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H: How many solutions does the equation have?
(a) Sketch the graph of y=(x−p)^2 × (x−q) where p < q.
(b) How many solutions does the equation (x−p)^2 × (x−q)=k have when k > 0?
I have understood part(a) of the question and have got it correct. But I do not understand part(b) correctly. I thought the answer would be ze... |
H: Limit $\lim_{x\to\infty} x(\arctan(a^2x)-\arctan(ax))$
I have a limit $\lim_{x\to\infty} x(\arctan(a^2x)-\arctan(ax))$ and I know the solution $\frac{a-1}{a^2}$, but I dont have any Idea, how to calculate this limit or at least how to start. Any idea?
AI: I assume that we are near $+\infty$.
If $ a=0 $, the limit i... |
H: Calculate the curvature $k(t)$, for the curve $r(t)=\langle 1t^{-1},-5,3t \rangle$
I have that $k(t)=\frac{\mid r'(t)\times r''(t) \mid}{\mid r'(t)\mid^3}$.
So first, $r'(t)=\langle -\frac{1}{t^2},0,3 \rangle$.
$r''(t)=\langle \frac{2}{t^3},0,0 \rangle$.
$\mid r'(t)\mid = \sqrt{t^{-4}+9}$.
Then I did $r'(t) \times... |
H: Limit of $\frac{1}{r}\ln\left(1+r\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)\right)$
Let be $n \in \mathbb{N}$ arbitrary but fixed, $\sum\limits_{i=1}^n p_i =1$ and $\forall ~ 1\leq i \leq n$ we assume: $x_i \in \mathbb{R}$.
What is the limit of
$\lim\limits_{r\to 0}~\frac{1}{r}\ln\left(1+r\sum\limits_{i=1}^n p_... |
H: A type of set used in convergence in measure theory
This is not a specific problem, but a general question. Often when we're showing convergence of functions (particularly pointwise) or even of sets in certain cases, a set of the following form appears:
$$
\bigcap_{N \in \mathbb{N}} \bigcup_{n \geq N} E_{n}
$$
Of... |
H: Find a ring homomorphism $\theta$ s.t. Ker $\theta = \mathbb{Z}_6 \times \{[0]\}$.
Find a ring homomorphism $\theta: \mathbb{Z}_6 \times \mathbb{Z}_{14} \to \mathbb{Z}_6 \times \mathbb{Z}_{14}$ for which Ker $\theta = \mathbb{Z}_6 \times \{[0]\}$.
Attempt: I know that Ker $\theta = \{(x, y) \in \mathbb{Z}_6 \times... |
H: How to quantify asymptotic growth?
Specifically, my research question is to find operator $A: (\mathbb{R}^+\rightarrow\mathbb{R}^+)\rightarrow\mathbb{S}$, where $\mathbb{S}$ is some totally ordered set, such that for $f, g: \mathbb{R}^+\rightarrow\mathbb{R}^+$ A(f) > A(g) iff $\lim_{x \to \infty} \frac{f(x)}{g(x)} ... |
H: Grothendieck group "commutes" with direct sum
The Grothendieck completion group of a commutative monoid $M$ is the unique (up to isomorphism) pair $\langle \mathcal{G}(M), i_M\rangle$, where $\mathcal{G}(M)$ is an abelian group and $i_M\colon M\to\mathcal{G}(M)$ is a monoid homomorphism, satisfying the universal pr... |
H: I'm stuck trying to factor $x^2-4$ to $(x-2)(x+2)$
I am trying to understand each step in order to get from $x^2-4$ to $(x-2)(x+2)$
I start from here and got this far...
$x^2-4 =$
$x*x-4 =$
$x*x+x-x-4 =$
$x*x+x-2+2-x-4 =$
$x*x+x-2+2-(x+4) =$
After this I try
$x(x-2)+2-(x+4) =$
and this clearly does not even equal... |
H: 'Locally' Convex Function
I have a continously differentiable function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ which I am trying to prove is globally convex. Computing the Hessian directly is very difficult as it is a somewhat complicated function of a matrix, other methods of proving global convexity have proved i... |
H: Let $f:\Omega\to\textbf{R}^{m}$ be a function. Then $f$ is measurable if and only if $f^{-1}(B)$ is measurable for every open box $B$.
Let $\Omega$ be a measurable subset of $\textbf{R}^{n}$, and let $f:\Omega\to\textbf{R}^{m}$ be a function. Then $f$ is measurable if and only if $f^{-1}(B)$ is measurable for every... |
H: Closed form of $\sum _{i=1}^n\:\frac{\left(m-i\right)!}{\left(n-i\right)!}$
I am interested in a closed form of the following sum:
$$
S_n := \sum _{i=1}^n\:\frac{\left(m-i\right)!}{\left(n-i\right)!}\;\; n,m \in \mathbb{N},\ n < m$$
Amongst other strategies I have also tried to compress the sum in order to gain som... |
H: SIR Model Specifics
I read on Wikipedia (under "Compartmental models in epidemiology") that the differential equations for the SIR Model was the following,
$$S'(t)=-\frac{\beta}{N}I(t)S(t)$$
$$I'(t)=\frac{\beta}{N}I(t)S(t)-\gamma I(t)$$
$$R'(t)=\gamma I(t)$$
It never states the range of $\gamma$ and $\beta$. Howeve... |
H: Finding a fixed polynomial under the multiplicative inversion automorphism
Can anyone find a polynomial $f ∈ ℚ\left(X+\frac{1}{1-X} + \frac{X-1}{X}\right) ⊆ ℚ(X)$ that is fixed under the automorphism $(X ↦ \frac{1}{X})$? $f = X+\frac{1}{X}$ would be nice, but I don't know how to check if it's in the given (sub)fiel... |
H: Need help understanding two steps in the proof that limit point compactness implies sequential compactness
So I want to learn the proof that a compact metric space $(X,d)$ is also sequentially compact. The proof goes as follows:
(X,d) is compact, so it is also limit point compact. Let $\{x_k\}_{k=1}^\infty$ be a se... |
H: Does $(\frac{n}{4})^{\frac{n}{4}}$ have a higher asymptotic growth than $4^{n^{4}}$
I'm trying to determine how these 3 functions should be ordered in terms of asymptotic growth:
$$f(n) = \left(\frac{n}{4}\right)^{\frac{n}{4}}$$
$$g(n) = n^{\frac{n}{4}}$$
$$h(n) = 4^{n^{4}}$$
$f(n)$ seems to be somewhat similar to ... |
H: Inequality on integrals of $L^1$ functions
Let $\lambda \geq 0$ and $(X,d,\mu)$ be a $\sigma-$finite measure space. Then for $f, g \in L^1(X,\mu)$
$$ \left| \int_X (|f|-\lambda)^{+} d\mu - \int_X (|g|-\lambda)^{+} d\mu \right| \leq \int_X ||f|-|g|| d\mu$$
holds (where $(x)^{+} = \text{max}(x,0)$).
I tried dividing... |
H: Find inverse matrix
I have a matrix:
$$A = \begin{pmatrix}
1 & 1 & 1 & \cdots & 1\\
0 & 1 & 1 & \cdots & 1\\
0 & 0 & 1 & \cdots & 1\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & 1
\end{pmatrix}$$
I need to find $A^{-1}$. How could I do that, knowing that matrix A has ... |
H: Critical points of a multivariable function
Critical points of Z(x,y)=x^3+3xy^2-15x-12y
enter image description here
I get confuse on how you get x^4-5x^2+4=0
AI: They only substituted $y=\frac{2}{x}$ in the equation above and then they multiplied it by $x^2$ to get better-looking equation so it can be solved bette... |
H: Find the arclength curve of $r(t)=i+3t^2j+t^3k$ for $0\leq t\leq \sqrt{12}$
I asked a question similar to this one, but I'm still confused on how to integrate this.
I have $r'(t)=\langle 0,6t,3t^2\rangle$. and so this gives you the integral from $0$ to $\sqrt{12}$ of $\sqrt{36t^2+9t^4}dt$. Step by step would be he... |
H: Question regarding surjectivity of induced homormophism in an old version of Hatcher's proof of Prop. 4.13
So I am currently trying to understand the given proof of Hatcher's proof of proposition 4.13.
It's this particular part (in the middle of the screenshot) I don't understand:
The extended $f$ still induces a... |
H: Are perpendicular vectors always in different subspaces?
So I understand when two subspaces are considered perpendicular and what it means for vectors to be perpendicular/orthogonal.
The question I have is, if two vectors are perpendicular, do they always have to exist in orthogonal subspaces such as the nullspace ... |
H: Establish the inequality for an analytic function.
So, I am self studying analysis. I came across this question. It asks us to establish the following inequality:
$\int_{-1}^{1} |f(x)|^2 dx \leq \frac{1}{2} \int_{0}^{2\pi} |f(e^{it})|^2 dt, $
for $f(z)$ analytic on the open unit disk and continuous on the closed u... |
H: Homeomorphism between $\mathbb{R}^2$ and $\mathbb{R}^2-B(0,1)$
Are $\mathbb{R}^2$ and $\mathbb{R}^2-B(0,1)$ homeomorphic? I'm trying to find a homeomorphism between the half of a hyperboloid of one sheet, $$C=\{(x,y,z)\in \mathbb{R}^3:x^2+y^2-z^2=1,\textrm{ }z\geq 0\},$$ and the plane $\mathbb{R}^2$. I've already s... |
H: Shilov Chapter 4 Problem 16
I am working on the captioned problem which is reproduced below.
And the hint for this problem is the following:
But I have no idea of how to use Chapter 3 Prob 12 for this problem. That problem is reproduced here:
I have no problem in finding out that three equations for unknown elem... |
H: Proving limits in terms of epsilon delta questions
$ f(x)= {x^3}-2x+1. $
We want to show that $ \lim_{x\to 2} ({x^3}-2x+1)= 5 $.
So here,
$ \lvert f(x)-f(2)\rvert = \lvert {x^3}-2x+1-5\rvert$ = $ \lvert {x^3}-2x-4\rvert $.
$ \lvert ({x^3}-2x)+(-4)\rvert \leq \lvert {x^3}-2x\rvert + \lvert -4\rvert = \lvert {x^3}... |
H: Calculate $ \int_0^{2\pi} \ln(2-2\cos(t)) \ln(2-2\cos(t+\theta)) dt$
I'm trying to evaluate
$$ \int_0^{2\pi} \ln(2-2\cos(t)) \ln(2-2\cos(t+\theta)) dt$$
but I'm not sure the best way to proceed. I've been trying to factor the inner terms in to rational functions of $e^{it}$ and use logarithm identities to split t... |
H: Minimum of $\sqrt {{x^2} + \frac{1}{{{x^2}}}} + \sqrt {{y^2} + \frac{1}{{{y^2}}}} + \sqrt {{z^2} + \frac{1}{{{z^2}}}} $?
I have been ponder around a difficult Vietnamese University entrance exam of the year 2003
that is to find the minimum of $\sqrt {{x^2} + \frac{1}{{{x^2}}}} + \sqrt {{y^2} + \frac{1}{{{y^2}}}} ... |
H: Average value of the orders of all elliptic curves over the finite field of p-elements
Is true that the average value of the orders of all elliptic curves over $\mathbb F_p$ is $p+1$?
More precisely, fix a prime $p$ and let $\mathbb F_p$ be the field of $p$ elements. Consider the set $S=\{(a,b)\in\mathbb F_p\times\... |
H: Example of non compact sets whose union and intersection is compact
Give an example of two non compact sets $A$ and $B$ such that $A \cup B$ is compact and $A \cap B$ is compact.
My attempt:
Let $A=\{1/n:n=1,2,...\}$ and $B=\{0\} \cup (1,2]$.
Then $A \cup B$ is closed and bounded so it is compact and $A \cap B$ is... |
H: The probability of choosing a 5 element subset from the set {1,2,...,20}, with 1 element from{1,4,6,8,9} and 1 from{11,13,17,19} but 0 from {2,3,5,7}
I need to make a 5 element subset using at least one of {11, 13, 17, 19} and at least one of {1, 4, 6, 8, 9} from the set of the first twenty integers but I cannot us... |
H: Find polynomial of degree n+1 for n+1 data points.
Generally, if we have n+1 data points, there is exactly one polynomial of degree at most n going through all the data points. What can we tell about existence of polynomial of degree n+1? How can we find polynomial of degree n+1 for a table:
x: 0 1 2 3
y: 1 0 -... |
H: Prove that $L \geq M$.
In the above proof, it is taken that for both of the sequences $\exists N$ s.t $|a_n - L|<\epsilon $ and $|b_n - M|<\epsilon$. Is it correct to take the same $N$ for both sequences?
After that they have assumed the inequality which is true for $n\geq N_0$ is also true for $n\geq N$. Is that ... |
H: It's possible to construct a disjoint sequence of open balls with center elements of a sequence in a metric space
Let $(X,d)$ a complete metric space, if $(x_n)_{n \in \mathbb{N}}$ is a non constant sequence such that $x_n \to x$. I try to construct a sequence of disjoint open balls with center in $x_n$ for all $n... |
H: Show that $ d(x, A) = 0 $ if, and only if, $ x \in \overline{A} $
Let $ X $ be a metric space with distance $d$ and let $ A \subset X $ not empty.
(a) Show that $ d(x, A) = 0 $ if, and only if, $ x \in \overline{A} $
(b) Show that if $ A $ is compact, $ d (x, A) = d(x, a) $ for some $ a \in A $
(c) Define the $ \ep... |
H: How to evaluate $\int_{|z|=2} \frac{1}{z^2+z+1} dz$
I found the poles of $\displaystyle f(z) = \frac{1}{z^2+z+1}$ given by the solutions for $z^2 + z + 1$, $ \displaystyle z_1 = -\frac{1}{2}+i\frac{\sqrt{3}}{2}$ and $\displaystyle z_2 = -\frac{1}{2}-i\frac{\sqrt{3}}{2}$.
But when I use the Residue's Theorem, I get ... |
H: How to solve this implicit equation? (use trial and error)
I want to solve this implicit equation and find $f$ When $Re$ is constant: $$\frac{1}{\sqrt{f}}=2\log({Re.\sqrt{f})}- 0.8$$
I tried to make the equation simple By using: $\sqrt{f}=t>0,Re=a$:
$$\frac{1}{t}=2\log({at})-0.8$$
$$0.8t=2t\log(at)-1$$
I can not ... |
H: Prove that $\lim_{y \to 0^+}\frac{\sin(yt)}{y}$ Uniformly converges to $t, t \in [0,R], R > 0$
I need to show that
$$
\lim_{y \to 0^+}\frac{\sin(yt)}{y}
$$
Uniformly converges to $t \in [0,R], R > 0$
The answers show that:
$$
\forall t \in [0,R]: 0 < yt < \delta \Rightarrow |t-\frac{\sin(yt)}{yt}| < \varepsilon
$$... |
H: Surface integral of piecewise volume boundary?
How should I go about solving the surface integral
$$\iint_{\partial V} 2(x^2+y^2) \, dS,$$ where $V$ is the region bounded by the paraboloid $$z=\frac12-x^2-y^2$$ and the cone $$z^2=x^2+y^2?$$
I have done surface integrals before, but I am unsure how to proceed with f... |
H: Implicit differentiation of a function with 3 variables
The problem says: If the equation $x^2 +y^2 +z^2 = G(ax+by +cz)$ defines $z=f(x,y)$, $f$ and $G$ being differentiable, $a$, $b$ and $c$ constants, find $\partial z/\partial x$.
Is this correct?:
$$
\frac{\partial z}{\partial x} = -\frac{Gx}{Gz} = - \frac{2ax}{... |
H: Weil divisors associated to Cartier divisors
Let $X=\{x_3^2=x_1^2+x_2^2\}\in \mathbb{P}^3$, let $L_1=Z(x_2,x_1+x_3)$, and $L_2=Z(x_2,x_1-x_3)$.
I don't quite understand how to get that $\operatorname{div}(x_2)$ is associated to $[L_1]+[L_2]$ and $\operatorname{div}(x_1+x_3)$ is associated to $2[L_1]$.
I think ... |
H: Is the function with prescribed Fourier coefficients bounded a.e.?
Consider a function $F\in L^2(0,1)$ whose Fourier coefficients are:
\begin{align*}
\widehat{F}(n)=
\begin{cases}\frac{1}{n},\quad &n=1,2,3,\dots,\\
0, \quad &n\le 0.
\end{cases}
\end{align*}
$\textbf{Question}:$ Is $F$ a bounded function on $(0,1)$... |
H: a countable open interval cover of the irrationals must also cover the rationals?
let $A:=[0,1]\setminus \Bbb Q$. Is it necessary for a countable open interval cover $(I_k)_{k=1}^\infty$ of $A$ also covers $[0,1]\cap \Bbb Q$, therefore, $A$?
I think this is true, due to the fact that $\Bbb Q$ and $\Bbb R \setminus ... |
H: Deriving a polynomial with certain properties...
What is an algebraic expression for a polynomial, $q$, with the following properties:
1. $q$ has real coefficients.
2. The only real zeros of $q$ are $-2$ with multiplicity $3$ and $1/4$ with multiplicity 2.
3. A complex zero of $q$ is $i$ with multiplicity $1$.
... |
H: An identity of Arithmetic Functions
Problem: Show that for all positive integers $n$,
$$ \sum_{a=1, (a,n)=1}^{n} (a-1, n) = d(n)\phi(n)$$
where $(a, b)$ stands for $\text{gcd}(a, b)$ and $d, \phi$ are the divisor and Euler's totient function, i.e., number of numbers co-prime to n and less than n = $\phi(n)$.
I fin... |
H: Prove or disprove that the ellipse of largest area (centered at origin) inscribed in $y=\pm e^{-x^2}$ has the equation $x^2+y^2=\frac12(1+\log2)$.
I can show that $x^2+y^2=\frac12(1+\log2)$ is the equation of the circle of largest area inscribed in $y=\pm e^{-x^2}$:
The minimum distance $r$ (which will be the radi... |
H: Prove $\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$
I have seen this equation:
$$\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$$
As you can see the numerator switched with the denominator and I wonder how. I know power laws and yet I can ... |
H: How to find Pythagorean triples with one side only?
I was using the formula to find triples, but I can only find two of them.
the pythagorean triple associate with 102 are
102 136 170,
102 280 298,
102 864 870,
102 2600 2602,
$a = m^2 - n^2$
, $b = 2mn$
, $c = m^2 + n^2$
let $a = 102
= (m+n)(m-n)$
since m and n are... |
H: Elements of the subgroup for Galois group over $\mathbb{Q}$ (Cyclotomic extension)
Well... this question looks like silly though, I have a curious about the below.
Let $K= \mathbb{Q}(\omega)$ for $\omega=e^{2\pi i \over n}$
My book said $G(K/\mathbb{Q}) = \{\sigma_i \vert \sigma_i : \omega \to \omega^{i}$ for $(i,... |
H: Calculate the integrals for $f(x) = x^2 + 1$
Let $ f \colon \ \mathbb{R} \to \mathbb{R}$ so that $ f(x) = x^2 + 1$. Then compute $$15\int_0^1 f^7(x) \,dx - 14\int_0^1 f^6(x) \,dx$$
There is the direct calculation option for me: using Newton binomial, exploiting $(x^2+1)^7$ and $(x^2+1)^6$ and then integral all term... |
H: Prove that $\operatorname{rank}(L_{A})=\operatorname{rank}(L_{A^\top})$
I want to prove $\operatorname{rank}(L_{A})=\operatorname{rank}(L_{A^\top})$ and $A\in M_{m\times n}(\Bbb F)$ by using two facts. 1. If $W$ is a subspace of $V$ which is a finite dimensional vector space, then $\dim(W)+\dim(W^{\circ})=\dim(V)$.... |
H: How to show the following cubic graph is Hamiltonian?
Suppose we have a complete graph $K_{2n}$, where $n$ is an integer, and we number vertices counterclockwise as $1,2,\cdots,2n.$ Then we give weight to the edge connecting $u,v$ as $|u-v|$.
Now, delete all the edges with weight less than $n-1$, that is, keep all ... |
H: Question on $L^p$ convergence.
Suppose $q \in [1, \infty)$, I'm trying to find a sequence of functions $g_n \in L^p$ (for all $p \in [1,\infty]$) such that $g_n$ converges in $L^p$ when $p \in [1, q]$ but doesn't converge into $L^p$ when $p \in (q, \infty]$.
I tried the sequence of functions $g_n(x) = \displaystyle... |
H: Question about Vector Calculus
Mine is probably a very trivial question. I didn't get good training on vector calculus and I am not able to find the correct materials to learn to understand the following equation.
Given
$F = \int \frac{D^2}{8\pi\epsilon} dV$
where $\vec{D}$ is a vector, the book gives:
$\delta F=\... |
H: is a matrix $A*A^{t}$ or $A^{t}* A $ Symmetric?
if $\mathbb{K}$ is a field and $A\in M_{m\times n}(K)$ proof or give a counterexample that $A\cdot A^t $ and $A^t\cdot A $ are Symmetric matrix
AI: Yeah they are symmetric
If A is an m × n matrix and A^t is its transpose, then the result of matrix multiplication with... |
H: Solving an equation having trignometric functions on RHS and an number in LHS
If $$\cos\frac\pi{2n}+\sin\frac\pi{2n}=\frac{\sqrt n}2, n\in\Bbb N,$$ find $n$.
I haven't been able to get an answer to this problem but I know that since the range of $$\sin x+\cos x$$ is $1,√2$
So the value of $n$ must lie between $4,... |
H: Vector spaces uniqueness proof: If $z_1$ and $z_2$ are two such elements, then $z_1 + z_2 = z_1$ and $z_1 + z_2 = z_2$; thus, $z_1 = z_2$.
I am currently studying Introduction to Hilbert Spaces with Applications, by Debnath and Mikusinski. Chapter 1.2 Vector Spaces says the following:
(a) $x + y = y + x$;
(b) $(x ... |
H: Proof of an inequality (maybe using mean value theorem)
Suppose that $f:[a,b]\rightarrow \mathbb{R}$ is a twice differentiable function. The inequality is given by
$$|f(x+h)-f(x)-f'(x)h|\leq \frac{h^2}{2}|f''(\eta)|$$
where $x,x+h\in [a,b]$. The aim is to prove that there is a $\eta \in[a,b]$ such that above inequa... |
H: Do we need to rigorously prove why a set is non-empty with non-negative integers?
For proofs using Well-Ordering Principle(WOP), can we prove the set is a non-empty set of nonnegative integers simply by "stating"?
Eg of what I mean by "stating":
Given any integer $n$ and any positive integer $d$, there exist intege... |
H: Uniform convergence of $\tan(x)^n$
I am considering the uniform convergence of $(\tan(x))^n$ in the interval$[0,π/4)$. This function sequence is pointwise convergent everywhere in the given interval and converges to $0$.
Is this function uniform convergent in the given interval? In my opinion, it is uniform conver... |
H: Show that basis $\mathscr{A}$ equals the intersection of all topologies on $X$ that contain $\mathscr{A}$
Attempt:
Let $\mathcal{T}$ be the topologie generated by $\mathscr{A}$. If $\{ \mathcal{T}_{\alpha} \}$ is a collection of all topologies on $X$ that contain $\mathscr{A}$, we prove that
$$ \mathcal{T} = \bigc... |
H: Asymptotic rate of decrease of error function
The complementary error function is defined as
$$
\text{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}}\int_0^{x} e^{-t^2} dt
$$
and is related to the Gaussian (Normal) distribution. Is there an approximation of the form $\exp(g(x))$ that converges to $\phi(x)$ asymptotically? i.e.... |
H: Calculate $\sum_{n=1}^\infty\frac{n^x}{n!}$
I want to evaluate function defined by following sum: $$\sum_{n=1}^\infty\frac{n^x}{n!}$$ I was thinking about writing Taylor series expansion for it. However my try resulted in sum that looks even harder to calculate:
$$\sum_{n=1}^{\infty}\frac{\ln^k(n)}{n!}$$Thanks for ... |
H: How to show that $\Bbb Q$ is incomplete under this special metric?
Fix a prime $p$ and a positive rational number $r$ can be written uniquely in a form $r=p^e\frac{b}{c}$, with $e\in \Bbb Z$ and $p,b,c \in \Bbb N$ being pairwise coprime.
Define $|\cdot|_p$ as
$|r|_p=0$, if $r=0$,
$|r|_p=|-r|_p$, if $r<0$,
$|r|=p^{... |
H: Two disjoint sets of positive measure everywhere.
Background
This question was asked a few minutes ago and then deleted after another user exhibited what he believed to be a duplicate but I fail to see the link between the two.
Here is the statement of the deleted question, $m$ being the Lebesgue measure :
Can we... |
H: Proof for arithmetic progression with different indices
How do I show that $\left(a_n\right)_{n\geq1}$ is an arithmetic progression if and only if $a_i+a_j-a_k = a_{\left(i+j-k\right)}$.
I tried by using the different definitions for arithmetic progressions and equating them, but I have no clue how to solve this. A... |
H: Homotopic functions-proof
Let $f(x)$ and $g(x)$ be continuous mappings from topological space $X$ to $S^n$, such that $f(x)$ is different from $-g(x)$ for all $x$. Prove that $f$ and $g$ are homotopic.
So, my idea was to start off with the definition of homotopy:
Let $F: X \times I\to S^n$
$F(x,0)=f(x)$, $F(x,1)=g... |
H: Can a real 2 by 2 matrix have one eigenvalue with geometric multiplicity 2?
Given the a real matrix $A=\begin{bmatrix} a & b \\ c& d\end{bmatrix}$, we assume that it has only one real eigenvalue $\lambda$. I am wondering if it is possible that the eigenvalue $\lambda$ has geometric multiplicity 2, but it seems like... |
H: Verification of power series solution to differential equation.
Verify that $$y=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}x^n$$ is the solution of the differential equation $(x+1)y+y’=0$. So we differentiate $y$ to get $$y’=\sum_{n=1}^{\infty} (-1)^{n+1}x^{n-1}$$ and substitute into the differential equation but I ca... |
H: finding solution of differential equation $y''-9y=(x^2-2)\sin(4x)$
Using method of undetermined coefficient finding solution of $y''-9y=(x^2-2)\sin(4x)$
What i try: First we will find characteristic solution
$r^2-9=0\Longrightarrow r=\pm 3$
So our characteristic solution
$y_{c}(x)=C_{1}e^{3x}+C_{2}e^{-3x}$
But i... |
H: Quadratic variation of $ X_t = t W_t $
Let $W = \{W_t : t \geq 0\}$ be the standard Brownian motion and consider
$$X_t=t W_t$$
Show that the quadratic variation of $X_t$ is $\frac{t^3}{3}$
I know this question has been answered here but I would like to do it by definition, ie, i would like to show that
$$\li... |
H: Well-ordered sets
I would like to ask how many well-ordering of set $\mathbb N$ exist. And they shouldn't be isomorphic to each other. I found out that the set of well ordering of set $A$ is the subset of $A \times A$. And that the set can be always well-ordered, because it is enough to do an injection between thes... |
H: Finding orthogonal projection on Hilbert space
Let $H = L^2(−1, 1)$ and $L \subset H$ be the set of all continuous functions such that $f(0) = 0$. Find the orthogonal projection $P : H → \bar{L}$.
My thoughts on this:
We say $g=P_{\bar{L}}(f)$ iff $f-g \perp \bar{L}$, for $g \in \bar{L}$ and $\forall f \in H$. So w... |
H: Proving that $ \sup E[X_n] \geq E[X]$
Consider the sequence of random variables $\{X_n\}_{n\geq1}$ such that $X_n$ are non-negative and $X_n \rightarrow X$ almost surely, with $\sup E[X_n]<\infty$.
Prove $E[X]\leq \sup E[X_n].$
My attempt
$$E[X]=E[\lim X_n]=\lim E[X_n]\leq \sup E[X_n]$$
Is this correct? I would ap... |
H: if M is a maximal ideal of R such that every element in 1 + M is a unit, then R is a local ring: Counter-example?
I think I have a counter-example to the theorem
If $M$ is a maximal ideal in $R$ such that for all $x \in M$ , $x+1$ is a unit, then $R$ is a local ring with maximal ideal $M$. $R$ has a unique maximal... |
H: Evans and Murthy: if $\sum_{i=0}^r a_iA^ib=0 , a_i > 0 , i = 0,1,\dots,r$ then $x$ can be expressed as a linear combination of $A^ib$
In the article of Evans and Murthy (1977) the following lemma is given:
If $A,b$ satisfy the relationship $$\sum_{i=0}^r a_iA^ib=0 \quad \quad a_i > 0 \quad i = 0,1,\dots,r$$ then a... |
H: How many equivalence classes does the following equivalence relation (over permutations) have?
Let $n\in\mathbb{N}$. Consider the set of all permutations over $n$.
Two permutations $\pi_1 = i_1..i_n$ and $\pi_2 = j_1..j_n$ are not equivalent if either $i_n i_1$ appears in $\pi_2$ (that is, $\exists k: j_k=i_n \land... |
H: Can someone help me solve this quartic equation?
$$x^4+5x^3-18x^2-10x+4=0$$
I cannot solve this quartic equation - is there any way to solve it apart from the quartic equation? It has no integer roots, and a hint given on the worksheet says to first divide through by $x^2$. Can somebody help?
AI: Like Quadratic sub... |
H: Obtain value of variable through inverse function
I am trying to implement a controller for a strict feedback system and I am reading a paper regarding the procedure. My problem has to do only with some math involved so this is why I am posting here. Suppose I have a function defined as:
$$ S_1(ε_1) = \frac{e^{2ε_1... |
H: If $x \not= 0$ and $\lambda x = 0$, then $\lambda = 0$.
I am trying to use the definition of vector spaces to prove that, if $x \not= 0$ and $\lambda x = 0$, then $\lambda = 0$.
Expressing the proposition differently, it seems to me that it is saying the following: If $x \not= 0$ and $\lambda x = 0$, then there exi... |
H: prove that $ \frac{\sigma(1)}{1}+\frac{\sigma(2)}{2}+\dots+\frac{\sigma(n)}{n} \leq 2 n $
[HMMT 2004] For every positive integer $n$, prove that
$
\frac{\sigma(1)}{1}+\frac{\sigma(2)}{2}+\dots+\frac{\sigma(n)}{n} \leq 2 n
$
If $d$ is a divisor of $i,$ then so is $\frac{i}{d},$ and $\frac{i / d}{i}=\frac{1}{d} .... |
H: Proving $xRy\iff xy^{-1}\in\ker(f)$ is equivalent relation, where $f:(G,.) \to (H,.)$ is homomorphism of groups $G$ and $H$.
I know for sure it is but I failed to prove how. I tried to use the homomorphism definition where $f(x*y^{-1}) = f(x)*f(y^{-1})$ but I didn't see a pattern.
AI: Note that $ {\rm Ker}(f) = \{x... |
H: Counter example with series $\sum_{k=2}^{\infty} \frac{1}{k}$ to show that $ \sum_{k=2}^{\infty} \frac{2^k}{\lfloor \frac{k}{2} \rfloor!} $ diverges?
I am inspecting the convergence of different series and I ran across the following series:
$$ \sum_{k=2}^{\infty} \frac{2^k}{\lfloor \frac{k}{2} \rfloor!} $$.
I know... |
H: How to show that two Euler-Lagrange-equations have the same solution?
Let $\phi:\mathbb R^n \to \mathbb R$ be a twice continuously differentiable function and let
$$L_{\phi}(t,x,\dot x) =\nabla\phi(x)^T\dot x = \sum_i\frac{\partial \phi}{\partial x_i}(x_1,..,x_n)\dot x_i.$$
Let $L:[a,b] \times \mathbb R^n \times \m... |
H: Infinite series of sequences
Let S be the set of sequences whose series converge absolutely. We define 2 norms on S: $$\| \{ a_n \}_{n=0}^{ \infty } \|_1 = \sum_{n=0}^\infty | a_n |$$ and, $$\| \{ a_n \}_{n=0}^\infty \|_{\sup} = \sup \{ |a_n|_{n=0}^\infty \} $$
Note: S is the set of sequences such that $\| a \|_1 <... |
H: What exactly does it mean for a vector to have a direction?
Vectors are defined as having magnitude and direction. If I understand it correctly, their magnitude is their length, meaning they have the properties of a line segment. What does it mean for a vector to have a direction? Let me be more specific:
Let $\vec... |
H: probability of non smokers equals probability of smokers
Suppose that $10$ % students are smokers.In a random sample of $10$ students, the probability that number of nonsmokers equals the number of smokers is
What i try: If $10$% students are smokers. Then $90$ % students are non smokers.
Let $p$ be the probabili... |
H: Find zero of function $\sum_{n=0}^{\infty}\frac{cos(x(n+1))}{n!}$
I'm interested in finding the smallest positive zero of function
$$\sum_{n=0}^{\infty}\frac{\cos(x(n+1))}{n!},\qquad x\in\mathbb R$$
It is approximately equal to $0.832$. I've calculated Taylor series expansion of this sum, which is:
$$\sum_{k=0}^{\... |
H: Reverse the order of integration.
Reverse the order of integration:
$$\int_{0}^{1}\int_{2\sqrt{x}}^{2\sqrt{x}+1} f(x,y)dydx$$.
This is my solution:
$$\int_{0}^{1}\int_{0}^{\frac{y^2}{4}}f(x,y)dxdy+\int_{1}^{2}\int_{\frac{(y-1)^2}{4}}^{\frac{y^2}{4}}f(x,y)dxdy+\int_{2}^{3}\int_{1-\frac{(y-1)^2}{4}}^{0}f(x,y)dxdy$$
B... |
H: Automorphism of commutative groups.
For every group G there is a natural group homomorphism G → Aut(G)
whose image is the group Inn(G) of inner automorphisms and whose
kernel is the center of G. Thus, if G has trivial center it can be
embedded into its own automorphism group
The inner automorphism group of a... |
H: Direct sum and determinant
Consider the vector space $V = V_1 \oplus V_2$, where $V_1,V_2$ are subspaces. Let $f_i\in\mathcal{L}(V_i)$ be a linear map for $i=1,2$, and define $f\in\mathcal{L}(V)$ by $f(v)= f_1(v_1)+f_2(v_2)$, where $v=v_1+v_2$ and $v_i\in V_i$, for $i=1,2.$
Prove that $\det(f) = \det(f_1)\cdot\det(... |
H: Show that the (conditional) probability of a triple occupancy of some cells equals $1/4$.
Seven balls are distributed randomly in seven cells. If exactly two cells are empty, show that the (conditional) probability of a triple occupancy of some cells equals $1/4$.
Let $H$ be the event that exactly two cells are em... |
H: How would I solve this, considering I have no values whatsoever?
I think that SQ is straight and so have tried to use Pythagoras, which leaves me with $a + b + c = ac/b$, but I don't see any values. How could I find values?
AI: Notice the shorter sides of the rectangle are equal and hence $a+b=c$. Using Pythagoras... |
H: $f=g \implies \nabla f = \nabla g$?
I want to disprove the following statement: $$f=g \text{ on } S \implies \nabla f = \nabla g \text{ on } S$$ where $S$ is some smooth closed surface and $f,g$ are smooth.
I don't understand why this shouldn't be the case, assuming $S$ is more than a single point. In the simplest ... |
H: Upper and lower darboux sum
I have recently been going through some questions on Riemann integration.I got stuck in one of those questions which happen to be a multiple-select question. It says that:
Let $f$ be a continuously differentiable real-valued function on $[a,b]$ such that $\left\lvert f'(x)\right\rvert\l... |
H: The problem of rebalancing an investement portfolio
I bought $30$ of stock $A$ at a price of 1 USD each. And 2 of sock $B$ at a price of $15$ USD each. Now 50% of my portfolio is stock $A$ ($\frac{30 \times 1}{60}$) and 50% of it is stock $B$ ($\frac{15 \times 2}{60}$). I want to keep this balance, so that each sto... |
H: De Morgan’s law: Wikipedia proof, cannot follow part 1, step 3.
I would like to prove De Morgan’s laws and have tried to follow the Wikipedia proof. However, I am stuck in part 1 of this proof, line 3:
1: Let $x \in (A \cap B)^c $. Then, $ x \notin A \cap B $.
2: Because $ A \cap B = \{y | y \in A \wedge y \in B ... |
H: Describe all martingales that only take values in $\{−1, 0, 1\}$.
Describe all martingales that only take values in $\{−1, 0, 1\}=:\Omega$.
In the first instance i would try to find a filtration of
$$P(\Omega)=\{\emptyset,\{0\},\{1\},\{-1\},\{1,-1\},\{1,0\},\{-1,0\},\Omega\}.$$
Candidates are:
$P(\emptyset),... |
H: Row sum of square matrix $\mathbf{A}^T\mathbf{A}$ relation to row/column sum of $\mathbf{A}$
I have a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ with positive entries. Its column sum is $k$ and its row sum is $1$, i.e. $\sum col(\mathbf{A}) = k$ and $\sum row(\mathbf{A}) = 1$.
Now looking at $\mathbf{A}^T\math... |
H: Using the Maclaurin series for $\frac{1}{1-x}$ to find $\frac{x}{1+x^2}$
Suppose I know the Maclaurin series for $$\frac{1}{1-x}=1+x+x^2+x^3+...= \sum_{n=0}^{\infty}x^n \tag{1}$$
then I can find the Maclaurin series for $\frac{1}{(1-x)^2}$ by the substitution $x\to x(2-x)$, which is obtained by solving the followin... |
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