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H: Simplifying the inequality
I have a simple question but couldn't get it at all.
How can I simplify this therm?
$$\frac{x}{x^2-1} \ge \frac 1x$$
AI: Simplify $\frac{x}{x^2-1} \ge \frac 1x$.
\begin{align*}
\frac{x}{x^2-1} &\ge \frac 1x\\
\frac{x}{x^2-1} - \frac 1x &\ge 0\\
\frac{x^2-(x^2-1)}{x(x^2-1)} &\ge 0\\
\frac... |
H: In a metric space, compact implies sequentially compact
I'd like to know if this demonstration is correct.
Let $X$ be a metric space and $K \subseteq X$. Show that if $K$ is compact, then $K$ is sequentially compact.
$K$ is compact, therefore every open cover has a finite subcover. Consider then a sequence $\{x_n... |
H: If $0 \in \sigma(N)$ and $A = NN^\ast$, then $0 \in \sigma(A)$.
I am going through a longer proof of a theorem which states the following as an intermediate result:
Let $N$ be a bounded normal operator on a Hilbert space $H$ with $0
\in \sigma(N)$. Then the self-adjoint operator $A = NN^\ast$ has $0
\in \sigma(A... |
H: Number of elements mapped to $f(a)$ where $f$ is a group homomorphism
Let $G$ and $G'$ be any two groups.
Let $f$ : $G$ $\rightarrow$ $G'$ be a group homomorphism.
Let $K$ denotes the kernel of $f$.
Let $a$ be any arbitrary element of $G$.
I need to show that there are total $m$ elements mapped to $f$($a$) where $m... |
H: If we pick a sequence of numbers $(a_k)$ at random, what is the expected radius of convergence of $\sum_k a_k x^k$?
Suppose we pick a sequence of positive integers independently and identically distributed from $\mathbb{N}^+$: call it $(a_k)=(a_0,a_1,a_2,a_3,\ldots)$. If we consider the corresponding generating fun... |
H: Number of points of discontinuity of $1/\log|x|$
I was solving a few questions from limits continuity and discontinuity when I came across a question asking for the number of points of discontinuity of $f(x)=1/\log|x|$.
I could easily observe that at $x=±1$, the limits tend to different infinities so the function w... |
H: A questions about the Big $\mathcal{O}$ -notation
In my Analysis textbook, the author writes $f(x)=\mathcal{O}(g(x))$
But in a video the person said $f(x)\in\mathcal{O}(g(x))$ is the correct interpretation, and even said, the other notation doesnt make any sense.
Is one considered better? Or is one really wrong? Do... |
H: Infinite sums involving factorials
My question is
$$\sum_{n=1}^{\infty}\frac{2}{n!}= 2e$$
and $$\sum_{n=1}^{\infty}\frac{n^2}{n!}= 2e$$
But each term in the series
$$\sum_{n=1}^{\infty}\frac{n^2}{n!}$$
except the first one is greater than each term in
$$\sum_{n=1}^{\infty}\frac{2}{n!}$$
So why isn't that
$$\sum_{n=... |
H: Integral of $\int\limits_0^{2\pi } {{a^{\frac{{b\cos (x - c)}}{d}}}dx} $?
I am trying to find the integral of,
$\int\limits_0^{2\pi } {{a^{\frac{{b\cos (x - c)}}{d}}}dx} $
Where, $a,b,c,d \in R$.
I am trying to find the definite integral in wolfram alpha online but it does not provide me any result. Does that mean ... |
H: Find the Taylor series for $f(z)=e^z$ about $z_0=1+i$.
Find the Taylor series for $f(z)=e^z$ about $z_0=1+i$.
I know that I want to use the geometric series for $e^z$ which goes $1+z+\frac{z^2}{2!}+\frac{z^3}{3!}...$, but this is centered around $z_0=0$. How would I go about changing this for $z_0=1+i$?
AI: Since $... |
H: Better quantifying $\exists B \subseteq \mathbb{R}^{n+k} \ \ \exists f:B \to \mathbb{R}^k \ \ \forall x \in B: a \in B \ \text{and} \ F(x, f(x)) = 0$
How could I express this result better, quantificationally? For a pre-specified $a$,
$\exists B \subseteq \mathbb{R}^{n+k} \ \ \exists f:B \to \mathbb{R}^k \ \ \foral... |
H: Validity of Proof of Sum of First $n$ Natural Numbers
Background
Lately I've been self-studying Tom M. Apostol's Vol. 1 Calculus to make my understanding of the subject more rigorous after taking the actual class. I came across a proof for what the sum of the squares of the first $n$ natural numbers was - $$\sum_{i... |
H: Transformation of PD matrix with rank deficient other matrix
This should be easy for you.
I have an intuitive feeling for why the following should be correct, but I would like something more rigorous than my feeling :)
Consider the $m\times m$-matrix $B$, which is symmetric and positive definite (full rank).
Now th... |
H: Can we extend the monoid $(\mathcal P(A),\cup,\emptyset)$ to a group?
Natural numbers are famously a way to build up "the rest" of the numbers: integers as pairs of natural numbers modulo the correct equivalence relation, and similarly for rational numbers, etc.
The powerset of some set $A$ has a similar structure ... |
H: Orthogonal Complement of quadratic functions in L²([-1,1])
How can we characterize $V^{\perp}$ where $ V= \{v \in L^2([-1,1]): v(x)=ax+bx^2,b\neq 0\}$ ?
I've tried looking for $ \{f \in L^2([-1,1]): \int_{[-1,1]}{fv\ d\lambda}=0, \ v(x)=ax+bx^2 \}$ but I cannot figure out an explicit characterization of this space.... |
H: Finding a domain where an integral of a function is '0'
Motivated from reading mr.fourier's tricks, I wanted to come with some of my own to solve some problems of mine.
consider,
$$ P(x) = a_o x^n + a_1 x^{n-1} ... + a_n x^0$$
Now, I multiply $ x^k$ on both sides with $ k \leq n $
I get,
$$ x^k P(x) = a_o x^{n+k} ... |
H: Differentiate $\left(x^6-2x^2\right) \ln\left(x\right) \sin\left(x\right)$
Differentiate
$$\left(x^6-2x^2\right)\ln\left(x\right)\sin\left(x\right)$$
with respect to $x$
My work so far
$\class{steps-node}{\cssId{steps-node-1}{\tfrac{\mathrm{d}}{\mathrm{d}x}\left[\left(x^6-2x^2\right)\ln\left(x\right)\sin\left(x\rig... |
H: Proof of non diagonalizibility of higher matrix power
So the proof is to show that if a regular matrix A is not diagonalizable in $ M_n(\Bbb C) $ then no power of $A^k$ for $k \in \Bbb N$ so i started it with a proof by contradiction suppose $A^k$ is diagonalizable then there exists a minimal polynomial such that $... |
H: Let $G$ be a group with $33$ elements acting on a set with $38$ elements. Prove that the stabilizer of some element $x$ in $X$ is all of $G$.
I'm trying to figure out this old qualifying exam question:
Let $G$ be a group with 33 elements acting on a set with 38 elements. Prove that the stabilizer of some element $x... |
H: Probability that piecewise continuous $X(\omega) \ \in A, \ A \in \mathcal{B}$
I solved this problem, but since my understanding of Borel sets and $X^{-1} \in \mathcal{B}$ is still not polished, I decided to ask it.
On a probability triple with Lebesgue measure on $[0,1]$, a random variable is defined such that
$$
... |
H: Understanding connection of $=$ and $>$ relations in proofs.
I would like to ask about a certain pattern I see in some proofs. This is an example taken from the book: Lang, Serge & Murrow, Gene. "Geometry - Second Edition" (p. 69)
An obtuse angle is an angle which has more than 90°. Prove (in a sentence or two) t... |
H: Definite integral of the following question
Evaluate:
$$\int_{0}^{\infty}\frac{4x\ln (x)}{x^4+4x^2+1}dx$$
I took $x^2$ common from the denominator and then substituted $\ln (x) =u$, and then I was stuck. The result turns out to be
$$\int_{-\infty}^{\infty}\frac{4u}{(e^u+e^{-u})^2+2}dx$$
AI: This integral cannot b... |
H: Study the convergence of the Series $\sum_{n=0}^{\infty} e^{-\sqrt{n}}$
Study the convergence of the Series $\sum_{n=0}^{\infty} e^{-\sqrt{n}}$ The only thing I know is that $e^{-\sqrt{x}}$ is strictly decreasing. I also know that the only method here to use is the Comparison or limit criteria, but i don't know to ... |
H: Does every non-compact Tychonoff space admit an unbounded continuous function?
Let $X$ be a completely regular Hausdorff space. Such a space is also known as Tychonoff space, or a $T_{3.5}$-space. Furthermore, let's assume that $X$ is not compact.
Question. Does $X$ admit a continuous function $f: X\to \mathbb{R}$ ... |
H: Need the result of composing an infinite number of smooth functions be smooth?
$f$ is a smooth function from a manifold to itself. So is $f\circ f$, and $f\circ f\circ f$ and so on...
If this sequence is extended forever, and supposing that it converges to some function, need the function it converges to be smooth ... |
H: Dijkstra's algorithm for a single path only
I'm looking to create maps for a board game with some specific properties, but my knowledge of graph theory is essentially negligible so I'd love some help. The maps will consist of territories which border each other in a 2D plane, I'm looking for a method for creating g... |
H: Extension of a differentiable function $f$ to an open superset
This is a question the book Munkres-Calculus on Manifolds pg.144(Exercise 3-b)
If $f :S\to \mathbb R$ and $f$ is differentiable of class $C^r$ at each point $x_0$ of $S$,then $f$ may be extended to a $C^r$ function $h: A\to \mathbb R$ that is defined on... |
H: Finding the closed form of $\int _0^{\infty }\frac{\ln \left(1+ax\right)}{1+x^2}\:\mathrm{d}x$
I solved a similar case which is also a very well known integral
$$\int _0^{\infty }\frac{\ln \left(1+x\right)}{1+x^2}\:\mathrm{d}x=\frac{\pi }{4}\ln \left(2\right)+G$$
My teacher gave me a hint which was splitting the in... |
H: Solving least squares with QR factorization
I'm looking at the notes on https://www.cs.cornell.edu/~bindel/class/cs3220-s12/notes/lec11.pdf.
On the first page, we have the following steps
\begin{align}
||Ax-b||^2&=||Q^T(Ax-b)||^2\\
&=\left|\left|\begin{bmatrix}R_{11}\\0\end{bmatrix}x-\begin{bmatrix}Q_1^Tb\\Q_2^Tb\e... |
H: Prove there is only one $2$-form $p^*\omega = dx\wedge dy$
I am new with forms and pullbacks and admit differential geometry is not my best area. I'm trying to solve the next problem.
Let be $(x, y)$ coordenates on $\mathbb{R}^2$. Let $p:\mathbb{R}^2\rightarrow\mathbb{R}^2/\mathbb{Z}^2=\mathbb{T}^2$ the projection.... |
H: Is there such a thing as a free quasivariety?
I have heard in universal algebra there is such a thing as a free variety, but is there such a thing as a free quasivariety? I would assume, that, for instance, in the language of a single binary operation symbol $*$, a free quasivariety is a free variety where addition... |
H: A coin is flipped 15 times. How many possible outcomes contain exactly four tails? contain at least three heads?
A coin is flipped 15 times where each flip comes up either heads or tails. How many possible outcomes (a) contain exactly four tails?, (b) contain at least three heads?
Hello everyone, I am currently a b... |
H: Perturbation on sequences and their limit points
I believe this is really simple, but I can't figure it out alone. Let $\{x_k\}_{k\in \mathbb{N}}\subset \mathbb{R}^n$ and be a sequence such that $\|x_k\|\to \infty$, and let $v\in \mathbb{R}^n$ be arbitrary.
Is the set of limit points of $\left\{ \frac{x_k}{\left\|... |
H: Sufficient condition for a flow to be symplectic
Suppose $(M,\omega)$ is a symplectic manifold and $X$ a smooth vector field defined on $M$ such that its corresponding flow $\{g_s\}_{s\in\mathbb{R}}$ is defined for all $s$ in sone some $U_s\subset M$. Is it true that if the Lie derivative $\mathcal{L}_X \omega=0$, ... |
H: Analogous version of $\operatorname{var(X+cY)} = \operatorname{var}(X) + c^2\operatorname{Var}(Y)$ for vectors of uncorrelated random variables?
Is there an analogous version of $\operatorname{var(X+cY)} = \operatorname{var}(X) + c^2\operatorname{var}(Y)$ for vectors of uncorrelated random variables ($X$ and $Y$ ar... |
H: $f^*(U_1 \times ... \times U_k) = \bigcap_{i=1}^k f^*_i(U_i)$
I’m trying to prove this result and I would really appreciate if you could give some feedback in my proof.
Result: Let $A,A_1,...,A_k$ be sets, for some positive integer $k$, let $f: A \rightarrow A_1 \times ... \times A_k$ be a function and let $U_i \s... |
H: How can we erase graph of $f(x)=-10a((x/a)-[x/a])$ from specific parts?
The function
$$f(x)=-10a(\frac{x}{a}-[\frac{x}{a}])$$
I want to erase some parts of graph from $x=a$ to $2a$ and from 3a to 4a ... And so on
How can i accomplish that? I have no idea of how can we do that also please suggest any resource from w... |
H: what is the projective dimension of $ (x,y)\mathbb{C}[x,y]_{(x,y)}$?
For the local ring $R = \mathbb{C}[x,y]_{(x,y)}$ and its maximal
ideal $M = (x,y)\mathbb{C}[x,y]_{(x,y)}$.
What is the projective dimension $\operatorname{pd}_R(M)$ of M?
My thought:
I tried to construct a minimal free resolution of $M$, $\cdots \... |
H: Solve the following non-linear system of equations $x = \alpha \log(y/(1-y)), y = \alpha \log(x/(1-x)) - \beta$ in terms of $\alpha, \beta$
I have the following non-linear equations,
$$x = \alpha \log(y/(1-y))$$
$$y = \alpha \log(x/(1-x)) - \beta $$
where $\alpha, \beta$ are constants, $\log$ is the natural logarit... |
H: Understanding an exercise (Ahlfors' Complex Analysis)
I have two questions about the solution of the following exercise, taken from Ahlfors' Complex Analysis.
In the first integral of the first equality, why is it equivalent to integrate along the curve $|z|=2$ to integrate along the curve $|z+1|=1$? And similarly... |
H: Does $i(n) < \log (n)$ imply $\frac{\log i(n)}{n} \in o \left( \frac{\log n}{n} \right)$?
$i(n)$ is a sequence of nonnegative numbers (integers) indexed by $n$. I think it only implies $ ... \in O\left(\frac{\log n}{n} \right)$, yet the other assertion was made in some paper I am reading. Just wanted to confirm.
He... |
H: $\operatorname{Hom}(\operatorname{Hom}(\mathbb{Q}$/$\mathbb{Z}$, $\mathbb{Q}),\Bbb{Z})$ is isomorphic to $0$
$\operatorname{Hom}(\operatorname{Hom}(\mathbb{Q}$/$\mathbb{Z}$, $\mathbb{Q}),\Bbb{Z})\cong\{0\}$ as $\mathbb{Z}$-modules.
Not sure how to see it. Any help would be appreciated!
AI: Notice that by the univer... |
H: If Cauchy's functional equation is continuous at some point, how to prove that it is continuous at every point? (Darboux Weakening)
Let $f$ be Cauchy's functional equation i.e:
$$f(x_1 + x_2) = f(x_1) + f(x_2) \quad (1)$$
Wiki states that
Cauchy proved that (1) is continuous. This condition was weakened in 1875 by... |
H: Given positive real numbers $a, b, c$ with $ab + bc + ca = 1.$ Prove that $ \sqrt{a^{2} + 1} + \sqrt{b^{2} + 1} + \sqrt{c^{2} + 1}\leq 2(a+b+c).$
Given positive real numbers $a, b, c$ with $ab + bc + ca = 1.$ Prove that $$ \sqrt{a^{2} + 1} + \sqrt{b^{2} + 1} + \sqrt{c^{2} + 1}\leq 2(a+b+c).$$
I have no idea to prov... |
H: A falling object does not keep accelerating indefinitely but, due to air resistance, reaches a terminal speed. What is the terminal speed?
Suppose that the speed of such an object, t seconds after the fall commences is vm/s where v=
$$\frac{200}{3}(1-e^{-0.15t})$$
Find the speed of the object after five seconds.
I ... |
H: Convergence of $\sum_{n=0}^\infty(z^n+\frac{1}{2^nz^n})$
I want to find the domain of convergence of $\sum_{n=0}^\infty(z^n+\frac{1}{2^nz^n})$.
My first thought was to use the ratio test, but that doesn't yield anything fun. So, I was wondering if we could play with Laurent series? I know $\sum_{n=0}^\infty z^n=\... |
H: Prove the following sequence converges
I am fairly confident that if $\alpha\in \mathbb{R}$ is such that $0<\alpha<1$, then the sequence
$$(a_n):a_n=n\alpha^n$$
converges to $0$. I created a generalization of a method found in Prove $ne^{-n}$ converges to zero for $0<\alpha<1/2$ in which you argue
$$n\alpha^n\leq \... |
H: Convergence of $\sum_{n=0}^\infty \frac{(-1)^n}{z+n}$
I want to find the domain of convergence of the series $\sum_{n=0}^\infty \frac{(-1)^n}{z+n}$
I recently posted a similar question here: Convergence of $\sum_{n=0}^\infty(z^n+\frac{1}{2^nz^n})$ where I was able to use Laurent series and find an annulus in which ... |
H: Compactness without using Heine-Borel in $L^p$ spaces
Consider the set of functions $S=\{\sin(2^nx):n\in\mathbb{N}\}$ in $L^2[-\pi,\pi]$ with the metric $d(f,g)=\left(\int_{\pi}^{\pi}|f(x)-g(x)|^2dx\right)^{\frac1{2}}$. Then is $S$ both closed and bounded in $L^2[-\pi,\pi]$ but noncompact in it?
I think yes. The p... |
H: Probability axioms does not make sense?
Assume a unit square to be sample space (infinite points inside it being its elements). Let the points are $\{p_1, p_2, ...\}$
then, by probability axioms,
$$1 = Pr(p_1 \cup p_2 \cup \cdots ) = Pr(\{p_1\}) + Pr(\{p_2\}) + \cdots + Pr(\{p_n\}) = \\
= Pr(p_1) + Pr(p_2) ... |
H: How does Synthetic Division for linear divisors $ax + c$ with $a>1$ work?
I used this guide from Mesa Community College to learn synthetic division. However it does not seem to work if $a>1$ in the divisor $ax + c$.
For example for this problem $\frac{3x^3-5x^2+4x+2}{3x+1}$ from the same website when I expand the s... |
H: How to prove this equation about calculation of matrix determinant?
How to prove the equation about the determinant of Matrix $M$, i.e.,
$|M|=\frac{(M \cdot a) \times (M \cdot b) \cdot (M \cdot c)}{a \times b \cdot c}$
where $a$, $b$ and $c$ are arbitrary vectors.
This euqation is encountered in An introduction to ... |
H: How to prove that the first derivative of $ \left| ln(x) \right| $ exists?
I am trying to prove that the first derivative of $ \left| ln(x) \right| $ exists.
$$ \lim_{h \to 0} \frac{f(x_o-h) -f(x_0)}{h} = \lim_{h \to 0} \frac{\ln(x_o-h) -\ln(x_0)}{h} = \lim_{h \to 0} \frac{\ln(\frac{x_o-h}{x_0})}{h} = \lim_{h \t... |
H: Convolution - Heaviside
I'm having a hard time seeing how $\int_0^t f(u)H(t-u-1)du$ where H is the Heaviside function, is equal to $0$ for $t<1$ and $\int_0^{t-1}f(u)du$ for $t>1$. I know of course that $H(x)$ is generally zero for $x<0$ and $1$ for $x>0$ but I don't see what happened here. Thank you!
AI: Let's put... |
H: Calculating the mean of a simple birth process
Consider a population in which each individual gives birth after an exponential time of parameter $\lambda$, all independently.
If $i$ individuals are present then the first birth will occur after an exponential time of parameter $i\lambda$.
Then we have $i + 1$ indivi... |
H: Construct a homeomorphism between $S^1/\rho$ and $S^1$
Construct a homeomorphism between $S^1/\rho$ and $S^1$ (the unit circle)
where $S^1=\{(x,y)\in \mathbb{R}^2|x^2+y^2=1\}$ and the equivalence relation is $$(x',y')\rho(x'',y'') \iff y''\leq 0 \text{ and } y'\leq 0.$$
I get it intuitively, I know this equivalen... |
H: Doubt about solution to Axler's Linear Algebra Done Right problem
I am confused about a solution by Stanford's MATH113 class to a problem in Sheldon Axler's Linear Algebra Done Right, 3rd Ed. I have seen solutions elsewhere (on Slader) that are very similar.
The question (3.A.11, pg 58) is below, where $\mathcal{L}... |
H: Is $2^{2^m-2}+1$ always a composite number for $m>2$?
Is $2^{2^m-2}+1$ always a composite number for $m>2$ ?
I really don't have any idea how to prove or disprove this , it is given to me that it is true . Since nothing else is coming to my mind , I tried to prove it by induction but it was not useful .
Could som... |
H: Looking for a paper in the game theory literature
I am looking for a paper I've read several years ago, but I cannot find it using google. I think it is quite well known.
It is about prices and their indication about quality. There are informed and uninformed buyers, as the informed buyers know the value of goods, ... |
H: Calculate $Y$ based on new$(X,Y)$ coordinates (from horizontally and vertically translated $x,y$)
For this quadratic function, I need to translate the $(x,y)$ coordinate axes horizontally and vertically, so that the new origin of the coordinate system is located at the point with old coordinates $(2,3)$.
The new $(... |
H: Inequality on time to reach absorption for Markov chain
Take any Markov chain on the state set $\{0,1,...,n\}$ with the condition that the transition probability $P_ij$ to go from state $i$ to state $j$ is zero whenever $j>i$.
Define the random variable $T_n$ to be the number of steps before the process reaches the... |
H: $\int_{0}^{1}\frac{x}{\sqrt{1-x}}dx$
Solvable fixing $1-x=t$, I have a doubt about the integration's extremes. If $x=1\rightarrow t=1-x=0$, while if $x=0\rightarrow t=1-x=1$, so we have $-\int_{1}^{0}\frac{1-t}{\sqrt{t}}dt=\int_{0}^{1}\frac{1-t}{\sqrt{t}}dt$?
Thanks in advance.
AI: To interchange the limits on the ... |
H: How to prove that two empty lists have the same elements in the same order using vacuous implication?
I know that to some extent this question may be insignificant, but I'm a little bit uncomfortable with the vacuous implication in this situation.
We know that two lists are equal iff they have the same length and t... |
H: Move a function from the integrand into the differential in a Stieltjes-Integral
If I have an integral like this
$$\int_{0}^{\infty} e^{-st}f(t)d(\alpha(t)),$$
then is it possible to transform it into a "classic" Laplace-Stieltjes-Integral of the form
$$\int_{0}^{\infty}e^{-st}d(\alpha_2(t))?$$
My idea would be to ... |
H: The natural map $V^* \times W \to \text{Hom}(V,W) $ Is bilinear.
I want to know how to show that $$V^*\times W \to \text{Hom}(V,W) $$ $$(\varphi,w) \mapsto (V \ni v \mapsto \varphi(v) w \in W) $$
is bilinear.
I am currently leaning things again for my exam that is comming up. I know that you need to show bilinear... |
H: Mean value theorem for integrals proof
Can you give me a proof of Mean value theorem for integrals without using Fundamental theorem of calculus (because I want to prove FTC using MVT for integrals).
AI: Let $f:[a,b]\rightarrow \mathbb{R}$ be continuous. By the extreme value theorem there exist $x_{m},x_{M} \in [a,... |
H: Possible orders of special element in a group
Let $G$ be a group. Let $x$ be an element of order $3$ and $y(\neq e)$ be an element of $G$ such that $xyx^{−1} = y^3$. Then what are the all possible order of the element $y$?
My attempt:
Since the order of the element $y$ and $y^3$ are same then if order $y$ is fini... |
H: Turning point of equation
Using differentiation, find the turning points of
$$
(x^{2}+y^{2}-x)^2=x^{2}+y^{2}
$$
Thanks!
AI: We have
\begin{equation}
(x^{2}+y^{2}-x)^2=x^{2}+y^{2}\tag1\label{eq:1}
\end{equation}
Differentiate both sides and put $y'=0$.
\begin{align*}
&(x^{2}+y^{2}-x)^2=x^{2}+y^{2}\\
\implies &2(x^2+... |
H: Doubt on the definition of a dynamical system
I am studying control theory and I am focusing on dynamical system. In the introdution of the notes of my professor, it is defined a dynamical system as a system defined by three elements:
time
a set of functions defined on the time interval, W
the behaviour of the sys... |
H: Starting with $\frac{X+A}{a}=\frac{B}{b}$ and $X-A=B$, derive $A=\frac{a-b}{a+b}X$ and $B=\frac{2b}{a+b}X$.
I've tried a few approaches, to no avail. Thanks in advance.
AI: The first equation gives $b(X+A)=aB$ and substituting $B=X-A$ gives $b(X+A)=a(X-A)$.
Re-arranging gives $X(b-a)=-A(a+b)$ so $A=\frac{b-a}{-(a+b... |
H: In Category Theory can id be considered an isomorphism?
The definition from isomorphism
$f$ and $g$ are isomorphisms iff $f.g=id$ and $g.f=id$
Well
$id.id=id$
Does this make $id$ an isomorphism?
My intuition says that not, because that would break the terminal object definition, right?
Terminal object $t$ is a o... |
H: Diffie Hellman - bad choice of parameters
In the diffie helman algorithm a key is generated, A and B choose a random number a,b resp only they know. A prime number p and a generator g is given. The key is defined as $g^{a*b} \mod p$. I have to justify that $(a,b,g,p)=(3,4,15,31)$ is a bad choice, but the reason is... |
H: Is the set $V=U\cap-U$ balanced?
Let $E$ be a topological vector space and $U$ be an arbitrary neighborhood of $0$. I would like to know if $V=U \cap -U$ is balanced, that is $\lambda V \subset V$ for all $\lambda \in \mathbb{C}$ such that $|\lambda|\leq 1$.
I used this neighborhood to proof that every connected to... |
H: $16=m^{19} \mod 143$ - what is $m$
The background is RSA encryption. Can I use some theorem to exploit this situation?
I thought about fermats theorem but I don t know how to use it here
fermats theorem (If a and p are coprime numbers such that $a^{p−1} − 1$ is divisible by p, then p need not be prime.)
AI: Hint:
W... |
H: Is $f^2 \circ f=f \circ f^2$ true?
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a function from the set of real numbers to the same set with $f(x)=x+1$.
We write $f^{2}$ to represent $f \circ f and f^{n+1}=f^n \circ f$.
Is it true that $f^2 \circ f = f \circ f^2$?
Why?
AI: Since composition of functions is asso... |
H: What is the expected number of additional rolls of dice to get $n$ even numbers consecutively, if we got $m$ ($0\le m
Basically how many times we should roll the dice to get $n$ even numbers back to back? But the catch is we have already started the trial and we got $m$ even numbers, where $m \ge 0$ and $m < n$.
Th... |
H: Plotting the Vertices of a Rotated Ellipse with Non-Origin Centre (MATLAB)
I'm trying to plot the vertices of an ellipse of the form:
$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
Here's my attempt:
A = -0.009462052409440;
B = 0.132811666715687;
C = -0.991096125887092;
D = 1.450474988439371;
E = -10.108254824293347;
F = -... |
H: If a and b are transcendent and algebraically dependent
If $a$ and $b$ are transcendental numbers and algebraically dependent, then for any $\alpha$ and $\beta$ algebraic, it follows that the linear combination of $a$ with $\alpha$, and the linear combination of $b$ with $\beta$ are also algebraically dependent.Thi... |
H: A question about Orthogonal Decomposition Theorem
In a textbook that I'm reading the "Orthogonal Decomposition Theorem" is given as below:
However, there is no proof for equation (2). Can anyone prove it?
Thanks
AI: Since $\{u_1,\ldots ,u_p\}$ form a basis of $W$, each $\hat y$ can be written as
$$\hat y = c_1u_1 ... |
H: Prove that $|V_\alpha|=|\operatorname{P}(\alpha)|$ if and only if $\alpha=\{2,\omega+1\}$ or $\alpha=\kappa+1$, $\kappa=\beth_\kappa$
$\kappa$ is a cardinal, $V_\alpha$ belongs to the Von Neumann hierarchy $\begin{cases} V_0=\emptyset \\ V_{\alpha+1}=P(V_\alpha) \\ V_\lambda=\underset{\gamma<\lambda}{\bigcup}V_\ga... |
H: Exercise question pertaining to Unions and Intersections of 3 sets
I have spent a day and a half trying to answer this question.
I do not know how to prove that C is a subset of A using the given equality.
AI: You can use, that $C \subset A \Leftrightarrow A \cup C = A$ and then simplify, for example, first membe... |
H: Using Cauchy convergence criterion to prove that, "if convergent series contains only finitely many negative terms then it is absolutely convergent"
This question is asked already here
Proof verification: convergent series with a finite number of negative terms is Absolutely Convergent
But, answer to this question ... |
H: Equivalent sets. Are they interchangeable?
Freshman question, really, but the more I think about it, the more I doubt.
Suppose that two sets belong to the same equivalence class. Are they in effect interchangeable? (I understand that there is no axiom of `interchangeability' in the definition of an equivalence rela... |
H: Simple notation question about independent sigma algebras
Let the random variables $X$, $Y$ be independent, i.e. for the sigma algebra generated by those variables holds $\sigma(X,Y) = \sigma(X)\sigma(Y)$.
If $ \omega \in \sigma(X,Y)$, then we have also $\omega \in \sigma(X)\sigma(Y)$.
My question: How can I move o... |
H: Example of a function whose second derivative does not exist but limiting formula for the second derivative holds
Here's Exercise 11 in Baby Rudin:
Suppose $f$ is defined in a neighborhood of $x$, and suppose $f^{\prime\prime}(x)$ exists. Show that
\begin{equation}\label{11.0}
\lim_{h \to 0} \frac{f(x+h)+ f(x-... |
H: Find a plane that passes through a given point and is orthogonal to a given plane
"Let $\pi = x+y+z=0$ be a plane. Let $\rho$ be a plane.
The projection of $\rho$ on $\pi$ is a line, and $\rho$ passes through the origin.
Find the plane $\rho$."
What I got:
$\rho$ and $\pi$ are orthogonal because the projection of $... |
H: Center of a topological group is closed?
Is it true that the center $Z$ of a topological group $G$ is closed?(maybe we need the space to be Hausdorff or something like that...) I was thinking I can just show it is opened. So if I pick $x\in Z$ then I need to find an open $U \ni x$ such that $U\subset Z$. But I am n... |
H: Is it possible to write every real polynomial in two variables like ths one in this form?
Is it possible to write every real polynomial in two variables $x$ and $y$ with this form:
$$a x^2+b x y +c y^2$$
with general coefficients $a,b,c$ into the form
$$(d x + e y)^2$$
for some, possibly complex, $d$ and $e$?
From ... |
H: Countinuity and openness of $f:(\mathbb{N},\varepsilon) \rightarrow (\mathbb{Z},\tau): f(x)=2x $, $\tau=\{A_n,\emptyset, \mathbb{Z}\}$
Consider the family $A_n=\{x \in \mathbb{Z}| -n \leq x \leq n\}, n\in \mathbb{N}$
Let $\tau=\{A_n,\emptyset, \mathbb{Z}\}$ be the topology over $\mathbb{Z}$
Let $f:(\mathbb{N},\vare... |
H: What does $\{g:\mathbb{R}\rightarrow\mathbb{R}\mid g\circ f=f\circ g\}$ mean?
Is the set $\{g:\mathbb{R}\rightarrow\mathbb{R}\mid g\circ f=f\circ g\}$ infinite?
Why?
Let's suppose $f(x)=x^2$
$g(x)$ is an inverse function of $f(x)$.
$g(f(x))=(x^2)^{\frac12}=x$
Therefore, since composition of functions are associati... |
H: Context free grammar for language $\{ \{a,b\}^*$: where the number of $a$'s is unequal to the number of $b$'s$\}$
I've seen many solutions for when the number of $a$'s and $b$'s ARE equal but how should the grammar be for the time when the numbers are unequal?
So far I have this but it can't produce many things lik... |
H: Sorgenfrey topology
$B$ is the base of the Sorgenfrey topology ($\mathcal{T}_{S}$ ), being $ \mathcal{T}_{u}$ the usual topology.
AI: If you show that $[5,\infty)$ is open in $\mathcal{T}_S$, then your set will be open in any product space $X\times Y$ where $X=(\mathbb{R},\mathcal{T}_S)$, because $U\times Y$ is ope... |
H: Modulus and Congruences, odd example.
Hey guys I am reading a math book and I got a bit confused on the congruence chapter.
I have just seen that $a \pmod n$ = remainder of n|a.
However as an example of " a (mod n) = remainder ", they wrote:
1 = 15 (mod 7)
The peculiar example was: "The integer 29 is 5 mod 6"
Wh... |
H: Study convergence of $\int_{0}^{\infty} \frac{e^{\sqrt{x}}}{e^x + 1}$
Study convergence of $$\int_{0}^{\infty} \frac{e^{\sqrt{x}}}{e^x + 1}$$
First of all, I can only use the comparison test or the limit comparison test, but I don't know to witch series compare it.
Is known that polynomials of grade $n \gt e^{x}$ a... |
H: Proof verification: $f$ is convex iff $f'$ is monotonically increasing
This is (the first half of) exercise 14 in Baby Rudin
Let $f:(a, b) \to \mathbb{R}^1$ be differentiable. Prove that $f$ is convex iff $f'$ is monotonically increasing.
($\Rightarrow$) Assume $f$ is convex in $(a, b)$. Fix $0 < \lambda < 1$ and... |
H: How to prove $ E - ( A \cap B ) $ = $ (E- A) \cup (E - B) $ where for the purpose of this exercise E is a set that all other sets are a subset of.
I understand what the left hand side is stating. The set of elements containing E (where E is a set that all other sets are a subset of for the purpose of this exercise... |
H: Why is it important the manifold has codimension $1$ in order to prove this identity for $\operatorname{div}fV$ on $\partial M$?
I've seen the following claim in some lectures notes which let me think that I might have a major misunderstanding:
The claim is that if $M$ is an embedded submanifold of $\mathbb R^d$ wi... |
H: Sines Fourier series
There is a basic principle I don't get. Say I want to find the Sines Fourier series of $e^x$ on $[0,1]$. Why do I in this case treat a "continuation" of $e^x$ from $[0,1]$ to $[-1,1]$ so it is an odd function? I always see it being done but I don't know why. Of course the Fourier series should ... |
H: Evaluate $\displaystyle\sum_{r=0}^8 (-1)^r \binom{20}{r} \binom{20}{8-r}$
Please help me with this question
$$\sum_{r=0}^8 (-1)^r \binom{20}{r} \binom{20}{8-r}$$
AI: Evaluate
$$\sum_{r=0}^8 (-1)^r \binom{20}{r} \binom{20}{8-r}$$
I would evaluate $$\sum_{r=0}^8 (-1)^r \binom{20}{r} \binom{20}{12+r}={20\choose 16}$$... |
H: Sigma index (Induction)
I was wondering how the index that's left is measured.
When you prove this via induction,
$$\sum_{k=1}^{2^n} \frac{1}{k} \geq \frac{n}{2} $$
$$\sum_{k=1}^{2^{n+1}} \frac{1}{k}= \sum_{k=1}^{2^{n}} \frac{1}{k} + \sum_{k=2^n+1}^{2^{n+1}} \frac{1}{k} $$
you will come across this part
$$\geq\frac... |
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