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H: Permutations: Given $P^4$, how many $P^1$s are possible?
Let $P^0$ be the identity tuple $(1,2,...,N)$
Let $P^{i+1}$ be the tuple after a permutation $P$ is applied to $P^i$.
For example, if $P$ is $(2,1,3,6,4,5)$ than:
$$\begin{align}
P^0 &= (1,2,3,4,5,6) \\
P^1 &= (2,1,3,6,4,5) \\
P^2 &= (1,2,3,5,6,4) \\
P^3 &= (... |
H: Determine convergence of $\sum_{n=1}^\infty\frac{\sin(na)}{n^2} $
I want to detemine the convergence of the next series:
$$\sum_{n=1}^\infty\frac{\sin(na)}{n^2} $$
I've solved the limit:
$$\lim_{n->\infty}\frac{\sin(na)}{n^2}=\frac{[ -1,1]}{\infty}=0$$
The series has the necesary condiction of convergence (limit=0)... |
H: Can there be a function that's even and odd at the same time?
I woke up this morning and had this question in mind. Just curious if such function can exist.
AI: Others have mentioned that $f(x)=0$ is an example. In fact, we can prove that it is the only example of a function from $\mathbb{R}\to \mathbb{R}$ (i.e a f... |
H: Solve the equation $3x+7=6$ in $\mathbb{Z}_{13}$
The title is exercise 2.2 in The Fundamental Theorem of Algebra.
The hint for the problem is: Find the value of $\frac{1}{3}$ in $\mathbb{Z}_{13}$
(please realize that my knowledge of the subject is what I read in Chapter 2)
I have gotten that $x = -\frac{1}{3}$... |
H: Partial Integration - Where did I go wrong?
For a Homework, I need $\int \frac{x}{(x-1)^2} dx$ as an intermediate result. Using partial integration, I derive $x$ and integrate $\frac{1}{(x-1)^2}$, getting: $$ \frac{-x}{x-1} + \int \frac{1}{x-1} dx = \ln(x-1)+\frac{x}{x-1} $$
WolframAlpha tells me this is wrong (it ... |
H: Sequence of Lipschitz functions
Let $\{f_{n}\}$ be a sequence of positive continuous functions on $\mathbb R$; $f_{n}:\mathbb R\to \mathbb R$, for all $n\geq 1$, with the folloing properties:
(1) $\{f_{n}\}$ is uniformly bounded by some constant $C>0$,
(2) $\{f_{n}\}$ is uniformly Lipschitz on $\mathbb R$ (so it is... |
H: Evaluation of $\Xi(z)=\sum_{t=1}^{\infty}\frac{t^z}{e^t}$
I would like to try and evaluate the following gamma function inspired sum.
$$\Xi(z)=\sum_{t=1}^{\infty}\frac{t^z}{e^t}$$
According to my computations, for large $z$,
$$\Xi(z)\approx\Gamma (z+1)$$
and perhaps even
$$\Xi(z) \sim \Gamma (z+1)$$
Does a close... |
H: $ A_0^a B_0^b + A_1 ^a B_1 ^b \leqslant (A_0 + A_1 )^a (B_0 + B_1 )^b$ given $A_0 , B_0 , A_1, B_1 \geqslant 0 $ ; $0 \leqslant a,b <1 $
Let $A_0 , B_0 , A_1, B_1 \geqslant 0 $ and let $0 \leqslant a,b <1 $. Then prove that $$ A_0^a B_0^b + A_1 ^a B_1 ^b \leqslant (A_0 + A_1 )^a (B_0 + B_1 )^b$$
AI: $A_0 = A_1 = ... |
H: What is the simplest proof that the mutual information $I(X:Y)$ is always non-negative?
What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)\ge0$
AI: By definition,
$$I(X;Y) = -\sum_{x \in X} \sum_{y \in Y} p(x,y) \log\left(\frac{p(x)p(y)}{p(x,y)}\right)$$
Now, negative logarithm... |
H: Did Structuralism influence the formulation of Category Theory?
Having only the a very cursory knowledge of Structuralism ( it's a movement generally held to have originated in linguistics, then moving on to philosophy & literature), there does appear to be some points of coincidence:
Structuralism (from wikipedia)... |
H: How to find every possible scalar product on $V$
I am given the following task:
"For which $a$, $b \in \mathbb{R}$ there exists a scalar product, such that
$$A = \left( \begin{matrix} 0 & 0 & a b \\ 1 & 0 & a \\ 0 & 1 & b\end{matrix}\right) $$
is a self-adjoint matrix".
A hint says, the task breaks down in finding ... |
H: Truncation in Lorentz spaces
I am reading a paper, whose author state the following: if $f \in L^{(q,\infty)}(\mathbb{R}^N)$, then $f_\delta \in L^p(\mathbb{R}^N)$ for every $p \in [1,q)$, where $\delta > 0$ and
$$
f_\delta = f\; \boldsymbol{1}_{\{x\in X: f(x) \geq \delta\}}.
$$
But is this really true?
AI: I will ... |
H: Rectangles Diagonal Calculation
I was having a problem with the following question, and could use some help:
If a rectangle with a perimeter of 48 inches is equal in area to a right triangle with legs of 12 inches and 24 inches, what is the rectangle's diagonal?
The answer to the above question is $12\sqrt{2}$. F... |
H: An Inequality involving integrations (Hölder-like).
Let $ p,r \geqslant 1, \; f \in L^r (\mathbb R), g \in L^p (\mathbb R), \; 2/q = 1/p + 1/r$. Here $ 2p / q > 1 $ and $ 2r / q > 1 $ . Also $ \frac{q}{2p} + \frac{q}{2r} = 1$. I want to prove following. $$ \sum_{k=1}^n \left( \int_{a_{k-1}}^{a_k} |g|^p \right)^{\... |
H: Proving the Möbius formula for cyclotomic polynomials
We want to prove that
$$ \Phi_n(x) = \prod_{d|n} \left( x^{\frac{n}{d}} - 1 \right)^{\mu(d)} $$
where $\Phi_n(x)$ in the n-th cyclotomic polynomial and $\mu(d)$ is the Möbius function defined on the natural numbers.
We were instructed to do it by the following ... |
H: Getting the shortest paths for chess pieces on n*m board
I originally posted this question of stackoverflow but I was suggested to post it here.
So:
I am stuck solving a task requiring me to calculate the minimal number of steps required to go from point A to point B with different chess pieces on a n*m board that ... |
H: ODE theory. Need someone to jog my memory
I have two questions regarding the same thing.
Let's say I have a homogeneous ODE (is that what they are called?) $ay'' + by' + cy = 0$
The trick in this problem is to multiply both sides by $e^{rt}$ and do some trick to get the homogeneous solution which usually includes... |
H: How to calculate all the four solutions to $(p+5)(p-1) \equiv 0 \pmod {16}$?
This is a kind of a plain question, but I just can't get something.
For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.
How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\eq... |
H: Convergence in the upper half-plane
I have a sequence $\{F_{n}(z)\}_{n=1}^{\infty}$ of analytic functions in the open upper half plane $\mathbb H$ and continuous on $\mathbb R$, such that $|F_{n}(z)|\leq 1$ for all $n\geq 1$, and all $z$ in the closed upper half plane $\overline{\mathbb H}=\mathbb H\cup \mathbb R... |
H: Linear transformation for projection of a point on a line
This is what my textbook wants me to do:
The matrix of the linear transformation $P_L$ that projects $\mathbb{R}^2$ on de straight line $l \leftrightarrow y = mx$ is:
\begin{pmatrix}
\frac{1}{1+m^2} & \frac{m}{1+m^2} \\
\frac{m}{1+m^2} & \frac{m^2}{1+m^2} \... |
H: Optimization of Unconstrained Quadratic form
So I'm learning about optimization of quadratic forms and this textbook goes through definiteness of matrices and principle minors etc. and then goes straight onto optimizing with constraints but never mentions how to solve the general problem of finding stationary point... |
H: $ \text{if} \;\;a^q \leqslant b^q + c^q \;\;\text{then}\;\; a \leqslant b+c. $
Let $a,b,c >0$ and $q >1$. Then $$ \text{if} \;\;a^q \leqslant b^q + c^q \;\;\text{then}\;\; a \leqslant b+c. $$ How can I prove this?
AI: Suppose to the contrary that $a \gt b+c$. Then $a^q \gt (b+c)^q \gt b^q+c^q$.
To prove that $(b+c)... |
H: Understanding Algebraic Multiplicity
Can you help me understand this statement:
An eigenvalue c has algebraic multiplicity $k$ if $(t-c)^k$ is the highest power of $(t-c)$ that divides the characteristic polynomial.
I am not sure, what does $t$ stand for. I have lifted this statement from the first statement under ... |
H: Sum of Sines Interval
Possible Duplicate:
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
How is it possible to show for integer $m$:
$$\frac{1}{M}\sum_{k=1}^{M}\sin(m\cdot y_{k})=0$$
Thank you very much
Interval $[-\pi,\pi]$ split into $M$ equal intervals, with the mid ... |
H: How you prove that $p^2 \mid m$ and $p^2 \mid n \Rightarrow p^2 \mid mn$?
We know that (Characterization of prime number) $p \mid ab \Rightarrow p \mid a$ or $p \mid b$, where $p$ is prime number. How you prove that $p^2 \mid m$ and $p^2 \mid n \Rightarrow p^2 \mid mn$?
AI: You surely do not mean to ask this, since... |
H: Finding the velocity of a rock given its height as a function of time
I'm trying to learn calculus here, but I know I have to set the $h$ equal to 0 and find the time at when it's equal to 0, but I have no idea what to do after. Here is the question. How do I find out the velocity at that time?
If a rock is thrown... |
H: Difference between $\left< x\right> \cap \left< x,y\right>^2$ and $\left< x,y\right>^3$
Consider the ideals $I = \left< x\right> \cap \left< x,y\right>^2 = \left<x^3,x^2y, xy^2\right>$ and $J=\left< x,y\right>^3=\left< x^3, x^2y, xy^2, y^3
\right>$ in $k[x,y]$.
What is the geometric difference between $I$ and $J$... |
H: Intermediate Value Theorem Confusion
I have a question on this online website I'm trying to learn calculus on. What I am really confused about the Intermediate Value Theorem is: it says I should set it to 0 but I'm totally lost at the steps used to approach and take this? Perhaps someone can guide me in the right d... |
H: Sum Cosine Mod?
interval $-\pi:\pi$ split into M equal intervals.
midpoints are $y_K$
but i dont understand how to show
$$
\frac{1}{M}\sum_{j=1}^{M}\cos(mx_{j})=\begin{cases} 1, & \ m \equiv 0\pmod{M}\\ 0, & \text{else} \end{cases}$$
thank you very much
AI: It is convenient to push the interval forward by $\pi$. S... |
H: Implication Laurent series to polynomial
Let $f$ be holomorphic on $\mathbb{C}$. We have $f(z)=\sum_{n=0}^\infty a_nz^n$. Let $g$ be defined on $\mathbb C\setminus\{0\}$ by the Laurent series $g(z)=\sum_{n=0}^\infty \frac{a_n}{z^n}$.
If $0$ is an essential singularity of $g$, then the coefficients satisfy $a_n \ne... |
H: Implication injective holomorphic function on the zeroes of derivative
We know the following things:
$f(z)$ is holomorphic on $\mathbb{C}$, $f(z)$ is injective and $f(z)$ is a polynomial, so $f'(z)$ is a polynomial too.
(1) Why is $f'(z) \neq 0$? Or why does $f$ have only one zero?
(2) Why is $f'(z)$ constant if (1... |
H: Natural deduction proof of $\forall x (\exists y (P(x) \vee Q(y))) \vdash \exists y (\forall x (P(x) \vee Q(y)))$
I'm trying to do a Fitch proof of
$$
\forall x (\exists y (P(x) \vee Q(y))) \vdash \exists y (\forall x (P(x) \vee Q(y)))
$$
Edit: using only the axioms on http://www.proofwiki.org/wiki/Category:Natural... |
H: Limit of $x^2\cos(1/x^2)$ when $x\to0$ by squeeze theorem
How can I argue that $$\lim_{x \to 0} x^2 \cos\left(\frac{1}{x^2}\right) = 0$$
I understand I have to use a squeeze theorem and that one piece goes to zero but I'm not sure how to tackle this problem to show on a test.
AI: Use $-1 \le \cos(\frac{1}{x^2}) \... |
H: Finding Area of a Triangle without Trignometric ratios.
Hi I need to figure out the area of the following triangle, without using Trigonometric ratios. Any suggestions on the best approach.
The answer is 12 square units
Edit:
I also think that the above triangle can't qualify for a $30-60-90$ triangle since it fai... |
H: Limit Find Value Question
If $f$ and $g$ are continuous functions, with $f(3) = 5$ and
$$\lim_{x \to 3} (2f(x) − g(x)) = 4$$
find $g(3)$.
I am confused at how to tackle this question, I understand I have to find $g(3)$ but do I plus in $3$ for $x$? How do I go about getting the solution because apparently someon... |
H: Tangent Line to $\sin x+\cos x$
When is the tangent line to
$y = \sin x + \cos x$
horizontal?
I have no idea how to solve this problem. Would I use the equation of a tangent line here? Because if so i have no idea how to apply that.
AI: Hint 1. A line is horizontal when its slope is $0$.
Hint 2. The slope of the ta... |
H: Indices in differential geometry
Often times in differential geometry it is convenient to use Einstein summation notation, and there it is presented to beginning graduates and advanced undergraduates alike that if you see two indices that are the same letter with one upper and the other lower, written next to each ... |
H: Directional derivative (muiltivariable calculus)
I have an encountered an example in my text book which I don't fully see the intuition of. I will write out the part of the example I'm struggling with:
A hiker is standing beside a stream on the side of a mountain examining her map of the region. The height of the ... |
H: Is there any geometric way to characterize $e$?
Let me explain it better: after this question, I've been looking for a way to put famous constants in the real line in a geometrical way -- just for fun. Putting $\sqrt2$ is really easy: constructing a $45^\circ$-$90^\circ$-$45^\circ$ triangle with unitary sides will ... |
H: Solve $y' = x + y$
I am suppose to use the substitution of $u = x + y$
$y' = x + y$
$u(x) = x + y(x)$
I actually forget the trick to this and it doesn't really make much sense to me. I know that I need to get everything in a variable with x I think but I am not sure how to manipulate the problem according to mathem... |
H: How to solve this recurrence $T(n) = 2T(n/2) + n\log n$
How can I solve the recurrence relation $T(n) = 2T(n/2) + n\log n$? It almost matches the Master Theorem except for the $n\log n$ part.
AI: Let us take $n = 2^m$. Then we have the recurrence $$T(2^m) = 2T(2^{m-1}) + 2^m \log_2(2^m) = 2T(2^{m-1}) + m 2^m$$
Call... |
H: Find the vertical and horizontal asymptotes of the function.
I am asked to find the vertical and horizontal asymptotes of the equation:
$$f(x)=(a^{-1}+x^{-1})^{-1}$$
I simplify this to
$$f(x)=\frac{1}{a^{-1}+x^{-1}}$$
$$f(x)=a^1+x^1$$$$f(x)=a+x$$Which is some constant, graphed as horizontal line - that will not hav... |
H: Rewriting a power series as a geometric series?
For this series, find the radius of convergence and write it as a geometric series and give a formula if $x>3$
$$\sum_{n=0}^{\infty} \frac{1}{2^{n+1}}(x-3)^n$$
Now finding the radius of convergence wasn't too difficult and I'll save you guys the trouble of doing it b... |
H: $K$ finite extension of $F$ s.t. for every 2 subextensions $M_1, M_2$, $M_1\subset M_2$ or $M_2\subset M_1$. Then there's $a\in K$ such that $K=F(a)$
Let K be a finite extension of a field F such that for every two intermediate field $M_1$, $M_2$ we have $M_1\subset M_2$ or $M_2\subset M_1$. I need to show that the... |
H: To understand some terminology of proof.
Let $|\sigma_{n}(x)|\leq K$
$\displaystyle \Rightarrow \frac{1}{\pi}\int_{-\pi}^{\pi}\sigma_{n}^{2}(x) dx\leq 2K^{2}.$
Now, $$\sigma_{n}(x)=\sum_{k=0}^{n}\left(1-\frac{k}{n+1}\right)(a_{k}\cos kx+b_{k}\sin kx).$$
By Parseval's identity,
$$\frac{1}{\pi}\int_{-\pi}^{\pi}\sigma... |
H: Diameter of wheel
If a wheel travels 1 mile in 1 minute at a rate of 600 revolutions per minute. What is the diameter of the wheel in feet ? The answer to this question is 2.8 feet.
Could someone please explain how to solve this problem ?
AI: We are told that every minute, the wheel made $600$ revolutions. When... |
H: A complex map with "bounded" derivative is injective
The exercise I try to solve states: "Let $\,f\,$ be analytic in $\,D:=\{z\in\mathbb{C}\;|\;|z|<1\}\,$ , and such that $$|f'(z)-1|<\frac{1}{2}\,\,\,\forall\,z\in D$$
Prove that $\,f\,$ is $\,1-1\,$ in $\,D\,$.
My thoughts: The condition $$|f'(z)-1|<\frac{1}{2}\,\,... |
H: Given a cubic function, and its quadratic derivative- can I recover the cubic from quadratic?
Background: I'm trying to learn how to work with cubic and quadratic bezier splines for various drawing libraries, and working through how to approximate a cubic spline with a quadratic spline. It's occured to me that it s... |
H: Piecewise Function Calc Confusion
Help me understand this question.
Consider The Function
$$f(x)= \begin {cases} \dfrac{2x-2}{x-1}&\text{if }x \leq2\\
\dfrac{8}{x}\
&\text{if }x \in_\ (2,4)\\
\sqrt x&\text{if }x \geq 4 \end {cases}$$
Where is the function continuous? If there are any removeable
discontinuities, ... |
H: How to calculate a linear transformation given its effect on some vectors
Im not sure if my question is worded very well, but I'm having trouble understanding how to tackle this problem.
Let $T\colon\mathbb{R}^3\to\mathbb{R}^2$ be the linear transformation such that $T(1,-1,2)=(-3,1)$ and $T(3,-1,1) = (-1,2)$. Fin... |
H: Question about a step in a proof (that nonempty open sets are unions of open sets)
Thm: Every nonempty open set $G$ of real numbers is the union of a finite or countable family of pairwise disjoint open intervals.
Proof: Let $x$ be a point in a nonempty open set $G$. There is an open interval $(y, z)$ such that $x\... |
H: $f$ closed iff $y\in N$ and open $V\supset f^{-1}\left(\{y\}\right)$ exists $U$ open such that $V\supset f^{-1}(U)\supset f^{-1}(\left\{y\right\})$
Prove that $f\colon M\to N$ (topological spaces) is closed if and only if for all $y\in N$ and all open sets $V\supset f^{-1}\left(\{y\}\right)$ in $M$ there exists an ... |
H: Showing that an algebraic number is not a root of a real
While answering this question on mathoverflow, I stumbled across a question that I expect may be easily answered by someone knowing a bit more algebra than me.
Let's make it really specific.
Consider the polynomial equation $X^4-X^3-X^2-X-1=0$. It has two rea... |
H: Compute Limit Question Confused
$$\lim_{r \to 0^+} \frac{\sqrt r}{(r-9)^4}\
$$
How do i compute this limit? I was told to see what 1/x is approaching and rewrite it but can someone guide me in the right direction?
How can i find which infinity it is approaching?
Also
What does it approach if the limit approach 9 ... |
H: Compute Limit Question with tangent
$$\lim_{x \to 0^+} \tan^{-1}\ \left(\frac{1}{x}\right)\
$$
I am not sure how to solve this limit either, it says i should first see what 1/x is approaching but im confused as how to do that, and how to solve it. How would i rewrite this or what rules would i use?
AI: This probl... |
H: compute area under two curve
suppose we are give task to calculate area of figure,which is bounded by two curve
$y=[x]$ and $y=(2-x^2)$, here $[x]$ denotes modulus,not ceiling or rounding of x.
i use wolframalpha to figure out what kind of location,intersection points has this two figure,here is lin... |
H: Characterizing all ring homomorphisms $C[0,1]\to\mathbb{R}$.
This is something I've been trying to work out this evening.
Let $R$ be the ring of continuous real-valued functions on $[0,1]$ with pointwise addition and multiplication. For $t\in [0,1]$, the map $\phi_t\colon f\to f(t)$ is a ring homomorphism of $R$ ... |
H: Numbers between real numbers
I wonder if there can be numbers (in some extended theory) for which all reals are either smaller or larger than this number, but no real number is equal to that number?!
Is there some extension of number which allows that? Under what conditions (axiom etc.) there is no such number.
AI:... |
H: $\gcd(n!+1,(n+1)!)$
The recent post didn't really provide sufficient help. It was too vague, most of it went over my head.
Anyway, I'm trying to find the $\gcd(n!+1,(n+1)!)$.
First I let $d=ab\mid(n!+1)$ and $d=ab\mid(n+1)n!$ where $d=ab$ is the GCD.
From $ab\mid(n+1)n!$ I get $a\mid(n+1)$ and $b|n!$.
Because $b\m... |
H: $G$ is a finite group. $ H,K \leq G $ and $ K \lhd G $. $G:H$ and $|K|$ are coprime. Show that $K \leq H $
Let $G$ be a finite group. $ H,K \leq G $ and $ K \lhd G $.
$G:H$ and $|K|$ are coprime. Show that $K \leq H $
I started like this:
$G:H = (G:KH)(KH:H)$
Therefore, both $(G:KH)$ and $(KH:H)$ are coprime to ... |
H: Chromatic Number Identity Involving Edges
I'm trying the prove the following:
Let $G$ be a simple graph with $m$ edges. Show that $\chi(G)\leq \frac{1}{2}+\sqrt{2m+\frac{1}{4}}.$
A very minute bit of algebraic manipulation shows that this is equivalent to proving $$\chi(G)(\chi(G)-1)\leq 2m.$$ From here I am a bi... |
H: Fourier-Series of a part-wise defined function?
I have a function f given as
$$
f(x) =
\begin{cases}
ax&\text{ if }\quad-\pi \leq x \leq 0\\
bx&\text{ if }\quad 0<x\leq\pi
\end{cases}
$$
I am supposed to develop the fourier series of this function, using the scalar product
$$
\varphi(f,g) = \frac{1}{\pi} \int_{-\... |
H: Differentiability of $(x,y)\mapsto|x|\cdot y$
Check the differentiability of the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ of two variables given by the formula $$f(x)=|x_1|\cdot x_2$$
I still have problems with this. I started by trying to count partial derivatives:
$\displaystyle\frac{\partial f}{\partial... |
H: find point with equidistant from two points
Suppose there is given two point $A=(-4;-2)$ and $B=(3,2)$ we have to find such $C$ point on
$OY$ axis, such that
a) $C$ is equidistant from $A$ and $B$
b) $ACB$ spline must be minimum.
As I know for solving part (a), we should write equation of line, which is p... |
H: $\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$ Evaluate Integral
Compute the following integral:
$$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$$
Any hint, suggestion is welcome.
AI: Yet a different approach: parametric integration. Let
$$
F(\lambda)=\int_{0}^{\infty} \frac{e^{-\lambda x} \sin(x)}{x}\,dx,\qquad\l... |
H: property to be exact a 1-form on $\mathbb R^2 -\{(0,0)\}$
(a) Let $\omega$ a $1-$form defined on the open set $ U \subset \mathbb R ^n$ and $ c:[a,b] \to U$ a $ C^1 -$differentiable curve such that $ |\omega (c(t))| \leq M \quad \forall t \in [a,b]$
Prove that
$$ \displaystyle{\Bigg| \int_c \omega \Bigg| \leq ML}$... |
H: Lebesgue integral vs area under a curve
Possible Duplicate:
Lebesgue measure on Riemann integrable function in $\mathbb{R}^2$
Is the Lebesgue integral of a positive real function of a real variable equivalent to the Lebesgue measure of the set (in $\mathbb{R}^2$) of all the points between the interval of integra... |
H: Continuous images of open sets are Borel?
Consider a Polish space $(X,d)$ and any metric space $(Y,e)$. If we have a continuous surjection $f:X\to Y$ then is the image $f(U)$ of any open subset $U\subset X$ a Borel set in $Y$?
I know that this is true if we allow $X$ to be compact, since every open subset of a met... |
H: Euler's product formula for $\sin(\pi z)$ and the gamma function
I want to derive Euler's infinite product formula
$$\displaystyle \sin(\pi z) = \pi z \prod_{k=1}^\infty \left( 1 - \frac{z^2}{k^2} \right)$$
by using Euler's reflection equation $\Gamma(z)\Gamma(1-z) \sin(\pi z) = \pi$ and the definition of $\Gamma(z... |
H: Automorphisms of the field of complex numbers
Using AC one may prove that there are $2^{\mathfrak{c}}$ field automorphisms of the field $\mathbb{C}$. Certainly, only the identity map is $\mathbb{C}$-linear ($\mathbb{C}$-homogenous) among them but are all these automorphisms $\mathbb{R}$-linear?
AI: An automorphism ... |
H: Symbol for finite
I understand there is a symbol for infinite. Is there one for finite?
I searched and found there is none. How is finite represented symbolically?
AI: I have never seen a notation for 'finite,' but what I do very often see is denoting something finite as simply being less than infinity. For exampl... |
H: Homomorphisms into complex numbers
Let $I$ be a set of huge cardinality, that is, let $|I|>\mathfrak{c}$. Consider the real product algebra $\mathbb{R}^I$ of all real functions defined on $I$. Can we determine:
1) all algebra homomorphisms $\varphi\colon \mathbb{R}^I\to \mathbb{R}$?
2) all algebra homomorphisms $\v... |
H: Lattices of Subgroups and Graph
In Dummit and Foote´s Abstract Algebra, when talking about the lattice of subgroups of $A_4$, the authors make the statement that, unlike virtaully all groups, $A_4$ has a planar lattice? My question is
What do they mean when they say virtually all groups? Is there a reference for t... |
H: Derivative of a function defined by the divided difference of another function.
Given a function $f$ of class $C$ $^{n+2}$ in an interval $[a,b]$ and $x_{0}=a<x_1<x_2 ... <x_n = b$ a subdivision of $[a,b]$ into $n+1$ points. Given another function $g$ defined in the same interval $[a,b]$ by the divided difference ... |
H: automorphisms of varieties with respect to a cover
Let $X$ and $Y$ be (smooth projective connected) varieties over $\mathbf{C}$.
Let $\pi:X\to Y$ be a finite surjective flat morphism.
Does this induce (by base change) a map $\mathrm{Aut}(Y) \to \mathrm{Aut}(X)$?
I think it does. Given an automorphism $\sigma:Y\to Y... |
H: Prove that $fg\in L^r(\Omega)$ if $f\in L^p(\Omega),g\in L^q(\Omega)$, and $\frac1 p+\frac1 q=\frac1 r$
Can anyone give me a hint for proving the following:
Let $\Omega$ be a measure space. Assume $f \in L^p(\Omega)$ and $g \in L^q(\Omega)$ with $1 \leq p, q \leq \infty$ and $\frac1p + \frac1q \leq 1$. Prove that $... |
H: How to handle the following percentage scenarios?
I have the following scenarios, but I am unclear on how to handle them correctly. I start off with a value like 200 and the following scenarios are:
Remove 10% from 200, and then remove a compound 20% from that value.
Remove 10% from 200, and then remove 20% from ... |
H: Prove that the function is constant
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a differentiable function such that for all $x\in\mathbb{R}^2$,
$$
\frac{\partial f}{\partial x_2}(x)=2\cdot\frac{\partial f}{\partial x_1}(x).
$$
Prove that for every $c\in\mathbb{R}$ function $f$ is constant on $$M_c=\left\{x... |
H: What is the length of a maximal deranged sequence of permutations
We were playing a home-made scribblish and were trying to figure out how to exchange papers. During each round, you'll trade k times and each time you need to give your current paper to someone who has never had it, and you need to receive a paper t... |
H: Does this multivariate function have only one maximum?
Let $X_1$ and $X_2$ be random variables (not of the same distribution and not independent). Both have a zero probability of being below $-1$. Their joint density is $\rho(x_1,x_2)$. Also, they both have finite expectations.
Now, define the region $A = \{ (t_1,t... |
H: Help with complex number phasor notation
I am having trouble understanding how $10jy$ is converted to $10 e^{j\pi/2}$. Here $x$ and $y$ are unit vectors:
(original image)
$$\large=\operatorname{Re}\left[(10\hat{x}-10j\hat{y})e^{-j10\pi z}e^{jwt}\right]$$
$$\large=\operatorname{Re}\left[(10\hat{x}-10e^{j\pi/2}\hat{y... |
H: Calculation of atan2
I am familiar with the basics of atan2. The doubt I have in the computation of atan2 came across from an image processing sofware.
This is a portion of the code segment when x>y. x and y are absolute values.
const_1 = 57.2836266;
const_2 = -18.6674461;
const_3 = 8.91400051;
const_4 = -2.539724... |
H: Why do mathematicians care so much about zeta functions?
Why is it that so many people care so much about zeta functions? Why do people write books and books specifically about the theory of Riemann Zeta functions?
What is its purpose? Is it just to develop small areas of pure mathematics?
AI: For one thing, the Ri... |
H: Probability of vertices in a complete bipartite graph being disconnected such that no path of length 2 remains between them?
My problem is the following. I have a set of vertices $N$ and a set of vertices $H$. Each vertex $n \in N$ is connected by means of an edge to each vertex $h \in H$. So the two sets of vertic... |
H: sum of even-valued and odd-valued Fibonacci numbers
I was solving the Project Euler problem 2
*By starting with 1 and 2, the first 10 terms of Fibonacci Series will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
Find the sum of all the even-valued terms in the sequence which do not exceed 4 million.*
here is my code in... |
H: Chain rule and inverse in matrix calculus
I am having trouble understanding the derivation of some seemingly simple matrix derivatives and am wondering if there is an intuitive (perhaps geometric) explanation. I am reasonably well-versed in multivariate calculus and linear algebra, but am not comfortable with tens... |
H: Sign of an inequality
I have the following:$$\log (0.46)^{k+1}<\log 0.018$$
I solved this by writing$$k<\frac{\log 0.018}{\log 0.46}-1,$$
so $k < 4.17$. The result should be $k > 4.17$. Why is that? Where am I getting wrong?
AI: Since $\log(0.46)$ is negative (because $0.46 \leq e$, or $0.46 \leq 10$, depending on ... |
H: How to prove that $\operatorname{lcm}\{1,\ldots,n\}\geq (\sqrt{n})^{\pi(n)}$?
Let $\operatorname{LCM}[n]:=\operatorname{lcm}\{1,\ldots,n\}$. It is easy to verify $\operatorname{LCM}[n]\geq 2^{\pi(n)}$, where $\pi(n)$ counts the number of distinct primes up to $n$.
But how can I prove the bound $\operatorname{LCM}[... |
H: Main differences between analytic number theory and algebraic number theory
What are some of the big differences between analytic number theory and algebraic number theory?
Well, maybe I saw too much of the similarities between those two subjects, while I don't see too much of analysis in analytic number theory.
AI... |
H: Orthogonal Trajectory of $x^2 + 2y^2 = k^2$
$x^2 + 2y^2 = k^2$
I first take the derivative like the instructions say.
$2x + 4y \frac{dy}{dx} = 0$
I am not entirely sure why a dy and dx appears but it does in the instructions so I go with it.
Now I need to solve for $y'$
$ + 4y \frac{dy}{dx} = -2x$
$ \frac{dy}{dx} ... |
H: A simple quadratic inequality
For positive integers $n\ge c\ge 5$, why does
$$c+2(n-c)+\frac{(n-c)^2}{4}\le\frac{(n-1)^2}{4}+1\text{ ?}$$
AI: To avoid fractions, we multiply the left-hand side by $4$, obtaining
$$(n-c)^2+8(n-c)+4c.$$
Complete the square. We get
$$(n-c+4)^2 +4c -16.$$
Now calculate $[(n-1)^2 +4]-[... |
H: norm for estimating the error of the numerical method
In most of the books on numerical methods and finite difference methods the error is measured in discrete $L^2$ norm. I was wondering if people do the in Sobolev norm. I have never see that done and I want to know why no one uses that.
To be more specific look ... |
H: Context-free grammar for words of a CF grammar starting with a certain symbol
Let $L$ be a context-free language on the alphabet $\Sigma$.
I need to show that for each $a \in \Sigma$ the language $L_a = \{ x \in \Sigma^* \; | \; a.x \in L\}$ is context-free as well.
I wanted to prove it by generating a context-fre... |
H: Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
There is a famous Theorem telling that:
For $n≥5$, $A_n$ is the only proper nontrivial normal subgroup of $S_n$.
For the proof, we firstly start with assuming a subgroup of $S_n$ which $1≠N⊲S_n$. We proceed until at the last part... |
H: Approximation of $\log(x)$ as a linear combination of $\log(2)$ and $\log(3)$
I wonder if it's possible to approximate $\log(n)$, n integer, by using a linear combination of $\log(2)$ and $\log(3)$.
More formally, given integer $n$ and and real $\epsilon>0$, is it always possible to find integer $x,a,b$ where:
$$\... |
H: Matrix Representation of the Tensor Product of Linear Maps
I'm trying to work out some examples of applying the tensor product in some concrete cases to get
a better understanding of it. Within this context, let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$
be a linear map with matrix $A$ and let $g:\mathbb{R}^2 \righ... |
H: Notation question. Piecewise function.
I have observed the below statement in a report.
$
f(x,y) = \left\{
\begin{array}{lr}
f<0 & : (x,y) \in A\\
f = 0 & : (x,y) \in B \\
f>0 & : (x,y) \in C
\end{array}
\right.$
I understand the meaning behind it, but is the notation correct?... |
H: Are There Any Symbols for Contradictions?
Perhaps, this question has been answered already but I am not aware of any existing answer. Is there any international icon or symbol for showing Contradiction or reaching a contradiction in Mathematical contexts? The same story can be seen for showing that someone reached ... |
H: What are operators that map a number to a probability distribution function called?
I'm a scientist who has stumbled upon an idea that I think might be helpful in my field. I'm looking for information about whether it has been treated in mathematics before – and I would be surprised if it hasn't – and if so, a poin... |
H: What is a good book to study linear algebra?
I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a $\circ$ are those which I'm mildly comfortable with. The ones that aren't mar... |
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