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H: Minimizing area of a triangle with two fixed point and a point on parabola
A triangle is made up of three points, $A, B$, and $P$.
$A(-1, 0)$
$B(0, 1)$
$P$ is a point on $y^2 = x$
Minimize the area of Triangle $ABP$.
My approach is far too complicated, which generalize $P$ as $(t^2, t)$, which makes the whole thi... |
H: Operators such that $T\circ S=I$ but $S\circ T\neq I$.
Suppose $S$ and $T$ are linear operators on a vector space $V$ and $T\circ S=I$ where $I$ is the identity map. It's easy to see that $S$ is one-to-one.
If $V$ is finite dimensional, rank-nullity implies $S$ is invertible, so a little manipulation shows that $S... |
H: A numerical analysis problem
I was looking at old exam papers and was stuck on the following problem:
I have hardly any idea how to progress with the problem. Can some give some explanation about how to progress with the problem?
AI: I am not sure what "large" and "small" mean precisely here, but I would imagi... |
H: Can't simplify this boolean algebra equation
So I've got an expression I have been trying to simplify and have the answer but I can't figure out how to get to it... can anyone help me out?
Equation: $(A\wedge \lnot B \wedge \lnot C \wedge \lnot D) \vee (C \wedge \lnot D) \vee (C \wedge \lnot D) = (A \wedge \lnot ... |
H: Given $\frac {a\cdot y}{b\cdot x} = \frac CD$, find $y$.
That's a pretty easy one... I have the following equality : $\dfrac {a\cdot y}{b\cdot x} = \dfrac CD$ and I want to leave $y$ alone so I move "$b\cdot x$" to the other side
$$a\cdot y= \dfrac {(C\cdot b\cdot x)}{D}$$
and then "$a$"
$$y=\dfrac {\dfrac{(C\cd... |
H: Is every field the field of fractions for some integral domain?
Given an integral domain $R$, one can construct its field of fractions (or quotients) $\operatorname{Quot}(R)$ which is of course a field. Does every field arise in this way? That is:
Given a field $\mathbb{F}$, does there exist an integral domain $R$... |
H: Find the product of the following determinants (involving logarithms with different bases)
Find the product of the following determinants:
$$\begin{vmatrix}
\log_3512 & \log_43 \\
\log_38 & \log_49
\end{vmatrix} * \begin{vmatrix}
\log_23 & \log_83 \\
\log_34 & \log_34
\end{vmatr... |
H: A choice question on determinants
If $A$ and $B$ are square matrices of order $2$, then $\det(A+B)=0$ is possible only when:
$(a)$ $\det(A)=0$ or $\det(B)=0$$(b)$ $\det(A)+\det(B)=0$ $(c)$ $\det(A)=0$ and $\det(B)=0$ $(d)$ $A+B=0$
I was sure that when $A+B=0$, $\det(A+B)=\det(0)=0$ So the answer is $d$.
But I ... |
H: Does this inequality have any solutions for composite $n \in \mathbb{N}$?
Does this inequality have any solutions for composite $n \in \mathbb{N}$?
$$\sqrt{2} < \frac{\sigma_1(n^2)}{n^2} < \frac{4n^2}{(n + 1)^2}$$
Note that $\sigma_1$ is the sum-of-divisors function.
AI: Note that,
$\displaystyle f(x)=\frac{4x^2}{(... |
H: what does this really mean
what exactly does this expression mean, i keep seeing it in statistics but i never really understood what its supposed to be, is it another way of writing the variance
$$\Sigma_{i=1}^{n}(Y_i-\bar{Y})^2$$
btw the variance expression i am familiar with is $Var(X)= E(X^2)- [E(X)]^2$, would ... |
H: How to prove that the implicit function theorem implies the inverse function theorem?
I can prove the converse of it, but I cannot do this one. Here is the problem:
Prove that the implicit function theorem implies the inverse function theorem.
AI: For $f : \mathbb{R}^n \to \mathbb{R}^n$, consider $F : \mathbb{R}^n... |
H: $X(G)=4$ then G contains $K_4$
This is a practice question in my text.
Its a true and false question and I have to prove it its true $X(G)=4$ then G contains $K_4$ where $X(G)$ is the chromatic number. I know this is true but how do I prove this. I have found a answer online http://book.huihoo.com/pdf/graph-theory-... |
H: How to simplify $\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$?
$$\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$$
I have been staring at it for ages and know that it simplifies to $x$, but have been unable to make any significant progress.
I have tried doing $(\frac{1-x}{1-2x})(\frac{1+2x}{1+2x})$ but that doesn... |
H: Does adding "monotone" change the meaning?
I wonder why math texts states "function is monotone increasing/decreasing" instead of "function is increasing/decreasing" without word "monotone". Nothing changes, right? Then, why?
AI: Saying monotone increasing/decreasing seems to me very imprecise. I'll discuss a diffe... |
H: Multiplication of rings is an abelian group homomorphism
Let $R$ be a ring without identity.
Suppose that the multiplication $ \cdot : R \times R \rightarrow R $ is an abelian group homomorphism.
For $a, b \in R$ what can we conclude about the product of $a \cdot b$ ?
AI: Let $m\colon R\times R\to R$ be the mult... |
H: Cauchy principal value and the "normal" definition.
Suppose that $\int^{\infty}_{-\infty}f(x)\, dx$ exist. How to prove that $\lim_{b\to\infty}\int^{b}_{-b}f(x)\, dx$ also exist, and $\int^{\infty}_{-\infty}f(x)\, dx=\lim_{b\to\infty}\int^{b}_{-b}f(x)\, dx$
AI: $\int_{-\infty}^{\infty}f(x)\, dx$ exist mean that bo... |
H: morphisms on topological spaces
In the category of topological spaces:
1.) Show that a morphism is monic IFF it is injective
2.) Show that a morphism is epic IFF it is surjective
3.) Are there any morphisms that are monic and epic but not invertible? Prove.
4.) Show that every idempotent splits.
NOTE: A morphism... |
H: Who introduced the term Homeomorphism?
Who introduced the term Homeomorphism?
I was wondering about asking this question on english.stackexchange but I think this term is strongly (and maybe solely) related to mathematics.
AI: An excerpt from Gregory H. Moore's The evolution of the concept of homeomorphism:
The evo... |
H: Limit calculation using Riemann integral
My task is to calculate limit: $$\lim_{n \rightarrow \infty} \sqrt[n^2]{ \frac{(n+1)^{n+1}(n+2)^{n+2}\cdots(n+n)^{n+n}}{n^{n+1}n^{n+2}\cdots n^{n+n}} }$$I denoted that limit as $a_n$. So:
$$\log a_n=\frac{1}{n^2} \left ( (n+1)\log \left (1+\frac{1}{n} \right )+\cdots+(n+n)\... |
H: Proving derived sets are closed
I am following a proof of the statement
The derived set(the set of accumulation points) $A'$ of an arbitrary
subset $A$ of $\mathbb{R}^2$ is closed.
in a book.
It starts with
Let $q$ be a limit point of $A'$. If it is proved that q $\in A'$, then the proof is done.
Let $G_q$ be ... |
H: Goldberg polyhedra coordinates
I would 3D-print some Goldberg Polyhedra importing in Sketchup, the coordinates provided on these links:
72 faces (2,1) - (coordinates)
132 faces (3,1) - (coordinates)
192 faces (3,2) - (coordinates)
252 faces (5,0) - (coordinates)
I noticed that they have pretty much the same volum... |
H: Largest domain on which $z^{i}$ is analytic.
Can anyone help me with this question:
What is the largest domain $D$ on which the function $f(z)=z^{i}$ is analytic?
AI: Rewriting as $f(z)=\exp(i\ln(z))$ suggests that $D$ contains at least the slit plane $\mathbb C\setminus(-\infty,0]$. Since the principle value of $... |
H: A strictly positive operator is invertible
Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is invertible? For example, I want to show that $\langle Tx , x\rangle$ is bounded belo... |
H: Show that "In a UFD, if $p$ is irreducible and $p\mid a$, then $p$ appears in every factorization of $a$" is false
This statement below is false, but I cannot find any counterexamples or explain why. When I tried to give some reasoning, I ended up showing the statement is true.
In a UFD, if $p$ is irreducible and ... |
H: Prove that $T$ is an orthogonal projection
Let $T$ be a linear operator on a finite-dimensional inner product space $V$. Suppose that $T$ is a projection such that $\|T(x)\| \le \|x\|$ for $x \in V$. Prove that $T$ is an orthogonal projection.
I can't understand well. The definition of orthogonal operator is $\|T... |
H: Why don't we include $\pm\infty$ in $\mathbb R$?
Why don't we include $\pm\infty$ in $\mathbb R$?
If we do so, many equations will got real solution (e.g. $2^x=0$), and $\mathbb R$ will be much more complete. Why don't we do so?
Thank you.
AI: A large reason why we don't include $\infty$ is because we can't reall... |
H: How to get out of this indetermination $\lim\limits_{x\to -1} \frac{\ln(2+x)}{x+1}$?
Well, is just that, I can't remember a way to get out of this indetermination (with logarithm), can someone help me?
I'm studying for my calculus test and this question is taking me some time.
$$\lim\limits_{x\to -1} \frac{\ln(2+x)... |
H: Order of the group generated by two matrices
I need to find the order of the group generated by the matrices
$$\begin{pmatrix}0&1\\-1&0\end{pmatrix},\begin{pmatrix}0&i\\-i&0\end{pmatrix}$$
under multiplication.
$\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}0&i\\-i&0\end{pmatrix}=\begin{pmatrix}-i&0\\0&-i\end... |
H: Extremal of a function -Euler equation
I have to calculate $J(t+h)-J(t)$ where $J(x)=\int_0^1 x'^3 dt$, $x=x(t)$, $h\in C^1[0,1]$, $h(0)=h(1)=0$
I have solution, I will write it below, and I will write my question.
$J(t+h)-J(t)
=\int_0^1 (t+h)'^3 dt - \int_0^1 t'^3 dt
=\int_0^1 (1+h')^3 dt - \int_0^1 dt
=... |
H: Understanding the proof for: $d(f^*\omega)\overset{!}{=}f^*(d\omega)$
Consider this
Proposition: Let $U\subset\mathbb{R}^n$ and $V\subset\mathbb{R}^n$ be open sets and $\phi:U\to V$ be differentiable. For all $k\in\mathbb{N}_0$ and $\omega\in \Lambda^k(V)$ it is true that
$$d(\phi^*\omega)=\phi^*(d\omega)$$
I am tr... |
H: The value of the $\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sqrt[k]{k!}}{k}$
What is the value of $$\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sqrt[k]{k!}}{k}?$$
AI: Let $x_k=\dfrac{k!}{k^k}$
Then $$\lim_{k\to\infty}\frac{x_{k+1}}{x_k}=\lim_{k\to\infty}\frac{(k+1)!k^k}{k!(k+1)(k+1)^k... |
H: The angle at which a circle and a hyperbola intersect?
$x^2 - 2y^2 = 4$
$ (x-3)^2 + y^2 = 25 $
How do you calculate the angle at which a circle and a hyperbola intersect?
If I express $y^2$ from the first equation and apply it to the second equation, I get the following:
$y^2 = -2 + \frac{x^2}{2}$
$(x-3)^2 + -2... |
H: Is $S=\sum_{r=1}^\infty \tan^{-1}\frac{2r}{2+r^2+r^4}$ finite?
Problem:
If $$S=\sum_{r=1}^\infty \tan^{-1}\left(\frac{2r}{2+r^2+r^4}\right)$$ Then find S ??
Solution:
I know that $\tan^{-1} x + \tan^{-1} y= \tan^{-1} \frac {x +y} {1-xy} $
But I have no idea how to such complicated question with it.
AI: HINT:
As $... |
H: Equivalent conditions for a measurable function
I am reading Stein and Shakarchi volume 3 and on page 28 they give the definition of a Lebesgue measurable (real - valued) function $f: \Bbb{R}^d \to \Bbb{R}$ to be on in which for any $a \in \Bbb{R}$, $f^{-1}(-\infty,a)$ is Lebesgue measurable. Now I am wondering it... |
H: is there a formula for working out the angles of a triangle to make the sides meet at the top?
I am doing a GCSE maths foundation paper for revision and one question has a triangle with the base side being 9cm and the other 2 sides 7.5cm. Is there a formula for finding the angles of the triangle given the lengths o... |
H: How do I solve such logarithm
I understand that
$\log_b n = x \iff b^x = n$
But all examples I see is with values that I naturally know how to calculate (like $2^x = 8, x=3$)
What if I don't? For example, how do I solve for $x$ when:
$$\log_{1.03} 2 = x\quad ?$$
$$\log_{8} 33 = x\quad ?$$
AI: The logarithm $\log_{b... |
H: How to express the powers of two by a Diophantine equation?
Let $P_2:=\{2^k | k\in\mathbb N\}$ be the set of powers of two. I would like to "see" a polynomial $p(z_1,\ldots,z_r)$ with integer coefficients for which $P_2 =\{n\in\mathbb N | n=p(z_1,\ldots,z_r)\text{ with }z_1,\ldots,z_r\in\mathbb Z\}$. This should be... |
H: Limit of sequence equals limit of convergent subsequences
Let $\{a_n\}$ be a bounded sequence. Prove that if every convergent subsequence of $\{a_n\}$ has limit $L$, then $\lim_{n\rightarrow\infty}a_n = L$.
I know that if the sequence has a limit, it must be $L$, because the limit must equal the limit of the subs... |
H: Is there some nomenclature to get the remainder of a value?
I want to write a formula where I can say that I have to get the remainder of a division by 4.
$y = \mathbf{remainder}(x\div4)$
Is there any math nomenclature I can use?
AI: You can say $y=x\pmod 4$
See modular arithmetic. |
H: Limit of $\cos(x)/x$ as $x$ approaches $0$
As the title says, I want to show that the limit of
$$\lim_{x\to 0} \frac{\cos(x)}{x}$$
doesn't exist.
Now for that I'd like to show in a formally correct way that
$$\lim_{x\to 0^+} \frac{\cos(x)}{x} = +\infty$$
I'm sure this is right since $\displaystyle\lim_{x\to 0^+} \... |
H: Is the set $\{(x, y) : 3x^2 − 2y^ 2 + 3y = 1\}$ connected?
Is the set $\{(x, y)\in\mathbb{R}^2 : 3x^2 − 2y^ 2 + 3y = 1\}$ connected?
I have checked that it is an hyperbola, hence disconnected am i right?
AI: Assuming, you mean
$$S=\{(x,y)\in\mathbb R^2:3x^2-2y^2+3y=1\}, $$
observe that $f\colon(x,y)\mapsto y-\fra... |
H: Natural transformations in $\textbf{Set}$
I am trying to understand the concept of a natural transformation by considering the following example, an exercise from Mac Lane's Categories for the working mathematician (p. 18, ex. 1):
Let $S$ be a fixed set and denote by $X^S$ the set of all functions $S\to X$. Show t... |
H: Find an analytic function with real part $\frac{y}{x^{2}+y^{2}}$
How do I find a analytic function such that $\displaystyle \mathfrak{Re}(f) =u(x,y)= \frac{y}{x^{2}+y^{2}}$.
I can call the real part $u(x,y)$ and by Cauchy-Riemann I will have $u_{x}=v_{y}$ and $u_{y}=-v_{x}$. So $$v_{y}=u_{x}(x,y)= -\frac{2x}{(x^{2... |
H: any open ball of radius $2$ is an infinite set?
Is it true that in an infinite metric space, any open ball of radius $2$ is an
infinite set?
for example $\mathbb{R}^2$ with discrete metric we have $d(x,y)=1\forall x\ne y$
so in this case also we have whole $\mathbb{R}^2$ within a ball of radius $2$ right? is my co... |
H: finding overlapping permutations
I have a data set $3\; 4\; 5\; 6\; 7\; 8\; 9$
I want to find all the permutations that can be formed using this such that neither $7$ nor $8$ is adjacent to $9$.
AI: We count the complement, the bad permutations where $7$ or $8$ is adjacent to $9$.
How many permutations have $7$ ad... |
H: Evaluating $\int_0^{2 \pi} \frac {\cos 2 \theta}{1 -2a \cos \theta +a^2}$
In order to evaluate $\int_0^{2 \pi} \frac {\cos 2 \theta}{1 -2a \cos \theta +a^2}$ we can define
$$
f(z) := \frac 1 z \cdot \frac { (z^2+z^{-2})/2}{1-2a( \frac {z+z^{-1}} 2) +a^2}
$$ I have $0 < a <1$ which gives singular points in $0$ and... |
H: Spectral radius of $A$ and convergence of $A^k$
I'm trying to understand the proof of first theorem here. Maybe it's very simple but I would like your help because I need understand this, I have no much time and my knowledge about this subject is very limited. Some users already have helped-me here and here but I s... |
H: In Levenberg–Marquardt, is forcing the Hessian to be positive definite OK?
I am often doing parameter estimation using the Levenberg-Marquardt method, which involves solving the following linear system at each step:
$$(H + \lambda I) \delta = r_{i}$$
where $H$ is a square Hessian matrix, $I$ is an identity matrix, ... |
H: Intersection of line and circle to evaluate integral
I have to evaluate $\int \int_{D}^{}x^2+y^2-1dA$ over $D = \left \{(x,y) \in R^2: 0\leq x \leq y,x^2+y^2\leq 1 \right \}$
My problem is defining the integral limits for $x$ and $y$.
I believe that the reagion is the intersection of the circle with the line. Doing... |
H: Exercice on periodic function
Let $f$ be a periodic function, $\mathcal{C}^1$ on $\mathbb{R}$ such that:
$$\displaystyle\int_0^{2 \pi} f(t) \, dt = 0$$
$$f(2 \pi) = f(0)$$
Prove that $$\forall t \in [0,2 \pi]: \int_0^{2 \pi} |f(t)|^2 dt \leq \int_0^{2 \pi} |f'(t)|^2 dt$$
How can we prove this please. I don't have a... |
H: How to calculate $\int_{0}^{2\pi} \sin(x)(\cos x+\sqrt{2-\cos(x)})\;\mathrm dx$ with substitution?
I have to calculate $$\int_{0}^{2\pi} \sin x \cdot(\cos x+\sqrt{2-\cos x})\,\mathrm dx$$ using the substitution $u(x)=2-\cos x$.
What I got so far is:
$$\int_{0}^{2\pi} \sin x\cdot(\cos x+\sqrt{2-\cos x})\,\mathrm dx ... |
H: The wikipedia proof of Bolzano Weierstrass theorem
I was going through the proof that has been written for Bolzano-Weierstrass theorem in the respective Wikipedia page.
http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem
I could understand the first part where it assumed the existence of infinite no. o... |
H: Calculate the $r$-derivative of the function $f$
Let $f$ be an analytic function defined by
$$f(s)=g(s)∑_{n=1}^{∞}a_{n}/n^{s}$$
where $∑_{n=1}^{∞}a_{n}/n^{s}$ is an absolute convergent series for $Re(s)>1$.
I have the following question:
Calculate the $r$- th derivative of the function $f$ for $Re(s)>1$, i.e., f^{... |
H: Conditional density function of gamma distributed R.V.'s, ${\Gamma(2,a)}$
I'm stuck on the following problem:
Let $X$ and $Y$ be independent $\Gamma(2,a)$-distributed random variables. Find the conditional distribution of $X$ given that $X+Y=2$.
So the problem is to find $f_{X\mid X+Y=2}(x)$ which from the definiti... |
H: Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subseteq \mathbb{R}$
Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subseteq \mathbb{R}$
(i) If A is connected, is $f^ {−1} (A)$ so?
(ii) If A is compact, is $f^{−1} (A)$ so?
(iii) If A is finite, is $f^ {−1} (A)$ so?
(iv) I... |
H: Diffrential equation of an integral
I read in a paper about a differential equation that I don't understand how.
let $I(y) = \int_0^\infty {g(x,\,y)f(x)dx} $ where $f(.)$ is an Probability Distribution Function and $g(.,\,.)$ is just a function of variables $x$ and $y$. Then we obtain
$$
\frac{{\partial I(y)}}{{\... |
H: $\vec{u}+\vec{v}-\vec{w},\;\vec{u}-\vec{v}+\vec{w},\;-\vec{u}+\vec{v}+\vec{w} $ are linearly independent if and only if $\vec{u},\vec{v},\vec{w}$ are
I'm consufed: how can I prove that $$\vec{u} + \vec{v} - \vec{w} , \qquad \vec{u} - \vec{v} + \vec{w},\qquad - \vec{u} + \vec{v} + \vec{w} $$ are linearly independen... |
H: What's the solution of a differential equation, when the "limit" is 0
Sorry if the questions sounds horrible to all the mathematicians' ears out there, but my math level is limited, and I just try to get a better intuitive idea of what happens in that case.
I "understand" the concept of differential. It's the chang... |
H: First isomorphism theorem for topological groups
Let $f$ a continuous homomorphism from a topological group $G$ onto a topological group $H$. We denote $K = Ker(f)$.
I already proved that $\overline{f}:G/K\to H$ defined by $\overline{f} (xK)=f(x)$ is an algebraic isomorphism and continuous.
Now, I suppose that $f$ ... |
H: How can i solve this separable differential equation?
Given Problem is to solve this separable differential equation:
$$y^{\prime}=\frac{y}{4x-x^2}.$$
My approach: was to build the integral of y':
$$\int y^{\prime} = \int \frac{y}{4x-x^2}dy = \frac{y^2}{2(4x-x^2)}.$$
But now i am stuck in differential equations, wh... |
H: If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?
Is the following true if $\theta\in\mathbb{Q}$?
$$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$
Is it true if $\alpha\in\mathbb{R}$? In each case, prove ... |
H: Integration exercise: $ \int \frac{e^{5x}}{ (e^{2x} - e^x - 20) }dx$
I have trouble integrating:
$$ \int \frac{e^{5x}}{e^{2x} - e^x - 20} dx$$
With $t=e^x$, I've rewritten it as:
$$\int \frac{t^5}{t^2 - t - 20} \frac{1}{t} dt$$
Then I tried integration by parts, but I am not any closer to the solution.
AI: $$\int ... |
H: Proving that a set over a field is a vector space
Given: $S$ is a nonempty set, $K$ is a field. Let $C(S, K)$ denote the set of all functions ${f}\in\ C(S,K)$ such that ${f}(s) = 0 $ for all but a finite number of elements of $S$. Prove that $C(S, K)$ is a vector space.
OK. I was thinking about using the simple a... |
H: algebraic-geometric interpretation of the principal ideal theorem
This quote is from Matsumura's Commutative Ring Theory, page 100: "The principal ideal theorem corresponds to the familiar and obvious-looking proposition of geometrical and physical intuition (which is strictly speaking not always true) that 'adding... |
H: Product-to-sum formulas?
My old pre-calculus book says:
$$\sin u\cos v=\frac{1}{2}[\sin (u+v)+\sin(u-v)]$$
and $$\cos u \sin v=\frac{1}{2}[\sin(u+v)-\sin(u-v)]$$ I don't understand why there is a difference, since multiplication is commutative. Can anyone help? Thanks!
AI: Note simply that $\sin (u+v) = \sin (v+u)$... |
H: Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$
If the order of $G$ is $p^2$ then how do I show that $G$ is isomorphic to $\mathbb Z_{p^2}$ or $\mathbb Z_p\times\mathbb Z_p$.
AI: Hint:
Argue that $G$ must be abelian (why?)
Then use the Fundamental Theorem of Finitel... |
H: Compactifications of limit ordinals
I thought I knew that but it seems I don't.
Let $\alpha$ be a countable, limit ordinal $\alpha>\omega$. Give $\alpha$ its order topology. What is the Stone-Čech compactification of $\alpha$? Is there any reason why it should be $\beta \omega$?
AI: If $\alpha>\omega$, then the sub... |
H: Automorphism that saves all subgroups of gruop.
Let $h\in$Aut($G$) so that it saves subgroups: $h(U)=U$ for each subgroup $U$ of $G$, and $\alpha$ is any automorphism. Is it true that $\alpha h \alpha^{-1}$ also saves subgroups?
AI: Hints:
$$\forall\,U\le G\;,\;\;\alpha^{-1}(U)\le G\implies h\alpha^{-1}(U)=\alpha^... |
H: The limit of $\lim\limits_{x \to \infty}\sqrt{x^2+3x-4}-x$
I tried all I know and I always get to $\infty$, Wolfram Alpha says $\frac{3}{2}$. How should I simplify it?
$$\lim\limits_{x \to \infty}\sqrt{(x^2+3x+4)}-x$$
I tried multiplying by its conjugate, taking the squared root out of the limit, dividing everythin... |
H: Inequalities between chromatic number and the number of vertices
I am currently doing exercises from graph theory and i came across this one that i can't solve. Could anyone give me some hints how to do it?
Prove that for every graph G of order $n$ these inequalities are true:
$$2 \sqrt{n} \le \chi(G)+\chi(\overlin... |
H: Finding distance in Hilbert space
How to calculate $d(e_1,L)$, where $e_1=(1,0,0,\ldots)$ and $L=\left\{x\in l^2\mid x=(\xi_j)_{j=1}^\infty,\sum_{j=1}^n\xi_j=0\right\}$.
Thanks in advance.
AI: For $n \in \mathbb N$ and let $x = (1, -\frac 1n, \ldots, -\frac 1n, 0,\ldots)$, then $x \in L$ and
\begin{align*}
\|x-e... |
H: Trying to show a measure is inner regular on open sets
Background:
Let $X$ be a locally compact Hausdorff space, $C_{00}$ the collection of continuous functions on $X$ with relatively compact support. Then let $I \in C_{00}^*$ such that $f \in C_{00}$ nonnegative implies $I(f)\geq 0$.
Define for nonnegative lower ... |
H: Contour integral for $\int _0^\infty \frac{t^2+1}{t^4+1} dt$
I know that the four singularities for $\int _0^\infty \frac{t^2+1}{t^4+1} dt$ are
$\pm \frac{\sqrt{2}}{2} \pm i \frac{\sqrt{2}}{2}$. Also, since the function is even, I can calculate $\frac{1}{2} \int _{-\infty}^\infty \frac{t^2+1}{t^4+1} dt$ instead.
I... |
H: $\lim_{n\rightarrow\infty}\sup(a_n+b_n) = \lim_{n\rightarrow\infty}\sup a_n + \lim_{n\rightarrow\infty}\sup b_n$ for all bounded sequences $(b_n)$
Let $(a_n)$ be a bounded sequence. Suppose that for every bounded sequence $(b_n)$ we have $\lim_{n\rightarrow\infty}\sup(a_n+b_n) = \lim_{n\rightarrow\infty}\sup a_n +... |
H: Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.
Prove $D_{8n} \not\cong D_{4n} \times Z_2$.
My trial:
I tried to show that $D_{16}$ is not isomorphic to $D_8 \times Z_2$ by making a contradiction as follows:
Suppose $D_{4n}$ is isomorphic to $D_{2n} \times Z_2$, so $D_{8}$ is isomorphic to $D_{4} \times Z_2$. If $... |
H: Question about finding minimum-Hilbert spaces
How to find $$\min_{a,b,c\in\mathbb{C}}{\int_0^{\infty}} |a+bx+cx^2+x^3|^2 e^{-x} dx = ?$$ Thanks in advance.
AI: $\langle f,g \rangle = \int\limits_0^{\infty} f \bar{g}e^{-x}dx$ is scalar product (for $f,g \in L^2(e^{-x}dx)$), so we have to find squared distances of v... |
H: How can I evaluate $\lim_{x\to7}\frac{\sqrt{x}-\sqrt{7}}{\sqrt{x+7}-\sqrt{14}}$?
I need to evaluate the following limit, if it exists:
$$\lim_{x\to7}\frac{\sqrt{x}-\sqrt{7}}{\sqrt{x+7}-\sqrt{14}}$$
How can I solve it without using differentiation or L'Hôpital?
AI: We can get a nice function whose limit will not be ... |
H: Complex function integral
I have the function $f : D_f \subset\mathbb{C} \rightarrow \mathbb{C} $ defined by
$$f(z) = \frac{1}{(z-1)(z^2+2)}, z \subset D_f$$
where $D_f$ is the domain of $f$.
How do I calculate
$$\oint_\gamma f(z)\,dx
$$
where $\gamma$ is circunference with center $-1$, radius $1$ and positive ori... |
H: Finding vectors in $\mathbb R^n$ with Euclidean norm 1
I have a couple of questions here which ask:
Find two vectors in $\mathbb R^2$ with Euclidean norm 1, whose Euclidean inner product with (3, -1) is zero.
and
Show that there are infinitely many vectors in $\mathbb R^3$ with Euclidean norm 1 whose Euclidean inne... |
H: Why is $X^4-16X^2+4$ irreducible in $\mathbb{Q}[X]$?
Determine whether $X^4-16X^2+4$ is irreducible in $\mathbb{Q}[X]$.
To solve this problem, I reasoned that since $X^4-16X^2+4$ has no rational roots hence irreducible.
But there is a hint to this question that uses different approach:
Try supposing it is reducible... |
H: "Rules of inference" when the last premise is a conditional?
Another very basic Discrete Mathematics homework problem. I don't want the answer as much as I want to understand the question:
Problem 7
For each of the following sets of premises, what relevant conclusion(s) can be reached? Explain which rules of in... |
H: Estimating the rate of convergence of an integral
I'm studying the integral $\displaystyle\int_0^w \frac{s\mathrm ds}{(e^s+1)\sqrt{1-(s/w)^2}}$ as $w\to\infty$. The intuition suggests that this integral converges to $\displaystyle\int_0^\infty \frac{s\mathrm ds}{ e^s+1 }=\frac{\pi^2}{12}$, because the singularity i... |
H: why is the expected value of a Wiener Process = 0?
This section of wikipedia says that the expected value of a Wiener Process is equal to 0.
Why is that?
AI: In the characterizations at Wikipedia,
W_t has independent increments with W_t−W_s ~ N(0, t−s) (for 0 ≤ s < t), .
(mean of the normal distributed increments... |
H: Show that there are irreducible polynomials of every degree in $\mathbb{Q}[X]$
There is this problem that I would like to ask for any verification whether my answer is correct.
Edited: Thanks @andybenji.
Show that for any $n\ge1$, there exists an irreducible polynomial $f\in\mathbb{Q}[X]$ of degree $n$.
My answer... |
H: p-norm Inequality
Let $x,y \in \mathbb{R}$, and $0<p<\infty$. Prove $$|x-y|^p\leq (1+2^p)(|x|^p+|y|^p) $$
The case $0<p\leq1$ is obvious, as it follows from the properties of the P-norm, where $P:=1/p$, $$|x-y|^p=|x-y|^{1/P}\leq |x|^{1/P} + |y|^{1/P} = |x|^{p} + |y|^{p} \leq (1+2^p)(|x|^p+|y|^p)$$. But I'm stumped ... |
H: Matrix computation from a given matrix
I have no idea how to go about this one, any hints on how to go about this one?
$$D = \begin{bmatrix}2 & 4 & 3\\0 & 1 & 0\\1 & 3 & 1\end{bmatrix}$$
Let:
$$T(\vec x) = D \vec x$$
Compute:
$$T\left(\begin{bmatrix}3 \\0 \\ -1 \end{bmatrix}\right)$$
Image of problem
AI: I suppose ... |
H: Convergence of the Series - $\sum_{n=1}^{\infty} \frac{\prod_{k=1}^{n}{(3k-1)}}{\prod_{k=1}^{n}(4k-3)}$
Prove that the following series is convergent.
$$\sum_{n=1}^{\infty} \frac{\prod_{k=1}^{n}{(3k-1)}}{\prod_{k=1}^{n}(4k-3)}$$
I don't know for where to begin.
AI: Hint: The Ratio Test is perfect for this. If $a_n$... |
H: Explicit generators of syzygies
Consider an $1\times n$ matrix
$$
\mathbf{A}=\begin{pmatrix}
f_1 &f_2 & \dots & f_n
\end{pmatrix}
$$
over $R=\mathbb{C}[X_1,\dots,X_r]$.
Let $M=\oplus_{i=1}^n R\mathbf{e}_i$ be the rank-$n$ free $R$-module.
We have the following Koszul complex:
$$
\wedge^2 M \xrightarrow{h} M\xrigh... |
H: Is there a Dihedral group of order 4?
If I use the notation $D_{2n}$, then does $D_4$ make sense?
If I showed that a group $G$ is isomorphic to $H \times D_4$ where $H$ is a group, then is $G$ not a group?
I am asking this because in my other question the answerer didn't directly address my question.
AI: As $$D_{2... |
H: Can finite theory have only infinite models?
I always thought, that when creating a theory (set of formulas of predicate logic of first order in some language) and when you want to have only infinite models, you must use infinite number of axioms. That's how Peano arithmetic or ZF are made.
But when I look at Robin... |
H: Newton's binomial problem
It is known that in the development of $(x+y)^n$ there is a term of the form $1330x^{n-3}y^3$ and a term of the form $5985x^{n-4}y^4$.
Calculate $n$.
So, I know that the binomial formula of Newton is: $\sum_{k=0}^n \binom{n}{k}a^kb^{n-k}$, but I can not understand how to establish the rela... |
H: What is computational group theory?
What is computational group theory?
What is the difference between computational group theory and group theory?
Is it an active area of the mathematical research currently?
What are some of the most interesting results?
What is the needed background to study it?
AI: There is ... |
H: Greatest Common Divisor of two numbers
If $ab=600$ how large can the greatest common divisor of $a$ and $b$ be?
I am not sure if I should check for all factor multiples of $a$ and $b$ for this question. Please advise.
AI: We have $600=2^3\cdot 5^2\cdot 3$. To make the gcd large, we give a $2$ to each of $a$ and $b... |
H: How to apply De Morgan's law?
If for De Morgan's Laws
$$( xy'+yz')' = (x'+y)(y'+z)$$
Then what if I add more terms to the expression ...
$$(ab'+ac+a'c')' = (a'+b)(a'+c')(a+c)?$$
AI: We'll apply DeMorgan's "twice", actually, and get:
$$(ab'+ac+a'c')' = (ab')'(ac)'(a'c')' = (a'+b)(a'+c')(a+c)$$
So, yes, you are cor... |
H: Finding the closest point to the origin of $y=2\sqrt{\ln(x+3) }$
Given $$y=2\sqrt{\ln(x+3) }$$
How do I determine a (x,y) pair satisfying the above relation which is the closest to the origin (0,0)?
AI: Minimize the square of the distance from the curve to the origin (and hence, the distance from the curve to the o... |
H: How to show that the limit of compact operators in the operator norm topology is compact
When I read the item of compact operator on Wikipedia, it said that
Let $T_{n}, ~~n\in \mathbb{N}$, be a sequence of compact operators from
one Banach space to the other, and suppose that $T_n$ converges to $T$
with respe... |
H: Every primitive matrix is irreducible?
$A$ is reducible if there is some permutation matrix $P$ such that
$$ PAP^T =
\begin{bmatrix}
B & C \\
O & D \\
\end{bmatrix}
$$
And, if $A^k > O$ for some k, then $A$ is called primitive.
Then, how can I show that every primitive matrix is irreducible?
AI: Your definitio... |
H: Support vs range of a random variable
Is there any difference between the two? I have not met any formal definition of the support of a random variable. I know that for the function $f$ the support is a closure of the set $\{y:\;y=f(x)\ne0\}$.
AI: The support of the probability distribution of a random variable $X$... |
H: Conditional Expected Value and distribution question
The distribution of loss due to fire damage to a warehouse is:
$$
\begin{array}{r|l}
\text{Amount of Loss (X)} & \text{Probability}\\
\hline
0 & 0.900 \\
500 & 0.060 \\
1,000 & 0.030\\
10,000 & 0.008 \\
50,000 & 0.001\\
100,000 & 0.001 \\
\end{array}
$$
Given t... |
H: Concrete Mathematics Iversonian Set Relation Clarification
Sorry for asking a very dumb question, but in Concrete Mathematics(Graham,Knuth,Patashnik), chapter 2 section 4, Knuth talks about this formula called "Rocky Road".
This is the formula to use when you want to interchange the order of summation of a double s... |
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