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H: How to rationalize multiple terms with fractional exponents
I'm trying to derive the derivative of $f(x) = x^{2/3}$ using the limit definition:
$$f'(x)=\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
$$=\lim_{h \to 0} \frac{(x+h)^{2/3} - x^{2/3}}{h}$$
I suspect I have to rationalize the numerator in order to cancel an $h... |
H: Fatou's Lemma applied to simple functions
Show that the sequence of measurable functions $f_i: \mathbb{R} \rightarrow \mathbb{R}$
defined via \begin{align*}f_i(x)=
\begin{array}{cc}
\{ &
\begin{array}{cc}
-1 & i \leq x \leq i+1 \\
0 & o.w. \\
\end{array}
\end{array}
\end{align*} (o.w. sta... |
H: If $|a_n| \to |a|$ and $|\frac{a_n}{|a_n|}-\frac{a}{|a|}|\to0$ can we conclude $a_n\to a$?
If $|a_n| \to |a|$ and $|\frac{a_n}{|a_n|}-\frac{a}{|a|}|\to0$ can we conclude $a_n\to a$?
I am not sure. I tried various algebraic manipulations but could not figure it out. This problem came up when trying to prove somethi... |
H: Solve $ny(x)^2=\sqrt{1+y'(x)^2}$ and determine the range of $x$ where $y(x)$ is real-valued
I have the following differential equation:
$$ny(x)^2=\sqrt{1+y'(x)^2}$$
I know that $n$ is a real number, and that the intial condition is $y(a)=b$, where $a$ and $b$ are also real numbers.
The questions I have are:
What i... |
H: Is there an ideal scoring term for determining two numbers?
Let's say I have a known vector of two numbers: c(A,B)
Is there a scoring term, or a combination of scoring terms, that can measure the unique closeness of a random vector c(a,b) to the known vector? In other words, is there a scoring term that can be used... |
H: From 7 inputs and 1 output, approximate a possible function?
I'm trying to approximate a car insurance quote algorithm/function:
It takes 7 input variables that I can change (Vehicle Cost, Post Code, Gender, Persons Age, Licence type, Licence age, and excess) and outputs a single numerical solution (Cost/week). I'v... |
H: Math operator for safe division
Is there a math operator for a safe division, returning a pre-set value, usually 0, when divide by zero is encountered? If not, in a computer science or mathematics paper would one just say before an equation that all divisions are "safe"?
AI: This is equivalent to defining an operat... |
H: If $a+b+c=\pi$ and $\cot t=\cot a+\cot b+\cot c$, show $\sin^3t=\sin(a-t)\sin(b-t)\sin(c-t)$
I have a problem from a mathematics book:
If $\alpha + \beta +\gamma = \pi \tag{1}$ and
$$\cot \theta = \cot\alpha + \cot \beta + \cot \gamma, 0 < \theta < \frac{\pi}{2}\tag{2}$$
show that
$$\sin^{3}\theta = \sin(\alpha - ... |
H: Characteristic functions and convergence of complex sequence
I'm trying to solve the following question, but I have no idea why the hint was given as it was:
My attempt: I'm not really able to make use of the hint so far, so I'm a bit lost:
The assumptions give that $e^{i t x_n} \rightarrow c(t) + i s(t) \equiv z... |
H: Understanding the roots of the irreducible factors of the 15-th cyclotomic polynomial modulo $7$
We consider the 15-th cyclotomic polynomial over $\mathbb{Z}$ first:
$$\Phi_{15} = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1.$$
If we reduce it modulo $7$, we obtain two irreducible factors of $\Phi_{15}$ over $\mathbb{F}_7[x... |
H: Are two distance regular graphs with the same intersection array also cospectral for their Laplacian matrices?
So we know that two DRGs with the same intersection array must be co-spectral on their adjacency matrices, i.e. their adjacency matrices have the same set of eigenvalues.
But is this necessarily true also ... |
H: Need help with limit proving.
Is it safe to say that $\frac{\sqrt{n+1}}{\sqrt{n}}\rightarrow1$ if $$\lim_{n\to\infty}(\sqrt{n+1}-\sqrt{n})=0$$?
Because I want to prove that $\sqrt{n+1}\sim\sqrt{n}\ $when $n\to\infty$, but I don't know how to approach from $\frac{\sqrt{n+1}}{\sqrt{n}}\rightarrow1$. So I was thinking... |
H: Proof of Equality of Null Spaces
Let ${\bf A}, {\bf B} \in \mathbb{C}^{n \times n}$ be a pair of matrices such that:
$$
{\rm Null}\left({\bf A}\right) \subseteq {\rm Null}\left({\bf B}\right)
$$
Furthermore, it is known that the null spaces of both $\bf A$ and $\bf B$ have the same dimension.
Is it true then that $... |
H: Question on the given proof of $(\det A)(\det B) \leq [(\operatorname{tr} AB)/n]^n$
I was reading through this paper and on page 6, there is a lemma proving $(\det A)(\det B) \leq [(\operatorname{tr} AB)/n]^n$ for two positive semideifinite matrices $A$ and $B$.
I get every single line until the part that concluded... |
H: $P(X_1 > 0 \mid X_1 + X_2 > 0)$ for IID $X_1, X_2 \sim \mathcal{N}(0,1)$
Given IID $X_1, X_2 \sim \mathcal{N}(0,1)$, we want to determine $P(X_1 > 0 \mid X_1 + X_2 > 0)$.
This is what I think is the approach for this problem:
\begin{align}
P(X_1>0\mid X_1 + X_2 > 0) = P(X_1 > 0 \mid X_1 > -X_2) \\
P(X_1 \le... |
H: Why bother with the space $\mathcal{L}^1$ for integration when we can abstractly deal with the completion of a semi-normed space
I'm studying the Bochner-Lebesue integral, and while I understand the general construction, I have a few questions about the way it is being presented. Typically, the story goes like this... |
H: Show that the transformation $w=\frac{2z+3}{z-4}$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$
Question:
Show that the transformation $w=\frac{2z+3}{z-4}$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$.
My try:
$$\begin{align}\\
&x^2+y^2-4x=0\\
&\implies (x-2)^2+y^2=4\\
&\implies... |
H: Does $ \lim_{n \to \infty}\sum_{k = 1}^n \zeta\Big(k - \frac{1}{n}\Big)$ equal the Euler-Mascheroni constant?
Let $\zeta(s)$ be the Riemann zeta function and $\gamma$ be the Euler-Mascheroni constant. I observed the following result empirically. Looking for a proof or disproof.
$$
\lim_{n \to \infty}\sum_{k = 1}^n ... |
H: Parameterization of a curve within a path integral?
I have a question about the following problem:
Find an appropriate parametrization for the given piecewise-smooth curve in $\mathbb{R}^{2}$, with the implied orientation.
The curve $C$, which goes along the circle of radius 3, from the point $(3, 0)$ to the poi... |
H: Absolutely continuous functions that fix zero and satisfies $f'(x)=2f(x)$
A past question from a qualifying exam at my university reads:
Let $f$ be a continuous real-valued function on the real line that is differentiable almost everywhere with respect to the Lebesgue measure and satisfies $f(0)=0$ and
$$ f'(x)=2f(... |
H: Prove that there are infinite number of mapping from $\mathbb{R}$ onto $\mathbb{Q} ?$
Prove that there are infinite number of mapping from $\mathbb{R}$ onto $\mathbb{Q} ?$
My attempt : If i take $f(x) = [x]^{x} $ where $[.]$ denote the greatest integer function .then it will be satisfied the onto mapping
Im confu... |
H: how (a!)/(b!) = (b + 1)×(b + 2)×⋯×(a − 1)×a
I was solving a problem in which i need to figure out the prime factorization of $\frac{a!}{b!}$ and i did that by computing (a!) and then (b!) by looping ((1 to a) & (1 to b)) and then derived n by dividing them ($n = \frac{a!}{b!}$) and then prime factors of n, but it g... |
H: Proving $|x|+|y|+|z| ≤ |x+y-z|+|y+z-x|+|z+x-y|$ for all real $x$
We need to prove that for all real $x,y,z$
$$|x|+|y|+|z| ≤ |x+y-z|+|y+z-x|+|z+x-y|$$
Source ISI entrance examination sample questions
I don't know how to solve in mod form so I thought about squaring and removing the mod, but the mod on RHS still pe... |
H: On a variant of Farkas Lemma
The version on Mordecai Avriel’s book Nonlinear Programming is:
Let A be a given $m\times n$ real matrix and $b$ a given n vector. The inequality $b^Ty≥ 0$ holds for all vectors $y$ satisfying $Ay ≥ 0$ if and only if there exists an m vector $\rho ≥ 0$ such that $A^T\rho=b$.
It seems... |
H: Convergence of the sequence $ a_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\int_{1}^{n} \frac{1}{x} d x$
Let
$$
a_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\int_{1}^{n} \frac{1}{x} d x$$
for all $n \in \mathbb{N} .$ Show that $\left(a_{n}\right)$ converges.
Actually $
a_{n}=1+\frac{1}{2}+\frac{1}{3}+\c... |
H: Well-ordering on natural numbers
Let $\omega=\{0,1,2,3,\ldots\}$. We say that $\omega$ is a well-ordered set. But I can't understand why. By the definition of well-ordering, there should be no infinite descending chain, but if I start from infinity, how can I reach 0 in finite descents? Or is this not allowed? Is t... |
H: Weak compactness of nonnegative part of unit ball of $L^1$
Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and let $B$ be the unit ball of $L^1(\Omega,\mathfrak{A},\mu;\mathbb{R})$. Suppose that
$$
B_+=\big\{u\in B\;|\;u\geq 0\;\:\text{$\mu$-a.e.}\big\}
$$
is weakly compact.
My question: Does the weak compactnes... |
H: Groups up to isomorphism
Does there exist an answer to the next question:
How many groups of order $n$ ($|G|=n$, $n\in \mathbb{N}$) are exist up to isomorphism?
(The groups are not necessarily Abelian)
I am curious about this question.
If the answer ($\forall n\in\mathbb{N}$) does not exist,does the answer exist ... |
H: Volume of convex body as an integral of its radial function
Let $C$ be a compact convex set in $\mathbb{R}^d$. Let the origin $O$ by in the internal of $C$. The gauge function $\gamma_C(.) : \mathbb{R}^d \to [0, \infty]$ of $C$ is defined as
$$
\gamma_C(x) = \inf\{t : x \in t \cdot C\}.
$$
The radial function is de... |
H: Lots of doubts abot the surface area of a cylinder.
I recently started to study parametric surfaces, and I come across this exercise that I try to solve but I have a lot of doubts reganding the correctness of my resolution, and also I don't find similar examples on the internet.
I need to find the surface area of ... |
H: How to prove that any matrices have their own generalized inverse.
Let $A$ be a matrix with a form $(m.n)$, and $X$ be a matrix with a form of $(n,m)$.
If $AXA = A$, $X$ is called a generalized inverse of $A$.
How can we prove that any matrices have their own generalized inverse?
\begin{eqnarray}
\\
\end{eqnarr... |
H: Is annihilator of principal ideal comparable?
Is annihilator of principal ideal comparable or intersection is zero?
It seems to me there is no reason to believe this is true. But I couldn't found a counter example yet.
Is it true under some conditions?
AI: Let $R=\mathbb{Z}[x]/(12,2x)$. Then $(2)$ is a principal ... |
H: What are the subdifferentials $\partial f(0)$ and $\partial f(1)$?
Let $ f: \mathbb{R} \to \mathbb{R} $ given by
\begin{equation*}
f(x) = \left\{
\begin{array}{rl}
x \log x -x & \text{if } x \geq 0\\
\infty & \text{if else}\\
\end{array} \right.
\end{equation*}
What are the subdifferentials $\partial f(0)$ and $\pa... |
H: Is the dimension of a Noetherian local ring equal to its associated graded ring?
For a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$, let $I$ be a primary ideal in $A$, the associated graded ring is
$$ \bigoplus_{n=0}^{\infty} I^n/I^{n+1}$$
AI: Yes. First we deal with the case $I=\mathfrak m$.
If we d... |
H: $(\forall n \in \mathbb{Z}):n^{3} \equiv n$ (mod $6$)
[This is not a duplicate, since I am seeking for an alternative proof for this problem]
This is a problem from Proofs and Fundamentals, by Ethan D. Bloch.
Show that, for all $n \in \mathbb{Z}$, $n^{3} \equiv n$ (mod $6$).
I wrote my proof as follows and I woul... |
H: Non-real numbers in system of equations
Given $a^2+b^2=1$,
$c^2+d^2=1$,
$ac+bd=0$
To prove
$a^2+c^2=1$,$b^2+d^2=1$,$ab+cd=0$
Now this can be easily done by trigonometric substitution if it was given that $a,b,c,d$ are real numbers.
I have a solution using matrix which I think is valid even if the given numbers are ... |
H: Shift invariance and Krylov subspaces
Let $A \in \mathbb{R}^{n \times n}$ a matrix, and $r_0 \in \mathbb{R}^{n}$. Also, let $(\sigma)_i$ a sequence of complex scalars.
Consider the Krylov subspace $K_n(A,r_0)=\text{span} \{r_0,A r_0, \ldots, A^{n-1}r_0 \}$.
I want to show that $$K_n(A + \sigma_j I,r_0) = K_n(A+\sig... |
H: Importance of the 'prime' condition
Prime avoidance theorem: Let $A$ be a ring (commutative with unity) and $p_1,...,p_n\subset A$ prime ideals. Let $a\subset A$ be an ideal such that $a\subset (p_1\cup p_2\cup\cdot\cdot\cdot\cup p_n)$, then $a\subset p_k$ for some $1\leq k\leq n$.
Now, I have no problem in proving... |
H: Area of a triangulation of a non-planar polygon
Does the area of a triangulated simple non-planar polygon in 3D space depend on the triangulation, or is it the same for any triangulation of the points?
I would suppose it is not the same, but when I try to come up with simple examples it looks like it's the same for... |
H: Dual space of continuous functions on an open set of $\mathbb{R}^m$
Let $V \subset \mathbb{R}^m$ be an open subset and define
$$
C(V) := \{f:V \rightarrow \mathbb{C}| f\text{ is continuous}\}.
$$
We can make $C(V)$ into a topological vector space as follow. Let $Q_1 \subseteq Q_2 \subseteq \cdots \subset V$ be comp... |
H: Trace norm equality in proof
Consider the following theorem in Murphy's '$C^*$-algebras and operator theory':
Why is the marked equality true? This seems to boil down to showing that
$$tr(uw^*) = tr(w^*u)$$
Myrphy already showed that $tr(uv) = tr(vu)$ when one of the operators $u,v$ is trace class or they both are... |
H: Why are we using combination in this problem instead of permutation?
What is the number of non-negative integers of at most $4$ digits whose digits are increasing?
The answer to this problem is $10\choose 4$.
But, I want to know why are we using combination instead of permutation when the order matters in this q... |
H: Serge Lang - Introduction to linear algebra, Linear Mappings
I have these problems in Serge Lang's Introduction to linear algebra's Linear Mappings section.
(a) What is the dimension of the subspace of $R_n$ consisting of those vectors $A = (a_1, ... ,a_n)$ such that $a_1 + ... + a_n = 0$?
I did the following.
A i... |
H: Find the determinant of a matrix $A$, such that $A^4 + 2A = 0$
Find the determinant of an revertible, $6 \times 6$ matrix $A$, such that $A^4 + 2A = 0$
This question seems odd to me, because when I tried to solve it:
$A^4 + 2A = O$
$(A^3+2I)A = O\det$
$|(A^3+2I)||A| = |O|$
But I know that $|A|$ is revertible, th... |
H: Variance of sum of independent random variables - case of undefined density
When the density $f_{X, Y}$ is not defined for independent random variables $X, Y$, is it possible to say anything beyond
\begin{align}
Var(X + Y) &= Var(X) + Var(Y) + 2 \int_{\Omega} (X - E(X))(Y-E(Y)) dP,
\end{align}
in terms of simplifyi... |
H: Linear function preserving the Gram determinant
In Euclidean space $X$ the Gram's determinant of a system of vectors $x_1,...,x_k\in X$ is called the determinant of $k\times k$ matrix $ [\langle x_i,x_j \rangle]$:
$
G(x_1,..,x_k)=\det[\langle x_i,x_j \rangle].
$
In $n$ dimensional Euclidean space $X$, let $f: X\r... |
H: Is this Factorization?
I'm doubtful about the some parts of the solution to this question:
Suppose that the real numbers $a, b, c > 1$ satisfy the condition $$ {1\over a^2-1}+{1\over b^2-1}+{1\over c^2-1}=1 $$ Prove that $$ {1\over a+1}+{1\over b+1}+{1\over c+1}\leq1 $$
The solution says that noticing $a\geq b\ge... |
H: Why is the blow up of a submanifold of $\mathbb{P}^n$ again projective
I saw somewhere that the blow up (at any point) of a submanifold of $\mathbb{P}^n$ is still projective. I have the feeling that this is a consequence of the Kodaira embedding theorem, any thoughts?
AI: Yes, this follows from Kodaira embedding. I... |
H: How to simplify $\frac x{|x|}
Is there a simplification for this relation?
$$ \frac{x}{\left| x\right| } $$
where $x=a+i b$, $a$ and $b$ are reals.
AI: By polar form $x=|x|e^{i\theta}$ we obtain
$$\frac{x}{\left| x\right| }=e^{i\theta}$$
with $\theta = \operatorname {atan2} \left(b, a \right)$ (see atan2). |
H: If every two-dimensional (vector) subspace of a normed space is an inner product space, then so is that normed space
Let $\big( X, \lVert \cdot \rVert \big)$ be a (real or complex) normed space. Suppose that, for every two-dimensional (vector) subspace $Y$ of $X$, the norm on $Y$ (i.e. restriction of the norm of $X... |
H: Other absolute value definitions in $\mathbb R$
I know these definitions for the absolute value (or module): given a real number $x$, then
$$\bbox[yellow]
{|x|=\begin{cases}x & \text{if } x\geq 0\\ -x& \text{if } x< 0\end{cases}}$$
or
$$\bbox[yellow]
{|x|=\max\{x,-x\}}$$
Are there other definitions in $\mathbb R$ (... |
H: Epsilon delta for infinite limits
The limit:
$$\lim_{x \to \infty} f(x) = -\infty$$
iff
$$\big\{ \forall M>0, \exists N >0 \ s.t\ \forall x > N \implies f(x) <-M \big\}$$
This means for all $x \in (N,\infty)$ , $f(x)$ lies in $(-\infty,M)$, however this doesn't account for when $f(x)$ is not defined so it is not $... |
H: prove that this function isn't Lipschitz continuous
I'm trying to prove that this function isn't Lipschitz continuous:
$f:[-1,1]\to \Bbb R:x\mapsto x^{2}\sin1/x^{2}$ when $x$ not equal to zero and $0$ if $x=0$.
Now I just can't find the right $x$ and $y$ to prove it.
Take $M$ from $\Bbb R^+$. Chose then $x$ and $y... |
H: Finding Expected Value of Conditional Poisson Distribution
Consider a random variable X ~ Poisson (1). Namely, $P(x=k) = \frac{e^{-1}}{k!}$ , k=0,1,2,...
I'm trying to solve for $\mathbb{E}\{X|X\geq 1\}$.
My approach:
Given that $X\geq1$ then we know that at least one person has arrived (using arrival / no arrival ... |
H: In an $n\times(n+1)$ nonnegative matrix, there is a positive pivot at which the row sum is greater than the column sum
Let $A$ be a matrix of shape $n\times (n+1)$ with non-negative real entries, which has at least one positive entry in each column. Show that there is an $a_{ij} > 0$ such that the sum over the $i$... |
H: Let $f(n) = an^2 + bn + c$ be a quadratic function. Show that there's an $n ∈ N$ such that $f(n)$ is not a prime number.
Let $f(n) = an^2 + bn + c$ be a quadratic function, where $a, b, c$
are natural numbers and $c ≥ 2$. Show that there is an $n ∈ N$ such
that $f(n)$ is not a prime number.
I figure this might ha... |
H: Evaluate right hand limit $\lim_{x\to 2^+} (x^2 + e^{\frac{1}{2-x}})^{-1}$
$$\lim_{x\to 2^+} \frac{1}{x^2 + e^{\frac{1}{2-x}}} = \frac{1}{\lim_{x\to 2^+} x^2 + \lim_{h\to 0} e^{\frac{1}{h}}} =\frac{1}{4+\lim_{h\to 0} e^{\frac 1h}}$$
How do I solve further ?
AI: Only a small mistake, it should be
$$L=\frac{1}{4+\li... |
H: Can an orthogonal matrix that represents a linear transformation from $\mathbb{R}^n \to \mathbb{R}^n$ have no eigenvalues?
I know that if the matrix is normal and represents a transformation in a unitary space then it can be unitarily diagonalized, so it must have eigenvalues. Plus, its characteristic polynomial is... |
H: Find the last two digits of $7^{100}-3^{100}$
Find the last two digits of $7^{100}-3^{100}$
From Euler's theorem one gets that $\phi(100) = 40 \Rightarrow 7^{40} \equiv 1 \pmod{100}, 3^{40} \equiv 1 \pmod{100}.$
I couldn't really work this out without using a calculator to compute the powers. How can I continue f... |
H: Inverse of multiple projections of the same point
I have a vector $\vec{x}$ in 3D space that is unknown. I do know $\vec{p_1}$, $\vec{p_2}$, $\vec{p_3}$ which are orthogonal projection vectors of $\vec{x}$ onto lines $P_1$, $P_2$, $P_3$, all going through the origin and not parallel to each other. From $\vec{p_1}$,... |
H: Choose correct answers on eigen value:
A =
[0 0 L 0 1
1 0 L 0 0
0 1 L 0 0
M M N N M
0 0 L 1 0]
a) eigenvalue are purely real
b) 0 is the only eigenvalue
c) eigenvalues are n-th roots of unity Exp(2πi/n) for i = 0,1,...,n-1
d) none of these
What can be the easiest way to get answers in these type of questions?
... |
H: Derive the change of coordinates without using differentials
Please forgive me for using the following lousy notations. I'm just a beginner in DG. Given two charts $(\psi,\{x_i\}_{i=1}^n),(\phi,\{y_i\}_{i=1}^n)$ around a point $p$ in an n-dimensional manifold $M$, I'd like to derive the change of coordinates
$$\fra... |
H: Is this formula equal to 1?
Is this formula equal to $1$?
$$\frac{\left| \eta \right| }{\sqrt{\eta } \sqrt{\eta ^*}}$$
where $\eta$ is complex. And if so, how can I prove that?
AI: By definition for $\eta=a+bi\in \mathbb C$
$$|\eta|=\sqrt {a^2+b^2}$$
$$\eta\cdot \eta^* =(a+bi)(a-bi)=a^2+abi-abi+b^2=a^2+b^2=|\eta|^2... |
H: Inequality in proof concerning trace class norm
Consider the following fragment from Murphy's '$C^*$- algebras and operator theory':
Why is the marked inequality true? I.e. why is $\Vert w'' \Vert \leq \Vert v \Vert?$
AI: $$\|w''\|=\|w^{\prime*}vw\|\le\|w^{\prime*}\|\|v\|\|w\|=\|v\|$$
P.S. $\|w\|=1$ for partial iso... |
H: Is the product of a maximal subgroup and a cyclic subgroup a group?
$G$ is a finite group. I wanted to show that for a maximal subgroup $M$ of $G$, if $g\in G\setminus M$, then $M\langle g\rangle=G$. It it true? My argument is that if $M\subset M\langle g\rangle\neq G$, then it is a contradiction to the maximality ... |
H: Is this answer on eigenvector diagonalisation wrong?
This is the question and answer from MIT OCW 18.06 on eigenvectors and diagonalisation:
Two things I don't understand:
Shouldn't the eigenvector for $\lambda = -0.3$ be $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$ because the second column is the free variable?
Ass... |
H: Quadratic closure of $Z_2$
Find or disprove the existence of a minimal quadratically closed field extension of $Z_2$
I.e. $$\forall b,c \in S, \exists x\in S, x^2+ bx+c = 0$$
Note: Because $S$ is a field we can reduce every polynomial to a monic one by multiplying by $a^{-1}$.
Attempt 1
In $Z_2$, $x^2+x+1$ is irr... |
H: If $xyz=32$, find the minimal value of
If $xyz=32;x,y,z>0$, find the minimal value of $f(x,y,z)=x^2+4xy+4y^2+2z^2$
I tried to do by $A.M.\geq M.G.$:
$\frac{x^2+4y^2+2z^2}{2}\geq\sqrt{8x^2y^2z^2}\to x^2+4y^2+2z^2\geq32$
But how can I maximaze 4xy?
AI: Your application of AM-GM is wrong. The statement for AM-GM state... |
H: Bijection between topology and topological space
I've been working with a class of topological spaces $(X,\tau)$ with the following property:
There exists a bijection $f:X\to \tau\setminus\varnothing$ such that for all $x\in X$, $x\in f(x)$ (And, $|X|>1$ to avoid discrete topological spaces).
Is there a name for su... |
H: Negate this statement: There exist $x, y ∈ \Bbb{R}$ such that $x < y$ and $x^2 > y^2$
Negate this statement:
There exist $x, y ∈ \Bbb{R}$ such that $x < y$ and $x^2 > y^2$
From my understanding,
"there exists" becomes "for all"
"and" becomes "or" by De Morgan's laws
equality signs "reverse"
So, our negated state... |
H: $|p(y)| > |y| + |\delta y|$
Let $p(y)$ be a complex polynomial of degree at least two and $\delta$ some complex number. I want to show
$$|p(y)| > |y| +|\delta y|$$
for $|y| > \kappa$ sufficiently large. I have tried applying the triangle or reverse triangle inequalities but I can't seem to get the absolute values t... |
H: proving stochastic process is independent
Guys can anyone help me with this question?
On a probability space let be filtration $F = (F_n)_{n \in N_0}$ and a real valued adaptive stochastic process $(X_n)_{n \in N_0}$ for all the Borelsets $ A \in B(R) $ we have
$P[X_{n+1} \in A | F_n ] = P [X_{n+1} \in A ]$ P-almo... |
H: Let $X \sim P_1$, $Y \sim P_2$, can we find $f$ such that $f(X,Y) \sim P_1 P_2$?
Let $X \sim P_1$, $Y \sim P_2$ be two random variables with respective probability laws $P_1,P_2$.
We define $P(A)=P_1(A)P_2(A)$ for all $A$'s in the sigma algebra. $P$ is a probability law.
Can we find a function $f$ such that $f(X,Y)... |
H: cardinality of centralizer of an element
Suppose $G$ is finite group of order $p^dn$ where $d$ and $n$ are positive integers and $p$ is a prime that does not divide $n$. Show that $G$ contains an element of order $p$ such that the cardinality of its conjugacy class divides $n$
Cardinality of conjugacy class of an... |
H: What am I doing wrong? Differentiation.
I was given the following function:
$$ f(t) = \frac{t}{t^2 + 1} $$
And I attempted to derive it and came up with this answer:
$$ \frac{-2t^2}{(t^2 + 1)^2} $$
So I was incorrect obviously with this answer, and I am unsure now if you can obtain the correct answer using the chai... |
H: Question on semialgebra
This exercise comes from A First Look At Rigorous Probability (Exercise 2.7.19):
Let $\Omega$ be a finite non-empty set, and let $\mathcal{J}$ consist
of all singletons in $\Omega$, together with $\emptyset$ and $\Omega$. Show that $\mathcal{J}$ is a semialgebra. Definition is semialgebra is... |
H: Show that $f\in \mathcal{L}^p$ and $||f||_p \leqslant M.$
Let $(X, \mathcal{B}, \mu)$ be a measured space, $1<p<\infty$, and $q$ the conjugate exponent of $p$ : $\left(\dfrac{1}{p}+\dfrac{1}{q}=1\right)$.
Show that $f \in \mathcal{L}^p \implies$ $\displaystyle{||f||_p=\sup\left\{ \left|\int_X f(x)g(x)d\mu(x) \righ... |
H: Proving the continuity of the inverse of a continuous, stricly monotonic function
Please tell me if the following is correct.
We have a continuous, strictly monotonic, increasing function on some closed and bounded interval $I$, and $x_0\in I$. Let $g(y)$ be its inverse, and $f(x_0)=y_0$. I want to show that $|g(y)... |
H: Justification for the calculation of expectation on the first indicator random variable of a series of mutually dependent experiments.
I am not asking for a proof of linearity of expectation, which is available in multiple posts.
Instead I would like to develop a more robust understanding as to why the calculation ... |
H: Convergence in measure implies alternating sequences converges to zero in measure
This problem came up in studying for a qual.
Suppose $\{f_{n}\}$ is a sequence of measurable functions that converges in measure to $f$. Prove that the sequence $\{g_{n}\}$, where $g_{n} = (−1)^{n}f_{n}$, converges in measure to a fun... |
H: Prove that $ \operatorname {dom} G \setminus \operatorname {dom} H \subseteq \operatorname {dom} ( G \setminus H ) $.
im trying to learn some basics from set theory, and I got stuck in this proof.
Prove that
$$ \operatorname {dom} G \setminus \operatorname {dom} H \subseteq \operatorname {dom} ( G \setminus H ) \t... |
H: If $(M,g)$ is a Riemannian manifold and $S$ is a regular level set of $f:M\to \Bbb R$ then $\text{grad}f|_S$ is nowhere vanishing
I have a question reading a proof of the following theorem.
Theorem. Let $M$ be an oriented smooth manifold, and suppose $S\subset M$ us a regular level set of a smooth function $f:M\to ... |
H: Surjective ring morphism $f:R\to R$ satisfies Ker$(f^{n+1})\subset $ Ker$(f^n)$ then $f$ is injective.
As in the title, the set-up of the problem is as follows: $f: R\to R$ is a surjective ring homomorphism and $R$ is a commutative ring. Suppose that for some $m\in \mathbb{N}$, Ker$(f^{m+1})\subset$ Ker$(f^m)$. Pro... |
H: Why is $\text{tr}(\sqrt{\sqrt A B \sqrt A})=\text{tr}(\sqrt{A B }) $ for positive semidefinite matrices $A,B$
Let $A,B$ be two real positive semidefinite matrices of same size. Denote the trace operator by tr. We define the square root of a positive semidefinite matrix $A$ by the unique matrix $\sqrt A$ such that $... |
H: Calculate :$\int_{0}^{2\pi }e^{R{ {\cos t}}}\cos(R\sin t+3t)\mathrm{d}t$
Calculate: $$\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)\mathrm{d}t$$
My try:
$\displaystyle\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)dt\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\cos(\mathfrak{I\textrm{z}}+3(-i\log z)dz\\... |
H: Is there a preference between writing complex numbers as $z=a+bi$ or $z=a+ib$?
This is probably just a minor notational issue, but I am unsure whether I should write $z=a+bi$ or $z=a+ib$ when denoting complex numbers. Though the former notation seems more common, Euler's identity tends to be written as
$$
e^{i\pi}+... |
H: Quick way to determine characteristic subgroups
Reminder: A characteristic subgroup of a group $G$ is a subgroup which is stable under all elements of $\mathrm{Aut}(G)$. This is a stronger property than being normal.
A while ago there was a question here, which unfortunately got deleted, about a group having two d... |
H: How are you able to factorise a determinant like this?
I'm aware of many different properties of determinants, like two rows being equal implies determinant is zero, or if a row is multiplied by a constant, then you factor out that constant. I'm not aware of being able to do this though. What property am I missing ... |
H: "Finite-Dimensional-Type-Spectral-Theorem" for Orthogonal Projections
Let $H$ be a Hilbert space, not assumed separable, and $p$ and $q$ (bounded - not sure if that is important) orthogonal projections.
Question 1: Is it the case that $p$ has an orthonormal eigenbasis for $H$/is diagonalisable?
Question 2: Is it t... |
H: Correspondence between binary numbers and $ab = \Pi_{i=1}^m p_i^{k_i}$ s.t. $a, b$ have no common factors
I'll note that $p_i$ is prime and $k_i \in \mathbb Z^+.$ I read in a textbook that the number of integers $ab = \Pi_{i=1}^m p_i^{k_i}$ s.t. $a, b$ have no common factors is $2^m$ which is the number of binary n... |
H: Let $G$ be a group and let $a, b$ be elements of $G$, show that $|ab|=l.c.m (|a|,|b|)$
Let $G$ be a group and let $a, b$ be elements of $G$ such that:
i) $\langle a\rangle\cap\langle b\rangle={\{1}\}$
ii) $ab=ba$
II) $|a|=m, |b|=n$
Show that $|ab|=l.c.m (|a|,|b|)$
Idea: Note that $a^m=1,\; b^n=1,$ let's do $k=l.c.m... |
H: Proving relation between exponential generating functions whose coefficients count certain types of graphs
Let's there be two exponential generating functions:
$A(x)=\sum^\infty_{n=1}\frac{a_n}{n!}x^n$
$B(x)=\sum^\infty_{n=1}\frac{b_n}{n!}x^n$
Sequence ${\{a_n\}}^\infty_{n=1}$ defines number of all possible simple ... |
H: Weird Sum Identity
I came across a weird sum identity that I would like to prove:
$$
\sum_{i=1}^{k-2}\frac{(-1)^i}{(i-1)!(k-2-i)!(n-k+2+ij)}=\frac{-\Gamma\left(1+\frac{n-k+2}{j}\right)}{j\Gamma\left(k-1+\frac{n-k+2}{j}\right)}.
$$
I would like to know how one can prove this. It seems like direct computations, induc... |
H: Proving a result related to $100^{th}$ roots of unity.
If we take the $100^{th}$ roots of unity ie. all complex roots of the equation $z^{100}-1=0$ and denote them as $\alpha_{1},\alpha_{2},...,\alpha_{100}$ then we are required to prove that $$\alpha_{1}^r+\alpha_{2}^r+...+\alpha_{100}^r=0$$ for $r\neq100k$ where ... |
H: What were the steps taken to get from point A to point B in this forced vibrations problem?
I'm working through the derivation of the forced response (vibration) of a cantilevered beam. I have a basic understanding of the derivation until this point.
screenshot of derivation
I can see how the summation(s) are equiv... |
H: Is there an infinite amount of primes in base n made from an equal amount of even and odd digits.
Is there an infinite amount of primes in base n made from an equal amount of even and odd digits?
A list of primes that have this property is this sequence
$$23,29,41,43,47,61,67,83,89,1009,1021,1049,1061,\small\dots$$... |
H: Markov Chain: Conditional distribution at time $t$, given $t-1$ and $t+1$
For a Markov process given by
$$x_t = \mu +\kappa(x_{t-1} - \mu) + \sigma \cdot \varepsilon_t $$
where $\varepsilon_t \sim N(0,1)$ and $\mu$, $\kappa$, $\sigma^2$ are the parameters, how would I find the conditional distribution $p\left(x_t \... |
H: Simplification of an algebraic determinant
After studying some analytic geometry, I came across this step in a solution, however, I am not how they managed to simplify the determinant in this way.
When I tried to evaluate this, I got:
$\frac{bc-ad}{2}+\frac{ad-bc}{2b-2d}$, but didn’t see how this got to the desire... |
H: Show a Cubic Diophantine Equation has no solutions in coprime integers
Show that $x^3+2y^3 = 7(z^3+2w^3)$ has no solutions in coprime integers $x,y,z,w$.
I've been stuck on this problem for a bit and haven't made any progress or found any strategies that might work. Any help would be greatly appreciated.
AI: Oh: if... |
H: $\frac{2}{3}^{\text{th}}$ roots of $4 + 4\sqrt{3}i$
A textbook, in a section about roots of unity, poses the problem
Find the values of
$$ \left( 4\sqrt{3} + 4i \right)^{\frac{2}{3}} $$
The answer key says
$$ w \in \left\{ 4e^{-100^\circ i}, \quad 4e^{20^\circ i}, \quad 4e^{140^\circ i} \right\} $$
and I know e... |
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