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H: Exercise: show that if $U$ is a connected open subspace of $ R^2$, then $U$ is path connected.
Show that if $U$ is a connected open subspace of $ R^2$, then $U$ is path connected.
The idea was to show that given $x_{0} \in U$, the set of points that can be joined to $x_{0}$ by a path in $U$ is open and closed in $U... |
H: Find all points $(a,b,c)$ where the graph $z = f(x,y)$ has a horizontal tangent plane.
Given $f(x,y)=x^2 + 4xy + y^2 - 2x + 2y + 1$
I tried setting $F(x,y,z) = f(x,y) - z$ and found the partial derivatives Fx, Fy, and Fz.
Fx = 2x + 4y -2
Fy = 4x + 2y + 2
Fz = -1
Apparently the gradient is perpendicular to the tang... |
H: Find $19\Diamond 98$, given rules $(xy)\Diamond y=x(y\Diamond y)$, $(x\Diamond 1)\Diamond x=x\Diamond 1$, and $1\Diamond 1=1$
Given any two positive real numbers $x$ and $y$, then $x \, \Diamond \, y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \, \Diamond... |
H: Localization and nilradical
I am trying to answer a question that has already been posted in here (About Nilradical and Localization). I did not have much success with the first two answers, and the other two mention sheafs, which is something I am not acquainted with.
Basically, I want to show that $\mathcal{N}(R_... |
H: Find matrices given sums of each row and column with bounded integer entries: maximize zero-valued entries
I want to find solutions for the following problem. It seems to be a classic problem in integer programmimg and logistics, but I don't know its name.
Find a matrix of m rows and n columns, with non-negative en... |
H: Nonlinear vector calculus problem
Let $A$ be a vector field on $\mathbb{R}^3$. I am interested in finding solutions of
$$
\nabla^2 A \times {\rm curl} A = 0,\\
\quad {\rm div} A = 0.
$$
Are there any exact solutions with nonzero $\nabla^2 A$ and ${\rm curl} A$? If there are, what are they?
These equations are a t... |
H: Understanding Fraleigh's proof of that the set $R[x]$ obeys associativity w.r.t. multiplication
I am trying to understand Fraleigh's proof of the fact that the set $R[x]$ of all polynomials in an indeterminate $x$ with coefficients in a ring $R$ obeys the associative law for multiplication. Here are the first few s... |
H: Using a function f(x) with sigma for summation bound
If I have $\sum_{ i = 0 }^{f(x) 1, \space\space(\forall \space1 \leq x \leq 4)$ where $ f(x) = 2$, then does this mean that this summation is counted 4 separate times? So for when x = 1, x = 2, x = 3, x = 4?
For example, is this equivalent to $\sum_{ i = 0 }^{f(1... |
H: Find all sequences that has $\sum_{i=1}^\infty a_i$ converges, where $a_i = \sum_{k=i+1}^\infty a_k^2$.
Find all sequences that has $\sum_{i=1}^\infty a_i$ converges, where $a_i = \sum_{k=i+1}^\infty a_k^2$.
My intuition is that the only sequence of this form is the zero sequence.
Here's what I have so far:
$a_n ... |
H: Let $x$, $y$ $\in$ $\mathbb{R}$, find all the complex numbers $z=a+bi$ satisfying $|z+x|$ $= y$
Let $x$, $y$ $\in$ $\mathbb{R}$, find all the complex numbers $z=a+bi$ satisfying $|z+x|$ $= y$.
Because $x$ and $y$ are real numbers and $z=a+bi$ then, $|z+x|=|(a+x)+bi|=\sqrt{(x+a)^{2}+b^{2}}$.
Therefore, $\sqrt{(x+a)^... |
H: Finding the expected value of a Random variable that is dependent on another Random Variable
The following problem is from the book "Probability and Statistics" which is part of the Schaum's outline series. It can be found on page 104 and is problem number 3.50.
Problem:
Let $X$ and $Y$ be random variables having j... |
H: shortest path metric and induced usual euclidean metric
Path metric is defined on path space $T_{x,y}(X)$ where $x,y\in X\subset\mathbb{R}^{n}:$
$$d(x,y)=\inf\{L(\sigma):\sigma\in T_{x,y}(X)\},$$ and $L(\sigma)=\sup\{\sum_{k=1}^{m}||\sigma(t_{k-1})-\sigma(t_{k})||:0=t_{0}<t_{1}<\cdots<t_{m}=1\}$ and $||(\cdot)||$ i... |
H: Prove that $\{a_n\}$ is convergent.
Prove that $\{a_n\}$ is convergent.
$$\{a_n\} = \frac{n^2-5n+6}{n^2+n+1}\:\forall n\in\mathbb{Z_+}$$
I know $\{a_n\}$ approaches $1$ when n goes to $\infty$. I tried to prove it using limit definition.
$$\forall\epsilon > 0\:\exists N>0\:\: s.t \:\:n>N \implies \left|\frac{n^2... |
H: Show there do not exist functions $f$ and $g$ such that $f(x+y) = g(x) - y$ for all $x$ and $y$.
As the title says, the problem is:
Show there do not exist functions $f$ and $g$ such that $f(x+y) = g(x) - y$ for all $x$ and $y$.
I think the exercise is wrong/has a typo, since it seems that $f(x) = 3 - x = g(x)$ wor... |
H: Olympiad Minimization Problem
I've been struggling to find a solution to this problem that I found in the archive of my country's Olympiad questions.
I'm particularly interested in a solution that doesn't involve the use of calculus since I know that Olympiad questions do not require the knowledge of calculus to s... |
H: Regular expression for languages with limit on repeated letters
I'm working through some mathematical Regex questions and I was wondering if you could review some of my answers.
(1) L={w ∈ {0,1}* | w contains at least three repeated 1s}
(0|1)*111(0|1)*
(2) L = {w ∈ {0,1,2}* | w cannot have 4 repeatd 2s}
I'm still... |
H: Is there a rule for evaluating a sum like $\sum_{x=1}^{10} (x + c)^3$ where $c$ is a constant?
I'm kind of new to summations and wonder if there is a rule to sum cubic terms. For example
$$\sum_{x=1}^{10} (x + c)^3$$
where $c$ is a constant.
I know how to calculate this sum, but I wonder if there is a rule such a... |
H: Is $ax+b=0$ still a linear equation if $a$ and $b$ are both $2$?
Is $ax+b=0$ still a linear equation if $a$ and $b$ are both $2$, like below?
$$2x+2=0$$
AI: Yes.$\text{ }$ |
H: $ \text{why} -\int_{0}^{1}{({1 - t})^{n} - 1 \over t}d t = \int_{0}^{1}{t^{n} - 1 \over t - 1}d t ?$
Question link : Proving Binomial Identity without calculus
i have one doubt in the given answer below ,my doubts mark in red colour
My doubt is that $$ \text{why} -\int_{0}^{1}{({1 - t})^{n} - 1 \over t}d t =
... |
H: Prove formula for $\int \frac{dx}{(1+x^2)^n}$
I was reading a calculus book and I saw this reduction formula:
$$\int \frac{dx}{(1+x^2)^n} = \frac{1}{2n-2}\frac{x}{(x^2+1)^{n-1}}+\frac{2n-3}{2n-2}\int\frac{1}{(x^2+1)^{n-1}}dx$$
Out of curiosity I attempted to prove it, but I got stuck near the end of it.
My attempt:... |
H: Does the following statement hold true ? (asymptotics)
Given any $\alpha > 0$ and $c > 0$, and $x \in \mathbb{R}$, does the following statement hold as $x \rightarrow \infty$? If true, is there a way to prove it ?
$$
x^{\alpha} \sim (x+c)^{\alpha}
$$
where $\sim$ means that given functions $f(x)$ and $g(x)$, we hav... |
H: determinant of an orthogonal matrix
The question goes like this,
For a square matrix A of order 12345, if det(A)=1 and AA'=I (A' is the transpose of A) then det(A-I)=0
(I have to prove it if it is correct and provide a counterexample if wrong).
So all that I know is that the given matrix is an orthogonal matrix. I... |
H: If $A$ is a connected subspace of $X$, can it be said that $Int(A)$ and $Bd(A)$ are connected? Is the reciprocal true?
If $A$ is a connected subspace of $X$, can it be said that $Int(A)$ and $Bd(A)$ are connected? Is the reciprocal true?
Clearly the converse fails:
Take $A = \mathbb{Q}$, then $Bd (\mathbb{Q}) = \ma... |
H: Finding minimum values of a function
I am trying to find the minimum values for the following
function:
$f(x)=(x-3)^4 + (x-5)^4 + (x-9)^4 + (x+10)^4$
Any hint is greatly appreciated.
AI: $$f(x)=(x-3)^4 + (x-5)^4 + (x-9)^4 + (x+10)^4$$
$$f'(x)=16 x^3-84 x^2+2580 x+476=0$$
If you follow the steps given here for one r... |
H: How many numeric strings of length 8 have exactly three 3's or have exactly 2 digits
I've been practising some combinatorics questions but am finding this one a bit difficult.
I recognise that we can split the question into 2 i.e.
Find how many strings have three 3's
Find how many strings have exactly 2 digits
Add... |
H: $A,B\neq0$ then $\langle A x_n, Bx_n\rangle\neq0\;\forall n\geq n_0)$?
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $(F, \langle \cdot,\cdot\rangle)$.
Let $(x_n)_n$ be a sequence in $F$ such that $\|x_n\|=1$ and $A,B\in \mathcal{B}(F)$.
Is the following assertion tru... |
H: Prove that for any point on the unit circle $\exists t \in \mathbb{R}$ such that $(\cos(t), \sin(t)) = (x, y)$
I'm being asked to prove this using the continuity of $\sin(t)$ and the identity $\sin(t)^2 + \cos(t)^2 = 1$.
Since the question specifically mentions continuity I was thinking I should divide the unit cir... |
H: Solve many linear equations of similar structure
Given
G: real and symmetric square matrix
v: real column vector
I need to solve n linear systems of the form
\begin{align} A = \begin{pmatrix} G & v \\\ v^T & 0 \end{pmatrix}\end{align}
\begin{align} Ax = b\end{align}
Where
n is large
G: real and symmetric square ... |
H: The equivalent condition for improper integral converges.
There is a question in my mathematical analysis book.
Suppose $f(x)$ is a monotonic function on $[0,+\infty)$, and $g(x)\not\equiv 0$ is a continuous and periodic function with period $T>0$ on $\mathbb R$. Prove that
$\int_0^{+\infty} f(x) \,\text{d}x$ c... |
H: Can a regular graph have the same Laplacian spectrum with a non-regular one?
So the Laplacian matrix of an undirected graph $G$ is $L(G)=D(G)-A(G)$, where $D(G)$ is the diagonal degree matrix and $A(G)$ is the adjacency matrix, as usual.
I can easily prove the case when the other graph $H$ is regular but with a dif... |
H: Domain of $f(g(x)),g(f(x)),f(f(x))$ in composite function
If $\displaystyle f(x)=4/x$ and $g(x)=4x+1$. Then domain of $f(g(x)),g(f(x)),g(g(x)),f(f(x))$ are
What i try: $$f(g(x))=\frac{4}{g(x)}=\frac{4}{4x+1}$$
So domain is $\displaystyle x\in\mathbb{R}-\{\frac{1}{4}\}$
$$g(f(x))=4f(x)+1=\frac{16}{x}+1$$
So domai... |
H: Number of elements $a\in\mathbb{F}_{5^4}$ such that $\mathbb{F}_{5^4}=\mathbb{F}_5(a)$
Determine the number of elements $a\in\mathbb{F}_{5^4}$ such that $\mathbb{F}_{5^4}=\mathbb{F}_5(a)$, and find the number of irreducible polynomials of degree $4$ in $\mathbb{F}_5[x]$.
My thoughts: Once we know the number of el... |
H: How can we get the probability of the complement of $C$ that $P(C^{c})\geq 1-2c-1/n?$
Assume events $A, B, C$. If we have the probability of $P(A)\geq 1-c$, $P(B)\geq 1-c$, and
$$P(A\cap B\cap C)\leq 1/n$$
How can we get the probability of the complement of $C$ that $P(C^{c})\geq 1-2c-1/n?$
We know that $P(A^c... |
H: Is $\tau$ is a discrete Topology?
Consider $\tau = \{\emptyset,\{a\},\{b,c,d\},X\}$. where $X= \{ a, b, c, d\}$
Is $\tau$ is a discrete Topology ?
My attempt : I think yes because here $\emptyset,\{a\},\{b,c,d\} and X$ are open set
we know that in discrete , all set are open
AI: Discrete topology is one in which... |
H: Why are there multiple base cases in this strong induction?
My understanding of needing a base case, in general, is that after proving the induction step, we can assert that the proposition is true for all values from the base case.
This question ($∀n ∈ Z, n≥12$) $\implies$ ( $∃x, y ∈N$ such that $n= 4x + 5y$) use... |
H: Find the extreme values of an absolute function on a given interval
(Q) Find the extreme values for $f(x)=|3x-5|$ on $-3≤x≤2$
Because this is an absolute function $f'(x)=0$ does not exist.
There is a local minimum at $x=\frac{5}{3}$
For the interval $-3≤x≤2$, $$f(-3)=14$$ $$f(2)=1$$
Thus, the extreme values are loc... |
H: Convergence of $\int_0^1\frac{1}{t^{\beta}}\sin(\frac{\pi}{t})dt$
Question: Check if the integral converges $$\int_0^1\frac{1}{t^{\beta}}\sin\left(\frac{\pi}{t}\right)dt$$
I have checked that if $\beta<1$ then the integral is absolutely convergent and if $\beta=1$ it is convergent (but not absolutely). I got stu... |
H: intuition behind finding inverse of this matrix
NOTE: I was not able to upload the matrix in the normal way so I created the image and i think it is easy to view it now .
AI: You can write
$$\begin{bmatrix}A&0\\B&C\end{bmatrix}\begin{bmatrix}X&Y\\Z&W\end{bmatrix}=\begin{bmatrix}I&0\\0&I\end{bmatrix}$$
Now you hav... |
H: Show that $S = f(f^{-1}(S))$ if and only if $f$ is surjective (used contradiction)
First I am aware of similar posts such as this one, however my question falls under the solution-verification tag and I used a totally different method.
Complete question:
Let $f : X → Y$ be a function from one set $X$ to another s... |
H: Find $f(x)$ to which the given sequence of functions converges
$$f_{n}(x) = \begin{cases} \sin^{2}\pi x, & n≤ |x|≤n+1, \\
0,& |x| < n \text{ or }|x|≥ n+1.\end{cases}$$
How can I find $f(x)$ to which $f_{n}(x)$ converges?
I do always have problems with finding $f(x)$ for the sequence of functions given in the form... |
H: How to describe an affine hyperplane by an equation?
Suppose I have an affine hyperplane of $\mathbb{R}^n$, say of the form $x + H$ for some vector $x$ and some hyperplane $H$. How would I go about describing this affine hyperplane by an equation $a_1x_1 + ... + a_nx_n = b$? I am aware that I can describe the hyper... |
H: Convergence $\int_0^{\pi/2}\frac{e^{\alpha \cos x}-\sqrt{1+2\cos x}}{\sqrt{\cos^5x}}\mathrm{d}x$
Check if the following integral converges:
$$\int_0^{\pi/2}\frac{e^{\alpha \cos x}-\sqrt{1+2\cos x}}{\sqrt{\cos^5x}}\mathrm{d}x$$
So, the point at we should study the integrand is $\pi/2$, but how to do this? This fun... |
H: Density of product space $2^X$ for arbitrary cardinalities
Take the discrete space $2=\{0,1\}$ and form the infinite product $2^X$ for some infinite index set $X$. The Hewitt-Marczewski-Pondiczery (HMP) theorem (Engelking 2.3.15 for example) says in this case:
Theorem (HMP): If $|X|\le2^\kappa$, then $d(2^X)<=\ka... |
H: Cusp forms with different orders at infinity
Let’s assume we have cusp forms $f_1,...,f_n\in S_k(\Gamma_1(N))$ which they are zero at infinity with different orders. Then, how can we show that they are linearly independent?
AI: You can write the cusp forms as
$$f_i(q)=q^{c_i}g_i(q)$$
where the $c_i$ are distinct, a... |
H: Proving that some integer multiple of a real number is within $\frac{1}{k}$ of an integer.
So I'm trying to prove that for every real number $a \in \mathbb{R}$, the set $M = \{a,2a,\dots,(k-1)a\}$ contains at least one element that is within $\frac{1}{k}$ of an integer. (Note that $k \in \mathbb{N}$)
So far I found... |
H: $F(\kappa, \nu)$ with $\kappa = \frac{x_1}{x_2}$ and $\nu = \frac{x_1}{x_3}$. Chain rule to $F_x$
Let $\kappa = \frac{x_1}{x_2}$ and $\nu = \frac{x_1}{x_3}$. Consider a function $F(\kappa, \nu)$
How to apply chain rule to the following $$\frac{\partial F}{\partial x_1}$$
I am confused that since both $\kappa$ and $... |
H: How to find the mean of this standard distribution?
Question:
The probability distributions for 2 variables are defined as follows$ X$ ~ $N$$(120, σ^2)$ and $Y$ ~ $N$$(μ, 2σ^2)$ and $P(X < 124)$ = $P(Y > 124$). Calculate $μ$.
I tried this for hours and no result. Since there are 2 unknowns, I tried forming a simul... |
H: Doubt in Construction of Real Numbers
I cannot understand why we say "The Completeness axiom" (that $\mathbb{R}$ has least upper bound property) when it can be proved from Dedekind's construction?
AI: We could describe axiomatically $\Bbb R$ as a Dedekind-complete (A) ordered field (B) $(R,+,\cdot,<)$. It turns out... |
H: Let $f$ be continuous on $X$ and $(x_n)$ be a Cauchy sequence on $X$. Show that $(f(x_n))$ doesnt have to be a Cauchy sequence...
Let $f$ be continuous on $X$ and $(x_n)$ be a Cauchy sequence on $X$. Show that $(f(x_n))$ doesnt have to be a Cauchy sequence. And show that $(f(x_n))$ always has to be a Cauchy sequenc... |
H: Why the same position u for two different functions applying mean-value theorem?
I don't know why the function in the nominator and denominator as shown in the figure (Red Arrow) has the same position u after the application of Mean Value Theorem. It is because they are not the same function, so they should have di... |
H: Sum $ \sum_{n=0}^{\infty}\frac{n}{n+1}x^{n} $
I want to find the sum function of the series
$$ \sum_{n=0}^{\infty}\frac{n}{n+1}x^{n} $$
and I would like to get your assessment of my work cause I'm really not sure about its validity?
I define the sum $$s(x) = \sum_{n=0}^{\infty}\frac{n}{n+1}x^{n}$$ and multipy with ... |
H: I'm struggling with finding variance of Ito integral.
Find $D \int_{0}^{t}W^{2}_{s}dW_{s}$ My solution is next:
variance = $E(\int_{0}^{t}W^{2}_{s}dW_{s})^{2} - (E\int_{0}^{t}W^{2}_{s}dW_{s})^{2}$
Which is equal to (using Ito's isometry principle) = $\int_{0}^{t}EW_{s}^{4}dW_{s} - (\int_{0}^{t}EW_{s}^{2}dW_{s})^{2}... |
H: Solve an Itô Integral by Itô calculus
I saw an example where the following Itô integral was solved by Itô calculus:
$\int^{T}_{0}W(t)dW(t)$. They say: let's take the stochastic process $X(t) = W(t)$, which means that $dX(t) = 0 dt + 1 dW(t)$. If we now apply Itô's lemma to g(X(t)) = $X^2(t)$, this would result in t... |
H: Solving a system of equations involving the floor function.
I have the following system of equations that I am stuggeling with:
$$ax\lfloor y\rfloor=k,by\lfloor x\rfloor=d$$
And I know that $x$ and $y$ are bigger than zero and all the other constants are natural numbers. Given is that $a$ divides $k$ and $b$ does... |
H: How is the integral form of Ramanujan theta function derived?
Ramanujan theta function defined as-$$f(a,b)=\sum_{n=0}^\infty a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}$$
And it's integral representation:$$f(a,b)=1+\int_0^\infty \frac{2ae^{-t^{2}/2}}{\sqrt{2\pi}}\left[\frac{1-a\sqrt{ab}\operatorname{cosh}(\sqrt{\opera... |
H: how to prove this recurrence of determinants
the question is :
Let Dn = $[a_{ij}]$n × n be a (n × n) determinant with the following conditions:
$$a_{ij} = 4 ;i=j$$
$$a_{ij} = 2 ;|i-j|=1$$
$$a_{ij} = 0 ; otherwise$$
then we have to prove that :
$$ D_n = 4D_{n-1} - 4D_{n-2}$$
my approach:
I found $D_1 =4 ,D_2=12, D_3... |
H: How to solve $0=1+\frac{x}{3!}+ \frac{x^2}{5!}+ \frac{x^3}{7!}+...$?
When I take some calculation, I find an equation
$$
0=1+\frac{x}{3!}+ \frac{x^2}{5!}+ \frac{x^3}{7!}+...
$$
Obviously, when $x=0$, the right part is positive. And when $x=-1$, the right part is negative, so, there should be a root in $[-1,0]$. Bu... |
H: Basis of annihilator of subspace in $\mathbb{R}^3$
I want to find a basis for the annihilator, $U^{\circ}$, of the subspace $U:=Span((1,0,0))\subseteq \mathbb{R}^3$. As far as I have understood, in $\mathbb{R}^n$ the linear functionals can be seen as the dot-product between two vectors. Then, the annihilator can be... |
H: How to determine $E(X(XZ))$?
The question in the book was:
Give an example of a pair of dependent and joinlty continuous random variables $X,Y$ for which $E(XY)=E(X)E(Y)$.
Example given: Let $X$ and $Z$ be independent. $X$ has normal distribution with mean $0$ and variance $1$. $Z$ takes the value $1$ or $-1$ wit... |
H: Each of eight, consecutive, three digit numbers is divisble by its last digit. What is the sum of digits of the smallest number?
Each of eight, consecutive, three digit numbers is divisble by its last digit. What is the sum of digits of the smallest number?
My approach:
Each number we can denote like that:
$\over... |
H: Weak convergence result in Levy's Continuity Theorem
I quote a part of Levy's Continuity Theorem and its proof.
Theorem
Let $\left(\mu_n\right)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$, and let
$\left(\hat{\mu}_n\right)_{n\geq1}$ denote their Fourier transforms (aka characteristic funct... |
H: How to get $1$ if number is postitive or $0$, -1 otherwise
I have a number $x$ which can be anything (positive, negative, zero, fraction). And I need to get $-1$ if it's negative and $1$ otherwise. Basically the following function:
$$
f(x)=
\begin{cases}
1, & x=0\\
\frac{|x|}{x}, & x \neq 0
\end{cases}
$$
But the... |
H: Is there an easy way to compute a finite number of sequential equal powers in some modulo?
For example, is there a easy way to compute
$$32^{A_1^{A_2^{\ \cdots^{{}^{A_{32}}}}}} \mod 7$$
whith $A_i=32 \, , \ i=1,\cdots,32$ ?
AI: Looking at the sequence of powers of $32$, reduced mod $7$, we get
\begin{align*}
32^0&... |
H: Is the p-norm on $l^p$ submultiplicative?
Consider a set $X$ and $l^p(X)$ for $p > 1$. In my notes, it is claimed that
$$\Vert fg \Vert_p \leq \Vert f \Vert_p \Vert g \Vert_p$$
However, I can't prove it. I tried to apply Minkowski-like inequalities but it did not work out. I'm beginning to suspect this is not even ... |
H: Why does the inverse of this mapping from a square to a trapezoid not exist?
I am trying to compute an inverse mapping. I shall start with the forward mapping.
Consider the parametric coordinates $(r,s)$:
$r \in \mathbb{R}\wedge[-1,1]$
$s \in \mathbb{R}\wedge[-1,1]$
Then consider the Euclidean coordinates:
$X=(x,y,... |
H: Variance of a linear transformed standard normal r.v.
I have the sequence $X_i$ that converges to $N(0, 1)$ in distribution and $$Y = 2X_i + 1$$
I was able to find $E(Y) = 1$, but I am struggling at $Var(Y)$.
From the general variance formula $E[(X-\mu)^2]$ I get $E[Y^2]$, since $\mu = 0$ by standard normal definit... |
H: Questioning the discrete nature of $\mathbb{Q}$
Background for question:
I recently heard someone say : " The measure of the set of rational numbers $\mathbb{Q}$ is $0$. ".
And I thought, well that makes sense because $\mathbb{Q}$ is made up of points. So its measure should naturally be sum of $0$ lengths.
But the... |
H: Proving $ 4\operatorname{arccot}(2)+\arctan\left(\frac{24}7\right)=\pi $. What am I doing wrong?
$$
4\operatorname{arccot}(2)+\arctan\left(\frac{24}7\right)=\pi
$$
original image
To prove the above result, I tried to equate the original expression to some constant $a$ such that $0<a<2.5\pi$ (from the range of the i... |
H: Show that the projection is $\in End(V)$
Consider a direct sum $V=U\oplus W$ where $U,W \subseteq V$.
In my lecture notes I have been given the definition of the projection on U along W:
$E\in End(V), E(v)=u$ with $v\in V$ and $u\in U$. I have then been given the task to prove that $E$ is indeed in $End(V)$, howev... |
H: How many different passwords of length 4 can be made with atleast 1 digit in it.
Consider the character set of total 62 characters : - 26 capital letters 26 small letters 10 digits
I know that i can solve this by
Total passwords - Total passwords only containing characters
That is $62^4 - 52^4$.
But i want to k... |
H: show that there are infinitely many integers $n$ such that $φ(n)\equiv2\pmod4$, where $φ(n)$ is Euler's totient function
Let $p$ be a prime and $k$ be an integer greater or equal to $1$. Then $φ(p^k) = p^k - p^{k-1}$.
AI: Hint:
For any prime $p\equiv 3\mod 4$, you have $\varphi(p)=p-1\equiv2\mod4$. |
H: Why this does not have a dot product on the right: $\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$
My main concern is because, on the right side of the expression, there is a vector $\nu$, but there is not any dot product.
$$\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$$
Where $\ph... |
H: How can I show binomial series converges to $\sqrt {2}$?
$\sum_{n=0}^{\infty} 2n\binom{-\frac{1}{2}}{n}(-\frac{1}{2})^n = \sqrt{2}$
From wolfram alpha, it says that above series including binomial term $\binom{-\frac{1}{2}}{n}$ converges to $\sqrt{2}$.
I tired to convert it to $(1+x)^{-\frac{1}{2}}$ but I can't bec... |
H: Solutions which start in right half plane (x>0) must remain there all time
How do I show that all solutions $x(t)$ and $y(t)$ of
$$\frac{dx}{dt}=y(e^x - 1),$$
$$\frac{dy}{dt}=x + e^y$$
which start in the right half plane $(x > 0)$ must remain there for all time.
I thought that looking at $\frac{dy}{dx}$ can help b... |
H: Genus $3$ curves with a couple of distinct points $P,Q$ such that $4P \sim 4Q$
Let $C$ be a smooth curve of genus $3$ over $\mathbb{C}$. Is it true that there exist $P\neq Q \in C$ such that $4P \sim 4Q$ ? ($\sim$ denotes linear equivalence)
Notice that if $C$ is hyperelliptic then this is true (just take two diffe... |
H: Simplifying $\sum_{m=1}^{N+1}e^{-m}$
I am currently trying to simplify $$\sum_{m=1}^{N+1}e^{-m}$$ I set $n = m-1 \implies m = n + 1$ and get $$\sum_{n=0}^{N+1}e^{-(n+1)} = \sum_{n=0}^{N+1}e^{-n-1} = \frac{1}{e}\sum_{n=0}^{N+1}e^{-n}$$ Here my question is if I have to also reduce the sum $N+1 \to N$ when I change t... |
H: Limit proof by contrapositive
I'm writing a proof for the statement : "If a function $f$ has a limit at $p$, then for every $\epsilon >0$ there is a $\delta>0$ such that $\vert f(x_1) - f(x_2)\vert < \epsilon$, whenever $x_1,x_2 \in U'(p,\delta)$."
I'm doing this by contrapositive and I wanted to make sure I have ... |
H: How to factor a polynomial with complex roots of the form $a+bi$, where $a\neq 0$? (Not just find the root)
For example, the quadratic formula reveals that the roots of $x^2 - 4x + 5$ are $x = 2\pm 2i $
But how do we use these roots to actually factor $x^2 - 4x + 5$?
My best guess was that, since $x = 2\pm 2i $, w... |
H: $(\mathbb{Z}/18\mathbb{Z})/(6\mathbb{Z}/18\mathbb{Z})\cong\mathbb{Z}/6\mathbb{Z}$ Proof
Use the homomorphism theorem or the first or/and second isomorphism theorems to show that $(\mathbb{Z}/18\mathbb{Z})/(6\mathbb{Z}/18\mathbb{Z})\cong\mathbb{Z}/6\mathbb{Z}$.
I was wondering if it is necessary to show that $6\mat... |
H: Solve a quadratic function passing through 2 defined points
I'm trying to find the values a, b and c that would validate y = ax^2 + bx + c
with the following parameters:
For x = 1; y = 1
For x = T; y = S
Essentially, I would like the function to pass through the (x, y) coordinates (1, 1) and (T, S) for any given... |
H: Some obscure points about the proof of Hahn Decomposition Theorem
The Hahn Decomposition Theorem. If $\nu$ is a signed measure on $(X,\mathcal{A})$, there exist a >positive set $P$ and a negative set $N$ for $\nu$ such that $P\cup N=X$ and $P\cap N=\emptyset.$
Proof Without loss of generality, we assume that $\nu... |
H: How come frontier points exist?
Logically speaking, either a point is an interior point to a given set or it is an interior point to its complimentary set. Then how in earth does a third possibility arise?
PS: I'm not a topology student. Was just going through different concepts in topology when this question struc... |
H: Proving $xH=yH$ if $x^{-1}y \in H$
Given $G$ is a group and $H <G$ and if for arbitrary $x,y \in G$ if $x^{-1}y \in H$ Prove that $xH=yH$
I started like this:
Since $x^{-1}y \in H$ we have $x^{-1}y=h$ for some $h \in H$
Then $y=xh$ $\implies$ $y \in xH$
Also $y \in yH$
How to proceed from here?
AI: We will show th... |
H: calculate $\sum_{n=0}^\infty \frac{3^n}{n!(n+3)}$ using power series
let $f(x)=\frac{e^x-1-x-\frac{x^2}{2}}{x}$, because $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$, $f$ can be expressed as $$f(x) = \frac{\sum_{n=0}^\infty \frac{x^n}{n!}-1-x-\frac{x^2}{2}}{x}=\frac{\sum_{n=3}^\infty \frac{x^n}{n!}}{x}=\sum_{n=0}^\inft... |
H: Proving fundamental trigonometrical inequalities geometrically
For $0 < \theta < {\pi}/{2} $,
$$0 < \cos \theta < \frac{\sin \theta}{\theta} < \frac{1}{\cos \theta}$$
I understand that proving this requires an advanced knowledge of calculus, so I was looking for the geometric intuition behind this. The only part th... |
H: Why is $1,a,a^2,...,a^{n-1}$ linearly independent?
I have a basic question about the proof of "Every finite field extension is algebraic".
Given the extension $K\subset L$ with $n:=[L:K]$ and $a \in L$, the proof says, that we have a linearly dependent set $1,a,a^2, ...,a^{n+1}$, because their quantity $n+1$ is b... |
H: Finding a generating function for number of existing integers
Let $k$ be a positive integer. $\left \{ a_r \right \} _{r=0}^{\infty}$ is the number of integers which exist between $0$ and $10^k$ (i.e integers with no more than $k$ digits), such that the sum of their digits is no more than $r$.
Find the generating... |
H: Proof checking - $\mathbb{R}^{\mathbb{N}} \sim \mathcal{P}( \mathbb{N})$ - Discrete Mathematics
I need to prove that:
$$\mathbb{R}^{\mathbb{N}} \sim \mathcal{P}( \mathbb{N})$$
We know that
1. $\mathbb{N} \times \mathbb{N} \sim \mathbb{N}$
2. And that $A^{B^C} \sim A^{B \times C}$
And so I start the proof:
I s... |
H: Can we use row operations in between matrices?
I know we can multiply a matrix $A$ on the left by some elementary matrix $E$ to get $EA$, which corresponds to an elementary row operation. This preserves a lot of things, such as rank, invertibility, null space, etc.
However, I'm wondering what happens if we try to i... |
H: Multiplication of Square roots in the form of (a+b) (a-b)
Question: What is the value of :
$$\left(\sqrt{24+5\sqrt{23}}\right)\left(\sqrt{24-5\sqrt{23}}\right)$$
I tried to apply $(a+b)(a-b) = a^2 - b^2$ but that is wrong.
Please suggest another method.
AI: Let $a=24, b=5\sqrt{23}$.
Then our expression is $\sq... |
H: Probabilities of Survival
So I have these estimation of survival probabilities for men:
The probability that a man lives at least $70$ years is $80\%$.
The probability that a man lives at least $80$ years is $60\%$.
The probability that a $80$ years old man lives at least $90$ years is $40\%$.
The first question ... |
H: Simple proof by induction problems
I just started learning proof by induction and I have come across 2 problems that I am not sure if am doing right. The first one is
Prove that $11^n - 1$ is dividable by $10$.
I started with
$ n = 0, 11^0 - 1 = 0 $, is dividable by $10$
I did the same for $1$ and $2$, ... |
H: Show $E[X]-1\leq \sum_{n=1}^\infty P[X\geq n]\leq E[X]$
Let $X$ be any nonnegative random variable. Then show that
$$E[X]-1\leq \sum_{n=1}^{\infty}P[X\geq n]\leq E[X].$$
I was able to show the second inequality by
$$\sum_{n=1}^{\infty}P[X\geq n]=\sum_{n=1}^{\infty}\int_{ X\geq n}dP=\int_\Omega \sum_{1\leq n\leq ... |
H: Suppose $r(t)$ and $s(t)$ are vector functions. $r(2) = \langle 1,2,−1 \rangle, r'(t) =\langle 3,0,4\rangle$, and $s(t) =\langle t,t^2,t^3 \rangle$.
Suppose $r(t)$ and $s(t)$ are vector functions with $r(2) = \langle 1,2,−1 \rangle$, $r'(t) =\langle 3,0,4\rangle$, and $s(t) =\langle t,t^2,t^3 \rangle$.
(a) Find th... |
H: If ($x * 10^q$) - ($y * 10^{r}$) = $10^r$, where q, r,x and y are positive integers and $q>r$, then what is the units digit of $y$?
Q. If ($x * 10^q$) - ($y * 10^{r}$) = $10^r$, where q, r,x and y are positive integers and $q>r$, then what is the units digit of y?
My approach:
$10^r$ (x * $10^{(q-r)}$ - y ) =... |
H: An inequality on Hausdorff distance of union
Definitions: Let $X,Y$ bet sets in $\mathbb{R}^n$ then we define:
$$B(X,\epsilon):=\{r\in\mathbb{R}^n|\exists{}x\in{X},\text{ s.t. }{}d_{\mathbb{R}^n}(r,x)<\epsilon\}$$
$$d_H(X,Y):=\inf\{\epsilon>0|X\subset{}B(Y,\epsilon),Y\subset{}B(X,\epsilon)\}$$
I am trying to show t... |
H: Taylor series of $\ln(1+x)$ when $x=1$
I know what the Taylor Series of $\ln(1+x)$ is, but, I don't know why it's true also for $x=1$.
If I decide to use the method of integration starting from $\frac{1}{1+x} \ =\ \sum ( -1)^{n} x^{n}$
then, that is true only for $-1< x< 1$.
How can I prove it's also for $x=1$?
Th... |
H: bookstore, ordering of books
A bookstore has 3 copies of the calculus book, 4 copies of the algebra book, and 5 copies of the statistics book. How many different orders can they be arranged on a shelf?
My thinking:
12 books, so 12!
AI: Let us start with a simpler example. Say we have the letters A,A,B. We need to ... |
H: Probability and correlation of two uniform discrete and independent random variable
I have the following problem:
We have two random variables, discrete, uniform and independent
$x=(1,2,3,4,5,6,7,8)$
$y=(1,2,3,4,5,6,7,8,9,10)$
Find:
$P(X+Y=5)$ where I have calc $P(X+Y=5)=\frac{1}{20}$
$P(X>Y)$
Correlation between ... |
H: Proving that removing any vector of the linearly dependent set gives a linearly independent set
Consider the matrix representing 6 linearly dependent vectors:
$$\left(\begin{array}{llllll}
1 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 1
\end{... |
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