in \mathbb{Z}_{p}$ and $Tr$ is $\mathbb{Z}_{p}$- linear tr... |
H: How to find integer solutions of the equation $x(x+9)=y(y+6)$ where x,y are integers?
I found that the equation becomes $$(2x+9)^2-(2y+6)^2=45$$. And $45 = 9^2-6^2 = 7^2-{2^2}$. From here I found these solutions $(x,y)=(0,0), \ (0,-6), \ (-9,0),\ (-9,-6),\ (-1,-2),\ (-1, 4), \ (-8,-2), \ (-8,-4)$.
Does this equatio... |
H: Eigenvectors of $A \in SO(2n)$ and $A \in SO(2n+1)$
Does every matrix $A \in SO(2n)$ have an eigenvector?
Does every matrix $A \in SO(2n+1)$ have an eigenvector?
I think that you can answer both questions with yes, is that true?
AI: Every matrix has eigenvectors over the complex numbers. If the matrix is real, the... |
H: Solving $AX=B$ recursively where $X$ and $B$ are matrices
It's possible to solve $Ax=b$ recursively where $x, b \in \Re^n$ are vectors and $A \in \Re^{m*n}$ where $m > n$, buy using Recursive Least Squares(RLS).
But what if $AX=B$ where $A \in \Re^{m*n}$ and $X, B \in \Re^{n*k}$ where $k > 1$ and $m > n$. How can I... |
H: Conformal mapping of the disk $| z |
could you help me with the following please:
Find the conformal mapping of the disk $| z | <R_1$ to disk $| w | <R_2$ such that $w (a) = b, Arg(w´(a)) = \alpha $, $(| a | <R_1, | b | <R_2)$.
I have tried the following, but it does not match the solution that comes in the book:
... |
H: Showing that $Z[\sqrt{-n}]/\sqrt{-n}\approx Z_n $ and other similar isomorphisms.
I found the isomorphism here: Show that $\sqrt{-n}$ and $\sqrt{-n} +1$ are not prime in $\mathbb{Z}[\sqrt{-n} ]$
First I would like to show the isomorphism $\mathbb Z[\sqrt{-n}]\simeq\mathbb Z[X]/(X^2+n)$. This is my attempt:
Consider... |
H: Matrix exponential of a particular block matrix
Can anyone compute the following matrix exponential
$$\Phi = \exp\begin{pmatrix} -\alpha I_3 & \alpha A \\ I_3 & O_3 \\
\end{pmatrix}$$
where $A$ is an arbitrary $3 \times 3$ matrix, $I_3$ is the $3 \times 3$ identity and $O_3$ is the $3 \times 3$ zero matrix?
Thanks ... |
H: Evaluate the integral $\int_{0}^{10} 2^x dx$ using the limit definition
How do I evaluate a the integral $\int_{0}^{10} 2^x dx$ using the limit definition of an integral? We can get the values:
$\Delta x = \frac{10-0}{n}$ and $f(\frac{10i}{n}) = 2^{\frac{10i}{n}}$, thus setting up
$$\int_{0}^{10} 2^x dx = \lim_{n ... |
H: Find $n$ such that $1-a c^{n-1} \ge \exp(-\frac{1}{n})$
I am trying to find the integer $n$ such that
\begin{align}
1-a c^{n-1} \ge \exp(-\frac{1}{n})
\end{align}
where $a>0$ and $c \in (0,1)$.
I know that finding it exactly is difficult. However, can one find good upper and lower bounds it.
It tried using lower bo... |
H: Question on the colored Jones polynomial (from Wikipedia)
I'm trying to understand how "coloring" the component of a link changes the link. I'm looking at the picture for the section on the colored Jones polynomial on the link provided, and was wondering if somebody could tell me what the contents of the rectangle ... |
H: How to construct a function $f(x)$ such that $f(x)e^{-px}$ wouldn't tend to $0$ as $x$ tends to infinity
How to construct a function $f(x)$ such that $f(x)e^{-px}$ wouldn't tend to $0$ as $x$ tends to infinity?
This question is motivated when studying Laplace Transform when I encountered the following result
Suppo... |
H: A sum of powers of $2$ or $4$ that is or isn't divisible by $3$
Let $n\in\mathbb{Z^+}$ (a positive integer), and define $E(n)=2^{2n}$ and $O(n)=2^{2n+1}$.
Alternatively we can define $E$ to be $4^n$ and $O$ to be $2\cdot4^n\\$.
Let $a$ and $b$ be some arbitrary positive integer where $a>b$.
I empirically found out ... |
H: Properties of the solutions of the ODE $y'' + p(x)y' + q(x)y = 0$
Let $u(x)$ and $v(x)$ be solutions of $y'' + p(x)y' + q(x)y = 0$, $p$ and $q$ continuous in $\mathbb{R}$, such that $u(0) = 1$, $u'(0) = 0$, $v(0) = 0$ and $v'(0) = 1$. Prove that, if $x_{1} < x_{2}$ are such that $u(x_1) = u(x_2) = 0$ and $u(x) \neq... |
H: How to solve a value of K for a root locus?
I'm using the book "Control Systems Engineering - Norman S. Nise" and in the Root Locus chapter most of the exersices are solved using software.
I'm aware that this is a practical solution to real problems, however I'm curious how can I get a closed form solution to probl... |
H: Isometric image of an open ball.
Let $(X,d)$ be a compact metric space, where $f: X \rightarrow X$ is a distance preserving map, ie, $\forall x, y \in X$ we have that $d(f(x),f(y)) = d(x,y)$.
a) Show that f is injective.
b) Show that $\forall x \in X$ and every open ball of radius $\epsilon$ centered at $x$, $B_{\e... |
H: Defining a mapping
For a supbspace $S$ of $V$, prove $L:V \rightarrow V$ such that $\ker(L)=S$. I am reviewing the answers posted in this link:
For $\mathbb{S}$ subspace of $\mathbb{V}$, prove $L:\mathbb{V}\to\mathbb{V}$ such that Ker$(L)=\mathbb{S}$
How would we represent the linear mapping $L$ mathematically? Rat... |
H: Calculate the volume of the solid generated by two regions
How can I calculate the volume of the solid generated by the S1 and S2 regions by rotating around the Ox axis and around the axis Oy?
S1 :
$0 ≤ x ≤ 2$
$0 ≤ y ≤ 2x − x^2$
S2:
$0 ≤ x ≤ π$
$0 ≤ y ≤ \sin^2(x)$
AI: $S_1$
$$\int\limits_{x=0}^2 \pi (2 x - x^2)^2... |
H: If a sequence converges to $c$, then $c$ is the only limit point of that sequence
I'm currently on chapter 6.4 of Analysis I by Terence Tao and am stuck on this proposition which was left as an exercise:
Let $\left(a_{n}\right)_{n=m}^{\infty}$ be a sequence
which converges to a real number c. Then c is a limit poi... |
H: Determine a rational number $r$ such that $0\lt \sqrt2 - r \lt 10^{-5}$
Determine a rational number $r$ such that $0\lt \sqrt2 - r \lt 10^{-5}$.
Any hint, please?
AI: Manipulating the inequality gives
$$-\sqrt2\lt -r\lt10^{-5}-\sqrt2$$
$$\sqrt2\gt r\gt\sqrt2-10^{-5}$$
So we need to find a number within $10^{-5}$ of... |
H: Intuitive meaning of Diffeomorphism
Let $U\subset\mathbb{R}^n$, $V\subset\mathbb{R}^m$ and a bijection
$f:U\to V$ is a diffeomorphism if $f$ and $f^{-1}$ are differentiable.
I would like to know the intuitive meaning of two open sets being diffeomorphic.
For example, if two spaces are homeomorphic, these spaces s... |
H: Suppose $\mathbb{F}$ is a field of characteristic $p$. Show that if $a, b \in$ $\mathbb{F}$ and $a^{p}=b^{p}$, then $a=b$
I'm trying to do Exercise 2.6.12 from textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell. Could you please confirm if my attempt is f... |
H: Is there a way to approximate the terms of $\frac{\left(2n\right)!}{\left(2^nn!\right)^2}$ for successive $n$ as $n$ becomes large?
I have encountered the ratio of the product of the first n odd numbers to the product of the first n even numbers and want to chart its ultimate convergence to zero. If a white noise s... |
H: Help in understanding integrating a function with an absolute value
My math is rusty, and although I initially thought I understand the solution, upon further examination I think I don't:
That's the original function:
$$ \Psi(x,t) = A \mathrm{e}^{-\lambda|x|} \mathrm{e}^{-\mathrm{i} \omega t} $$
\begin{align*} \... |
H: How to prove $|\sin x| \leq 1$, $|\cos x| \leq 1$ in a simplistic way without resorting to pictures?
So my question is looking at if there is a way to show in a less advanced way than this question:
Is it possible to prove $|\sin(x)| \leq 1$, $|\cos(x)| \leq 1$ and $|\sin(x)| \leq |x|$ algebraically? if it is poss... |
H: Prove that if $p_1,...,p_k$ are distinct prime numbers, then $\sqrt{p_1p_2...p_k}$ is irrational
Prove that if $p_1,...,p_k$ are distinct prime numbers, then $\sqrt{p_1p_2...p_k}$ is irrational.
I do not usually prove theorems, so any hint is appreciated. I have taken a look at this and tried to repeat that argum... |
H: $\ A+A=\{3i,3+2i,3+4i\}$. find A
Can someone help me and answer this exercise of complex analysis? I don't know how to try to resolve this because I have never seen an exercise like this.
AI: Hint Let $z = a+bi \in A$. Then $z+z \in A+A=\{3i,3+2i,3+4i\}$. This gives you 3 choices for $z$.
Since $A$ must have at lea... |
H: Let $K,L$ be two rings and $g:K\to L$ be a homomorphism. Now $a\in K$ is zero divisor if and only if $g(a)$ is a zero divisor.
Let $K,L$ be two rings and $g:K\to L$ be a homomorphism.
Now
$a\in K$ is zero divisor if and only if $g(a)$ is a zero divisor.
For the right direction:
Let $a$ be a zero divisor $\exists x\... |
H: System of one quadratic and two linear equations over the positive integers
Find all solutions $(x, y, z)$ of the system of equations
\begin{align*}
x y + z &= 2019 ,\\
x − y + 2 z &= 1 , \\
x + y − 7 z &= 2
\end{align*}
in positive integers.
I am thinking of solving this system equation. The only proble... |
H: Matrix equation as a Tensor
I am new to Tensor algebra and still getting used to many of the terms. I have the below matrix equation and I wish to write it in Ricci calculus notation but am struggling:
$$(A \otimes_k B)(C \otimes_k D)$$
Where $\otimes_k$ is the Kronecker product. I understand that for a matrix, say... |
H: Solving for exact solution: $0.5 = 1-e^{-x}-xe^{-x}$
Is it possible to solve the following equation for an exact solution, and if so how?
$$0.5=1-e^{-x}-xe^{-x}$$
My textbook (Introduction to Mathematical Probablity) jumps from this equation directly to the solution $1.678$.
I was unable to solve this myself, and c... |
H: $p(x).y''+p'(x).y'+\frac{1}{p(x)}.y=0$
Is there a general solution I can give to the differential equation $\; p(x).y''+p'(x).y'+\frac{1}{p(x)}.y=0\;$ (the function $p(x)$ wasn't given) ?
AI: $$\; p(x).y''+p'(x).y'+\frac{1}{p(x)}.y=0\;$$
You can easily reduce the order:
$$(p(x).y')'=-\frac{1}{p(x)}.y$$
$$p(x)y'(p(x... |
H: How can you play a game that requires an 15 sided die with a 6 sided die
How can you play a game that requires a 15 sided die with a 6 sided die?
AI: Roll the 6-sided die once, discarding any 6s until you get a roll that's not a 6.
Then roll the 6-sided die again, and halve it (divide by 2, rounding up).
The formul... |
H: Bijection between subgroups $X$ satisfying $U\leq X\leq G$ and $U$-invariant subgroups $Y$ satisfying $U\cap N\leq Y\leq N$.
Hi: this problem is from Kurzweil and Stellmacher, The Theory of Finite Groups, An Introduction, page 20:
The homomorphism theorem gives a bijection between subgroups of the image and subgro... |
H: Complex polynomial whose roots contain the fifth roots of another complex number
Let $\alpha,\beta$ be two complex numbers with $\beta\ne 0$ and $f(z)$ a polynomial function on $\mathbb C$ such that $f(z)=\alpha$ whenever $z^5=\beta$. What can you say about the degree of the polynomial $f(z)$?
It is very clear that... |
H: A tangent property of an incircle
In $ \triangle{ABC}$ with $ AB = 12$, $ BC = 13$, and $ AC = 15$, let $ M$ be a point on $ \overline{AC}$ such that the incircles of $ \triangle{ABM}$ and $ \triangle{BCM}$ have equal radii. Let $ p$ and $ q$ be positive relatively prime integers such that $ \tfrac{AM}{CM} = \tfra... |
H: Question about $\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}$
Let $(X,B,\mu)$ be a complete measure space,Show that $$\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}, \quad \forall f \in \bigcup_{p} \bigcap_{p \leqslant q<\infty} L^{q}$$
So,$\lim _{q \rightarrow \infty}\|f\|_{q}$ , $\|f\|_{\infty}$ are... |
H: The condition and proof about the integral test for convergence
The proof about the integral test:
Suppose $f (x) $ is nonnegative monotone decreasing over $[1,\infty)$, then the positive series $\sum_{n=1}^{\infty}f(n)$ is convergent if and only if $\lim_{A\rightarrow +\infty}f(x)dx$ exists.
Proof:
Since $f(x)$... |
H: Find the equation of the plane in $\mathbb{ℝ}^{3}$ perpendicular to the subspace $S = \{(\alpha, 3\alpha, -4\alpha):\alpha\in\mathbb{R}\}$
I'm totally lost on how to do this. I know if I was given a normal $Ax + By + Cz = D$ plane, I would just have to find the normal vector to a point to find the perpendicular pla... |
H: Showing that a complex function is constant
The question is:
Let $f:\Omega\rightarrow\mathbb{C}$ be analytic and $\Omega$ be a domain in $\mathbb{C}$. Prove that if there exists $c\in\mathbb{C}$ such that $f(z)=\overline{cf(z)}$ for every $z\in\Omega$, then $f$ is a constant function.
my thoughts: Since $f$ is ana... |
H: Show that a sequence $a_n$ is a solution of the given recurrence relation
Show that the sequence $a_n=3^{n+4}$ is a solution of the recurrence relation $a_n=4a_{n-1}-3a_{n-2}$.
I'm stuck on this question as I'm having trouble figuring it out when $a_n=3^{n+4}$.
After substituting $3^{n+3}$ for both n's in $4a_{n-1}... |
H: Counting the number of rooted $m$-ary trees.
I know that the Catalan number $C_n$ is the number of full (i.e., 0 or 2 children per node) binary trees with $n+1$ leaves. I am interested in the generalization.
Note that I do not care about any labeling, ordering, or number of leaves. I just want the tree to be rooted... |
H: Multiplicity of each eigenvalue in a minimal polynomial of a matrix
It is well known that for a $n \times n$ matrix $A$ ,
the charicteristic polynomial $p(x)$ satisfies
$p(x)=\prod_{\lambda : eigenvector} (x-\lambda)^{a(\lambda)}$
where $a(\lambda )$ is the algebraic multiplicity of $\lambda$
The minimal polynomi... |
H: Define $k:R[x,y]\to R[x,y]$ by $f(x,y)\mapsto\frac{\partial^2f}{\partial x\partial y}$. What is the kernel of $k$?
Let $R[x,y]$ be a set of all real polynomial with two variables, $x$ and $y$. Define
a homomorphism $k:R[x,y] \to R[x,y]$ by $f(x,y) \mapsto \frac{\partial^2 f}{\partial x \partial y}$. Find $Ker \ k$.... |
H: Is a function $f\colon C\to\mathbb{R}$ bounded if $C$ is compact but $f$ is not necessarily continuous?
If I have a function mapping a compact set to the real numbers, is that function bounded? I know that this is true if the function is continuous. But is it true even if the function is not continuous? If so, how ... |
H: Finding $\lim_{n \to \infty}(n^{1\over n}+{1\over n})^{n\over \ln n}$
$$\lim_{n \to \infty}\left(n^{1\over n}+{1\over n}\right)^{n\over \ln n}$$
The limit is same as
$$e^{\displaystyle{\lim_{n \to \infty}}{n^{(1+{1\over n})}-n+1\over \ln n}}$$
But I am stuck here , I noticed that if I take $n$ common from the num... |
H: On the number of invariant Sylow subgroups under coprime action -Antonio Beltrán, Changguo Shao
I'm reading the papers of Antonio Beltrán, Changguo Shao. The article is On the number of invariant Sylow subgroups under coprime action:
https://www.researchgate.net/publication/318675516_On_the_number_of_invariant_Sylo... |
H: Rings Trapped Between Fields
Some Background and Motivation: In this question, it is shown that an integral domain $D$ such that $F \subset D \subset E$, $E$ and $F$ fields with $[E:F]$ finite, is itself a field. However, a significantly more general result holds and seems worthy, of independent address; hence,
Le... |
H: Can we define the distance from $x$ to boundary $A$ when $A$ is open? (Michael Spivak "Calculus on Manifolds" on p.64)
I am reading "Calculus on Manifolds" by Michael Spivak.
On p.64 in this book, Spivak wrote "distance from $x$ to boundary $A$" when $A \subset \mathbb{R}^n$ is open.
Intuitively, I think we can def... |
H: Dealing with negative values for $d$ in RSA Encryption.
I worked through an a RSA Encryption example, where I am given $p,q,e$ and I have to work out $n,\phi(n),d$. I don't have any difficulty determining all the of other items, however I get a negative value for $d$. Namely, $d=-37$. I've heard suggestions from my... |
H: Prove each linear transformation can be written as a matrix
I'd like to show that Any linear transformation between two finite-dimensional vector spaces can be represented by a matrix. I've seen a proof for linear transformation from and to $\mathbb{R}^n$ but I want to generalize it to any finite-dimensional vector... |
H: Abelian extensions $E|F$ and $F|K$ $\implies E|K$?
Let $E,F,K$ fields such that $K\subset F \subset E$
If $F|K$ and $E|F$ are abelian extensions, then $E|K$ is an abelian extension?
I can't find a counter example
AI: $K=\Bbb Q$, $F=\Bbb Q(\sqrt2)$, $E=\Bbb Q(\sqrt[4]2)$. |
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