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H: 2-dimensional regular submanifold of $GL(2, \Bbb R)$
I wish to show that the subset $A$ of $GL(2 ,\Bbb R)$ consisting of matrices of the form
$$\begin{bmatrix}
a &b\\0 &a
\end{bmatrix}$$
where $a >0$ is a regular submanifold of dimension $2$. The easy way to show that it's a regular submanifold is to invoke Cartan'... |
H: If the number of units of a ring is odd, then the ring has cardinality as a power of two
If the number of units of a finite ring is odd, then does the ring has cardinality as a power of $2$?
I think yes. For fields, it is trivial. For non-fields, it is a hard question for me. I saw a paper here that sates that an ... |
H: Let $G$ be a group and $n\in \Bbb Z$. $\forall x,y\in G$, $x^n=y^n \Rightarrow x=y$ and $xy^n=y^nx$. Thus, prove that $G$ is abelian.
Let $G$ be a group and $n\in \Bbb Z$. $\forall x,y\in G$, $x^n=y^n \Rightarrow x=y$ and $xy^n=y^nx$. Thus, prove that $G$ is abelian. Working backwards, I get $$xy=yx$$$$(xy)^n=(yx)^... |
H: Can you do this question without matrices?
Can I use the formula for a trapezium? I'm not really sure where to start
AI: Refer to the graph:
$\hspace{2cm}$
$$\small{A=\frac{b_1+b_3}{2}\cdot (a_3-a_1)+\frac{b_3+b_2}{2}\cdot (a_2-a_3)-\frac{b_1+b_2}{2}\cdot (a_2-a_1)}$$
I leave the rest work to you. |
H: A graph with $n$ vertices has distinct degrees except for one degree, say $x$ which occurs twice. Find $x$ and prove it.
I find out that when $n$ is an odd number, then $x$ equals to $\frac{n-1}{2}$. When $n$ is an even number, then $x$ has two possible values, one is $\frac{n}{2}$ and another is $\frac{n}{2} -1$.
... |
H: Comparing elements of $L^p$ spaces
I was studying the answers to the following question: About two functions whose Lebesgue integral on all sets of a $\sigma-$algebra are equal. Now I am wondering how to interpret the sets $\{f>g\}$, $\{f=g\}$,... and so on.
In the case where $f$ and $g$ are continuous (and thus Bo... |
H: Representing $n!$ as a Polynomial
For $n\in\mathbb N$, $n!$ could, theoretically, be expanded into a polynomial of degree $n$ as $$\underbrace{n(n-1)(n-2)(n-3)\cdots \left(n-(n-2)\right) (n-(n-1))}_{n \ \text{factors}} =\sum_{k=0}^n a_k n^k $$ How can I determine the coefficients $a_k$?
For the $n^n$ term, there i... |
H: Areas of two polygons with same centroid
Give a polygon $P$ in $\mathbb{R}^2$, the centroid of $P$ is $(v_1+\cdots+v_n)/n$, where $v_1,\ldots,v_n\in\mathbb{R}^2$ are the vertices of $P$.
Suppose $P$ and $Q$ are two polygons in $\mathbb{R}^2$ satisfies:
They have the same centroid.
The ratios of their shaows in eve... |
H: I got a problem in indices
$3^{(2x+3)} - 2.9^{(x+1)} =1/3$
Please help me with this problem
Its my elementary mathematics indices problem
AI: Using the index laws, transform $3^{2x+3} = 3^{2x} \times 3^3$ and $9^{x+1} = 3^{2(x+1)} = 3^{2x}\times 3^2$, so that the LHS becomes $3^{2x}(3^3 - 2\times 3^2) = 3^{2x}(3\ti... |
H: Does there exist a symmetric matrix $A$ such that $2^{\sqrt{n}}\le |\operatorname{Tr}(A^n)|\le2020 \ \cdot 2^{\sqrt{n}}$ for all $n$
Does there exist a symmetric matrix $A$ such that $2^{\sqrt{n}}\le |\operatorname{Tr}(A^n)|\le2020 \cdot2^{\sqrt{n}}$ for all $n$?
I think no. The trace of $A^n$ equals $\sum\limits_... |
H: If $\sqrt{1-a}\leq\sqrt{1-b}+\sqrt{1-c}$ would it imply $\sqrt{1-a^2}\leq\sqrt{1-b^2}+\sqrt{1-c^2}$?
Question: If $\sqrt{1-a}\leq\sqrt{1-b}+\sqrt{1-c}$ would it imply $\sqrt{1-a^2}\leq\sqrt{1-b^2}+\sqrt{1-c^2}$
That is, $a,b,c\in [-1,1].$
Would this inequality necessarily be true?
I tried to break up $\sqrt{1-a^2}=... |
H: There is only one positive integer that is both the product and sum of all its proper positive divisors, and that number is $6$.
Confused as to how to show the number 6's uniqueness. This theorem/problem comes from the projects section of "Reading, writing, and proving" from Springer.
Definition 1. The sum of divis... |
H: why $(M_1^{\perp})^{\perp} \subset [T^*(H_2) ]^{\perp} ?$
I have some confusion in subset sign given below , my confusion marked in red circle as given below
It is given that $T^*(H_2) \subset M_1^{ \perp}$ then why $(M_1^{\perp})^{\perp} \subset [T^*(H_2) ]^{\perp} ?$
I think its should be $[T^*(H_2) ]^{\pe... |
H: Understanding the definition of a differential operator on manifolds
In Christian Bar's "Geometric Wave Equations" notes it has this definition of a differential operator. I know what $\frac{\partial f}{\partial x^i}$ means when $f:M\rightarrow \mathbb{R}^n$ is a smooth function. But I don't understand what is mea... |
H: Let $A,B\in\mathbb{R}^{n\times n}$, where $A$ is PSD and $B$ NSD. If $\mathrm{tr}(AB)=0$, show that $AB=0$.
Let $A,B\in\mathbb{R}^{n\times n}$, where $A$ is a positive semidefinite matrix and $B$ a negative semidefinite matrix. If $\mathrm{tr}(AB)=0$, show that $AB=0$.
AI: If you are using a definition in which "po... |
H: What is the actual meaning of second derivative?
I am confused why we use second derivative to find the maxima and minima. I cannot understand what is the meaning of second derivative. Also i have come across some formulae that is
if second derivative is greater than zero then it is minima.
if second derivative ... |
H: When is the sum of two uniform random variables uniform?
Suppose that $X$ and $Y$, two random variables, are both uniformly distributed over $[0,1]$. Let $Z=\frac{1}{2}X+\frac{1}{2}Y$.
I know that in general, $Z$ is not uniform. For instance, $Z$ is not uniform if $X$ and $Y$ are independent.
On the other hand, if ... |
H: Relation between the eigenvalue of $T$ to the eigenvalue of $p(T)$
Let $V$ be a vector space over a field $\mathbb{F}$. Suppose that $T: V \rightarrow V$ is a linear operator with an eigenvalue $\lambda$, and $v$ is an eigenvector of $T$ corresponding to $\lambda$.
Why is it true that, for every $p(x) \in \mathbb... |
H: Show that a function is invertible
I have to show that the function $f(x) = \frac{ax + b}{cx + d}$, where $ad - bc\neq 0$, has an inverse function.
I've tried some ways to go around it, i.e. checking if $g(f(x))$ has $x$ as an identity, but the algebra got really difficult and I could not get anywhere.
Any hints on... |
H: What is the expression for the centroid of an arbitrary parameterized space curve?
Let $\gamma:t\in[a,b]\rightarrow (x(t),y(t),z(t))\in \mathbb{R}^3$ be a parametrized curve
I am looking for the expression of the centroid of the curve $\gamma$ in a good reference. (I didn't find a good one.)
AI: You can use the sta... |
H: Evans' PDE Exercise 6.6: Weak solution of Dirichlet-Neumann boundary value problem
The exercise is the exercise 6.6 from Evans' PDE.
Suppose $U$ is connected, and $\partial U$ consists of two disjoint, closed sets $\Gamma_1$ and $\Gamma_2$. Define what it means for $u$ to be a weak solution of Poisson equation wit... |
H: Use polar coordinates to compute volumes, via the change of variables theorem
So there is a question in my Anaylis book (with 3 subquestions) that I think understand, but I can not seem to understand the approach used in the solution. I've tried all subquestions, and for each of them I seem to make a mistake somewh... |
H: Is the set of monotone functions $f:[a,b] \to [0,1]$ compact in $L^2([a,b])$?
Is the set of equivalent classes of monotone functions $f:[a,b] \to [0,1]$ compact in $L^2([a,b])$?
AI: Yes. Let $f_n$ be a sequence of such functions. By the Helly selection theorem, there is a subsequence $f_{n_k}$ converging pointwis... |
H: Euler's method to approximate a differential equation $\frac{dy}{dx} = x - y$
Question: Use Euler's method to find approximate values for the solution of the initial value $-$ problem
$$\frac{dy}{dx} = x-y$$
$$y(0)=1$$
on the interval $[0,1]$ using five steps of size $h = 0.2$.
My attempts:
I know that the recurre... |
H: Intuition of subsets which are not in the sigma algebra
I am studying probability theory and from what I have understood is that when the sample space is uncountable, the probability measure cannot assign probabilities to every possible subset of the sample space, hence we build another set containing the subsets o... |
H: Show that $\mathrm{Cov}[g(X), h(X)] \ge 0$ whenever $g$ and $h$ are nondecreasing.
Intuitively, the covariance of two nondecreasing functions of a random variable should be nonnegative. However I can't seem to come up with a proof for this.
Here is the formal setup:
Let $X: (\Omega, \mathcal A)\to (\mathbb R, \math... |
H: Probability problem related to 2 rooks on a 8×8 chessboard
Two distinct squares are chosen uniformly at random on an $8\times 8$ chessboard, and rooks are placed on these squares. What is the probability that they will attack each other?
Edit : thanks everyone for their concern. I solved it :p. No matter where I pl... |
H: Graph of Topologist's Sine Curve
I'm looking for whether the graph of topologist's sine curve and closed topologist's sine curve are closed or not. But due to some misconception, I'm facing problems with this.
$\underline{\text{Question} : 1}$ Here it proves that if $f\colon X\to Y$ continuous and $Y$ is Hausdorff,... |
H: Using implicit function theorem to solve a system of equation
I have the following question:
Consider the set $\Gamma \subseteq \mathbb{R}^3$ of solutions of the system
\begin{equation*}
\begin{cases}
x+\ln{y}+2z-2=0 \\
2x+y^2+e^z-1-e=0
\end{cases}
\end{equation*}
Describe $\Gamma$ in a neighborhoo... |
H: A ferris wheel completes 2 revolutions in 30 seconds. Determine how far it has travelled in 15 seconds. The radius of the ferris wheel is 10 m.
If the Ferris wheel completes two revolutions in $30$ seconds, how many revolutions does the Ferris wheel complete in $15$ seconds? The radius of the Ferris wheel is 10 m. ... |
H: Why a hyperplane is a subspace?
Given a nonzero vector $a \in \mathbb{R}^n$ and a scalar $b \in \mathbb{R}$, we define the hyperplane
$$
H = \{x \in \mathbb{R}^n \; | \; a^T x = b\}.
$$
Let $x$ and $y$ be any two vectors that belong to $H$, clearly $a^T (x - y) \neq b$ (unless $b = 0$), that is, $x - y$ is not in $... |
H: Clarification on definition of a Sheaf
On Wikipedia, the gluing and locality properties of a Sheaf are defined in terms of elements $s$ of the object $S$ associated with $\mathscr{F}(U)$.
I have two points of confusion.
I thought objects in a category don't necessarily have elements so does this definition even ma... |
H: What equation best represents this set of data?
Here is the graph. (It is the same as below.)
The points are symmetrical over the $y$-axis, but I cannot find an equation that accurately represents this graph.
AI: Well, observe that $y-x$ for $x>0$ are almost in arithmetical progression: $0, 3, 4.5, 6, 9, 12, 15, 3... |
H: Overloading binary operation symbols
In computer science I'm used to using overloaded operators. Is this also valid in mathematical notation? Concretely, I have the following example:
Definition:
Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be two graphs.
We define $\phi = \{(v,w) \in V_1 \times V_2 | v \text{ is mapp... |
H: Order of a subgroup $H$ and $\langle H,b\rangle$
Let $G$ be a finite abelian group, and $H$ a subgroup. Suppose $H$ contains an element $a$ where there is some $b \in G$ with $a \in \langle b \rangle$ and $|b|/|a| = p$, some prime $p$. Do we necessarily have $[\langle H,b\rangle : H] = 1 \text{ or } p$ ?
Note I can... |
H: Basic graph theory proof verification
Let $G(V, E)$ be an undirected final and simple graph. $\bar{G}(V, E')$ a simple graph on the same vertices, while $e \in E'$ iff $e \notin E$
Prove that if G is not a connected graph, then $\bar{G}$ is a connected graph.
This is my idea:
Let $v_1, v_2$ $(v_1 \neq v_2)$ be ve... |
H: Advantage of fast Fourier transform in programming
Someone asked me about the advantage(s) of fast Fourier Transform in civil engineering programs?!
or
What is the application of the Fourier series in engineering programming?
Can you help me or give me a clue? I've been searching google, but do not find an explana... |
H: Pigeonhole Principle Proof and Existence
So, I’m going through a textbook on combinatorics, and I came across this exercise question.
Let $n$ be odd, and suppose $(x_1, x_2, \dots, x_n)$ is a permutation of $[n].$ Prove that the product of $(x_1-1)(x_2-2) \cdots (x_n-n)$ is even.
So far, I have this:
in order for... |
H: Gradient and laplacian of a function defined on Riemannian manifold in local coordinates.
I was trying to derive an expression of the gradient of a riemannian manifold.
Let $M$ be a Riemannian manifold of dimension $n$ and $f : M \to \mathbb{R}$ and let's define $grad f(p) : M \to\mathcal{X}(M)$ a vector field such... |
H: conditional probability problem with two random events
This the problem:
In police station 1 there are 3 cars of type A and 8 of type B. In police station 2 there are 5 of type A and 2 of type B. In each station one of the cars is randomly chosen and damaged by an outsider!( a damaged car can not move). Some even... |
H: find explicit expression for the function $f(x)= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)(x+1)^{2n}}$
I got this in one of my assignments:
Let $$\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)(x+1)^{2n}}$$
(a) find the domain of convergence
(b) let $\alpha=\arctan(\frac{1}{2})$, consider the function defined by $$f(x)=\sum_{n... |
H: Unitary operators and representations of Von-Neumann algebras
I found the following assertion and I would like to know why does it hold:
Let $\pi$ be a representation of a von-Neumann algebra such that $\pi=\pi_1 \oplus \pi_2$ where $\pi_1,\pi_2$ are irreducible representation. and let {$x_1,x_2$} linearly independ... |
H: (c) Find the area contained between the curve, the y-axis, the line t = 1 and the asymptote to the curve which is parallel to the t-axis.
Part (a) and (b) are fine, and I believe (c) is an integral, but i'm not quite sure how to go about solving said integral with the given parameters. Mainly the vertical limits, ... |
H: Binomial Expansion Of $\frac{24}{(x-4)(x+3)}$
Can somebody help me expand $\frac{24}{(x-4)(x+3)}$ by splitting it in partial fractions first and then using the general binomial theorem?
This is what I've done so far:
$$\frac{24}{(x-4)(x+3)}$$
$$=\frac{24}{7(x-4)}-\frac{24}{7(x+3)}$$
Now I know I have to find the bi... |
H: How to prove a norm identity for a Banach space and its dual
Is the following claim true? It feels like it should be true, but I don't really know how to show it.
Let $X$ be a Banach space, and $x \in X$ an element of it. Then there exists a functional $\phi \in X^*$ such that $\| \phi \| = 1$ and $\| x \| = | \ph... |
H: area inside the curve $\phi(t)=(a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t)))$
I tried using
$$(1)A=\int_0^{2\pi}x(t)y'(t)\,dt=\int_0^{2\pi}a(2\cos(t)-\cos(2t))a(2\cos(t)-2\cos(2t))\,dt=6\pi a^2$$
and
$$(2)A=\frac{1}{2}\int_0^{2\pi}r^2(t)\,dt=\frac{1}{2}\int_0^{2\pi}(a(2\cos(t)-\cos(2t)))^2+(a(2\sin(t)-\sin(2t)))^2\,d... |
H: Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also contains a free subgroup of rank $2$.
I am having difficulties in solving the following problem.
Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also... |
H: What's with this strange sequence?
We have the sequence : $$V_n=\frac{n(n+1)}{2V_{n-1}}\text{ with } V_1 = 1$$
This sequence appears really similar to this sequence :
$$
a_n =
\begin{cases}
n & \text{if n odd}\\
\frac{n}{2} & \text{if n even}
\end{cases}
$$
like $a_{n+1}=V_{n}$ when $n\ge1$.
How is this possible ?... |
H: why can we factor a polynomial using its solutions
Can someone please explain why we are able to factor an $n$ degree polynomial function using only it roots?
What I mean is this:
Lets say we have a function defined like so:
$$f(x) = ax^4 + bx^3 +\dots$$
It can supposedly be factored like so:
$$f(x) = a(x−p)(x−q)(x... |
H: Perform NSGA 2 without variables
I have a data set with two columns. The variable names are cost 1 and cost 2. I want to minimize both cost 1 and cost 2 using the Pareto optimization method. So, while implementing NSGA II I have two objective functions i.e. cost 1 values and cost 2 values but I don't have any varia... |
H: Why is $\text{Gal}(K/\mathbb{Q}) \cong G_{\mathbb{Q}}/{\{\sigma \in G_{\mathbb{Q}}: \ \sigma|_K=id_K \}}$?
Here, in page $1$, the absolute Galois group is defined by $$G_{\mathbb{Q}}:=\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})=\{\sigma: \bar{\mathbb{Q} }\to \bar{\mathbb{Q}}, \ \text{field automorphism} \}$$ is a profi... |
H: Limit as a Function on Sequence Spaces
The question is motivated by the following posts: A and B. Let $X$ be a metric space (probably only Hausdorff is needed but I'm being safe) and let $X_0$ be the subspace of the sequence space $X^{\mathbb{N}}$ (equipped with the product topology) whose elements $(x_n)_{n \in \... |
H: homology groups of a torus with a disk glued to it in a certain way
I am trying to study for my qualifying exams and I was trying to solve this problem. So the idea is to form a topological space X by attaching a disk $D^2$ along its boundary to the torus $T^2$ so that the boundary is attached to a loop representin... |
H: Inverse function in $\mathbb R^2$
How do I find the inverse function of $f: X\to Y$ where $X,Y$ both are subsets of $\mathbb R^2$ and $f$ is defined as $f(x,y)=(x+y,x-y)$.
AI: Hint:$$\underbrace{\begin{bmatrix}x \\y \end{bmatrix}}_f\mapsto\begin{bmatrix}1 &1 \\1 & -1 \end{bmatrix}\begin{bmatrix}x \\y \end{bmatrix}$... |
H: A surprisingly simple determinant
Let $a_k^{(n)}$ be the $n$-vector whose components are the first $n$ non-null coefficients of the Taylor expansion of $\sin(k x)$ around $0$. Define the matrix $A^{(n)}$ as the matrix whose rows are the vectors $a_1^{(n)};a_2^{(n)}\dots a_n^{(n)}$, i.e. the $k$-th row of $A^{(n)}$ ... |
H: Finding the Möbius transformation when $z= \infty$
The following is available:
$ T(2i) = \infty $
$ T(0) = -i $
$ T(\infty) = i $
So I've got:
$ \frac{a(2i)+b}{c(2i)+d} = \infty \Rightarrow d=-2ic $
$ \frac{b}{d}=i \Rightarrow b = -2c $
$ \frac{a \cdot \infty -2c}{c \cdot \infty + -2ic} = i $
how do I continue when... |
H: Dual of the subspace of sequences with finite non-zero entries
I found the following question in the book Introdução à Análise Funcional, by César R. Oliveira.
Let $\mathcal{N}_p \subset \mathcal{l}^p(\Bbb{N})$, $1 \le p \le \infty,$ the subspace of all sequences with finitely many non-zero entries. Show that $\ma... |
H: Why left side is negative on Number line?
Yeah, probably it is a stupid question If so sorry about that. But I wonder why left side is negative on number line and who had proofed that?
AI: Copied from my comment:
It's just convention. The convention being this way rather than the other way sort of makes sense in so... |
H: Can we prove that in any ring $a+a=2a$?
Before I get to the question itself I want to clarify a few things:
Definition for a ring from my textbook (translated (not that well) to English and then shortened by yours truly,so please pardon any mistakes):
Set $R$ is called a ring if it has two operations defined on it,... |
H: Ferris wheel Trig Question
Question: Suppose you wanted to model a Ferris wheel using a sine function that took 60
seconds to complete one revolution. The Ferris wheel must start 0.5 m above ground.
Provide an equation of such a sine function that will ensure that the Ferris wheel’s
minimum height of the ground is ... |
H: Trigonometric Equation $\cos^{2}{x} - \sin{x} = 0$
Hi I tried everything I know with this equation but, I can not solve it.
$$\cos^{2}{x} - \sin{x} = 0$$
I know it has a solution because I made a graph and it cuts the $x$ axis.
Do you have advice?
AI: Turn this into a polynomial using the hint @AnginaSeng gave:
$\c... |
H: Determine the value of “c” using the mean value theorem
For the function $F(x) = Ax^2 + Bx + C$ determine the value of $c$ (critical point) at which the tangent line is parallel to the secant through the endpoints of the graph on the interval $[x1,x2]$. Not sure how to start this or do it at all so any help would b... |
H: Is $e$ arbitrary? If not, how is it derived?
Probably a stupid question but where did the constant $e$ come from? How did it come about? How is it derived mathematically other than $e^{i\cdot \pi} = -1$? What exactly does natural growth mean? Or is Euler's constant arbitrary?
AI: Some definitions of $e$:
The numbe... |
H: Getting three of a kind in a game of yahtzee
A.I am trying to calculate getting three of a kind in a game of yahthzee but I am not sure what I am doing wrong.
So we have five tossed dices so our possible outcomes are $6*6*6*6*6=7776$ We then have the form of three of a kind $AAABC$ where A can be chosen 6 different... |
H: Prove that for any sets $A$ and $B$, if $\mathscr P(A)\cup\mathscr P(B)=\mathscr P(A\cup B)$ then either $A\subseteq B$ or $B\subseteq A$.
Not a duplicate of
Suppose $\mathcal{P} (A) \cup \mathcal{P} (B) = \mathcal{P} (A \cup B) $. Then either $A \subseteq B$ or $B \subseteq A$.
Prove that if $\mathcal P(A) \cup \m... |
H: How to subtract IEEE754 floating point?
I have two numbers represented in floating point:
$A: 10101001001110000000000000000000$
$B: 01000011011000000000000000000000$
For $A$ I know $e=82$ and for $B$, $e=134$ ($e$=exponent), but I don't know how to subtract these numbers in floating point.
How should we make the ... |
H: Suppose $x,y \in V$ are vectors such that $\lVert x\rVert=\lVert y\rVert=1$ and $\langle x,y\rangle=1$. Show $x=y$
Let V be a vector space over $\mathbb{R}$. Let $(x,y) \mapsto \langle x,y\rangle$ be an inner product on $V$ with induced norm $\lVert x\rVert=\sqrt{\langle x,y\rangle}$. Suppose that $x$ and $y$ are ... |
H: Find $\sum_{n=1}^{\infty} \frac{1}{\prod_{i=0}^{k} \left(n+i\right)}$
Original question is $$\sum_{n=1}^{\infty} \frac{1}{\prod_{i=0}^{k} \left(n+i\right)}$$
I got it down to $$\sum_{n=1}^{\infty} \frac{(n-1)!}{(k+n)!}$$
Here I am confused. Possible fraction decomposition but its ugly! Maybe this approach is not ... |
H: Why do we need to make use of the random variable concept if we already know the measure $P$?
I have a doubt involving the concept of random variable.
Let us consider that $\Omega = \{\omega_{1},\omega_{2},\omega_{3}\}$, $\mathcal{F} = 2^{\Omega}$ and $P(\{\omega_{i}\}) = 1/3$ for $1\leq i\leq 3$.
Having said that,... |
H: How to find the number of groups of 5 with 2 defective modems.
A store has 80 modems in its inventory, 30 coming from Source A and the
remainder from Source B: Of the modems from Source A; 20% are defective.
Of the modems from Source B; 8% are defective. How many groups of 5
modems will have exactly two defective m... |
H: What Lipschitz function can tell me about?
If a function is said to be "Lipschitz", what kind of informations it can give?
I know that is about continuity but, I think maybe it can give more informations about.
AI: It is stronger than simple continuity. It is one of its strongest form. The greatest disadvantage I c... |
H: Let $G$ be a group of order $p^n q$, where $p$ and $q$ are distinct prime. ,
Assume $q \not| p^i - 1$ for $1 \leq i \leq n - 1$. Prove that $G$ is solvable.
Since if $G$ has a solvable normal subgroup $N$ such that $G/N$ is solvable, and if $r$ is prime, every $r$-group is solvable, we know that if $G$ has a normal... |
H: Undetermined or indeterminate forms: $\frac{0}{0}, \frac{\infty}{\infty}, 0\cdot\infty, 1^\infty, 0^0, +\infty-\infty$
I wanted to know who has decided that for the calculation of the limits of the following forms,
$$\color{orange}{\frac{0}{0},\quad \frac{\infty}{\infty},\quad 0\cdot\infty,\quad 1^\infty,\quad 0... |
H: sums and differences of perfect powers
We have
$1=3^2-2^3$
$2=3^3-5^2$
$3=2^7-5^3$
$4=5^3-11^2$
$5=3^2-2^2$
and it is unknown if $6$ is representable as a difference of two perfect powers. Next such undecided example is $14$. More: http://oeis.org/A074981
However, I found that
$6=64-49-9=2^6-7^2-3^2$
and
$6=27+4-2... |
H: Bipartite-Graph GCD question
Let $G$ be a bipartite graph with bipartition $(A, B)$. Suppose every vertex in $A$ has degree $k_a$, and every vertex in $B$ has degree $k_b$. Prove that if $G$ has a bridge, then $\operatorname{gcd}(k_a,k_b) = 1$.
AI: HINT: Take a look at the answer to this question, which proves that... |
H: Identities involving hyperbolic functions.
I came across the following identity,
$$
\int_{-\infty} ^\infty dx \frac{e^{-i kx}}{e^{-ax} +1} = \frac{2\pi i}{a} \sum_{n=0}^{\infty} e^{-\frac{(2n+1)\pi k}{a}} = \frac{\pi i }{a\mathrm{sinh}\frac{k \pi}{a}}
$$
I can to some extent see the first equality by doing the cont... |
H: Minoration of max function
I am wondering if we have the following minoration of the $\max$ function :
$$ \forall a, b, c \in \mathbf{R} ~~~~~ \max(a, b, c) \geq \dfrac{1}{3} ( a+b+c) $$
AI: $$\dfrac{1}{3} ( a+b+c) \leqslant \frac{\max(a,b,c)+\max(a,b,c)+\max(a,b,c)}{3} = \max(a,b,c)$$
$$\min(a,b,c) = \frac{\min(a,... |
H: Why is $\mathbb{Z}_{m} \otimes_{\mathbb{Z}} \mathbb{Z} = \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}_{m} = \mathbb{Z}_{m} $?
Why is $\mathbb{Z}_{m} \otimes_{\mathbb{Z}} \mathbb{Z} = \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}_{m} = \mathbb{Z}_{m} $?
Could anyone show me the proof of this, please?
I have read this que... |
H: small distances between powers of irrationals
The value of $$\inf \left\{ |\pi^m-e^n|: m,n\in\mathbb{N} \right\}$$ is a known unsolved problem. But transcendental numbers are known to cause problems of this sort.
Is the value of $$\inf \left\{ |\sqrt{2}^m-\sqrt{3}^n|: m,n\in\mathbb{N} \right\}$$ known? Or at least... |
H: Show that there exists a neighborhood $U$ of $(0,1)$ such that the restriction $g:U \rightarrow g[U]$ is invertible
Let $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be defined by $g(x,y)=(2ye^{2x},xe^y)$. Show that there exists a neighborhood $U$ of $(0,1)$ such that the restriction $g:U \rightarrow g[U]$ is inverti... |
H: Simultaneous convergence of sequence with weak- and weak$^*$-convergent terms
Suppose I have a Banach space $X$ with continuous dual. In that space, I have a sequence $(x_n)_{n = 1}^\infty$ in $X$ converging weakly to $y$, and a sequence $(\phi_n)_{n = 1}^\infty$ in $X^*$ satisfying $\phi_n \to \psi$ in the weak$^*... |
H: Separation axiom implied by semidecidability of comparison
I am studying computable analysis. What I'm fascinated by is the analogy between computable analysis and general topology: a Wikipedia article
Semidecidable sets are analogous to open sets.
So I treat them essentially the same.
Discrete sets in topology ... |
H: Is it true that $\operatorname{meas}(\partial(\operatorname{supp}(f)))=0$?
Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $C_c^{m}(\Omega)$ the space of $m$-times continuously differentiable functions with compact support with $0 \leq m \leq \infty$. Denote by $\partial X$ the boundary of the set $X$ and by... |
H: Usage of linear operator $T$ on basis
Let $T: V \rightarrow V$ linear operator and $V$ is finite vector space. Let $$\varepsilon=\left\{\varepsilon_{1}, \ldots, \varepsilon_{n}\right\}$$ be basis for $V$. so if I have $\vec{v} \in span(\left\{\varepsilon_{1}, \ldots, \varepsilon_{n}\right\})$ why is it true that $$... |
H: Solve Complex Equation $z^3 = 4\bar{z}$
I'm trying to solve for all z values where $z^3 = 4\bar{z}$.
I tried using $z^3 = |z|(\cos(3\theta)+i\sin(3\theta)$ and that $|z| = \sqrt{x^2+y^2}$ so:
$$z^3 = \sqrt{x^2+y^2}(\cos(3\theta)+\sin(3\theta))$$ and $$4\bar z = 4x-4iy = 4r\cos(\theta)-i4r\sin(\theta)$$
but I have n... |
H: Let $b \in [0,1)$. Prove that $\frac{b}{1-b} \in [0,\infty)$
Can someone check my solution for this problem? It seems to me that it’s incomplete, and I’m not sure.
Problem: Let $b \in [0,1)$. Prove that $\frac{b}{1-b} \in [0,\infty)$.
Solution: We know that $b \in [0,1)$, so $0 \leq b < 1$. From here we can also... |
H: Show that there do not exist functions $f(x)$ and $g(h)$ such that $\cos{(x + h)} − \cos{x} = f(x)g(h)$ for all $x, h \in \mathbb{R}$
Show that there do not exist functions $f(x)$ and $g(h)$ such that $\cos{(x + h)} − \cos{x} = f(x)g(h)$ for all $x, h \in \mathbb{R}$.
So far, I have tried following the same logic a... |
H: Find an upper bound for a modulus of a complex number
I need to find an upper bound for the modulus $ |3z^2+2z+1| $ if $ |z| \leq 1$
My solution is:
$$ |3z^2+2z+1| = |(3z-1)(z+1)+2|$$
Using the next 2 equations from triangle inequality
$$(1).|z_{1}+z_{2}| \leq |z_{1}|+|z_{2}| $$
$$(2).|z_{1}-z_{2}| \geq ||z_{1}|-|... |
H: Why do we need arithmetical operations instead of teaching arithmetic based on the successor function?
Since we can construct the set of natural numbers off of the peano axioms with the operations of addition, subtraction, multiplication and division following, why do we even need 4 arithmetical operations? Since t... |
H: Can an injective function have unmapped elements of the domain?
I know that a function is injective if every element of the domain maps onto at most one element of the co-domain, that is, if $f(x_{1}) = f(x_{2})$ implies $x_{1} = x_{2}$, or the contrapositive. However, is it allowable for an injective function to h... |
H: Neighbourhood of infinity
In the extended complex plane, does there exist a neighbourhood of $\infty$ which contains the origin?
My feeling is that there is no neighborhood of $\infty$ containing the origin. But this defies my intuition since for example a neighborhood of any point in the complex plane can be made ... |
H: How to prove that this construction is a group homomorphism?
Let $\phi:G \rightarrow H$ a group homomorphism such that $M=\phi(G) \neq H$ and $M$ having at least 3 different cosets in $H$. Take $K$ as being the group of all permutations of $H$. Choose 3 different cosets
$M$, $Mh'$, $Mh''$ of $M$ in $H$ and define $... |
H: Help with a differential equation system
Given $x' = -x$ and $y' = -4x^3+y$, we want to linearize and show phase portrait at origin.
So I make system $\vec{Y}' = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\vec{Y}$ by just scrapping the $-4x^3$ term. But now we have repeated $0$ eigenvalue, so I try to find an eig... |
H: How many elements of order $2$ does Sym $6$ have?
First, I will answer the following question:
''How many elements of order $2$ does Sym $5$ have?''
The answer is:
$(12),(13),(14),(15),(23),(24),(25),(34),(35),(45),(12)(34),(12)(35),(12)(45),(13)(24),(13)(25),(13)(45),(14)(23),(14)(25),(14)(35),(15)(23),(15)(24),(1... |
H: Contraposition with assumptions
I was just doing a practice problem, and found myself in the following scenario, which I've abstracted to get at the logical question that I have.
We want to prove: Given an assumption $A$, $B$ and $C$ cannot both be true at once.
If we can show "$B\implies\text{not }C$", I would usu... |
H: How do we prove that $\sup_{n\geq 1}f_{n}$ is a measurable function when each term $f_{n}$ is measurable?
Proposition
For each $n\in\mathbb{N}$, let $f_{n}:(\Omega,\mathcal{F})\to(\overline{\mathbb{R}},\mathcal{B}(\overline{\mathbb{R}}))$ be a $\langle\mathcal{F},\mathcal{B}(\overline{\mathbb{R}})\rangle$-measurabl... |
H: Proving $(Y\cap Z)\cup (X \cap Z ) \cup (X \cap Y )= ((Y \cup Z) − (\bar{X} \cap \bar{Z})) \cap (X \cup Y)$
I'm trying to prove that $$(Y\cap Z)\cup (X \cap Z ) \cup ({X} \cap {Y} )= ((Y \cup Z) − (\bar{X} \cap \bar{Z})) \cap (X \cup Y)$$ using set identities.
I have been using mainly de morgans law and gotten the ... |
H: For $n \in \Bbb{Z}_{>0}$, let $(3+i)^n = a_n+ib_n$, where $a_n, b_n \in \Bbb{Z}$.
Find expressions for $a_{n+1}$ and $b_{n+1}$ as linear combinations of $a_n$ and $b_n$ with coefficients independent of n.
With some of your comments, I see $a_{n+1} +ib_{n+1} = (a_n+ib_n)(3+i) = 3a_n + ia_n + i3b_n-b_n$. So the imag... |
H: Question on order of perfect shuffles
Imagine you have a stack of $n$-even chips where the bottom half is blue and the top half is red. You split the stack equally and perform a perfect shuffle where the lowest blue chip remains on the bottom and hence the top red chip stays on the top. How many shuffles does it t... |
H: Metric Space where every real continuous functions are Bounded, but does not attain its Bound.
Are there any metric space in which every continuous function are bounded but there exists one such function who does not attain its bound.
AI: Assuming that you are talking about real-valued functions, there is not. A sp... |
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