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H: A problem on separable extension on field of characteristic $p>0$
Problem: Let $k$ be a field of characteristic $p>0$. Let $\alpha $ be algebraic over $k$ then show that $\alpha$ is separable if and only if $k(\alpha)=k(\alpha^{p^n})$ for all positive integer $n$.
I have a solution below but I am not convinced if... |
H: Compact condition for base elements
Suppose $X$ is a topological space and $\{B_i\}$ form a base for the topology on $X$, where the $i$ run over some index set $J$.
$X$ is said to be compact if every open cover of $X$ contains a finite subcover of $X$.
Suppose you know that for every covering of $X$ by base element... |
H: Kan extenstion and left adjoint
This is a continuation of the question asked here: Kan extension "commutes" with a certain left adjoint.
Let $\mathcal{A},\mathcal{B}$ be small categories and $\mathcal{C},\mathcal{D}$ an arbitrary category. Consider functors $F:\mathcal{A}\rightarrow\mathcal{B}$, $G:\mathcal{A}\righ... |
H: Rank of $R^n$ in characteristic $n$
I'm reading Deligne-Milne's introduction to Tannakian categories, and I noticed a troubling consequence of the definition of rank in a rigid ACU tensor category $(\mathcal{C}, \otimes)$. Specifically, the rank of an object $X$ is defined as the trace of its identity morphism, and... |
H: Problem with extension of a continous function
Let $X$ a first-countable Topological Space, let $Y$ an Hausdorff Topological Space, let $A\subset X$ a subset ot $X$ and let $f:A\rightarrow Y$ a continous function.
Prove that, if there is an extension
$$\overline{f} :\overline{A}\rightarrow Y$$
$\overline{f}$ is so... |
H: Almost everywhere pointwise convergence
I am trying to solve this problem
Let $a_n$ be a sequence of numbers so that $\lim_{n \to \infty} \sin(a_nx)$ exists pointwise almost everywhere on $\mathbb{R}$. Show that $\lim_{n \to \infty}a_n$ exists.
I tried to use Egoroff's theorem and other things, but I could not sol... |
H: What is the mathematical notation for dependency length calculation algorithm?
I'm doing a computational linguistic research in python programming. I have written an algorithm that calculate dependency length of any sentence, but I won't to describe it in a simple statistical notation. The idea is simple:
Any sente... |
H: What operation does this algorithm do on graphs?
I have to find the solution that this algorithm gives about graphs, knowing that the Graph is given by $ G = (V, E) $, with arcs labeled by $ w $ and that $ G $ \ $ e $ indicates that the arc $ e $ of the graph. I mean $ G $\ $ e $ is the elgraph with vertices $ V $ ... |
H: True /False question based on quotient groups of $S_{n} $ and $A_{n} $.
I am trying assignment questions of Abstract algebra and I need help in following True/ False question.
Which one of following is true?
Every finite group is subgroup of $A_{n} $ for some $n\geq 1.$
Every finite group is quotient of $A_{n} $... |
H: Show that $\binom{p}{0} + \binom{p+1}{1} + \binom{p+2}{2} +\dots+\binom{p+q}{q}$=$\binom{p+q+1}{q}$
How can I prove that $\binom{p}{0} + \binom{p+1}{1} + \binom{p+2}{2} +\dots+\binom{p+q}{q}$=$\binom{p+q+1}{q}$ using a combinatorial argument? The left part is all the permutations of $p$ white balls that have $0\le... |
H: How many digits are there in the product of $(3698765432123456789)$ and $(345678909876543)$?
How many digits are there in the product of $(3698765432123456789)$ and $(345678909876543)$? I could not find any formula to solve it and I stuck in it. Can you suggest any formula or way for it?
AI: The first number has ... |
H: 3.85 of LADR by Sheldon Axler
I'm a little confused on the proof of the implication of $(c)\to(a)$. In his proof, Axler says to take two elements $u_1, u_2$ of $U$ such that $v+u_1=w+u_2$. Whatever follows after that, I understand, but this beginning I don't. How can we just suppose this from $(c)$?
Here is the the... |
H: Let $X$ be a standard normal random variable and $Z$ be a random variable taking values $\{-1,1\}$ with probability $\frac{1}{2}$
Let $X$ be a standard normal random variable and $Z$ be a random variable taking values $\{-1,1\}$ with probability $\frac{1}{2}$
Let $Y=XZ$, determine whether $X$ and $Y$ are independen... |
H: Find Probability Of First Rolling an even number and then rolling a
The sides of a cube show numbers $2, 3, 3, 4, 4, 4$. Alice is rolling this cube three
times. Find the probability that the first roll results in an even number, and the sum
of the numbers obtained from the second and third rolls is six.
My Work: $\... |
H: Lebesgue dominated convergence counterexample
I'm working on the following problem:
Given a sequence of integrable functions $f_n: \mathbb{R} \to \mathbb{R}$ with $f_n \to 0$ pointwise and $|f_n(x)|≤ \frac{1}{|x|+1}$ for all $x$ and $n≥1$,
prove or find a counterexample of the following assertion: $$\lim_{n \to \in... |
H: Number of functions $f: X \to X$ with $k$ being the minimal such that $f^k(a) = b$
With some notations added, I am trying to calculate $\Psi_X$ where:
Given finite set $X$ define $\Psi_X: X \times X \times \mathbb{N} \to \mathbb{N}$
where $\Psi_X(a, b, k)$ is the number of functions $f: X \to X$ such that $k$ is th... |
H: Find the derivative of $f(x)= \int_{\sin x}^{\tan x} \sqrt{t^{2}+t+1}\, \mathrm d t$
Find the derivative of $$f(x)=\int_{\sin x}^{\tan x} \sqrt{t^{2}+t+1}\,
\mathrm d t$$ with respect to $x$
So from may understanding, I need to apply the fundamental theorem of calculus and then differentiate. I think the upper ... |
H: Is there a metric space on $\omega^\omega$ such that $\alpha+n\to\alpha+\omega$ as $n\to\infty$?
Is there a metric space on $\omega^\omega$ such that $\alpha+n\to\alpha+\omega$ as $n\to\infty$?
Let $\omega^\omega$ be the set of all ordinals less than $\omega^\omega$ then I seek a function:
$d:\omega^\omega\times\... |
H: If $B_1\subseteq\mathbb R^d$ and $f$ is a diffeomorphism of $B_1$ onto an open subset of $\mathbb R^d$, then $B_1$ is open
Let
$d\in\mathbb N$
$B_1\subseteq\mathbb R^d$
$\Omega_2\subseteq\mathbb R^d$ be open
$f:B_1\to\Omega_2$ be a $C^1$-diffeomorphism (in the sense of equation $(2)$ in this question)
Why can we... |
H: Polynomial bijections from $\mathbb{Q}$ to $\mathbb{Q}$
Prove or Improve: Polynomials $f\in \mathbb{Q}[x]$ which induce a bijection $\mathbb{Q}\to\mathbb{Q}$ are linear.
The question of existence of a polynomial bijection $\mathbb{Q}\times\mathbb{Q}\to \mathbb{Q}$ is open, as discussed in this MO thread, this post ... |
H: Let $a, u$ be vectors in $\mathbb{R}^n$ where $|u| = 1$. Show that there is exactly one number $t$ such that $a - tu$ is orthogonal to $u$.
Let $a, u$ be vectors in $\mathbb{R}^n$ where $|u| = 1$. Show that there is exactly one number $t$ such that $a - tu$ is orthogonal to $u$.
My attempt:
I tried expanding $(a - ... |
H: Why doesn't u-Substitution work for $\int \ln({e^{6x-5}})\,dx$?
I was trying to evaluate the indefinite integral $\int \ln({e^{6x-5}})\,dx$. I know that the correct way to solve it is to use the following property of logarithms: $$\ln{e^{f(x)}}=f(x)\ln{e}=f(x)$$
Using this property, the integral becomes $\int 6x-5\... |
H: Having trouble determining the quotient group in an algebraic topology course
I am working on an algebraic topology course (hatcher's book) and it has been quite a time since I took akgebra. I have my exams soon and I want a suggestion for a chapter or a resource online that helps me understand this:
For example, i... |
H: Rank and Jacobian matrix of smooth $F:M\to N$ between manifolds
I'm currently reading about submersions and immersion's Lee's Introduction to Smooth Manifolds (p.77), and I'm slightly confused about what is meant when he says that the rank of $F$ and $p$ is "the rank of the Jacobian matrix of $F$ in any smooth char... |
H: modulus question!!
I just have a question that is it:
I want to know the equation that finds an unknown number which is a number that when we will mod it with 17 it is equal to 3 and when we mod it with 16 it is equal to 10 and when we will mod it with 15 it will equal to 0.
in other words, I am a programmer and I ... |
H: Problem with showing that any nonempty open subset of plane is not contained in countable sum of segments and usage of this fact
I have a problem with showing the fact which states that none of nonempty subset of plane is not contained in countable sum of segments. It seems to be trivial, but maybe my intuition is ... |
H: Find a norm for $\mathbb{R}^d$
Let $B : \mathbb{R}^{d} \to \mathbb{R}^{d}$ be a linear isomorphism such that all eigenvalues have absolute value less than $1$. Show that there is some norm in $\mathbb{R}^{d}$ for which the operator norm of $B$ is less than $1$.
The operator norm is
$$\| B \| := \sup \, \left\{ \... |
H: Is $(\mathbb{Q}, +)$ an essential subgroup of $(\mathbb{R},+)$?
Given $H$ subgroup of $G$, we say that $H$ is a essential subgroup of $G$ if, for every non-trivial subgroup $K$ of $G$ we have that $H\cap K$ is not the trivial subgroup.
An example is $\mathbb{Z}$, which is an essential subgroup of $\mathbb{Q}$. I wo... |
H: General Method To Find All Of The Isomorphism Classes Of Groups Of A Particular Order
Ok, so bare with me here, there's quite a few questions. I am looking at this website: https://www.math.wisc.edu/~mstemper2/Math/Pinter/Chapter13F. It's basically explaining the method of going about finding all (two: $\mathbb{Z}_... |
H: Problem about uniform convergence of series of functions
Prove the following series are not uniformly convergent in $[0,1]$:
\begin{align*}
&1.\quad\sum\limits_{n=0}^\infty x^n\log x\\
&2.\quad\sum\limits_{n=0}^\infty \frac{x^2}{(1+x^2)^n}
\end{align*}
A common way to prove $\sum f_n(x)$ is not uniformly convergent... |
H: Expected number of steps needed until every point is visited in bounded simple symmetric random walk?
I was wondering how to calculate this. Say the state-space is $\{1, \dots, N \}$. Would it be correct to calculate the expected value of the first hitting time of $N$ starting from $1$ by using the coupon's collect... |
H: Pre image of product of ideal
Let $f$ be a surjective homomorphism from $R$ to $S$. How pre image of product of ideal $f^{-1}(I_1...I_n)$and product of pre images of ideals $f^{-1}(I_1)...f^{-1}(I_n)$ are related. I know they need not be equal, is any of the containment holds?
AI: Let, for each $1 \leq k \leq n$,... |
H: inverse of $y(x) = 1 - \exp( - (\alpha x + \beta x^2 )) $
I'd like to compute the inverse of $y(x) = 1 - \exp( - (\alpha x + \beta x^2 )) $. I used to know a method but I can't remember how to do it. I am stuck at the step where I have:
$$- \ln( 1 - y) = x ( \alpha + \beta x ) $$
The function is not itself invertib... |
H: Suppose $t, u, v, w \in \mathbb{R}^3$. If $(t \times u) \times (v \times w) = 0$, are $t,u,v,w$ on the same plane?
Suppose $t, u, v, w \in \mathbb{R}^3$. If $(t \times u) \times (v \times w) = 0$, are $t,u,v,w$ on the same plane?
My Answer
No. Let $p_1$ be the plane described by $x + y + z = 3$ and $p_2$ the plan... |
H: Proving a result for $\prod_{k=0}^{\infty}\Bigl(1-\frac{4}{(4k+a)^2}\Bigr)$
$$\prod_{k=0}^{\infty}\Bigl(1-\frac{4}{(4k+a)^2}\Bigr)=\frac{(a^2-4)\Gamma^2\bigl(\frac{a+4}{4}\bigr)}{a^2\Gamma\bigl(\frac{a+2}{4}\bigr)\Gamma\bigl(\frac{a+6}{4}\bigr)}$$
According to WA. I attempted using
$$\prod_{k=0}^{\infty}\Bigl(1-\fr... |
H: calculating Area inside intersection of circle and ellipse using line integral
Consider a circle parametrized as $(r\cos (t), r \sin (t))$ and an ellipse parametrized as $(a\cos (t), b \sin (t))$.
Assuming that $a>r>b$, you find the area of region of intersection of circle and elipse by setting up line integral an... |
H: The action of $SL(2,\mathbb R)$ on $T^1(\mathbb H)$ is transitive?
Let $\mathbb H$ to be the complex upper-half plane and let $SL(2,\mathbb R)$ act on $\mathbb H$ by
$$\phi(z)=\frac{az+b}{cz+d},$$
where $\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}\in SL(2,\mathbb R).$
A book I am reading says $SL(2,\mathbb R)$ ac... |
H: Find exact value of $f^{-1}(f(a))$
Given the function $$
f(x)=\frac{1}{4}\left((x-1)^{2}+7\right)
$$
The first part of the question asks to find the largest domain containing the value $x=3$ for which $f^{-1}(x)$ exists. I determined the domain to be $x≥1$.
The second part of the question is:
Let $a$ be a real num... |
H: Necessity of uniformity in "almost uniform convergence $\implies$ convergence a.e"
Let $(X,\mathcal{A}, \mu)$ be a measure space, and $E$ a Banach space (for this discussion, a metric space suffices I guess). We say a sequence of functions $f_n:X \to E$ is $\mu$-almost uniformly convergent if for every $\delta>0$, ... |
H: How to find the Number of Roots of a Polynomial in a Real Range
Is there a way to efficiently find the number of real roots of a polynomial $P$ in a range $[a,b]$ with $a,b \in \mathbb{R}$? You may/may not know much about the coefficients of the polynomial, so I want methods that work based on the fact that it's a ... |
H: What's the difference between $v = a\cdot t$ and $\vec{v} = \int \vec{a} \, \mathrm dt$
In highschool, I learned $v = at$ and in university, I am learning $\vec{v} = \int \frac{\vec{F}}{m} \, \mathrm dt = \int \vec{a} \, \mathrm dt$.
I understand one is for $v= at$ is for one-dimension and the latter for multiple d... |
H: Conditional Expectation Properties
Let $(\Omega, \mathcal{F}, P)$ be a probability space and $G$ a finite group of measurable, bijective maps $g: \Omega \rightarrow \Omega$ which are $P$ invariant, i.e. they have the property $P(g^{-1}(A)) = P(A) \quad \forall A \in \mathcal{F}$, and define $$\mathcal{C}_G \equiv \... |
H: How to factorize $a^2-2ab+a^2b-2b^2$?
I have been stuck on factorizing this:
$$a^2-2ab+a^2b-2b^2$$
I thought I could solve it by making $(a+b)$ as one factor but it didn't work then I tried to add and deduct some terms which that didn't lead me to anything either.
I don't really know what to do next.
AI: $a^2-2ab+a... |
H: Show that $\mathcal A_1$ $\cap$ $\mathcal A_2$ is also a $\sigma$-algebra
I am currently studying for a measure theory final and have come across a past short exam question that reads "Let $\mathcal A_1$ and $\mathcal A_2$ be a $\sigma$-algebra of subsets of a set X. Show that $\mathcal A_1$ $\cap$ $\mathcal A_2$ i... |
H: Derivative of $h(x,y) = \lambda f(x) + \mu g(y)$
Let $A \subseteq \mathbb{R}^m$ be open and $B \subseteq \mathbb{R}^n$ be open. Suppose $f: A \to \mathbb{R}^k$ is differentiable at $a$, and $g: B \to \mathbb{R}^k$ is differentiable at $b$. Define $h: A \times B \to \mathbb{R}^k$ by $h(x,y) = \lambda f(x) + \mu g(y)... |
H: Proving solution space for $y'+a_0y=0$ has $\{{e^{-a_{0}t}}\}$ as a basis
I am trying to prove that the solution space for $y'+a_0y=0$ has $\{{e^{-a_{0}t}}\}$ as a basis. (From Friedberg Linear Algebra Thm2.30)
First we can see that ${e^{-a_{0}t}}$ is a solution since
$y'(t)+a_0y(t)=-a_0e^{-a_0t} +a_0e^{-a_{0}t}=... |
H: Are there operations that can't be defined using a rule, and if they exist what is their significance?
Wikipedia:Operations(mathematics)
In mathematics, an operation is a function which takes zero or more input values (called operands) to a well-defined output value
What I took away from this fact was that operat... |
H: Example of a weak convergent sequence that is not strongly convergent in $\ell^p(\mathbb{N})$
Here $\ell^p(\mathbb{N})$ is the space of non-negative integer sequences $\{x_n\}_{n \in \mathbb{N}}$ where
$$\sum_{n \in \mathbb{N}} |x_n|^p < \infty$$
So I'm looking for an example of a sequence in this space that conver... |
H: Regular hexagon divided into triangles
Problem: Give an regular hexagon and an interior point of this, join this point with each vertex. The hexagon is divided in $6$ triangles, paint the triangles alternately. Show that the sum of the areas of the painted triangles is equal to that of the unpainted triangles
The s... |
H: Defining the natural almost complex structure on a complex manifold.
The definition of an almost complex structure is as follows. If $X$ is a differentiable manifold and $TX$ is its tangent bundle, then the endomorphism $I: TX \to TX$ defines an almost complex structure if $I \circ I = -1$ on all the fibers. If $X$... |
H: How to say a variable is invertible in Macaulay2?
I'm a very beginner in Macaulay2, so I apologize if this question is too trivial...
I'm using Macaulay2 for a computation involving over $30$ variables. Roughly speaking I have a $4\times 4$ matrix where entries are polynomials while coefficients are also variables.... |
H: Integrable function $f$ such that $\int_I f(x)dx=0$ for intervals of arbitrarily small length.
A past qual question from my university reads:
Let $f$ be an integrable function satisfying $\int_0^1 f(x)dx=0$. Prove that there are intervals $I$ of arbitrarily small positive length such that
$$\int_I f(x)dx=0$$
I'm no... |
H: A Rational Parameterization of Multiple Simple Expressions (Or the intersection of two rational parameterizations)
Context
I am interested specifically in all rational values of $x$ for which $\sqrt{1-x}$ and $\sqrt{1+x}$ are rational.
In general; however, I am curious if there is a method for taking any number of ... |
H: If $P$ is on the circumcircle of a triangle, show that the feet of the perpendiculars from $P$ to the side-lines of the triangle are collienar
Let $ABC$ be a triangle and $P$ be any point on its circumcircle. Let $X,Y,Z$ be the feet of the perpendiculars from $P$ onto lines $BC, CA$ and $AB$. Prove that points $X,... |
H: Show $e^{-tA}$ is a trace class operator, $t>0$
I have the next definition: $Tr(A)=\sum_n<u_n,Au_n>$, where $A$ is a positive linear operator on $H$ (Hilbert), and $\{u_n\}$ is an orthonormal base of $H$. And an operator is trace class if $Tr(A)<\infty$.
Let $A:D(A)\subset H\to H$ a positive, self-adjoint, densely ... |
H: What is wrong with this derangement argument $((n-1) !(n-1))$?
The number of derangements for $n$ objects is given by the recursive relation:
$$!n = (n-1) (!(n-1) + !(n-2))$$
This can be easily proved (for example, see the argument on Wikipedia page). Before looking at this argument, I thought along these lines: su... |
H: If G is a connected graph and C is a cycle from G, my question is: G-C is connected graph?
If G is a connected graph and C is a cycle from G, my question is: G-C is connected graph? This question is related with clasiffication surfaces theorem. If the Euler characteristic is lower than 2, then exist a simple curve ... |
H: Understanding why the answer is no?
Let $P (A) = 0.7$, $P(B^c) = 0.4$ and $P(B ~\text{and} ~C) = 0.48$
a. Find $P (A ~\text{or}~ B)$ when $A$ and $B$ are independent
$P(B) = 1 - P(B^c) = 1 - 0.4 = 0.6$
Seeing as $A$ and $B$ are independent $P(A ~\text{and}~ B) = 0.7 \times 0.6 = 0.42$
$P (A~\text{ or}~ B) = 0.7 + 0... |
H: Solve $ \frac{d^2y}{dx^2} \cos x + \frac{dy}{dx} \sin x - 2y \cos^3 x = 2\cos^5x $ by a suitable transformation
Consider $ \frac{d^2y}{dx^2} \cos x + \frac{dy}{dx} \sin x - 2y \cos^3 x = 2\cos^5x $.
By a suitable transformation, reduce this equation to a second order linear differential equation with constant coef... |
H: Evaluating limits of integrals
How to evaluate $$\lim_{n \to \infty}\sum_{m=1}^{\infty}\int_{0}^{\infty} \left(\frac{ m+x}{(m^n+x^n)^n} \right )dx$$
I made the substitution $$x = mt$$and factored out $$m^{-(n^2-2)}$$. I got this:
$$\lim_{n \to \infty} \left(\sum_{m=1}^{\infty}m^{-(n^{2}-2)}\right)\int_{0}^{\infty}... |
H: Inequality with a High Degree Constraint
This question-
Suppose that $x, y, z$ are positive real numbers and $x^5 + y^5 + z^5 = 3$. Prove that $$ {x^4\over y^3}+{y^4\over z^3}+{z^4\over x^3} \ge 3 $$
The inequality has a high degree constraint which can convert a $5$-degree polynomial to a $0$-degree term and mak... |
H: $f$ isn't necessarily bijective but still $f^{-1}$ shows up
If $A$ is compact, is then $f(A)$ compact?
The answer here by David Mitra uses $f^{-1}$, however we only know $f$ is continuous in its domain, so how do we come up with the inverse?
Its not mentioned to be strictly montone either. I do think it's a stupid ... |
H: Relative error when exact quantity is $e^{-200}$ and significant digits; what's going on?
Caveat: I've already searched for this topic here in MSE but also on other sites, but I have not still found anything that can answer my doubt.
I'm checking my own implementation of a code. The context is not important. The c... |
H: Finding the general solution of a system of Differential Equations
I need helping find the general solution to the following systen
$$ x'=2t^2+2-4x+6y $$
$$ y'=-2t^2-t+6-3x+5y $$
I know i need to turn the 2 equations into a matrix, but I can't figure out how to do it.
So far I have the matrix for the $x$ and $y$ va... |
H: Does weak continuity imply continuity?
I have come across the following excerpt from a mathematical Statistics book:
where $H$ and $J$ are Hilbert spaces and $H^{\star}$ is the dual space. For me, the statement after Definition 10 is unconvincing and I cannot, in general, show that weak continuity implies continui... |
H: Suppose $x_1,x_2$ and $x_3$ are roots of $(11 - x)^3 + (13 - x)^3 - (24 - 2x)^3$ . What is the sum of $x_1 + x_2 + x_3$?
Suppose $x_1,x_2$ and $x_3$ are roots of $(11 - x)^3 + (13 - x)^3 - (24 - 2x)^3$ . What is the sum of $x_1 + x_2 + x_3$ ?
What I Tried :- I expanded the expression and got :-
$$ \rightarrow (11... |
H: Is a dense subset in the domain of a closed, densely defined linear operator a core?
Let $X_0,X_1$ be Banach spaces. Let $A:D(A)\subseteq X_0\to X_1$ be a closable linear operator. Recall the definition of a core for such an operator:
A set $\mathcal D\subseteq D(A)$ is called a
core for $A$ if $\overline{A_{\math... |
H: Number of ways to select a target
In how many ways given 8 targets can be shot (one at a time), if no target can be shot until the target (s) below it have been shot ?
My approach :
${3 \choose 1}$ to select any group and $1$ way to shoot it. Followed by ${3\choose 1}$ to select it again and $1$ way to shoot it.... |
H: how to compute $\sum{\frac{(s+k)!}{s!k!}*x^k}$
For $\sum_{k=0}^{\infty}{\frac{(s+k)!}{s!k!}x^k}$, $0\leq x\leq1$. It is not binomial. So how can we simplify the factorial?
AI: It is binomial:
$\frac{(s+k)!}{s!k!} = \frac{1}{k!}(s+k)(s+k-1)\cdots(s+1) = (-1)^k\frac{1}{k!}(-s-1)(-s-2)\cdots (-s-k) = (-1)^k\binom{-s-1... |
H: Enclose open interval as $ x\to \infty$
Can I "close" an open interval $[0,\infty)$ as $x$ approaches infinity with some real number, if given that
$\displaystyle \lim_{x \to \infty }f(x)=f(0)$ ?
The final goal of the exercise is to prove that $f$ isn't one-to-one. So I thought I could use Weierstrass theorem to pr... |
H: Bounds for $\frac{\sigma(q^k)}{2\sigma(q^{k-1})}$ in terms of $q$ and $k$
If $q$ is a prime number and $k$ is a positive integer, does the following double-sided inequality hold?
$$\frac{q}{2} + \frac{q - 1}{2q^k} < \frac{\sigma(q^k)}{2\sigma(q^{k-1})} \leq \frac{q}{2} + \frac{1}{2q^{k-1}}$$
Here, $\sigma(x)$ is ... |
H: Factorisation of a polynomial.
Let $F$ be a field and $\operatorname{char} F = p$. If $x^p - x - a$ is reducible in $F[X]$, I am to prove that the irreducible factors of the polynomial have at most degree 1. The case for $p = 2$ is easy. I have not been able to progress any further. All I know is that $F[X]$ is an ... |
H: Prove that this set of functions is not a subspace
It was asked to prove that the set $X=\{f: f(x) = (f(x))^{2}\}$ it is not a subspace of continuous functions from $\mathbb{R}$ to $\mathbb{R}$.
I took $f(x)=1, \forall x$, and $g(x)=1, \forall x$. So both $f,g$ satisfy the condition. But, $f(x)+g(x)=1+1=2 \neq [f(x... |
H: Some point set topology regarding the set $[7, \infty)\setminus \mathbb{Q}$
Consider the set $A=[7, \infty)\setminus \mathbb{Q}$.
a) Determine $\operatorname{int}A$, $\operatorname{cl} A$, $A'$ and $\delta A$.
b) Is $A$ connected or compact?
Ok, so for a) I think that $\operatorname{int}A=(7, \infty) \setminus \mat... |
H: Are these norms equivalent in the product space?
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and let $p_1, p_2\geq 1$. Consider the product space $W:=W_0^{1, p_1}(\Omega)\times W_0^{1, p_2}(\Omega)$ equipped with the norm
$$\Vert (u, v)\Vert_W = \Vert u\Vert_{W_0^{1, p_1}} + \Vert v\Vert_{W_0^{1, p_2}}... |
H: Continuous inverse of an injective linear function
$X,Y,Z$ are Banach spaces, $A:X\to Y$ and $B:X\to Z$ are continous linear injective functions and $B$ is also compact. Moreover, there exists $C>0$ such that $\Vert x\Vert\leqslant C\Vert Ax\Vert+C\Vert Bx\Vert$. To show is there exists some constnat $D$ such that ... |
H: Question about strongly convergent nets.
Consider the following theorem in Murphy's book "$C^*$-algebras and operator theory":
Why do we need to truncate the net in order to conclude that $(u_\lambda)_{\lambda}$ is bounded below?
Would the following be correct? Fix $\lambda_0 \in \Lambda$ and consider $\Lambda':= ... |
H: Let $f:ℝ→ℕ$ be onto. Does there exist a $g:ℕ→ℝ$ such that $f(g(b))=b$ for all $b∈ℕ$?
I'm reading Classic Set Theory for Guided Independent Study, and they introduced ZF set theory (no axiom of choice still) and the construction of integers, real, rational and natural numbers. The books says that it's impossible to ... |
H: In a pretriangulated category, a morphism is an isomorphism if and only if its homotopy kernel and homotopy cokernel are zero
Let $\mathcal{T}$ be a pretriangulated category with suspension $\Sigma$ (assumed to be an automorphism) and a class of distinguished triangles. $v\colon Y\to Z$ is a homotopy cokernel of $u... |
H: Confusion with $U(1)$ and $SU(2)$
I was reading Physics from Symmetry from Jakop Schwichtenberg and I got confused by the definitions of groups $U(1)$ and $SU(2)$.
As far as I understood, unit complex numbers with the ordinary multiplication forms a group and it is called $U(1)$. $U$ for its being unitary ($U^*U =1... |
H: Solving $x^8 - x^5 + x^2 - x + 1 > 0$ over $\mathbb{R}$
I could not find any decent approach to solve this inequality. I would appreciate any help, and input if this is even possible to solve(without a computer).
$$x^8-x^5+x^2-x+1>0$$
AI: Using the AM-GM inequality, we have
$$x^8 + \frac{x^2}{2} \geqslant 2\sqrt{x^... |
H: A sequence $\{ x_{n}\}$ converges to $x \in (X, d)~$ if and only if $~\lim\limits_{n \to \infty} d(x_{n}, x) = 0$
$\blacksquare~$Problem: A sequence $\{ x_{n}\}$ converges to $x \in X~$ if and only if $$\lim_{n \to \infty} d(x_{n}, x) = 0$$
Where $(X, d)$ is a Metric Space.
$\blacksquare~$My approach:
$\bullet~$If... |
H: Prove that $p^n \nmid ((p - 1)n!)$ for all primes $p$
Prove that $p^n \nmid ((p - 1)n!)$ for all primes $p$ .
First I am thinking maybe modular arithmetic will help (although I am not sure) , and I don't know a quick and a general proof of this . Can anyone help ?
AI: It is well-known that the maximal $k$ with $p... |
H: Weak Topology and the induced topology
Given a normed space $E$ with a subspace $M$, it is known that the weak topology on $M$ is the same as the induced topology of the weak topology on $E$. Why is this the case? From the Hahn-Banach theorem, we can extend the linear functionals on $M$ to $E$. So my intuition is t... |
H: Number of conjugacy classes of maximal subgroups
$G$ is a finite group. If $G$ has, say, $n$ conjugacy classes of maximal subgroups, can we say that each subgroup of $G$ has at most $n$ conjugacy classes of maximal subgroups?
I tried some small groups, $S_4$ for example. Is it true for all finite groups?
AI: No. Fo... |
H: How to transform $z$ into $\hat z$
Assume that we have vector $a = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix} \in \mathbb R^n$, $b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} \in \mathbb R^n$, $c = \begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix} \in \mathbb R^n$.
Then, the v... |
H: Global minimum for $\frac{2(q - 1)(q^k + 1)}{q^{k+1} + q - 1}$, if $q \geq 5$ and $k \geq 1$
Let $q$ be a prime number, and let $k$ be an integer.
THE PROBLEM
Does the function
$$f(q,k) = \frac{2(q - 1)(q^k + 1)}{q^{k+1} + q - 1}$$
have a global minimum, if $q \geq 5$ and $k \geq 1$?
MY ATTEMPT
I tried asking Wol... |
H: Epsilon recursion and ZF-Inf+TC in the Inverse Ackermann Interpretation
In the paper On Interpretations of Arithmetic and Set Theory of Kaye and Wong they write:
Equipped with $\in$-induction, we obtain an inverse interpretation of
PA in ZF−Inf*. The plan is to define a natural bijection $p: V \to \text{On}$
betwe... |
H: Find what is span of 2 linearly independent vectors is
I have been trying assignment questions of linear algebra and I am unable to solve this particular question
Let $x=\left(x_{1}, x_{2}, x_{3}\right), y=\left(y_{1}, y_{2}, y_{3}\right) \in \mathbb{R}^{3}$
be linearly independent. Let $\delta_{1}=x_{2} y_{3}-y_{... |
H: Need help in finding limit.
I have solved the limit of an expression like this.
Solution
But the answer is 1/2 by using L Hospital rule. Why am I wrong?
AI: The pieces of a function in a limit have to go to their limits together. When you replace $\ln(1+x)/x$ by $1$, then you're letting one bit run on ahead of the... |
H: Paracompact Hausdorff Space with Dense Lindelof subset is Lindelof
Let $X$ be a Paracompact Hausdroff space with a dense subset $A$ which is Lindelöf. Then, $X$ is Lindelof
I've written down my attenpt below -
As per the hint in the problem, as a paracompact $T_2$ space is regular, all I have to do is show that e... |
H: prove that if $|f(z)|\geq |z|+|\sin(z)|$ then it cannot be an entire function
Problem: Prove that if $\forall z \in \mathbb{C}.|f(z)|\geq |z|+|\sin(z)|$ then it cannot be an entire function.
I thought about claiming that $f$ must be a polynomial because it has a pole in infinity, but I stuck why it polynomial can... |
H: Is there a formula for generating all positive integers that cannot be written as a linear combination over nonnegative integers?
Let $a,b \in\mathbb{Z}_+$ such that $a \leq b$. We know that when $a$ and $b$ are coprime, then the largest integer that cannot be written as $am+bn$ for some nonnegative integers $m$ an... |
H: Prove $\sum_{n=0}^{\infty} \frac{\Gamma(n+(1/2))}{4^n(2n+1)\Gamma(n+1)}=\frac{\pi^{3/2}}{3}$
Prove $$\sum_{n=0}^{\infty} \frac{\Gamma\left(n+\frac{1}{2}\right)}{4^n\left(2n+1\right)\Gamma\left(n+1\right)}=\frac{\pi^{\frac{3}{2}}}{3}$$
The original sum is multiplied by $\frac{\sqrt{\pi}}{2}$ and so it equals $\frac{... |
H: Canadian Mathematical Olympiad 1987, Problem 4
On a large flat field, $n$ people $(n>1)$ are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd, show that there is at... |
H: How to find the generators of a principal Ideal?
Suppose I have $\mathbb{Z}/24\mathbb{Z}$ and $I = \{0, 3, 6, 9, 12, 15, 18, 21\}$.
$I$ is a principal ideal.
Is there a method to find ALL the generators ?
Thanks in advance !
AI: Hands-on approach: We know that the order of $a\in G$ an element equals the order of th... |
H: If $15$ distinct integers are chosen from the set $\{1, 2, \dots, 45 \}$, some two of them differ by $1, 3$ or $4$.
$\blacksquare~$ Problem: If $15$ distinct integers are chosen from the set $\{1, 2, \dots, 45 \}$, some two of them differ by $1, 3$ or $4$.
$\blacksquare~$ My Approach:
Let the minimum element chosen... |
H: Derivative of $a^{T}Xb$ with respect to b, where $a, b$ is a $d$-dim vector and $X$ is a $d\times d$ matrix
Since the derivative of $a^{T}Xb$ with respect to a is $Xb$, I was wondering how do I solve the derivative of $a^{T}Xb$ with respect to $b$?
AI: It's easy to get confused, due to a lack of clear and uniform ... |
H: Evaluating 8th derivative of $(e^x-1)^6$ at $x=0$
8 distinct objects are distributed into 7 distinct boxes. Find the number of ways in which these objects can be distributed to exactly 6 boxes.
I have solved this question quite easily using method of division and distribution ${7 \choose 6}(\frac{8! \times 6!}{5!\t... |
H: Find the bilinear transformation which maps $z=(1, i, -1)$ respectively into $(w=i, 0, -i)$
Find the bilinear transformation which maps $z=(1, i, -1)$ respectively into $w =(i, 0, -i)$
My try:
Here, $w_1=i$, $w_2=0$, $w_3=-i$, $z_1=1$, $z_2=i$, $z_3=-1$
$\text{As the formula states,}$
$$\begin{align}\\
&{\begin{a... |
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