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H: Clues for $\lim_{x\to\infty}\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2^k-1)x^k}{k k!}$
Some clues for this questions?
$$\lim_{x\to\infty}\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2^k-1)x^k}{k k!}$$
AI: Hint: You're looking at $f(2x) - f(x) = \int_x^{2x} f'(t)\ dt$ where
$f(z) = \sum_{k=1}^\infty (-1)^{k+1} z^k/(k k!)$. ... |
H: Calculate limit sequence of roots equation $f(x) = g(x)$.
Considering the functions $\;f,g:(0,\frac{\pi}{2})\rightarrow \textbf{R}\;$ given by $f(x)=\tan x, \;g(x)=nx,\;$
where $n\in\textbf{N}, n\neq0\,$ defines the sequence $\,x_{n}\,$ given by the roots of the equation $f(x) = g(x)$, namely $f(x_{n}) = g(x_{n})$... |
H: Recommendations for website/journal/magazine in applied mathematics
Which website/journal/magazine would you recommend to keep up with advances in applied mathematics?
More specifically my interest are:
multivariate/spatial interpolation
numerical methods
computational geometry
geostatistics
etc
I am looking for ... |
H: Can all real/complex vector spaces be equipped with a Hilbert space structure?
Let $X$ be a vector space over $\mathbb K \in \{\mathbb R, \mathbb C\}$.
Does there exists a pairing $X \times X \rightarrow \mathbb K$ that induces a Hilbert space structure on $X$?
I have been thinking that one may choose an arbitrary... |
H: If $ 5x+12y=60$ , what is the minimum of $\sqrt{x^2+y^2}$?
I know this can be easily done by solving for $y$ and substituting, so that you only have to find the minimum value of the parabola $\large x^2 + \left ( \frac {60-5x}{12} \right)^2$ using standard techniques, but is there a less messy way to do this using ... |
H: eqution of the plane knowing two conditions
How can I find the equation of the plane $\pi$ such that:
$$\pi || \pi_{1}: x_{1}+3x_{2}-2x_{3}+15=0$$ and $$d_{1}: \frac{x_{1}+3}{4}=\frac{x_{2}-2}{2}=\frac{x_{3}-1}{5} \subset \pi.$$
thanks :)
AI: Since the normal vector for $\pi_1$ is $(1,3,-2)$, the equation of any ... |
H: Holomorphic function on $D\subset \mathbb{C}$ has to be $\mathcal{C}^\infty (D)$?
I'm confused about this, in my notes I have the following:
Theorem: Let $f$ be holomorphic inside and on the boundary ($C$, itself a contour) of a simply connected region $D$. Then $\oint_Cf(z)dz=0$.
Proof:
Assuming that $f'$ is cont... |
H: Question about Green function
How to find the Green's function associated to the operator $\frac{-d^2}{dx^2}$ and to the boundary Dirichlet conditions: $u(a)=u(b)=0$ ?
Please help me.
Thank you.
AI: This is a great set of notes on this exact green's function with identical boundary conditions. They're from an MIT m... |
H: How can one prove the inequality $(|r_1s_1|+\cdots+|r_ns_n|)^2\leq(r_1^2+\cdots+r_n^2)(s_1^2+\cdots+s_n^2)$ in $\mathbb{R}^n$?
In the inner product space $\mathbb{R}^n$, Cauchy's inequality tells us that
$$
(r_1s_1+\cdots+r_ns_n)^2\leq(r_1^2+\cdots+r_n^2)(s_1^2+\cdots+s_n^2).
$$
Apparently the inequality can be im... |
H: Show that the Beta Function $\beta (x,y)$ Converges When $x \gt 0, \space y \gt 0$
Given $ \beta (x, y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt$, Show that $\beta (x,y)$ converges when $x \gt 0, \space y \gt 0$.
$$\int_0^1 t^{x-1}(1-t)^{y-1} dt = \int_0^{0.5} t^{x-1}(1-t)^{y-1} dt + \int_{0.5}^1 t^{x-1}(1-t)^{y-1} dt$$
No... |
H: Find the equation of the tangent plane given a vector instead of point
Find the equation of the tangent plane at $\mathbf p = (0,0)$ on the surface $z=f(x,y)=\sqrt{1-x^2-y^2}$.
Give an intuitive geometric argument to support the result.
However $\mathbf p$ is a vector.
I see that the surface of $z$ represents a ... |
H: Finite union of compact sets is compact
Let $(X,d)$ be a metric space and $Y_1,\ldots,Y_n \subseteq X$ compact subsets. Then I want to show that $Y:=\bigcup_i Y_i$ is compact only using the definition of a compact set.
My attempt: Let $(y_n)$ be a sequence in $Y$. If $\exists 1 \leq i \leq n\; \exists N \in \mathbb... |
H: What is $A$ in the definition of atlas?
The definition of atlas I encountered is
An atlas for $M$ is a family {$\varphi_\alpha: U_\alpha \rightarrow U_\alpha^\prime: \alpha \in A $} of charts such that {$U_\alpha: \alpha \in A$} is an open cover of $M$.
Intuitively, I guess it is just the index set, a place where... |
H: If two maps induce the same homomorphism on the fundamental group, then they are homotopic
This is exercise 15.11(d) in C. Kosniowsky book A first course in algebraic topology:
Prove that two continuous mappings $\varphi,\ \psi:X\to Y$, with $\varphi(x_0)=\psi(x_0)$ for some point $x_0\in X$, induce the same homo... |
H: If $\langle v,s\rangle+\langle s,v\rangle\leq \langle s,s\rangle$ for all $s\in S$, why is $v\in S^\perp$?
Suppose you have a complex inner product space $V$, a subspace $S$, and some vector $v\in V$ such that
$$\langle v,s\rangle+\langle s,v\rangle\leq\langle s,s\rangle$$ for all $s\in S$. How can you determine $... |
H: A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$.
Let $ p_1<p_2 <\cdots <p_k < \cdots $ the increasing list in set $\mathbb{P}$ of all prime numbers .
By sum of infinite geometric series we have $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$, for $0<r<1$. For all $s>1$ an... |
H: Computing directional derivative of $f(x,y,z)$
Find the direction derivative of $f(x,y,z) = xy + yz + xz$ at the point $P(1,1,1)$ in the direction of $v = \langle7,3,-6\rangle$. I got the answer as $32/\sqrt{94}$. Is that right? If not what is the right answer please help
AI: $$
\left . f_v \right |_P = \left . \le... |
H: Gradient of a function in multi-variate calculus please help
Find the gradient of the function at the given point.
$g(x, y) = 3xe^{y/x}$, at point $(3, 0)$.
How do you compute the gradient of this function. Please help me.
AI: Initially OP wrote $g(x,y) = 3xy^{\frac{y}{x}}$ but realized it was incorrect. I have le... |
H: What is an effective and practical means to teach about natural logarithms and log laws to high school students?
My students are quite practically minded, and I have found that teaching them concepts in a practical manner to be very helpful (maths 'experiments'; modelling on the smartboard etc).
I am looking for a... |
H: why can't we divide by zero ?!
in arabic sites which is interested in maths , i find many topics like ,here is a proof that 0=2 .
and we answer that the proof is wrong as we can't divide by zero .
but i really wonder , why can't we divide by zero ?
i think the reason that mathematicians refused dividing by zero ... |
H: Real number construction : prove that $Q$ is a subfield of $R$
I am slightly confused about the proof presented in Rudin. It says that the ordered field $Q$ is isomorphic to the ordered field $Q*$ whose elements are the rational cuts. It is this identification of $Q$ with $Q*$ which allows us to regard $Q$ as a sub... |
H: Why inverse modulo exponentiation is harder than inverse exponentiation without modulo
I am new to number theory. I read in cryptography inverse modulo exponentiation is used because it is hard. But I couldn't understand the advantage of it over inverse exponentiation without modulo. Could someone please explain ... |
H: extrema and saddle points
Examine the following function for relative extrema and saddle points:
$$f(x, y) = 9x^2-5y^2-54x-40y+4.$$ I did this and got that the point should be at $(3, -4, 3)$. Is that right? Also, how do I know if it is a saddle point or a minimum?
AI: Hints:
Your solution is correct, the critical... |
H: $C⊆A$ and $D⊆B$ and A and B are disjoint, then C and D are disjoint.
Let A,B,C and D be sets. How to prove:
$C\subseteq A$ and $D\subseteq B$ and $A$ and $B$ are disjoint, then $C$ and $D$ are disjoint.
Could anyone please explain to me how to approach this problem? Thanks.
AI: Use the definition of disjoint set... |
H: equation of the tangent plane
Find an equation of the tangent plane to the surface at the given point. $g(x, y) = x^2-y^2$ at $(7, 2, 45)$. I know the answer is between $14(x-7)-4(y-2)+(z-45)=0$ or $14(x-7)-4(y-2)-(z-45)=0$. I think it is $14(x-7)-4(y-2)+(z-45)=0$. Am I right? If not, please explain why I am not.
A... |
H: Definition of differentiable manifold.
In the notes above, I am not quite sure what the up-side-down $\Pi$ is, nor the notation of $/ \sim$.
$\amalg$ means disjoint union as Peter Tamaroff advised.
As for the notation of$/ \sim$, I guess it means take away subsets in the manifold that are mapped for multiple tim... |
H: Determine whether $F(x)= 5x+10$ is $O(x^2)$
Please, can someone here help me to understand the Big-O notation in discrete mathematics?
Determine whether $F(x)= 5x+10$ is $O(x^2)$
AI: Let us give a definition for Big-O notation:
Suppose $g(x)\geq 0$. We say that $f(x)=O(g(x))$ as $x\rightarrow\infty$ if:
Loosely: ... |
H: Question about $\langle b\rangle$ in a Hausdorff Topological Group
Let $G$ be a topological Hausdorff group, if $b\in G$, does there exist an open neighborhood of $b$ such that $U\cap \langle b\rangle$ is finite.
I know this is a really odd question, but I would really appreciate to know whether this is true or no... |
H: Finding the angle of a moving target
I'm developing a submarine game and found a mathematical problem that exceeds my knowledge.
A submarine has $x$ and $y$ coordinates in the plane, a speed $v$, and two angles: one indicates the direction in which it moves and the other direction it shoots. If a submarine in motio... |
H: An inequality property of the Fibonacci sequence
Given the Fibonacci sequence $F_n$, Wikipedia says (http://en.wikipedia.org/wiki/Fibonacci_number#List_of_Fibonacci_numbers) $$ F_{2n-1} = F_n^2+F^2 _{n-1}$$ so that $$F_{2n-1}>F^2_n$$
What is the smallest such k for which $$F_{n+k}>F^2_n\,\,?$$
I'm not sure where to... |
H: Let A be a family of pairwise disjoint sets. Prove that if B⊆A, then B is a family or pairwise disjoint sets.
I know that for this problem I have to use contradiction. Could anyone check my work and guide me through the problem if it's wrong? So far, this is what I have.Thanks!!
Contradiction: $\mathcal B\subseteq ... |
H: Linear Differential Equation
So I asked this question a while back but the answer I recorded really did not help me solve it.
Find the general solution to:
$a_0 + a_1x + a_2y + a_3dy/dx = 0$
If $a_0 = 0$ this becomes quite easy to solve (just set y = vx and factor it) but separation of variables doesn't work otherw... |
H: Is a uniformly continuous functions bounded?
Let f be uniformly continuous on (a,b). How do you prove that it is bounded on (a,b)?
AI: Hint: Any uniformly continuous function on a dense set can be extended continuously on the whole set. |
H: finding probability density function
Let $x$ be random variable with cdf
$ F(x)=0, x\leq 0 $ and $f(x)=1-e^{-x}, x>0$
The value of $P(\frac{1}{4}\leq e^{-x}\leq \frac{1}{3})$ is
(A) $\frac{1}{12}$
(B) $\frac{1}{3}$
(C) $\frac{1}{4}$
(D) $e^{-3}-e^{-4}$.
I know that $P(a\leq X\leq b)=F(b)-F(a)+P(X=a)$ i am using t... |
H: Question about modular arithmetic and divisibility
If $$a^3+b^3+c^3=0\pmod 7$$
Calculate the residue after dividing $abc$ with $7$
My first ideas here was trying the 7 possible remainders and then elevate to the third power
$$a+b+c=x \pmod 7$$
$$a^3+b^3+c^3+3(a+b+c)(ab+bc+ac)-3abc=x^3\pmod 7$$
$$3(a+b+c)(ab+bc+ac)-... |
H: Monte Carlo Integration - determining if a random x,y coordinate falls within the circle or square
My textbook says you can take any random (x,y )coordinate between -1 and 1 like (-.3, .5) or (.4, -.7) and determine if the given coordinate falls within the circle if you calculate $\sqrt(x^2+y^2)$ < 1.
The part ... |
H: Mathematical Economics - Utility maximization
I am thankful to any hints:
What I have:
Simple log-utility form:
$u = \log c_1 + \beta \log c_2$
Budget constraints:
$c_1 + s \leq w$
$c_2 \leq R\; s$
Problem:
For utility maximization: $s = \frac{\beta}{1+\beta} \cdot w \; \; \; \; \;$ I am not getting this!
I have ... |
H: Finding Big-O with Fractions
I'd want to know how I can find the lowest integer n such that f(x) is big-O($x^n$) for
a) $f(x) = \frac {x^4 + x^2 + 1}{x^3 + 1}$
I've fooled around with this a bit and tried going from
$\frac {x^4 + x^2 + 1}{x^3 + 1} \le \frac {x^4 + x^2 + x}{x^3 + 1} \le \frac {x^4 + x^2 + x}{x^3}$... |
H: Double integrals question please?
We have the double integral$$\int_{0}^{1} dx\int_{0}^{x} {f(x,y)} dy$$
In my book it says that we change the order of the integral and we have
$$\int_{0}^{1}dy \int_{y}^{1} {f(x,y)} dx$$
How is this even possible? Can you explain this to me?
AI: $$\left\{\begin{aligned} 0\le x\co... |
H: Finite groups $G$ with epimomorphisms $\phi_1,\phi_2 : F_2 \rightarrow G$ s.t. $|\phi_1(x^{-1}y^{-1}xy)|\neq|\phi_2(x^{-1}y^{-1}xy)|$.
Can someone give an example of a finite (ideally nonabelian) group $G$ and two surjective homomorphisms
$\phi_1,\phi_2 : F_2 \rightarrow G$ (where $F_2$ is the free group on the gen... |
H: Finding mathematical expectation
Suppose that a distribution of a random variable $X$ is given by,
$P(X=-1)=\frac{1}{6}=P(X=4)$ and
$P(X=0)=\frac{1}{3}=P(X=2)$.
Then, Find the value of $E\left(\frac{X}{X+2}\right)$.
(A) $\frac{1}{36}$
(B) $\frac{1}{18}$
(C) $\frac{1}{9}$
(D) $\frac{1}{6}$.
I am finding $E(X)=\frac... |
H: Prove that $(3+5\sqrt{2})^m=(5+3 \sqrt{2})^n$ has no positive integer solutions?
Is my proof ok? I set $b=3+5 \sqrt{2}$, so that we have $b^m=(b+2-3 \sqrt{2})^n$ , or $b^m=(b+\sqrt2(\sqrt2- 3))^n$. Since $RHS<LHS$, $n>m$ . However, from what we know about binomial expansion, we will have a $b^n$ on the $RHS$ whic... |
H: Find $\lim_{n\to\infty}2^n\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt2}}}}_{n \textrm{ square roots}}$.
Find $\displaystyle \lim_{n\to\infty}2^n\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt2}}}}_{n \textrm{ square roots}}$.
By geometry method, I know that this is $\pi$. But is there algebraic method to find... |
H: Boston Celtics VS. LA Lakers- Expectation of series of games?
Boston celtics & LA Lakers play a series of games. the first team who win 4 games, win the whole series.
The probability of win or lose game is equal (1/2)
a. what is the expectation of the number of games in the series?
So i defined an indicat... |
H: Isomorphism between quotient rings of $\mathbb{Z}[x,y]$
I need to find the condition on $m,n\in\mathbb{Z}^+$ under which the following ring isomorphism holds:
$$
\mathbb{Z}[x,y]/(x^2-y^n)\cong\mathbb{Z}[x,y]/(x^2-y^m).
$$
My strategy is to first find a homomorphism
$$
h:\mathbb{Z}[x,y]\rightarrow\mathbb{Z}[x,y]... |
H: How Josiah Willard Gibbs introduced the notions of dot product and vector product
Before posting this question, I have realised that this question might be repeating the same topic which some users have asked before, but still I cannot find a satisfying answer. So please forgive me for repeating this question: How ... |
H: Polynomial whose only values are squares
Given a polynomial $ P \in \Bbb Z [X] $ such that, $ P (x)$ is the square of an integer for all integers x, is $ P $ necessarily of the form $ P (x)= Q (x)^2$ with $ Q \in \Bbb Z [X]$?
AI: Yes, see this answer, which gives a proof, and references this article, which deals wi... |
H: Is there a power series which pointwise convergent but not uniformly convergent on $(-1,1)$?
I was recently reading that power series of form $\sum_{n=0}^\infty b_n(x-a)^n$ converge uniformly to some uniform limit function on compact intervals $[a-r,a+r]$ if $r$ is less than the radius of convergence.
I was curious... |
H: Find a sum of geometric series with changing signs?
I need to find the sum of the following series:
$$\sum_{i=0}^\infty {3 * (-1)^{n+1} \over 2^{n}}$$
I started to simplify this:
$$3 *\sum_{i=0}^\infty {(-1)^{n+1} \over 2^{n}}$$
And then I stuck. any idea how to continue?
Thank-you.
AI: Recall that
$$\sum_{n=0}... |
H: Reference request for derivatives of complex functions
I have been searching for reference for derivatives of complex numbers. All I found so far were texts that were too convoluted for me to grasp. I was (and still am) searching for a reference that is aimed at beginners to complex analysis.
I have one more ques... |
H: Double integral question area?
Calculate the integral: $$^{} \iint_{D}dx dy$$ where $D :=\{(x,y):\ x^2+y^2\le a^2\}$.
So this is obviously a circle, but how do I use it to integrate?
AI: Without polar coordinates but suffering a little more. The integral is
$$\int\limits_{-a}^adx\int\limits_{-\sqrt{a^2-x^2}}^{\sqrt... |
H: Representation of discrete distributions
It is well-known, that any Borel probability distribution on $[0,1]$ can be obtained starting from the probability space $([0,1],\mathscr B([0,1]),\lambda)$ where $\lambda$ is the Lebesgue measure. I wonder, whether a similar result exists for the case of discrete distributi... |
H: A problem on limits of functions
Find two functions $ f(x) $ and $ g(x)$ such that
1) $\lim_{x\to 0} g(x)=10$; $\quad\lim_{x\to 10} f(x)=100$ but $\lim_{x\to 0} f(g(x))$ doesn't exist.
2) $\lim_{x\to 0} g(x)=10$; $\quad\lim_{x\to 10} f(x)=100$ but $\lim_{x\to 0+} f(g(x))=10 $ and $\lim_{x\to 0-} f(g(x))=100 $
3... |
H: Proof of $e^z \neq 0$
Proof: Let $a \in \mathbb{C}$ be s.t. $e^{a} =0$. Then $0=e^a e^{-a} = e^{-a+a} = e^0 =1$, contradicting the existence of $a$.
But why can we multiply by $e^{-a}$?? If $e^a=0$, then $e^{-a}=\frac{1}{e^a}=\frac{1}{0}$ which can't be defined?
AI: How are you defining $e^a$ over the complex numbe... |
H: The notations change as we grow up
In school life we were taught that $<$ and $>$ are strict inequalities while $\ge$ and $\le$ aren't. We were also taught that $\subset$ was strict containment but. $\subseteq$ wasn't.
My question: Later on, (from my M.Sc. onwards) I noticed that $\subset$ is used for general conta... |
H: Isomorphism between Hom and tensor product
I am looking for an explicit isomorphism $Hom(V,V^*)\rightarrow V^*\otimes V^*$ where $V$ is a vector space.
I thought of:
$\phi\rightarrow ((u,v)\rightarrow \phi(u)(v))$
But I'm not sure this works.
Does anyone have a suggestion?
AI: I normally think of it the other way a... |
H: Multiobjective optimization with two real functions over two real vector spaces
Question:
Does anyone know about a book, a paper or an algorithm for the following optimization problem? What are the sufficient conditions for the existence of the joint optimum, and how to find it?:
Specs:
Let $f : \mathbb{R}^m \times... |
H: f(z) can be expressed as u(x,y) , iv(x,y). Am I doing this right?
So, $f(z) = z^2 + 4z - 6i$ and I need to express this as $u(x,y)$ , $iv(x,y)$. So, I plug in $z = x+iy$ and simplify.
I am left with this:
$f(x+iy) = x^2 - y^2 + 2xy + 4x + 4iy -6i$.
Now, how do I express this as $u(x,y)$ and $iv(x,y)$
The reaso... |
H: Question about matrix discretisation numerical methods
Tomorrow I have an exam about Numerical Methods, and I came up with the following question. Let $$-\frac{d}{dr} \left ( \frac{1}{r} \frac{dy}{dr} \right ) = 1 $$with $r\in [1,2], y(1) = 1 \mbox{ and }y(2)=10$. Take $h = \frac{1}{n+1}$ and $r_i = ih + 1$. (We ha... |
H: How to solve this integral? - A Proof is needed
I am trying to solve this integral
$$\int_{0}^{\pi /2}\sin^n x\cdot dx.$$
I think we should solve it for:
a) odd numbers $2n+1$
$$\int_0^{\pi /2}\sin^{2n+1}x\cdot dx = \int_0^{\pi /2}\sin x\cdot \sin^{2n}x\cdot dx=\int_0^{\pi /2}\sin x\cdot (1-cos^2x) ^n\cdot dx$$... |
H: direct proportional
Mr Tan’s monthly savings (S) is directly proportional to the square root of his monthly income (I). His income in January and February 2011 is 3600 and 2500 respectively. His savings in January is 80 more than in February. Find the amount he saved in January 2011.
AI: $S_{Jan}$ $-$ January savin... |
H: Signal compression
I believe I have an extremely simple question but I can't seem to figure it out.
This image shows $x[n]$
and I have to draw $y[n]=x[2n-4]$ by first doing a compression and then a time shift.
The image from the compression $y[n] = x[2n]$ is this:
I understand how the time shift is done but can... |
H: Bounds of $\oint_{\partial R}\left((x^2-2xy)dx+(x^2y+3)dy\right)=\iint_{R}\left(2xy+2x\right)dxdy$
I'm just having some trouble figuring out the bounds and boundary of the following integral. Question as follows:
Evaluate $$\oint(x^2-2xy)dx+(x^2y+3)dy$$ around the boundary of the region contained by $y^2=8x$ and $x... |
H: Find a subdivision of K4 in the Grötzsch graph.
It is known that the Grötzsch graph is 4-coloring.
Hence it contains a subdivision of K4.
But where is this subdivision?
AI: a is joined to b, e, f.
b is joined to e via j, and to f via h and k.
e is joined to f via g.
So a, b, e, f is a $K_4$, subdivided at j, h, ... |
H: Calculating area of astroid $x^{2/3}+y^{2/3}=a^{2/3}$ for $a>0$ using Green's theorem
question as follows.
Show that for any planar region $\Omega$, $$\mathrm{area}\left(\Omega\right)=\frac{1}{2}\oint_{\partial\Omega}(xdy-ydx).$$
Use this result to find the area enclosed by the astroid $x^{2/3}+y^{2/3}=a^{2/3}$ for... |
H: When a limit does not exist, can its derivative be found?
I am learning derivatives of complex numbers (functions, actually) and what a learned community member pointed to me was that there is a subtle difference between finding derivatives of real numbers.
He said that the derivative of a complex function can ... |
H: Archimedean Property - The use of the property in basic real anaysis proofs
I've been looking for something like this in the previous answers on the topic, but I didn't enounter anything similar, so here there is my problem.
First of all, here there is the definition of Archimedean Property (AP) that I found in the... |
H: Proving that every finite group has a composition series
I'm having trouble understanding one step in the proof that every finite group has a composition series. We proved this with induction over $d = |G|$ in our algebra lectures. The case $d=1$ is trivial. For $d > 1$, let $H$ be a real normal subgroup of $G$ wit... |
H: Interval of convergence of a Laplace-Stieltjes transform
I have a two-sided Laplace-Stieltjes transform,
$$
\int_{-\infty}^{+\infty} e^{-xt}d\mu(t)
$$
that converges absolutely in $(a,b)$.
If the measure $\mu$ is finite,then
$$
\int_{-\infty}^{+\infty}d\mu(t)=\mu(\mathbb{R})<\infty
$$
can I conclude that $(a,b)$ M... |
H: Find number of solutions of $|2x^2-5x+3|+x-1=0$
Problem: Find number of solutions of $|2x^2-5x+3|+x-1=0$
Solution:
Case 1: When $2x^2-5x+3 \geq 0$
Then we get, $2x^2-5x+3+x-1=0$
x=1,1
Case 2: When $2x^2-5x+3 < 0$
Then we get, $-2x^2+5x-3+x-1=0$
x=1,2
In both cases, common value of x is 1
Hence solution is x=1
Am ... |
H: For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
Let $G$ be a group and $H$ be a subgroup of $G$ with finite index. I want to show that there exists a normal subgroup $N$ of $G$ with finite index and $N \subset H$. The hint for this exercise is t... |
H: Is the derivative the natural logarithm of the left-shift?
(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.)
I noticed something really neat the other day.
Suppose we define $L$ as... |
H: Show that $\frac{a^2+1}{b+c}+\frac{b^2+1}{a+c}+\frac{c^2+1}{a+b} \ge 3$
If $a,b,c$ are positive numbers then show that
$$\dfrac{a^2+1}{b+c}+\dfrac{b^2+1}{a+c}+\dfrac{c^2+1}{a+b} \ge 3$$
I am stuck at the first stage. Please give me some hints so that I can solve the problem. Thanks in advance.
AI: Hint: Note t... |
H: Proving that every finite group with $|G| = pqr$ for $p,q,r$ distinct primes is solvable
Without loss of generality, let $p < q < r$. Using Sylow's theorem, the amount of $r$-Sylow groups is 1 mod $r$ and is a factor of $pq$, so only $1$ and $pq$ are possible. Now the proof states that in the case that there only e... |
H: Uniform convergence of Fourier Series, how do I check it?
Let $f(x)=x(\pi-x)$, $x\in (0,\pi)$.
The function satisfies the Dirichlet conditions so its Fourier series, $S_f$ converges pointwise to $f$.
The definition of a Fourier series of $f$ on $[a,a+L]$ is:
$$S(x)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos{ \frac{2n\... |
H: Proof of formula of number of Power $\leq n$
I recently came across this OEIS sequence: http://oeis.org/A069623.
A Perfect Power refers to a number which can be expressed in the form of $x^y$ where $x > 0$ and $y > 1$. The sequence lists the number of distinct Perfect Powers less than or equal to $n$. Here, The seq... |
H: A vector field is a section of $T\mathcal{M}$.
By definition, a vector field is a section of $T\mathcal{M}$. I am familiar with the concept of vector field, as well as tangent plane of a manifold.
But such definition is not intuitive to me at all. Could some one give me some intuition? Thank you very much!
AI: Reme... |
H: Find a $2$-Sylow subgroup of $\mathrm{GL}_3(F_7)$
We have $|\mathrm{GL}_3(F_7)| = 7^3 \cdot 2^6\cdot 3^4\cdot 19$. I can find the $3,7,19$-Sylow subgroup of it, but failed to find a $2$-Sylow subgroup. Can one help?
AI: Short version: $
\newcommand{\GF}[1]{\mathbb{F}_{#1}}
\newcommand{\GL}{\operatorname{GL}}
\newco... |
H: Why is this integral zero for every $n \in \mathbb{Z}$? $\int_0^\pi x(\pi -x) \sin 2nx$.
The integral
$$\int_0^\pi x(\pi -x) \sin 2nx$$
evaluates to zero, but the function can't be said to be even or odd. What argument, other than pure calculation (as I did) would give the value $0$ immediately? The thing multiplyi... |
H: Determining the smallest possible value
If both $11^2$ and $3^3$ are factors of the number $a \times 4^3 \times 6^2 \times 13^{11}$, then what is the smallest possible value of a?
IS there any trick to answer this type question quickly?
AI: HINT:
As $(4,11)=(6,11)=(13,11)=1$
$11^2$ must divide $a$
Observe that th... |
H: Translation of; If X is any real number other than 1, then...
i've just started reading a book on number theory and am trying to follow along with the example proofs of theorems. I've not had too much trouble once I have managed to "translate" the mathematical notation into an English sentence.
Could somebody state... |
H: Why is this derivative correct?
Why is the following correct? Can't understand it.
$$\dfrac{d}{dz}\bigg(\dfrac{e^z-e^{-z}}{e^{z}+e^{-z}}\bigg)=1-\dfrac{(e^z-e^{-z})^2}{(e^{z}+e^{-z})^2}$$
AI: We use the quotient rule for taking derivatives:
$$\dfrac{d}{dz}\bigg(\dfrac{\overbrace{e^z-e^{-z}}^{\large f(x)}}{\underbra... |
H: The Poisson distribution?
Particles are suspended in a liquid medium at a concentration of 6 particles per ml. A large volume of the suspension is thoroughly agitated, and then 3 ml are withdrawn. What is the probability that exactly 15 particles are withdrawn?
AI: The usual model is that the number of particles in... |
H: What is the area of the shaded region of the square?
To find area of shaded portion in the below figure, the picture generate by following mathematica code.
Block[{cond = {x^2 + (y - 1/2)^2 < 1/4 && x > 0,
(x - 1)^2 + (y - 1)^2 < 1 && x < 1 && y < 1,
(x - 1)^2 + y^2 < 1 && y > 0 && x < 1}
},
RegionPlot... |
H: How to integrate $\int x\sin {(\sqrt{x})}\, dx$
I tried using integration by parts twice, the same way we do for $\int \sin {(\sqrt{x})}$
but in the second integral, I'm not getting an expression that is equal to $\int x\sin {(\sqrt{x})}$.
I let $\sqrt x = t$ thus,
$$\int t^2 \cdot \sin({t})\cdot 2t dt = 2\int t... |
H: How to calculate the number of banner appearance based on monthly page views.
Hello fellow mathematicians. I have a website that gathers more then 44.000 page views per month. In my website I have 1 rotating place of 4 banner positions, each time it rotates 4 new banners will appear. So the client asks me, how many... |
H: Does the Stone-Čech compactification respect subspaces?
That is, is it true that if $X$ and $Y$ are topological spaces (assume they are Tychonoff, if necessary), with $X \subseteq Y$, then $\beta X$ is homeomorphic to a subspace of $\beta Y$? If so, how does one prove this? If not, what would be a counter-example?
... |
H: How do we define $\sin(\theta)$ or $\cos(\theta)$
On the interval $[0,2\pi]$, how do we define either sine or cosine? Obviously if we have one, the other is straight forward to generate as a phase shift of the other one.
To expand a little, we know what properties we want these functions to satisfy: $2\pi$-periodic... |
H: Does the nerve of a category preserve directed colimits?
The nerve $N(C)$ of a category $C$ is a simplicial set and defines a functor
$$
N\colon\operatorname{Categories}\to \operatorname{sSets}
$$
from the category of small categories to simplcial sets.
It is given by $N(C)_n=\operatorname{Hom_{Categories}}(\Delta... |
H: Can we *ever* use certain log/exp identities in the complex case?
This article on Wikipedia points out that certain identities for the log and exponential functions which are familiar from the real case require care when used in the complex case. Failures in the following identities are pointed out
$$\log{z^w} \eq... |
H: Bernoulli Random Variables and Variance
The question is:
Suppose $Z_1, Z_2, \ldots $ are iid $\operatorname{Bernoulli}\left(\frac{1}{2}\right)$ and let $S_n = Z_1 + \ldots +Z_n$. Let $T$ denote the smallest $n$ such that $S_n = 3$. Calculate $\operatorname{Var}(T)$.
What I know is that $\operatorname{Var}(T) = E(... |
H: Probability question involving playing cards and more than one players
I've been struggling with this problem for a while, so I have to ask you guys for a little mental push.
My problem involves a full deck of playing cards and 4 players, but I'll simplify it to 4 cards and 2 players with the hopes to be able to ap... |
H: Solving differential equation first order
Please help me with calculation, or with method I can do it by myself.
Maybe $$y=uv?$$
$$
y'=\frac{x+y-2}{y-x-4}
$$
AI: Hints:
This is a first order nonlinear equation
Rewrite it as $M dx + N dy = 0$
Test to see if it is an exact equation
Solve to get:
$$y(x) = x \pm \sqr... |
H: Number of letters required to make three letter names
If a monster has 63 children and he wants to keep 3 letter names for each of them so that they are distinct,but with the condition that you can use the same letter more than once,how many letters at minimum does the monster need to name it's children?-source-BdM... |
H: A question about the composite function of a derivative
This may seem dumb, but, I'm trying to understand the proof of the chain rule, but here is my issue:
By definition, the derivative is the following:
$f'(a)=\lim\limits_{x\rightarrow a}(\frac{f(x)-f(a)}{x-a})$
So far, so good.
But then, if I were to do a compos... |
H: Is my application of Cauchy-Riemann right?
Question:
Given $f(z) = 3z^2 + 9z^3 -z$.
1. Find $f^\prime(z)$
2. Find $f(z)$ when $z = 3 + 2i$
3. Use Cauchy-Riemann to find if $f(z)$ is differentiable at $3 + 2i$
My Attepmt:
1. $f^\prime(z) = 6z + 27z^2 - 1$
2.
$$
\begin{align}
f(3 + 2i)
&= 3(3+2i)^2 + 9(3+2i)^3 -... |
H: Non-symmetric $A^T=A$
Wikipedia says that symmetric matrices are square ones, which have the property $A^T=A$. This assumes that one can have non-square $A^T=A$ and, because it does not satisfy the first property of symmetry, it is not symmetric. So, there can be non-symmetric $A^T=A$ matrices and the definition is... |
H: Hölder- continuous function
$f:I \rightarrow \mathbb R$ is said to be Hölder continuous if $\exists \alpha>0$ such that $|f(x)-f(y)| \leq M|x-y|^\alpha$, $ \forall x,y \in I$, $0<\alpha\leq1$. Prove that $f$ Hölder continuous $\Rightarrow$ $f$ uniformly continuous and if $\alpha>1$, then f is constant.
In order to ... |
H: Probability of of an event happening at least once in a sequence of independent events?
I want to find the probability of flipping heads at least once if you flip a coin two times. The possible outcomes (we don't care about the order) are (each equally likely) $TT$, $TH$, $HT$, $HH$. Three out of four have an $H$ ... |
H: practical arithmetic in prime factorizations
I am quite adept at doing arithmetic mentally or on paper, but I know little about the relatively sophisticated stuff that software experts use to crunch numbers. My question is whether the following idea is frequently used by such experts.
Say I'm trying to find the pr... |
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