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H: How to find roots of this equation?
I have this equation and want to find its roots.
$\left(a^2+1\right) \cosh (a (c -b))- \cosh (c a)=0 $.
Any comment is welcome.
AI: One has
\begin{equation}
1 \le f(a) = \frac{(a^2-1)^2}{(a^2+1)^2}\cosh(2\pi a) + \frac{4 a^2}{(a^2+1)^2}< \cosh(2\pi a)
\end{equation}
because this ... |
H: Given the solution of $Ax = b$, do we know whether $A$ spans $\mathbb{R}^3$?
Let $A$ be a $3 \times 4$ matrix such that all solutions to the equation $A\mathbf{x} = \mathbf{b}$ may be written as:
$$
\left[ \begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix} \right] + x_3 \left[ \begin{matrix} 3 \\ 2 \\ 1 \\ 0 \end{matrix}... |
H: Rudin's Real and Complex Analysis, Section 9.16
In Section of 9.16 from Rudin's RCA, it says
Let $\hat{M}$ be the image of a closed translation-invariant subspace $M \subset L^2$, nder the Fourier transfrom. Let $P$ be the orthogonal projection of $L^2$ onto $\hat{M}$ (Theorem 4.11): To each $f \in L^2$ there corr... |
H: What are the homology groups $H_k(\mathbb{R}\setminus \{0\})$?
Is there an easy way to calculate the homology groups
$$H_k(\mathbb{R}\setminus \{0\}), k\geq 0$$
I was able to calculate $H_k(\mathbb{R}^m\setminus \{0\}), m > 1$ because we have a homeomorphism onto the sphere $S^{m-1}$ whose homology groups I know, b... |
H: Exercise on finitely generated $A$-modules
Here is the exercise I'm trying to solve:
Let $M$ be a finitely generated $A$-module (where $A$ is a commutative ring) and let $g:M\rightarrow A^n$ a surjective $A$-module morphism. Prove that $\text{Ker}(g)$ is finitely generated.
Here is what I'd do:
Since $g$ is surje... |
H: How to solve this ODE with the Laplace transform?
I want to solve this ODE
$$ y'''-y''-y'+y= -10 \cos (2t-1)+5 \sin(2t-1) $$
with
$y( \frac12)= 1 $ , $ y'( \frac12 )=2 $, $y''( \frac12 )=1 $
$ t \in [ \frac12 , + \infty [ $
using the Laplace-Transformation.
usually I use the differential-approach:
$$ ( L( f^{(k)}))... |
H: Building a sequence that alternates for odd numbers without using cases
I was wondering, whether it is possible to build a sequence, that alternates for odd numbers, such as
$$a_n = \begin{cases}
1, & \text{if} \ n = 4k+1, k\in \mathrm{N} \\
-1, & \text{if} \ n = 4k+3, k\in\mathrm{N}
\end{cases}$$
but without usin... |
H: Can't prove $(B\oplus C)\cap D=(B\cap D)\oplus (C\cap D)$.
Prove $(B\oplus C)\cap D=(B\cap D)\oplus (C\cap D)$.
Note: $B\oplus C=(B\cup C)-(B\cap C)$ and $B-C=B\cap C^C$.
I have tried to prove it as below.
\begin{align*}
(B\oplus C)\cap D &= ((B\cup C)-(B\cap C))\cap D\\
&=((B\cup C)\cap(B\cap C)^C)\cap D\\
&=((B... |
H: Find gradient & angle of inclination of the tangent where x = 3
Differentiate f (x) = $x$$^2$ − 3$x$ − 6.Find the gradient and the angle of inclination of the tangent at the point where $x$ = 3
I have already differentiated the beginning and got
2$x$ -3 but I am not sure what to do next.
AI: The angle of inclina... |
H: Are all vertical asymptotes the points where the denominator is zero?
Question 1: Are all vertical asymptotes the points where the denominator is zero?
Question 2: For all rational functions ( fractions of polynomials), are all vertical asympototes the points where the denominator is zero?
Thank you so much.
AI: Fo... |
H: Hall and knight inequality question
If $a^2+b^2=1$ and $x^2+y^2=1$, show that $ax+by<1$
AI: This is probably what you're looking for:$$1=(a^2+b^2)(x^2+y^2)=(ax+by)^2+(ay-bx)^2\geqslant (ax+by)^2\implies \vert ax+by\vert\leqslant 1$$ |
H: Yet another Rouché's theorem example for $f(z) = e^z-3z^{2019}$ - solution verification
Use Rouché's Theorem to conclude that
$$
f(z) = e^z-3z^{2019}
$$
has exactly 2019 zeroes in $D(0,1)$.
My main question is, since we're working for $|z| = 1$ can I conclude that $|e^z| \sim e^1 = e \sim 2.71...$?
Because if that ... |
H: How to calculate distance run on athletics track
first time poster and definitely no maths expert.
I am trying to solve a basic problem using an athletics track. The total distance around a standard athletics track is 400m:
If you run in the first lane you run 400m, I am trying to work out the formula to estimate ... |
H: How to prove that $1^n+2^n+...+(p-1)^n \equiv 0\pmod p$?
I have a homework for the university and I am 'on this' for the entire week, so I really need help.
The question:
let $p>2$ be a prime number and $n\in \Bbb N$, $\ p-1\nmid n$.
Prove that $1^n+2^n+...+(p-1)^n \equiv 0\pmod p$.
I thought:
it is pretty clear t... |
H: Isomorphisms on $L^p$ and $\ell ^p$ spaces.
I'm studying real analysis and I have some questions about isomorphism on $L^p$ and $\ell ^p$ spaces.
First of all, I want to see whether there exist infinite sets like $A$ and $B$ such that $\ell ^p(B)$ is not isomorphic to a subspace of $\ell ^p(A)$, with $p \in [1, +... |
H: Find largest number divided by which each element of vector is integer
This is my first question on math.stackexchange, and as you will notice, I am not a mathematician at all, and this may be a very simple question. Apologies.
I also don't know if I used the right terms when asking the title, so here an example of... |
H: Complex projective plane: $\mathbb C^2$ vs $\mathbb C^3$
I have a question about Phillip A. Griffiths - Introduction to algebraic curves.
In Chapter I.1, it seems to say $\mathbb C^3 \cup L_{\infty} = P^2\mathbb C$, but later on it says $P^2\mathbb C \setminus L_{\infty} = \mathbb C^2$.
What's going on please?
Gues... |
H: Hartogs set of a well ordered set
so i understood that the hartogs set of a well ordered set $A$ is defined as $H(A)$ the minimal ordinal such that $H(A)\nleq A$ (there is no injection from $H(A)$ to $A$)
and i also uderstood the proof to the existance of an ordinal like that.
the only thing that im having trouble ... |
H: Find local extrema of the function.
I have the following function and I want to find its local extremas:
$f(x,y)=x^3+3xy^2-15x-12y+4$
Is it correct? Wolphram alpha says that the only solutions are -24 and 32 I am little bit confused.
I have done the following calculations:
$\nabla f(x,y)=[3x^2+3y^2-15,6xy-12]$
$\na... |
H: Least squares problem regarding distance between two vectors in $\mathbb{R}^3$
I'm solving an exercise problem and was facing some confusion regarding how to solve it. The problem is (roughly translated to English):
Given the following:
$$\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 1 & 1 \\ 0 & 1 \end{bmatrix},\ \math... |
H: Given a fixed perimeter, which shape will have the minimum area?
I understand that a circle will have the largest area for a given perimeter but I don't get The smallest area.
Is it a triangle because it has the least amount of sides so must be smaller?
AI: There is no lower-bound to the area of a shape with given ... |
H: $F(x,y,z)=3x{\bf{i}}-4y{\bf{j}}+5z{\bf{k}}$ is a vector field, $S: x^2+y^2+z^2 = a^2$ is an outward oriented surface. Find $\iiint\text{div}\ F\ dV$
$F(x,y,z)=3x\textbf{i}-4y\textbf{j}+5z\textbf{k}$ is the vector field and $S: x^2+y^2+z^2 = a^2$ is an outward oriented surface. Evaluate $\iiint\ \text{div}\ F\ dV.$
... |
H: Quadratic matrix bounds
Let A be a singular matrix with a simple (non-repeated) zero-eigenvalue.
Dose the following inequality hold?
$$\|Ax\|^2\geq\sigma_2\|x\|^2, \qquad \forall x\notin Null(A)$$
where $\sigma_2$ is the smallest nonzero singular value of the matrix $A$. If it is true, where can I find a proof??
AI... |
H: Do statement 1 and 2 imply statement 3?
Statement 1 : All locks are safety tools
Statement 2 : Some safety tools are chain
Statement 3 : Some chain are not locks
I tried to answer this myself but i'm having a problem to decide the correct venn diagram for the statement "Some B are C"
I tried to draw both venn dia... |
H: How to show $x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1 = 0$ have five different roots?
I have to prove this fact when determining the elementary divisors of a matrix. The numerical solution confirms that this equation indeed has five different roots (one real root, two pairs of conjugate complex roots), but how to show it ... |
H: Bounded operators on an Hilbert space
Let $H$ be an hilbert space and consider the set $L(H,H)$ of bounded linear operator $T: X\rightarrow X$. Now we know that this is going to be a Banach space since $X$ itself is a Banach space. Now is this going to be an Hilbert space ? I have tried getting a counterexample to ... |
H: Help with Lagrange multipliers
I need to find the absolute minima and maxima of the function
$f(x,y) = 12 x^2 + 12 y^2 - x^3 y^3 -5$ in the region bounded by the disk $x^2 + y^2 \le 1$.
I know that $f(x,y)$ has three critical points in its domain, but only one point of the three, namely $(0,0)$, fits in the regio... |
H: Non-decreasing function in $f:A\rightarrow B$
Let $f:A\rightarrow B$ be a function where set A contains 4 elements and set B contains 3 elements.
Then find the total number of non-decreasing functions.
As per the book the solution is "Number of non decreasing function is equal to number of non negative integral sol... |
H: Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}$
Question: Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}?$
For instance, $\cos^2(z)+\sin^2(z)=1$ is still true when we move to complex plane but a... |
H: How to prove Cauchy Schwartz Inequality for norms in Lebesgue Integration
I am self studying Apostol ( Mathematical Analysis) but I couldn't prove this particular theorem given in text despite the hint given .
So, I am asking here.
Its part (e) , I have no idea how to use RHS from the inequality to prove the CS i... |
H: Distribution function of $Y = \max\{X \sim \text{Exp}(2), 3\}$
Let $X$ be an exponential distributed random variable with $\lambda =2$ and $Y$ the random variable that takes the maximum of $X$ and $3$. What is the distribution function of $Y$?
So how can I calculate $P(Y\leq y) = P(\max\{X,3\} \leq y) = P(3\leq y, ... |
H: Probabilities in the expectation of a hypergeometric random variable
So I've read this explanation and the answer given by Andi R helped to give me some intuition of my problem. Now would like to understand it more mathematically rigorously.
Suppose we have a jar of marbles with $r$ red balls and $w$ white balls. W... |
H: Prove that the tangent $A$ and $B$ are perpendicular .
Let $f(x)=x^x$ and $g(x)=\Big(\frac{1}{x}\Big)^{x}$ .Let $A$ be the tangent of $f(x)$ at $x=1$ and $B$ the tangent of $g(x)$ at $x=1$ . Then A and B are perpendicular .
Proof without first derivative
Clearly since the equation $x^x=x$ have a unique solution whi... |
H: There exists $M \in \mathscr{M}(n \times n,\mathbb{R})$ such that $e^M=L^2$?
Problem:
Define $C=\left(\begin{matrix} a & -b \\ b & a \end{matrix}\right)$ and $\Delta=\left(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right)$.
Define $L=\left(\begin{matrix} C & \Delta & \\ &C & \Delta & \\ & & \ldots \\ & & & C &
... |
H: How can I find the interval of x? I apply ratio test and get indeterminate. $\sum_{n=1}^\infty({(n+1)(n+2)....(2n)\over n^n})\space (x-2)^n$
$$\sum_{n=1}^\infty({(n+1)(n+2)....(2n)\over n^n})\space (x-2)^n$$
Find the radius and interval of convergence of the power series given above.
Hi! I am trying to find the int... |
H: Number of irreducible factors of $x^{p^n + 1} - 1$ over $\Bbb F_p$
Let $p,n$ be two odd prime numbers. I want to show that the number of irreducible factors of $x^{p^n + 1} - 1$ over $\Bbb F_p$ is
$$N = \frac{p^n-p}{2n} + \frac{p-1}{2} + 2$$
I know that this is equal to $\sum_{d\mid (p^n+1)}\frac{\phi(d)}{\operator... |
H: A case where the Euler-Lagrange equations produces $1=0$
As a reference, I asked the same question (https://physics.stackexchange.com/questions/561449/equations-of-motions-of-mathcall-phi-x-phix) in the physics community but I am interested in the mathematical reason why this produces $1=0$.
Suppose a Lagrangian of... |
H: Compute number of elements in a cycle
Given numbers from 1 to N and a number K can we find the number of cycles beginning at number 1. K is the distance to the next number. Elements 1 to N are arranged in a circular order.(N and 1 are adjacent).
For eg : N = 7 and K = 2
then, 1 --> 3 --> 5 --> 7 --> 2 --> 4 --> 6 -... |
H: UC Berkeley Integral Problem: Show that $\int_0^{2\pi} \frac{\min(\sin x, \cos x)}{\max(e^{\sin x},e^{\cos x})}\ {\rm d}x = -4\sinh(1/{\sqrt2})$.
Show that $$\int_0^{2\pi} \frac{\mathrm{min}(\sin{x},\, \cos{x})}{\mathrm{max}\left(e^{\sin{x}},\, e^{\cos{x}}\right)}\ \mathrm{d}x = -4\sinh\left(\frac{1}{\sqrt{2}}\righ... |
H: Convolution with Dirac delta
How to solve this expression:
$$\int_{-\infty}^{\infty} \left[ \delta(k-k_0)f(k)\right]*f(k)dk=?$$
Here $\delta$ represents the Dirac delta function and $*$ represents the convolution over the $k$ variable.
What I think:
$$\int_{-\infty}^{\infty} \left[ \delta(k-k_0)f(k)\right]*f(k)dk=\... |
H: convergences of improper irrational Integral
Finding Convergence or Divergence of $$\int^{\infty}_{0}\frac{1}{\sqrt{x^6+1}}dx$$
What i Try:
I am trying to prove $(x^2+1)\leq \sqrt{x^6+1}$ for all $x\geq 0$
$(x^2+1)^2\leq (x^6+1)\Longrightarrow x^4+2x^2+1\leq x^6+1$
Getting $x^4+2x^2\leq x^6$ but which is false.
H... |
H: $(X,Y)$ on a triangle
Let $(X,Y)$ a random variable uniformly distributed on the triangle $(0,0)$, $(0,1)$, $(1,0)$.
Find the density of $(X,Y)$.
$\rightarrow f_{X,Y}(x,y)=2$
Determine if $X$ and $Y$ are independent or not, and find $cov(X,Y)$.
$\rightarrow$ $X,Y$ are not independent, so $cov(X,Y)=\mathbb{E}[XY... |
H: Convergence of a rearrangement of conditionally convergent series
$\{a_n\}$ is a sequence of real numbers.$\space\sum_{n=1}^{\infty} a_{2n}$ and $\sum_{n=1}^{\infty} a_{2n-1}$ are both conditionally convergent. Is there such $\sum_{n=1}^{\infty} a_{n}$ that is divergent?
I understand that $\sum_{n=1}^{\infty} a_{2n... |
H: How can I calculate the self-convolution-like integration?
Assume $f(x)$ is a real, differentiable and continuous function over $R$, I want to calculate an integration :
$$
\lim_{a\to \infty} \int_0^a f^m(x) \frac{df(a-x)}{d(a-x)}dx.
$$
where $m$ is any positive integer. The conditions are: $f(x) = f(-x)$, $f(0) =... |
H: Line $CD$ meets the leg $\overline{AB}$ at $P$ and line $BF$ meets the leg $\overline{AC}$ at $Q$. Prove $|AP|=|AQ|$.
$\triangle ABC$ is a right-angled triangle at $A$. There are squares $AEDB$ and $ ACFG$ described from the outside on the legs $\overline{AB}$ and $\overline{AC}$, respectively. Line $CD$ meets the... |
H: Infinitely iterating the cosine function yields the Dottie number
For simplicity’s sake, Let’s define our function to be cos(x).
For any value of x, iterating this function will yield some constant, take a calculator and try it.
But quite surprisingly, I recognized that number to be the solution to the equation cos... |
H: Which of these functions are in the range of $T$?
Let $C[0,1]$ be the vector space of continuous functions on the interval $[0,1]$, and let $T: C[0,1] \to C[0,1]$ be the linear transformation that takes a function $f(x)$ to $\int_0^x f(t) \ dt$. Which of the following functions are in the range of $T$? (Select all ... |
H: Creating new metrics by combination of other metrics
Once we have some metrics, e.g., $d_1$ and $d_2$, we can perform some operations to create new metrics $d$ based on the former ones. For example:
$d=\dfrac{d_1}{1+d_1}$
$d= \text{min}(d_1, 1)$ (See equivalent metrics in wikipedia)
$d= d_1 + d_2$ (See question he... |
H: question about limes inferior/superior
Does $\lim\limits_{n \to \infty}\sup x_n \leq \lim\limits_{n\to \infty} \inf x_n$ implies
$\lim\limits_{n \to \infty}\sup x_n = \lim\limits_{n\to \infty} \inf x_n=\lim\limits_{n \to \infty} x_n$?
AI: Yes, as by definition:
$$
\inf_{k \geq n} x_k \leq x_n \leq \sup_{k \geq n} x... |
H: Matrix differential equation set to zero
I have a bijective continuous function $f$, which maps an $(n\times1)$ dimensional column vector $t=[t_1,...t_n]'$ to another $(n\times1)$ dimensional column vector $f(t)=[f_1(t),...f_n(t)]'$. I define the matrix derivative $\frac{df(t)}{dt}$ as follows:
$$\frac{df(t)}{dt}=\... |
H: Prove that $a$ is primitive root modulo $p^2$
I really need to answer this question quickly for my homework due tomorrow:
Let $a,p \in \Bbb N$, $p$ is prime, $a$ is a primitive root modulo $p$ that $p^2\nmid (a^{p-1}-1)$.
Prove that $a$ is primitive root modulo $p^2$.
My thoughts: I proved that $a^{\phi (p^2)} = a^... |
H: in which bases we have $\frac 1x=0.\bar x$?
In base ten we obtain the fact:$$1/3=0.\bar3=0.3333333333333\dots$$I want to know in which other bases thre is a number like that and what is that number.
AI: You're actually asking for which base $a$ there is a number $b$ such that:$$\frac 1b=\sum_{n=1}^\infty ba^{-n}$$ ... |
H: Question about the definition of the right limit
Let $\lim_{x\rightarrow 5^+}f(x)=7$.Then:
a) exists $a>5: \forall x \in [5,a): f(x)>6$
b) exists $a\in [4,7): \forall x \in (5,a]: f(x)>6$
c) exists $a\in[5,7]: \forall x \in [5,a): f(x)<8$
d) exists $a>7: \forall x \in [7,a): f(x)<6$
for the definition of limit: $\f... |
H: Sets related combinatorics
How many subsets of the set {1, 2, 3, 4, . . . , 30} have the property that the
sum of the elements of the subset is greater than 232?
I really have no idea how to move ahead with this problem.
AI: Gathering comments and including the punchline:
Note that $1+2+3+\dots+30 = \dfrac{30\times... |
H: b) If $ det A = 0 $ then $ Kerf $ contains a vector different from the zero vector
Let $f : R^n → R^n$ be a linear mapping and let $A = [f]$ be the matrix of f. Decide whether the
following statements are true or not:
a) If $ Kerf $ contains a vector different from the zero vector, then $ det A = 0 $.
b) If $ det A... |
H: Calculate the distance between 2 points on a sinusoidal curve
I'm fairly novice at maths so please excuse my imperfect description of the problem at hand.
I'm trying to find the formula that would calculate the length of the $y = \sin(\pi x)$ curve contained between to x points.
See this curve for reference, here w... |
H: Let $p$ be a prime number. For all $1 \le k,r < p$, there exists $n \in \mathbb N$ such that $nk \equiv r \pmod{p}$
I'm solving a problem in group theory. A key argument in my attempt is to assess if the following statement is true.
Let $p$ be a prime number. For all $1 \le k,r < p$, there exists $n \in \mathbb N$... |
H: $\left(\sum_i a_i\right)^2\ge (n-1)\sum_i a_i^2 + b\implies 2a_i a_j \ge b/(n-1) \quad (\forall i\ne j)$
Let $a_1, \dots, a_n(n\ge 2), b$ be real numbers, and
$$
\left(\sum_{i=1}^n a_i\right)^2\ge (n-1)\sum_{i=1}^n a_i^2 + b,
$$
Then we have
$$
2a_i a_j \ge b/(n-1) \quad (\forall i\ne j)
$$
$n=2$, clear.
Suppose ... |
H: Primitive root modulo $2p$
The question: Let $a,p \in \Bbb N$,$ \ $ $p$ is an odd prime, $a$ is a primitive root modulo $p$. prove that:
if $a$ is odd, $a$ is primitive root modulo $2p$.
if $a$ is even, $a+p$ is primitive root modulo $2p$.
Thank you
AI: $\textbf{First one:}$
Since,$\phi(2p)=p-1$.
So,First we prov... |
H: Is this notation for a domain standard?
I guess they are defining domain in a very general way. So you can say a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is taking values (as a function) in $\mathbb{R}^{\mathbb{R}}$.
Is this a common thing? I read a lot of mathematics (applied generally) and I have not seen ... |
H: Area of $\{(x,y)\in \mathbb{R}^2:\ 1\leq x y\leq 2, x \leq y \leq 2x\}$
Trying to evaluate the area spanned by the set $\{(x,y)\in \mathbb{R}^2:\ 1\leq x y\leq 2, x \leq y \leq 2x\}$, I use the following reasoning:
\begin{equation}
1\leq x y \leq 2\ \Longrightarrow\ \frac{1}{y}\leq x\leq \frac{2}{y}
\end{equat... |
H: Prove that $\int |\nabla u||\nabla v|+|u||v|\leq |u|_{H^1}|v|_{H^1}$
Prove that $\int |\nabla u||\nabla v|+|u||v|\leq |u|_{H^1}|v|_{H^1}$. Here $H^1$ is Sobolev Space. Then my attempt is first (im not sure) apply Inequality'Holder so $
\int |\nabla u||\nabla v|+|u||v|\leq |\nabla u|_{L^2}|\nabla v|_{L^2}+| u|_{L^2}... |
H: A basic property of convex sequences
A sequence $\{a_n:n\in\mathbb{Z}_+\}\subset\{(0,\infty)$ is convex if
$$
a_{n-1}+a_{n+1}-2a_n\geq0,\quad n\geq1
$$
This is equivalent to saying that $\{a_n-a_{n+1}:n\in\mathbb{Z}_+\}$ is a monotone non increasing sequence.
Convex sequences that converge to $0$ (i.e. $\lim_na_n=0... |
H: Littlewood's Second Principle : Simple function is continuous on restricted domain
I am stuck on understanding one point Proposition 11 (Real Analysis by Royden and Fitzpatrick, 4th edition, Page-66)
Proposition 11: Let$ f$ be a simple function defined on a set E of finite measure. Then for each $ε>0$, there is a ... |
H: Probability of getting 1 Head and 2 Tails with a Fair Coin
If a fair coin is tossed three times what is the probability that heads shows once and tails shows twice?
I thought the answer would be $1/4 + 1/2$ because the probability of getting heads is $1/2$ and the probability of getting tails twice is $1/2 * 1/2 = ... |
H: Derivative of a multivariable, piecewise function
I was asked to show that the function $h(t)=f(g(t))$ is differentiable at $t=0$, where
$f(x,y)=\begin{cases}
\frac{x^2y}{x^2+y^2} & (x,y)\neq 0 \\ 0 & (x,y)=0 \end{cases}$
and $\vec{g}(t)=(kt,pt)$ for constants $k$ and $p$.
I made a diagram and said "$h$ is a functi... |
H: yes/ No : Is there exist disjoint subsets of $A$ and $B$ of $\mathbb{R}$ such that $m^*( A \cup B) = m^*(A) + m^*(B) ?$
Is there exist disjoint subsets of $A$ and $B$ of $\mathbb{R}$ such that $m^*( A \cup B) = m^*(A) + m^*(B) ?$
My attempt : If I take $A= [-1,1]$ and $B= [-2,2]$
Here $m^*([-1,1]) = 2, m^*([-2,2]... |
H: How can I evaluate: $\int_a^b \frac{1}{\sqrt{1-\cos \theta}}d\theta$?
I want to evaluate:
$$\int_a^b \frac{1}{\sqrt{1-\cos \theta}}d\theta$$
I get stuck if I try a $u$ substitution and I have tried changing the expression using trigonometric identities but still no luck.
How would I go about evaluating this ?
AI: U... |
H: Is it a necessary condition for an even function to have a local extremum (for $f(x)=k,$ derivative${}=0$) at $x=0$
Let $f(x)$ be an even function ($f(-x)=f(x)$) if $f(x)$ is continuous and differentiable at $x = 0$ will it be necessary for it to have a local extremum? Or more generally, have it's derivative $=0$ a... |
H: Is This Proof for "If $\sup A < \sup B$, show that there exists an element $b\in B$ that is an upper bound for $A$" correct?
This question is from Understanding Analysis (Stephan Abbot) Exercise $1.3.9$. The Question is If $\sup A < \sup B$, show that there exists an element $b \in B$ that is an upper bound for $A... |
H: $ Av=0 $ is $ r(A)
We now that $ n \times m $ matrix $ A $ and the vector $ v \in R^m $, $ v \neq 0$ $Av=0$. Is $ r(A)<m$ or $ r(A)<n$?
I think since $Av=0 $ has infinitely many solutions then $r(A)<m$ But I am not sure
Update: The question is active and there is no accepted answer
AI: $r(A)$ is the dimension of th... |
H: Let $f$ be a measurable function on $[0,1]$ such that $\int_0^1 f(x)\,dx<\infty$ Let $U_1,\dots$ be a sequence of i.i.d Uniform $(0,1)$
I have problems with this sequence, don't know how to start. Could anyone give me a hint, please?
Let $f$ be a measurable function on $[0,1]$ such that $\int_0^1 f(x) \, dx < \inft... |
H: Analogue of $(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2$ for vectors
The Brahmagupta–Fibonacci identity $(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2$ allows us to write a product of squares as a sum of squares. Is there an analogue of this identity when $a, b, c, d$ are vectors in $\mathbb{R}^n$ and ... |
H: For every $k \in \Bbb Z$ there is $0 \le x \le p-1$ such as $x^3\equiv k \pmod {p}$
This question is looking like an easy one but I have been trying to solve it for the last couple days and I haven't been able to prove it - so I need some help.
The question:
Let $p$ be a prime number, $p\equiv 5 \pmod{6}$ prove tha... |
H: Proof verification: $g$ is continuous iff $g^{-1}(B)$ is open whenever $B \subseteq \mathbb{R}$ is an open set.
I am trying to prove:
Let $g$ be defined on all of $\mathbb{R}$. If $B$ is a subset of $\mathbb{R}$, define the set $g^{-1}(B)$ by
\begin{equation*}
g^{-1}(B) = \{x \in \mathbb{R}: g(x) \in B\}
\end{... |
H: Recursive equation answer
Consider the following recursive equations:
$$3A_n = 2A_{n-1}+B_{n-1}$$
$$3B_n = A_{n-1}+2B_{n-1}$$
Let $A_0=2$ and $B_0=-1$. I know there are supposed to be different ways to solve this but I was thinking about solving one equation and replace it in the other to find the final answer I tr... |
H: Under what condition is a complete bipartite graph $K_{m,n}$ a regular graph
This is a quick question and I know all about these graphs what I am supposed to know. But I am unable to explain myself in words.
Like I know for regular graph the vertex must have same degree and bipartite graph is a complete bipartite i... |
H: Let $G$ be a finite group. Then the number of elements of prime order $p$ is divisible by $p − 1$
I'm trying to prove the following proposition from textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell.
Let $G$ be a finite group. Then the number of element... |
H: Show that if $A$ is a $m \times n$ matrix with $AA^T = 2J + 5I$ then $n \geq m$ ($J$ is matrix of ones)
I am studying for a qualifying exam and I got stuck on this question:
Show that if $A$ is a $m \times n$ matrix with $AA^T = 2J + 5I$ then $n \geq m$.
Here $J$ is the matrix of all $1$s. The hint for the problem ... |
H: Represent ${f(x) = \arctan(x) + x\ln(4+x^2)-x(1+\ln(4))}$ by a Maclaurin series.
I have the following problem:
Represent the function:
$${f(x) = \arctan(x) + x\ln(4+x^2)-x(1+\ln(4))}$$ by a Maclaurin series.
I do not know how to represent this part: ${x\ln(4+x^2)}$
Could you please help me?
AI: Okay, so notice that... |
H: Manipulating $C=\frac{C_a-C_b}{a-b}+\frac{aC_b-bC_a}{(a-b)m}$ into $C=\frac{1}{m}\Bigl(\frac{m-b}{a-b}C_a+\frac{a-m}{a-b}C_b\Bigr)$
I need to know the step to get this:
$$C=\frac{1}{m}\Bigl(\frac{m-b}{a-b}C_a+\frac{a-m}{a-b}C_b\Bigr)$$
from:
$$C=\frac{C_a-C_b}{a-b}+\frac{aC_b-bC_a}{(a-b)m}$$
Thank you!
AI: $$C=\fra... |
H: Compute $\pi_2(S^2 \vee S^2)$
As far as I know, there are two ways to calculate higher homotopy groups. One way is if we have a fibration then we get a long exact sequence in homotopy. The other is if we know a space is $(n-1)$-connected, then by Hurewicz Theorem, $\pi_n \cong H_n$.
I know $H_2(S^2 \vee S^2)=\mathb... |
H: How to find the inverse of the complex function $ f(z)=\dfrac{z}{\sqrt{1+|z|^{2}}}$
I'm trying to calculate the inverse function of :
$$ f(z)=\dfrac{z}{\sqrt{1+|z|^{2}}}$$ where $z\in$ $\mathbb{C}$.
Can someone help me?
AI: If you mean the inverse, $1-|f|^2=\frac{1}{1+|z|^2}\implies z=f\sqrt{1+|z|^2}=\frac{f}{\sqr... |
H: Continuity of this Piecewise function $f:\mathbb{R}^2\to \mathbb{R}$
I have shown that $f:\mathbb{R}^2\to\mathbb{R}$ given by $f(0,0) = 0$ and $\displaystyle f(x,y)=\frac{x|y|}{\sqrt{x^2+y^2}}$ if $(x,y)\ne (0,0)$ isn't differentiable at $(0,0)$, now I'm trying to show whether it is continuous or not.
My attempt: I... |
H: Computing singular points of curves, exercise 5.1 (Hartshorne)
I am just trying to cross check my answer as it slightly differs from https://math.berkeley.edu/~reb/courses/256A/1.5.pdf to be sure of any mistake I am making. Here $k$ is an algebraically closed field with ${\mathrm{char}}~k \neq 2$.
5.1(a) The curve ... |
H: How prove the finite induction
Theorem (Finite induction)
Let be P$(x)$ a property. So we suppose that
P$(0)$,
P$(n)\rightarrow$P$\big(S(n)\big)$, for all $n<k$
So P(n) is true for all $n<k$.
Unfortunately I can't formally prove the theorem. So could someone help me, please?
AI: Let $\pi:=\{h\in k:\mathbf{P}(h)\}... |
H: Matrix with spectral radius 1 that converges
Let $\mathbf{A}$ be a matrix of size $n \times m$ with $\sum_{j = 1}^m A_{ij} = 1$ and $\mathbf{E}$ be a matrix of size $n \times m$ with $\sum_{i = 1}^n E_{ij} = 1$ and $\forall i,j \ \ \ \ 1 \geq E_{ij} \geq 0, \ \ \ 1 \geq A_{ij} \geq 0$. Consider the following iter... |
H: Three-Body Problem - how to find the Figure-eight solution?
Suppose the coordinates of the Earth and the Moon are fixed and let $(u,v)$ be the coordinates of the satellite.
I'm looking for the numerical solution of the three-body problem:
$$u'' = 2v + u - \frac{c_1(u+c_2)}{((u+c_2)^2 + v^2)^\frac{1}{2}} - \frac{c_2... |
H: If $X$ and $Y$ are two ordered sets, how many orderings of $X \times Y$ exist that preserve the orderings of $X$ and $Y$?
Suppose $X$ and $Y$ are two totally ordered sets with $|X| = n_X$ and $|Y|=n_Y$. We'll say an ordering ($\preceq$) of $X \times Y$ preserves the orderings of $X$ and $Y$ if for any elements $x_1... |
H: Three series theorem - convergence of $\sum X_n$ with $f_n(x) = \frac{1}{\pi} \frac{n}{1+(nx^2)}$
Suppose X_s are independent r.v.s with densities:
$$
f_n(x) = \frac{1}{\pi} \frac{n}{1+(nx^2)}
$$
Does the following series converge with probability 1:
$$
\sum_{n=0}^\infty X_n
$$
Now, I've calculated the CDF to be
$$... |
H: If $M$ has only two definable subsets, must it have a transitive automorphism group?
I was told that a sufficient condition for a structure $M$ to have only the empty set and $M$ itself as parameter-free definable subsets, is for $M$ to have a transitive automorphism group. Is the converse true?
AI: No, it's not - ... |
H: find all $n$ such that $\varphi(\sigma(2^n)) = 2^n$
Problem: Find all positive integers $n$ such that $\varphi(\sigma(2^n)) = 2^n$, where $\varphi(n)$ is Euler's totient function and $\sigma(n)$ is the sum of all divisors of $n$.
I know that $\sigma(2^n) = 1+2+2^2+2^3+\dots+2^n = 2^{n+1}-1$, so we only need to find... |
H: Evaluate $\int_0^{\frac{\pi}{4}} \left( \frac{\sin^2{(5x)}}{\sin^2{x}} -\frac{\cos^2{(5x)}}{\cos^2{x}} \right)\mathop{dx}$
Evaluate $$\int_0^{\frac{\pi}{4}} \left( \frac{\sin^2{(5x)}}{\sin^2{x}} -\frac{\cos^2{(5x)}}{\cos^2{x}} \right)\mathop{dx}$$
I tried substitutions like $u=\frac{\pi}{4}-x$, and trig identities ... |
H: Soft Question Regarding the Divergence of $\sum\limits_{k=1}^{\infty}\frac{(-1)^k k}{3k+2}$
Determine whether $\sum\limits_{k=1}^{\infty}\frac{(-1)^k k}{3k+2}$ converges or diverges.
Consider the function $f(x)=\frac{x}{3x+2}$ that generates the unsigned terms of our series.
Taking the derivative of $f$ we have $f'... |
H: Why is $\mathbb{R}^2\setminus \mathbb{Q}^2$ not a topological manifold?
I would like to understand why $\mathbb{R}^2\setminus \mathbb{Q}^2$ endowed with the subspace topology is not a topological manifold. It seems to me it is Hausdorff and second countable. So I am wondering. Why is it not locally Euclidian?
AI: A... |
H: Misconception about basic mixed fraction
We know,
$$3 \frac12=3+\frac12$$
Then, if we have
$$3 \frac12 ÷ 3 \frac12$$
It means:
$a)\,\frac72 ÷ \frac72=1$ or
$b)\, 3+\frac12 ÷ 3+\frac12 =\frac{7}2$ or
$c)\, 3 + 1 ÷ 2 ÷ 3 + 1 ÷ 2=\frac{11}3$
Which one is true?
Sorry, maybe it appears on another question, but i want to... |
H: Does adding a constant or reducing all the numbers by 5% change the z score?
I have a set of numbers.
I have two cases:
If I add a constant to them, does the mean, standard deviation and z score change?
What I think: Mean changes, std deviation doesn't change and z score doesn't change.
If I reduce all the numbe... |
H: Central limit theorem with a biased dice
We want to know the prbabiltiy $p \in (0,1)$ that a biased die rolls a $6$. We roll the die $n$ independent times
$X_i=\begin{cases}
1 & \textrm{if the i-th roll is a 6} \\
0 & \textrm{else} \\
\end{cases}
$
calculate with the clt the approximate min. number of rolls s... |
H: calculate expected value and variance of $Y := 1 − X$
I'm struggling with the following exercise:
Given the random variable $X$ with expectation value $\mu$ and variance $\sigma^2$:
What is the expectation value and variance of $Y := 1 − X$
Isn't it just $E(X) = 1-\mu$?
Thanks in advance
AI: Comment: And the same v... |
H: Prove that every collection of partitions $T$, there exists $\inf{T}$ and $\sup{T}$
I am self-studying Hrbacek and Jech's Introduction to Set Theory (3rd edition), and I want to know if the following solution to problem 5.10 (c) is correct. Unfortunately the book contains no answers and the only solution manual for... |
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