text stringlengths 83 79.5k |
|---|
H: If $A=\begin{bmatrix}a&b\cr b&c\end{bmatrix}$ show $2|b|≤$difference of eigenvalues
If $A=\begin{bmatrix}a&b\cr b&c\end{bmatrix}$ and let $\alpha\ \&\ \beta$ be the eigenvalues where $\alpha\ge\beta$. Show $2|b|≤\alpha-\beta$
I first tried to find the eigenvalues by solving for:
$$(a-\lambda)(c-\lambda)-b^2=0$$
$$... |
H: Divergence of Infinite Series: ${e^{(tx)}\left(\frac{1}{2}\right)^{x}}$
I need to show whether the following infinite series converges, and if it does, to which value. Here, $t$ is a constant:
$$\sum_{x=1}^{\infty}{e^{tx}(\frac{1}{2})^{x}}$$
This is my solution:
$$\sum_{x=1}^{\infty}{e^{tx}(\frac{1}{2})^{x}} = \sum... |
H: Christoffel symbols, dual space
I am confused with the definition of Christoffel symbols for the dual space.
Let $M$ be some manifold, $x_i$ local coordinates
The Christoffel symbols are defines as
$\nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k$
where $\nabla$ is the Levi Civita connection on $M$.
Now I... |
H: What should the maximum possible rank of a matrix be under certain conditions?
I have been trying questions in Linear Algebra and I couldn't correctly solve this particular question.
Let $A$ be a non-zero $4\times 4$ complex matrix such that $A^2= 0$. What is the largest possible rank of $A$?
All eigenvalues mus... |
H: Show that $f(X) = X^2$, where $X \in M_n(\mathbb R)$, is smooth
Considering that $M_{n}(\mathbb{R})$ can be identified with $\mathbb{R}^{n^{2}}$, define $f(X) = X^2$. I want to prove that $f$ is a smooth function.
I don't know how can I relate the matrices with the concept of differentiability in Euclidean space. F... |
H: I need to prove that in a complete oriented graph , you can change the orientation of one edge so that the resulting graph is connected.
Prove that in a complete oriented graph, you can change the orientation of one edge so that the resulting graph is connected.
I tried to use induction.
but I dont know how
if $n=3... |
H: Proof that any number is equal to $1$
Before I embark on this bizzare proof, I will quickly evaluate the following infinite square root; this will aid us in future calculations and working:
Consider $$x=\sqrt{2+\sqrt{{2}+\sqrt{{2}+\sqrt{{2}...}}}}$$
$$x^2-2=\sqrt{2+\sqrt{{2}+\sqrt{{2}+\sqrt{{2}...}}}}=x \implies x^... |
H: Show that $|a|\leq \int_{-1}^{1}\,|ax+b|\,\text{d}x$
Show that $\displaystyle|a|\leq \int_{-1}^{1}\,|ax+b|\,\text{d}x$.
I did this problem but in a way that I do not like. I just divided it into many cases. Assumed $a>0$ and then did cases depending on where $-b/a$ is. However, this seems very computational and t... |
H: Constructing one-one functions under a constraint using derangements
This question has been asked before, but I wanted to approach it via derangements: and I haven't seen this answered satisfactorily on MSE using derangements.
Let $A=(1,2,3,4,5)$ and $B=(0,1,2,3,4,5)$. We need to find construct one-one functions $f... |
H: Generate two valid vertices of isosceles triangle, given one vertex, an angle, and a distance
Trigonometry question:
I want to find a way to randomly sample the coordinates of the two remaining vertices $C_2$ and $C_3$ of an isosceles triangle, given one initial coordinate $C_1$. I have the coordinate of one vertex... |
H: Intuition behind equation for volume of a cone without calculus
I have come back to study geometry a bit and I'm kind of stuck at deriving the volume formula for a cone. I have read the calculus-based derivation and it totally makes sense, but calculus has been around for 200+ years, cones have been around forever.... |
H: Problem about choice of balls from an urn with Bayes rule
An urn contains 2N balls numbered from 1 to 2N. An experiment with It consists of choosing, randomly and without replacement, two balls from the ballot box, consecutively. Calculate the probability that the first choice results in an even number under the hy... |
H: Show that $\int_a^be^{tf(x)}\text{d}x \underset{}{\sim} \int_{x_0-\delta}^{x_0+\delta}e^{tf(x)}\text{d}x$ under an hypothesis.
Let $a < b$, and let $f : [a, b] \mapsto \mathbb R$, $f \in C^\infty[a,b]$.
Suppose there exists a unique point $x_0 \in [a,b]$ where $f$ reaches its maximum, we have $a < x_0 < b$, $f''(x_... |
H: If the differential of a function is constant, does the function has to be affine?
Suppose $f:V \subset R^n \to R^m$ is a smooth function such that $D_f = A$ for some constant matrix $A$ in $V$, and $V$ is open. My question is, is it true that $f(x) = y + Ax$ for some constant $y \in R^m$? If so, why?
AI: This is n... |
H: Sigma Notation Equation?
So I came across this post on stackoverflow which discussed the ranges of integer variables in C++. And the last point of the top-voted response was the unsigned long long int which apparently ranges from 0 to 18,446,744,073,709,551,615 (18.5 quintillion?). But that response did not mention... |
H: What's $(1 2 3)(1 4 5)$? Everybody gives a different answer.
From my calculation: $(1 4 5 2 3)$.
From Joseph Gallian's Contemporary Modern Algebra, 9th edition, page 100:
From WolframAlpha:
AI: Technically, the book isn't mistaken but might be slightly misleading.
The order of the permutation $(1 2 3)(1 4 5)$ is e... |
H: $A\subseteq\mathbb{R}$ is closed iff $\sup(A\cap[a,b])\in A$ and $\inf(A\cap[a,b])\in A$.
Suppose that $A\subseteq\mathbb{R}$. Show that the following are equivalent:
(a) $A$ is closed.
(b) If $[a,b]$ is a closed interval for which $A\cap[a,b]$ is non-empty, then $\sup(A\cap[a,b])\in A$ and $\inf(A\cap[a,b])\in A$.... |
H: Uniform convergence of $f_n(x) = \frac{x^ne^{-x}}{(2n)!}$
I came across this question while studying for my qualifying exams and it was grouped together with problems involving the Weierstrass M-test.
Let $f_n(x) = \frac{x^ne^{-x}}{(2n)!}$. Does $(f_n(x))$ converge uniformly on $[0,\infty)$?
I don't think I can a... |
H: What does it mean to converge to a point if it is not clear what the point even is?
What does it mean for a sequence to converge to an element if the limit might not necessarily be defined or known, or not necessarily in the universe under consideration (whatever this means)?
I am not just talking about real number... |
H: Increase in argument of a smooth function (Gamelin V.III. section 2 exercise 8)
The following is question 8 in chapter 8.2 of Gamelin.
Let $D$ be a domain in the complex plane and $f(z)$ a smooth complex valued function on $D$. Suppose that for any $a,b\in \mathbb{C}$ and any circle $\partial D(z_0,\varepsilon)\sub... |
H: Continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f\big(f(x)\big)=rf(x)+sx$ and $r,s \in (0, 1/2).$
I wish to find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$f\big(f(x)\big)=rf(x)+sx\quad\forall x\in\mathbb{R}\,,$$ where $r,s \in (0, 1/2)$.
Here's my work so far:
Let $r_1 ... |
H: Factorial—Googol problem help
So over the past week, I encountered this interesting problem in which I need help with, so any help is highly appreciated!
Problem: What it is least integer $n$ such that $n!$ is greater than googol?
My work so far: We are trying to find the minimal n where $n!\ge 10^{100}$. With this... |
H: Given a bilinear map, how do I produce a cross-product on cohomology?
If $M$, $N$, and $P$ are $R$-modules, and there exists a bilinear map between $M\times N$ and $P$, how do I construct a cross product $$\times: H^p(X;M)\times H^q(Y;N)\to H^{p+q}(X\times Y;P)?$$
I say let $S^*(X;M)$ be the singular cochain comple... |
H: Show that a subset $I$ of $\textbf{R}$ is an interval if and only if whenever $a,b\in I$ and $0\leq t\leq 1$ then $(1-t)a + tb\in I$.
Show that a subset $I$ of $\textbf{R}$ is an interval if and only if whenever $a,b\in I$ and $0\leq t\leq 1$ then $(1-t)a + tb\in I$.
MY ATTEMPT
We say that a subset $I$ of $\textbf{... |
H: Conditions on an entire function that would make it a polynomial
Here is the question:
Suppose $f$ is entire.
a) Suppose $|f(z)|\leq A|z|^N+B$ $\forall z\in\mathbb{C}$ where $A,B<\infty$ are constants. Show $f$ is a polynomial of degree $\leq N$.
b) Suppose $f$ satisfies $f(z_n)\rightarrow\infty$ whenever $z_n\rig... |
H: Isometry between $k$-dimensional subspace of $\mathbb{R}^n$ and $\mathbb{R}^k$
If $W\subset\mathbb{R}^n$ is a $k$-dimensional subspace of $\mathbb{R}^n$ with the usual Euclidean norm, then is there a result that says that there is an isometry between $W$ and $\mathbb{R}^k$? Could someone possibly provide either an ... |
H: Proving a non-homogeneous inequality with $x,y,z>0$
For $x,y,z>0.$ Prove: $$\frac{1}{2}+\frac{1}{2}{r}^{2}+\frac{1}{3}\,{p}^{2}+\frac{2}{3}\,{q}^{2}-\frac{1}{6} Q-\frac{3}{2} r-\frac{2}{3}q-\frac{1}{6}pq-\frac{5}{3} \,pr\geqslant 0$$
where $$\Big[p=x+y+z,q=xy+zx+yz,r=xyz,Q= \left( x-y \right) \left( y-z \right)
... |
H: Questions about proving the statement: $\tau(\omega)$ is a stopping time, iff $\{ \tau(\omega) < t\}$ $\in \mathcal{F}_t$, for all $t \geq 0$.
Assume the filtration is right-continuous ($\mathcal{F}_{t+0} := \cap_{s>t}\mathcal{F_s} = \mathcal{F}_t$) and complete, then we have that $\tau(\omega)$ is a stopping time,... |
H: Let $a$,$b$ be positive elements in a C* algebra with $\|a\|,\|b\|\leq 1$, does $\|a-b\|\leq 1$ hold?
On Murphy's C*-algebra and operator theory page 89 it says
Now suppose that $a$ is positive and $\|a\| \leq 1$. Then $u_\lambda - a$ is hermitian
and $\| u_\lambda -a\|\leq 1$.
Where $u_\lambda$ is an arbitrary a... |
H: Given $f(x)$ is continuous on $[0,1]$ and $f(f(x))=1$ for $x\in[0,1]$. Prove that $\int_0^1 f(x)\,dx > \frac34$.
Let $f$ be a continuous function whose domain includes $[0,1]$, such that $0 \le f(x) \le 1$ for all $x \in [0,1]$, and such that $f(f(x)) = 1$ for all $x \in [0,1]$. Prove that $\int_0^1 f(x)\,dx > \fr... |
H: Group Characters and Centralizer of an element
Let G be a finite group. $\hat{G}$ donates a set of all irreducible representations of G over $\mathbb{C}$. $C_G(h)$ denotes centraliser of an element $h\in G$. I want to prove
$|C_G(h)|=\sum\limits_{\chi \in \hat{G}}\chi(h)\overline{\chi (h)}$.
I dont know how to get ... |
H: Why is $\mathbb{Z}_{6}$ a free $\mathbb{Z}_{6}$ module?
$\mathbb{Z}_{6}=\{\bar{0},\bar{1},\bar{2},\bar{3},\bar{4},\bar{5}\}$. I know that in order for a module to be free, it has to have a basis. Further, I know that if $b_{1},...,b_{n}$ is a basis for $\mathbb{Z}_{6}$, then $r_{1}b_{1}+...+r_{n}b_{n}=0$, $r_{1}=..... |
H: lamellar field if line intergral zero
I'm reading in the textbook Electromagnetics with Applications and it says (page 18) that if the line integral around a closed path is zero, $$\oint_C\vec{F}.d\vec{L}=0$$
then the vector field $\vec{F}$ is lamellar (conservative). Specifically, the book says, "Any field for wh... |
H: Prove the following by changing the order of integration
This question is from my university paper:
$$\int_{0}^{\infty} \int_{0}^{x} x e^{\frac{-x^2}{y}}\; dy\;dx = \frac{1}{2}$$
I tried using DUIS method but integration is getting complicated
$f(x) = \int_{0}^{x} x e^{\frac{-x^2}{y}}\; dy$
$ \frac{d}{dx}f(x) = ... |
H: Definition of convergence of real sequence
We say a sequence converges to a real number l if given any epsilon greater than 0 there exists a natural number N such that after the first N-1 terms all the remaining terms of the sequence lie in epsilon symmetric neighborhood of l i.e|xn-l|<epsilon after first N-1 terms... |
H: Prove that the closed ball $\overline{B}(0,1) \subseteq \mathcal{C}([0,1], \mathbb{R})$ can't be covered by countably-many compact sets.
For $\mathcal{C}([0,1],\mathbb{R})$ the space of continuous functions $f : [0,1] \to \mathbb{R}$, I'm asked to prove that $\overline{B}(0,1) = \{ f \in \mathcal{C}([0,1], \mathbb{... |
H: How to prove $n^{\log n}$ is $\mathcal{O}(2^n)$?
I've seen proofs here that help with $n\log n = \mathcal{O}(n^2)$. However, if we take it a step further, how could one prove $n^{\log n}$ is $ \mathcal{O}(2^n)$?
We are assuming $n\in\mathbb{N}$. Would it extend to $n\in (0,\infty)$?
If we apply limit of $x\righta... |
H: The convergence of sequence.
i'd like to show that $a_n=-\ln(n)+$$\sum_{k=1}^{n} {{1}\over {k}} $ converges to some $\alpha$ $\in$$[0,1]$
I found that the sequence monotone decreasing sequence, by mathematics induction.
So i'd like to show that the sequence $a_n$$\in$$[0,1]$, for concluding above statement.
But I'... |
H: Density of certain space in $L^\infty (\mathbb{R},\Sigma, \mu),$ for some finite positive measure $\mu.$
Let $\mu$ be a finite positive measure on $\mathbb{R}.$ Consider the measure space $(\mathbb{R},\Sigma, \mu), $ where $\Sigma$ is the collection of all Borel sets.
Q:1) Is it true that the space of all compactly... |
H: Does existence of a left or right inverse imply existence of inverses?
Suppose $G$ is a set with a binary operation such that:
(Associativity) For all $a, b, c \in G$, $(ab)c = a(bc)$.
(Identity) There is $e \in G$ such that, for all $a \in G$, $ae = ea = a$.
(Left inverse or right inverse) For all $a \in G$, $ba ... |
H: The tangent to the curve $y = ax^3$ at the point $(5, b)$ has a gradient of $30$. Find the values of the constants $a$ and $b$.
The tangent to the curve $y = ax^3$ at the point $(5,b)$ has a gradient of $30$. Find the values of the constants $a$ and $b$.
My working so far:
$$\frac{dy}{dx} = 3ax^2$$
tangent: $y = mx... |
H: Remainder when $\prod_{n=1}^{100}(1- n^{2} +n^{4})$ is divided by $101$
What is the remainder when the expression
$$\prod_{n=1}^{100}(1- n^{2} +n^{4})$$
is divided by $101$?
If $\zeta=\dfrac{-1+\sqrt{-3}}{2}$, then
$$1-n^2+n^4=(1-n+n^2)(1+n+n^2)=(-\zeta-n)(-\bar{\zeta}-n)(\zeta-n)(\bar{\zeta}-n).$$
We then have
$... |
H: How to prove that if f is continuous that the exponential function satisfies the functional equation?
Prove that if $f : \mathbb{R}→\mathbb{R}$ is continuous with the property $f(x+y)=f(x)·f(y)$ for all $x,y \in\mathbb{R}$ and $f(1) = a > 0$, then $f(x) = e^x$.
I'd be greatfull for any help!
AI: $f(x)=f(0+x)=f(0)... |
H: Are differentiable functions of bounded variation on [0,1] also absolutely continuous?
There was a similar question on this matter, but "continuous differentiable" is too strong an assumption, so I was wondering if this can be relaxed to "differentiable".
AI: I got an answer referencing Rudin'n Real and Complex Ana... |
H: $\lim_{x \to 0}\left(\frac{\sin^2(x)}{1-\cos(x)}\right)$ without L'Hopital's rule
I'm trying to calculate $\lim_{x \to 0}\left(\frac{\sin^2(x)}{1-\cos(x)}\right)$ without L'Hopital's rule.
The trigonometrical identity $\sin^2(x) = \frac{1-\cos(2x)}{2}$ doesn't seem to lead anywhere. I also attempted to calculate us... |
H: 5 digit even number and different digits
I have a machine that creates different numbers with $5$ digits.
Of course the first digit can't be $0$.
I would like to choose one them, what is the probability that the number is even?
And what is the probability that the number is even and all the digits differ?
My soluti... |
H: Cumulative Probabilities: What am I missing here?
Sorry if this question is a bit lower-level, yet complicated, but I feel like there is something wrong and I cannot put my finger on it. This scenario is adapted from something I read elsewhere on the internet, but simplified to get the basic point across.
Lets say ... |
H: How do I find the area of the region bounded by the curve and the tangent?
The diagram shows a sketch of the graph of the curve $\displaystyle y=\frac{1}{4}x-x^3$ together with the tangent to the curve at the point $A(k, 0)$.
Find the area of the region bounded by the curve and the tangent, giving your answer as a... |
H: Why does $\left|\frac{\sin(n+1)}{2^{n+1}}+...+\frac{\sin(n+p)}{2^{n+p}}\right|\leq\frac{|\sin(n+1)|}{2^{n+1}}+...+\frac{|\sin(n+p)|}{2^{n+p}}$ hold?
I trying to understand a proof (using Cauchy's general criterion of convergence) of why the series $\sum_{n=1}^{\infty }\frac{\sin (n)}{2^{n}}$ converges .
At the beg... |
H: I don't understand uniform convergence
Assume we have some series of functions defined for any $x\geq0$:
$$S(x)=\sum_{n=1}^{\infty} \frac{3x+n}{x+n^{3}}$$
Let's assume uniform convergence. Meaning
$$\forall{\epsilon\geq0}, \exists N s.t. \forall{n \geq N}, \forall{p}, \forall{x\geq0}$$
$$ \lvert \frac{3x+(n+1)}{x+(... |
H: Understanding the monotonous convergence theorem
I'm stuck at the next point in the proof of this theorem
Monotone Convergence Theorem: If $(f_n)$ is a monotone increasing sequence of nonnegative measurable functions which converges to $f$,
then $$\int f d\mu=\lim\int f_n d\mu$$
How the integral preserves order i... |
H: Find $2f(x)\cdot f(x-8) - 3f(x+12) - 2 = 0$
Function $f$ $\in \mathbb{R}$ is odd and has a period of $4$. On a $[0,2]$ segment function $f$ is defined as $f(x)= 4x - 2x^2$. Find the set of solutions for the equation: $$2f(x)\cdot f(x-8) - 3f(x+12) - 2 = 0$$
So, here's my attempt: function has a period of $4$ mea... |
H: On the functoriality of assigning a simplicial complex to is Stanley-Reisner ring
If $k$ is a field and $\Delta$ a finite simplicial complex with vertex set $x_1, \ldots, x_n$, the Stanley-Reiser ideal of $\Delta$ is
$$I_\Delta := \left\langle \prod_{i \in S}x_i : S \not \in \Delta \right\rangle \subset k[x_1, \ldo... |
H: Curve tangent line
A curve with the equation $x^2-x+1$ has two tangent lines $a$ and $b$ that intersects at $x=1$, what is $y$? can I determine $y$ when $a$ and $b$ are perpendicular, or the gradient of $a$ and $b$ are $-1$
I tried using the second point, but knowing that they have the same gradient means they won'... |
H: Convergence of the Series $ \sum_{n=1}^{\infty} \left(1-(\ln 2)^{1 / n^2}\right) $
Check the Convergence of the
Series
$$
\sum_{n=1}^{\infty} \left(1-(\ln 2)^{1 / n^2}\right)
$$
My attempt: I feel that
$$
0\leq 1-(\ln 2)^\frac{1}{x^{2}} \leq \frac{1}{x^{2}}~~~ \forall x \in \mathbb R$$
to use direct comparison tes... |
H: Questions in Theorem 6 of Chapter 6 of Hoffman Kunze Linear Algebra
While self studying Linear Algebra from Hoffman Kunze I am unable to understand some deductions in Theorem-6 on Page 204 .
Adding Image:
$(1)$ In 7th last line from below I am not able to deduce how $q- q(c_{j})$ equals $( x-c_{j} ) h$ despite the... |
H: A question about a proof of 'Cauchy integral formula'
Theorem: Let $U \subset \Bbb C$ be an open setand $f \in H(U) $, i.e. $f $ is holomorphic on $U$.
Let $z_0\in U$ be a point and $r>0$ such that $\bar{D}(z_0,r)\subset U $. Then $\forall z \in \bar{D}(z_0,r)$, $$f(z)=\frac{1}{2\pi i}\displaystyle \oint _{\lvert \... |
H: Odds of a specific dice distribution occuring when playing Catan (2d6)
So this is messing with me, because I have no idea how I would even go about starting calculating this.
I have played a game of Catan. For those that don't know, it's a board game that involves rolling two D6 per turn. So there are 12 possible o... |
H: How to calculate $\lim _{x\to \infty }\left(\frac{x^2+3x+2}{x^2\:-x\:+\:1\:}\right)^x$
I am trying to calculate
$$\lim _{x\to \infty }\left(\frac{x^2+3x+2}{x^2\:-x\:+\:1\:}\right)^x$$
My initial thought is that it is in exponential form $\left(1+\frac{a}{f(x)}\right)^{f(x)}$.
I tried to factor the polynomials $\fra... |
H: If $\exists f : (X,\tau) \rightarrow (\{ 0,1\}, \tau_\text{discr})$ continuous, non constant, $(X,\tau) $ is not connected
Let $(X, \tau)$ be a topological space and consider the discrete topology over $\{0,1\}$;
Prove
that if, there exists a continuous non-constant mapping $ f : X \rightarrow \{ 0,1\}$,
allora $(X... |
H: Hilbert space subspace of "equally projected elements"
Apologies for the title, bit of a struggle to come up with something non-generic.
Let $H$ be a Hilbert space and $p:H\rightarrow H$ an orthogonal projection.
Suppose $h_1,\,h_2\in H\backslash\{0\}$ and
$$\alpha:=\frac{\|p(h_1)\|^2}{\|h_1\|^2}=\frac{\|p(h_2)\|^2... |
H: Proving that $\mathbb Z$ with the finite-closed topology satisfies the second axiom of countability.
In my general topology textbook there is the following exercise:
A topological space $(X,\tau)$ is said to satisfy the second axiom of countability if there exists a basis $B$ for $\tau$, where $B$ consists of only... |
H: Prove that $\sup \{f(x)+g(x):\space x\in X\}\leq \sup \{f(x):\space x\in X\}+\sup \{g(x):\space x\in X\}$
I know that this question had been answered before here but I am asking to please check a method used by me which resulted in the wrong conclusion.
Let $X$ be a nonempty set, let $f$ and $g$ be defined on $X$... |
H: Matrixmultiplication of 2 invertible Matrices is not commutative
I need to proof that the set of all invertible n x n Matrices with n > 1 is not commutative under multiplication? How could I do this? Thank you.
AI: We typically refer to the group of all invertible $n \times n$ matrices over the real numbers with th... |
H: Prove a set is of full measure
This is a problem in my textbook:
$E\subset[0,1], E$ is Lebesgue measurable, if there exists $\delta\in(0,1)$ such that for any interval $I\subset[0,1]$, $m(E\cap I)\geq \delta|I|$, then $m(E)=1$
A hint of this problem is also given in the textbook: use the lemma below and prove the... |
H: If we didn't have examples of irrational numbers, would we know they exist?
Irrational numbers are very easy to find. Square roots require only a little bit more than the most basic arithmetic. So it might be that this question is impossible to answer because it presupposes a world where math looks completely diffe... |
H: Properties of centralization in group theory
Recall Let $C\subseteq G$ where $G$ is a group. If $x\in G$ for any $c\in C,$ $xc=cx,$ then $x$ centralizes $C$
$$C_G(C) := \{x \in G : \text{ for any } x\in C, xc=cx \}$$
Clearly,
$$C_G(C)=\bigcap_{c\in C}C_G(c)$$
Prove that
a) $C\subseteq C_G(C_G(C))$
b) If $C\subsete... |
H: On interiors of Jordan curves
Recently,while studying a problem in physics whose solution required an application of the Jordan curve theorem to the phase space in order to make it rigorous, I asked myself if its possible for two Jordan curves $E$ and $F$ on the plane to be contained in each other's interiors.I tri... |
H: Why does $G$ finite abelian group with $i_G$ involutions have no subgroups isomorphic to $K_4$ if $6\nmid i_G(i_G-1)$?
I was reviewing my answer here, and have realized that the provided solution ought to work for any finite abelian group. The generalization would then be:
Claim. Let $G$ be a finite abelian group a... |
H: calculus problem involving integration and mean value theorem
http://people.math.sc.edu/girardi/m555/current/hw/MVT-Flett.pdf
the first question of this pdf
$g(x)$ is a continous function in $[a,b]$ and $ g(a)=0$, $\int_a^b g(t) \ dt=0$
to prove there exit $ c\in (a,b) $ such that $g(c)(c-a)=\int_a^cg(t) ... |
H: Does absolute convergence imply conditional convergence?
My problem is $\sum_{n=1}^{\infty} a_{n} $ absolute converges does it imply that $\sum_{n=1}^{\infty} a_{n}^{2} $ converges?
My proof for $\sum_{n=1}^{\infty} a_{n} $ absolute converges does it imply that $\sum_{n=1}^{\infty} a_{n} $
$-|a_{n}| \le a_{n} \le ... |
H: Find the integral of the second kind for Bernoulli leminscate
I have tried to solve this integral of the second kind( with respect to $x,y$) but I stumbled on finding a correct path over which to integrate. I know from the definition that I should get at $ \int\limits_{C}{{Pdx\, + Q\,dy}} = \int\limits_{C}{{P\left... |
H: Model the constraints of an integer linear program
I have to model the costraints of an integer linear program. The objective function is not important here.
Suppose we have $C$ configurations (indexed by $i = 1, \dots, C$) and $T$ time intervals (indexed by $j = 1, \dots, T$). I wish that:
a) only one configuratio... |
H: Properties about topological vector spaces
Let $E$ be a topological vector space. First I want to prove that, given a $V \subset E$ balanced and $\lambda>0$ then
$$
\lambda V \subset \beta V, \: \forall \;\lambda< \beta. \tag{1}.
$$
For this I tried the following: let $\lambda,\beta>0$ such that $\lambda<\beta$ and... |
H: How many subsets of $\{0,1,\ldots,9\}$ have the property that there are at least two elements and the sum of the two largest elements is 13?
I am not sure how to start but I think I need to consider the amount of subsets where there are 2 elements, 3 elements, and 4 elements ... separately. How do I start?
AI: Sugg... |
H: Compactness, connecteness and Hausdorffness on $ (S^2 /\mathscr{R}, \tau_{e_{|S^2}}/\mathscr{R}) $
Consider over $S^2$ with the induced euclidean topology, the equivalence relation:
$(x,y,z)\mathscr{R}(x',y',z') \iff x+y=x'+y'$
Let $X=S^2 /\mathscr{R}$ be the quotient set with the quotient topology $\tau$, Prove th... |
H: Number of 5 digit numbers with the first three numbers decreasing and the last 3 increasing?
The question is "Compute the number of 5-digit integers ABCDE, with all digits distinct, such that the first 3 digits are strictly decreasing, and the last 3 digits are strictly increasing."
This is an old question for an A... |
H: Proving that $\mathcal B$ is a basis for a product topology
In my general topology textbook there is the following exercice:
Let $\mathcal B_1$ be a basis for a topology $\tau_1$ on a set $X$, and $\mathcal B_2$ be a basis for a topology $\tau_2$ on a set $Y$. Let $\mathcal B$ be a be a collection of subsets $X \t... |
H: Given matrix $A$, find matrix $X$ such that $e^X = A$
Given the following matrix
$$A = \begin{bmatrix} -2 & 2 & 1\\ 2 & -3 & -2\\ -5 & 6 & 4\end{bmatrix}$$
how can we show that there exists a complex matrix $X$ such that $e^X = A$.
I have struggled to find the information about workaround the problem. However, the ... |
H: Riesz representation theorem vs. natural duality for $L^2$
We know that the spaces $L^p(\Omega)$ and $L^q(\Omega)$ are isometric and isomorphic for $p,q$ conjugate and $p,q \neq 1,\infty$. Call the isomorphism $l\colon L^p(\Omega) \to L^q(\Omega)$.
Take $p=q=2$. $L^2(\Omega)$ is a Hilbert space so we have a isometr... |
H: Properties of rational function of two monotone functions
Consider two increasing functions $f,g:\mathbb{N}_{>0}\to \mathbb{N}_{>0}$ and let $h:=f/g$ (we assume $g(x)\neq 0$ for all $x>0$).
In other words, $f(x)< f(x+1)$, $g(x)< g(x+1)$ holds for all $x>0$ and $h(x)=f(x)/g(x)$.
We assume that $\lim_{x\to \infty}h(x... |
H: Show that the sequence $\left\{\frac{1}{1+nz}\right\}$ is uniformly convergent to zero for all $z$ such that $|z|\geq 2$.
Show that the sequence $\left\{\frac{1}{1+nz}\right\}$ is uniformly convergent to zero for all $z$ such that $|z|\geq 2$.
The definition for uniform convergence is:
We say $\left\{f_n\right\}$ c... |
H: Proving that $\mathbb{Z_{-}} \cap \mathbb{N}=\emptyset$
In assumption that $\mathbb{N}$ and successor function ($\overline{x}$) over $\mathbb{N}$ is defined by 5 Peano axioms:
$1\in\mathbb{N}$
$n\in\mathbb{N} \Rightarrow \overline{n}\in\mathbb{N}$
$\nexists n\in\mathbb{N}:\ \overline{n}=1$
$\forall n,m,k\in\mathbb... |
H: Why $\mathrm{Restriction}\circ \mathrm{Corestriction}$ is multiplication on cohomology?
Let $G$ be a group, and let $H$ be a subgroup of index $m$.
Let $A$ be a $G$-module. we have restriction $$\mathrm{Res}: H^n(G,A)\to H^n(H,A)$$ and co-restriction
$$\mathrm{Cor}: H^n(H,A)\to H^n(G,A).$$
It is known that
$$\mathr... |
H: Is $\left(\begin{smallmatrix}0&0&1\\1&0&0\\ 0&1&0\end{smallmatrix}\right)$ diagonalizable over $\mathbb{Z}_2$?
Is $A= \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix}$ diagonalizable over $\mathbb{Z}_2$?
I tried two approaches and got two different answers so I was hoping someone could point me to a ... |
H: series involving zeta function and cotangent
I have been recently finding the values for the even positive integers of the zeta function using fourier series, and it is well know that these are all of the form $\frac{\pi^{2n}}{a_{2n}}$ and so I thought about whether or not the series below would converge:
$$S=\sum_... |
H: Constructing entire function from two functions with the same simple pole
Given two complex functions $f,g : \mathbb{C}\backslash\{z_0\} \to \mathbb{C}$ which are analytic everywhere in $\mathbb{C}$ except in one simple pole, which they have in the same point $z_0$. Does there exist a function $h : \mathbb{C} \to \... |
H: In $\mathbb{Z}[X]$ what inputs can X take?
In the polynomial rings $\mathbb{Z}[X]$ or more generally $R[X]$ what inputs can we put in the place of X.
For example in $\mathbb{Z}[X]$. X can be replaced by any real or complex numbers.
But in general can we put any member of any set in the place of X ?
AI: In $R[X]$, $... |
H: Reference for a real algebraic geometry problem
Disclaimer: I am not a mathematician by training.
I encountered the following problem in my research. Assume that I have $N$ real variables $x_1, x_2, \dots, x_N$. I am given $N$ homogeneous polynomials in the $x_i$ unknowns, each with a different degree. More specif... |
H: Why can't I minimize the squared distance?
My question is "Why can't I minimize the squared distance?" It would be a lot easier, but it yields the wrong answer.
I set out to write an example for using Newton's Method for Multiple equations and decided to show the following problem. Given a line, such as $\space 4x+... |
H: Finding $a$ and $b$ such that $\lim _{x\to \infty}\left(\frac{x^2+1}{x+1}-ax-b\right)=0 $
I'm having trouble understanding limits at infinity. For instance,
If $$\lim _{x\to\infty}\left(\frac{x^2+1}{x+1}-ax-b\right)=0 $$
where $a$ and $b$ are some real constants; find $a$ and $b$.
As per the solution the value of... |
H: Show that the power series $\sum_{n=0}^\infty \frac{z^n}{n!}$ converges uniformly for all $z$.
Show that the power series $\sum_{n=0}^\infty \frac{z^n}{n!}$ converges uniformly for all $z$.
I know that the definition for uniform convergence is: We say ${_}$ converges to uniformly on a subset of Ω iff for every $>... |
H: Pontryagin dual of an inverse limit
For a locally compact Hausdorff abelian topological group $G$, let $G^\vee = \mathrm{Hom}_{cts}(G, \mathbb{R}/\mathbb{Z})$ denote its Pontryagin dual, endowed with the compact-open topology. It is more or less easy to show that, if $(M_i)_i$ is a directed system of such groups, ... |
H: Showing that there is always an unbounded $f:X \to \mathbb{R}$ if $X$ is infinite, without choice
Consider the claim "for any infinite set $X$, there exists an unbounded $f:X \to \mathbb{R}$". If we assume the axiom of choice, this claim is trivial to prove. Indeed, given choice we know there exists an $S \subset X... |
H: values of q for which tangent integral is converges
Finding value of $q$ for which $$\int^{1}_{0}\frac{1}{(\tan (x))^{q}}dx$$ converges
What i try::
Let $\tan x=t.$ Then $\displaystyle dx=\frac{1}{\sec^2 (x)}dx=\frac{1}{1+t^2}dt$
And changing limits
$$I=\int^{\tan (1)}_{0}\frac{1}{(1+t^2)t^{q}}dt<\int^{\tan(1)}_{... |
H: How can I prove that $16 \lt {1+\frac1{\sqrt2}+\frac1{\sqrt3}+\cdots+\frac1{\sqrt{80}}<18}$?
What i want to prove is this
$$16 \lt {1+\frac1{\sqrt2}+\frac1{\sqrt3}+\cdots+\frac1{\sqrt{80}}<18}$$
I haven't encountered any problem of this kind before, how do we proceed?
Making approximations dosen't seem feasible, so... |
H: Integrating factor for an ordinary differential equation
our teacher had asked a question in final exam and i did something and sent him. but my solution graded 0. so i have the question you may help me. i really appreciate your help.
$$ xdy -(x^2+y^2+y)dx = 0 $$ eq is given.
$$ \mu = (x^2+y^2)^\Omega $$ and this ... |
H: Solutions of biquadratic equation being successive members of arithmetic progression
What should be the relationship between $p$ and $q$, so that $x^4+px^2+q=0$ equations has four solutions which are successive members of arithmetic equation.
The answer is root from $q / p = -3/10$, but I have no clue what is going... |
H: Question about SVD and orthogonal matrices
Let $X$ be a $m \times n$ matrix. By SVD, I obtain $X = UDV^T$, where $U$ and $V$ are both orthogonal matrices, and $D$ is a diagonal matrix.
I think the following is true (but not sure why):
$(VDV^T + aI_n)^{-1} = V(D + aI_n)^{-1}V^T$
where $a\in \mathbb{R}$ is some scala... |
H: Does $L^1$ imply $L^p$ on finite measure spaces?
If $(\Omega,\mu)$ is a finite measure space, i.e., if $\mu(\Omega)<\infty$, then does $f\in L^1(\Omega)$ imply that $f\in L^p(\Omega)$ for every $p$?
This is just a statement that I feel like I've heard before, but I don't have a lot of great intuition for $L^p$ spac... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.