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H: Uniqueness of meets and joins in posets
Exercise 1.2.8 (Part 2), p.8, from Categories for Types by Roy L. Crole.
Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of upper bounds for $A$. A meet of $A$, if such exists, is a greatest element in ... |
H: A simple algebraic question
Show, as elegantly as possible, that
$2p_0(1-p_0^r) > (p_0+\Delta)[1-(p_0+\Delta)^r] + (p_0 - \Delta)[1-(p_0-\Delta)^r]$
for $r \in \mathbb{R^+}$, $0 < p_0 - \Delta < p_0 + \Delta < (r+1)^{-\frac{1}{r}}$.
This is trivial for $r=1$, but I believe it should also hold for general $r > 0$. I... |
H: Integral of exponential function
Consider $f$ being a measurable function on $R^n$ such that $$\int_{E} e^{|f|}=1$$ ($E$ measurable) and $f$ vanishes outside $E$ . Then $f\in L^p(R^n)$ for all $p\in (0,\infty)$.
I tried using that measure of $E$ cannot be bigger than $1$ and the formulae
$$\int|f|^p=p\int_0^\infty... |
H: Derivative of an indicator inside an integral
I have a fairly basic question that relates to understanding a particular derivation.
I have the following function $Q(x) = E\left[I(F(x+\varepsilon)>c)\right]$, where $x \in R$, $\varepsilon \sim N(0,\sigma^2)$ for a known $\sigma>0$, $c$ is a scalar constant, $F(\cdo... |
H: Given $\lambda$ and $A$, find $v$ such that $\lambda = v^{\intercal}Av$
If I know the values of $\lambda$ and $A$, how do I find a vector $v$ such that $\lambda = v^{\intercal}Av $?
This isn't a homework question; I just ran into this problem in Real Life and realized I couldn't solve it!
AI: If $A$ is positive def... |
H: Integral over circle
Consider the following integral:
$\displaystyle \int_{|z-3i|=1} \frac{e^{z^2+z}}{z} dz$.
Why is my approach wrong?
Set $f(z)=e^{z^2+z}$, then from the Cauchy integral formula for circles, we have:
$\displaystyle f(0)=\frac{1}{2\pi i} \int_{|z-3i|=1} \frac{f(z)}{z} dz$.
So we have: $\displaystyl... |
H: $<$ on a preorder is a strict partial order
Definition: Suppose $X$ is a preorder. Define $x < y$ as $x \le y$ and $y \not\le x$ for each $x, y \in X$.
Question: Show that this gives a strict partial order on $X$.
AI: We must show that the relation $<$ is irreflexive and transitive.
Irreflexive: Suppose $x \in X$. ... |
H: Determining if sets are nonempty
How to show that these sets are nonempty (here $\mid $ means "divides")?
Here N is an arbitrary large integer and q is some fixed integer.
$R = \lbrace k \in {\mathbb N}:(kN\mid k!) \wedge ((k - 1)N\mid k!) \wedge \cdots \wedge (N\mid k!) \wedge (k > Nq)\rbrace$
$S = \lbrace k \in... |
H: What could this sum possibly converge to?
Consider
$$\sum_{i=0}^{\infty} \dfrac{n-i}{n!}$$
For $n \geq i$
Consider $n$ to be any natural number. I know for sure it's going to converge, but how do I write a formula for the sum?
Possible interpretation:
Find:
$$\lim_{n\to \infty}\sum_{i=0}^{n} \dfrac{n-i}{n!}$$
AI... |
H: Generating Function of Even Fibonacci
I was posed the following question recently on an exam:
Determine the generating function of the even-indexed Fibonacci numbers $F_{2n}$ given that the generating function of Fibonacci numbers is $\frac{x}{1-x-x^2}$.
I could not think of how apply knowledge of the generating... |
H: Qubits and vector projections
In $\Bbb C^2$, how many real unit vectors are there whose projection onto $|1\rangle$ has length $\sqrt{3}/2$?
I would think zero as $\bigl(\frac{\sqrt{3}}{2}\bigr)^2 + x^2 = 1$, therefore there are no real values of $x$ to satisfy the equation. Please help.
AI: Write $\Bbb C^2$ as $\B... |
H: Evaluating $ \int_{|z|=1} \frac{e^z}{(z+3)\sin(2z)} \ dz$
Consider the following integral:
$\displaystyle \int_{|z|=1} \frac{e^z}{(z+3)\sin(2z)}dz$.
To apply the Cauchy integral formula, I rewrite it as:
$\displaystyle \int_{|z|=1} \frac{ze^z}{z(z+3)\sin(2z)}dz$ and take
$\displaystyle f(z)=\frac{ze^z}{(z+3)\sin(2z... |
H: Is it possible to solve a linear inequality system using SVD?
I have a large linear inequality system of the form $Ax ≤ 0$.
Is there a way to solve this system using linear algebra tools like SVD?
AI: No; loosely speaking this problem is more related to linear programming, which is not equal to linear algebra.
(Plu... |
H: Evaluating $\int_{-\pi}^{\pi} \cos(e^{it})dt$
We have to calculate:
$$\displaystyle \int_{-\pi}^{\pi} \cos(e^{it})dt.$$
Is there something more promising one could try instead of a subsitution $u=e^{it}$?
AI: Actually the Taylor series approach is fairly straightforward too...
$\int_{-\pi}^{\pi} \cos(e^{it}) \,dt =... |
H: what can we say about the SVD of a matrix with respect to another SVD?
I have two matrices, both are $n \times m$ where $m < n$. These two matrices are $A$ and $B$.
I also know the singular value decomposition of $AB^{\top} = U \Sigma V^{\top}$.
Is there anything I can say about the SVD of $A$ or $B$? (for example,... |
H: how to solve an ODE with boundary conditions $ y(0)=y(\infty) $ by shooting method
how could I solve the linear differential equation
$$ -y''(x)+x^{3}y(x)=0$$ with the boundary conditions $ y(0)=y(\infty) $ by the linear shooting method ?
If we had $ y(0)=y(1)=0$ then the interval is finite however how can i deal w... |
H: Constructing $\mathbb N$ from the set of factorials
Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example:
$$ 3 = 2! + 1!$$
$$ 4 = 3! - 2! = 2! + 1! + 0!$$
$$ 146 = 4!\cdot... |
H: If $g(x) := \int_1^2 f(xt)dt \equiv 0$ then $f \equiv 0$
Let $f \colon \mathbb R \to \mathbb R$ be a continuous function. Let's
define $$ g(x) := \int_1^2 f(xt)dt. $$
Prove that $g \equiv 0 \Rightarrow f \equiv 0$.
Well, I show you what I have done. First of all, I've noted that $g$ is differentiable for every... |
H: conformally equivalent riemann surfaces
Two riemann surfaces $S$ and $R$ are said to be conformally equivalent if there exist a holomorphic map $f:S\rightarrow R$ which is one-one and onto, and inverse is also holomorphic.
I have to show No two of them are conformaly equivalent:
$a$) $\hat{\mathbb{C}}$, $b$) $\math... |
H: Re-writing in sign basis.
$\newcommand\ket[1]{\left\vert #1\right\rangle}$ Let $\ket\phi = 12 \ket{0} + 1 + 2\sqrt{i2}\ket{1}$. Write $\ket\phi$ in the form $\alpha_0\ket{+} + \alpha_1\ket{-}$. What is $\alpha_0$?
I came across this problem in a course i am doing, i have been struggling writing things in sign basi... |
H: Compute $\sum \frac{1}{k^2}$ using Euler-Maclaurin formula
I read that Euler used the summation formula to calculate the value of the series $\sum_{k =1}^{\infty} \frac{1}{k^2}$ to high precision without too much hassle. The article Dances between continuous and discrete: Euler’s summation formula goes into the cal... |
H: Evaluating $\int_{|z|=1} \frac{\sin(z^2)}{ \left( \sin(z) \right)^2} dz.$
Consider $z \in \mathbb{C}$ and
$$\int_{|z|=1} \frac{\sin(z^2)}{ \left( \sin(z) \right)^2} dz.$$
How would we integrate this?
AI: The sine is an entire function, i.e. has no sigularities at finite points. So the only thing that could cause b... |
H: Inverse image of a union equals the union of the inverse images
I just wonder if my following solution is true.
Let $X,Y$ be sets, let $f:X\to Y$ be a function, let $\{Y_i\}_{i\in I}$ be a family of subsets of $Y$. (Note: I use equalities instead of mutual containment)
$$\begin{align}f^{-1}\left[\bigcup_{i\in I} Y_... |
H: Evaluating $\int_{|z|=1} \sin\left(e^{\frac{1}{z}}\right) \ dz$
Let
$$\int_{|z|=1} \sin\left(e^{\frac{1}{z}}\right) dz.$$
Is there an alternative to the residue theorem if we want to calculate the above integral?
AI: Sasha posted a terse comment suggesting the same substitution I was about to suggest. Here's why... |
H: Relating sum of a sequence to sums in subsequences.
I have been wondering about this question for some time now and would greatly appreciate if anyone had any leads on how to proceed.
Let $\{A_{i}\}_{i \in N}$ such that for, $i \neq j$, $A_{i} \cap A_{j} = \emptyset$ and $\bigcup{A_{i}} = N$. Let $(a_{i})_{i \in N}... |
H: Joint continuous random variables pdf
I have the following problem that I think I know how to solve, but I don't see why the given choices are as they are:
Suppose X and Y are jointly continuous random variables with joint
probability density function given by
f(x, y) = 1/c, x > 0, y > 0, x^2 + y^2 ≤ 2; or 0 othe... |
H: Complex contour integral
Let
$$\int_\gamma z(e^{z^2}+1)dz,$$
where $\displaystyle \gamma(t)=e^{it},t \in \left[0,\frac{\pi}{2}\right]$.
In order to apply Cauchy's integral formula, I'll set $f(z)=z^2\left(e^{z^2}+1\right)$ and rewrite
$$\int_\gamma z(e^{z^2}+1)dz=\int_\gamma \frac{f(z)}{z} dz.$$
$f(0)=0$, so we wou... |
H: Projective representations of loop groups
If $G$ is a Lie group and we take its loop group $LG$ why do we deal with projective representations of $LG$ and central extensions thereof? Where does the extra complexity come in to require us to consider this extra, very complicated, step?
AI: Projective representations ... |
H: The number $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$ is always an integer
For each $n$ consider the expression $$\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$$
I am trying to prove by inducti... |
H: why this map is injective?
let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and $[z:w]=[\lambda z:\lambda w],\lambda\in\mathbb{C}^{*}$ is a point in $\mathbb{CP}^1$, the map is $\phi:U_0\rightarrow\mathbb{C}$ defined by $$\phi([z:w])=w/z$$
$$\phi([z_1:w_1])=\phi([z_2:w_2])$$ which ... |
H: Laurent series of $f(z)=\frac{1}{z(z-1)(z-2)}.$
Consider $$f(z)=\frac{1}{z(z-1)(z-2)}.$$
I want to determine the Laurent series in the point $z_0=0$ on $0<|z|<1$.
Partial decomposition yields:
$$f(z)=\frac{1}{z(z-1)(z-2)}=(1/2)\cdot (1/z) - (1/(z-1)) + (1/2)(1/(z-2)).$$
Is the general strategy now, to try to use th... |
H: map from $\mathbb{C}$ to $\mathbb{C}/L$ is open map?
Let $w_1,w_2\in\mathbb{C}$ be linearly independent vectors and let$$L=\{m_1w_1+m_2w_2:m_1,m_2\in\mathbb{Z}.\}$$
How does one show that the projection map $\pi:\mathbb{C}\rightarrow\mathbb{C}/L$ is open map? Well, we define $U \subset \mathbb{C}/L$ is open iff $\... |
H: Conditional probablity with tree diagram
Could someone tell me if I got this right?
So I drew a tree diagram
Sorry if it is too messy, but i had to do this on PaintBrush.
(a) I just added all the branches $\mathbb{P}(S_4 = 17) = 0.6 \times 0.4+0.4\times0.4=0.4$
(b)This one is a bit tricky, so basically I have $\... |
H: Laurent series - $f(z)=\frac{1}{z^2-4}+\frac{1}{6-z}$
We have $f(z)=\frac{1}{z^2-4}+\frac{1}{6-z}$. I want to expand this as a Laurent series in $z_0=2$ on $\{4<|z-2|<\infty\}$.
The partial decomposition is:
$$f(z)=\frac{1}{4}\frac{1}{z-2}-\frac{1}{4}\frac{1}{z+2}+\frac{1}{6-z}$$
In my reference, they expand $\fra... |
H: Find the particular solution of the given differential equation that satisfies that indicated side condition
$$\frac{dy}{dx} = 3\sin(x/2)$$
$$y = 1 ,x= \frac{\pi}{3}$$
I just get stuck on the integration of the $3\sin(x/2)$.
AI: $$\int\sin kx\,dx=-\frac{1}{k}\cos kx+C\,\,,\,k\neq 0\,$$ |
H: map from $D$ to $\pi(D)$ is injective?
please follow this map from $\mathbb{C}$ to $\mathbb{C}/L$ is open map? ,let w be a non zero element of the lattice L so that |w|>2ϵ, fix such ϵ>0 and any $z_0$∈C and take an open disk of radius ϵ with centre at $z_0$, could you please tell me why $π:D\rightarrow π(D)$ is inje... |
H: Comparing money earned by two social games
Which game makes more money on a daily basis?
game A, which has 175,000 DAU, a 32% second-day retention rate, a $0.05 ARPDAU, and a 30-day lifetime
or
game B, which has 150,000 DAU, a 22% second-day retention rate, a $0.08 ARPDAU, and a 15-day lifetime
AI: Lets label Dau ... |
H: Diagonalizing Matrices over UFDs
Suppose $R$ is a UFD, $K$ its field of fractions. If a matrix $M$ with entries in $R$ has distinct eigenvalues, then it is certainly diagonalizable over $K$. If those eigenvalues are actually in $R$, must it be diagonalizable over $R$?
If it helps: the specific case I'm interested ... |
H: Integration by substitution in $n$ dimension
I just read the proof of Hardy-Littlewood-Sobolev inequality abaout fractional integral operator. Then I found the following identity (but don't understand it)
\begin{equation}
\int_{%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}\backslash B\left( ... |
H: Specifying plane waves
I'm having trouble understanding how to specify a transverse wave (including it's longitudinal axis and transverse direction) in 3d space.
I know this is called a "plane wave", and I know the formula for a plane wave along a unit vector $\hat{k}$ is:
$$
A( \vec{r}, t ) = A_0 cos( \vec{k} \cdo... |
H: Integrating $3\sin(x/2)$.
$$\int 3\sin\left(\frac{x}{2}\right)dx$$
can't figure this one out! I'm not sure if I'm supposed to substitute or not?
Here's where I'm at...
$$3\int\sin\left(\frac{x}{2}\right)dx$$
$$u = \frac{x}{2}$$
$$du = \frac{1}{2}dx$$
$$dx = 2du$$
$$3\int\sin\left(u\right)2du$$
and then...
$$-6cos\... |
H: A power-exponential congruence equation
Let $n \in \mathbb{N}$ with $(n,\varphi(n))=1$ , where $\varphi$ is the Euler-totient function. Prove the equation $x^x \equiv c \pmod{n}$ has integer solution for all $c \in \mathbb{N}$
My thought: By Euler's theorem one has $\varphi(n)^{\varphi(n)} \equiv 1 \pmod{n}$, so I ... |
H: Area of Triangle inside another Triangle
I am stumped on the following question:
In the figure below $AD=4$ , $AB=3$ , and $CD=9$. What is the area of Triangle AEC?
I need to solve this using trigonometric ratios however if trig. ratios makes this problem a lot easier to solve I would be interested in look... |
H: Subvariety of Product of Elliptic Curves
This is almost certainly known (and maybe written down somewhere?). Is there an example of two elliptic curves $C, E/k$ that are not isomorphic, yet there is an embedding $C\hookrightarrow E\times E$ as an abelian subvariety?
If this is known, is there a reference that talk... |
H: Average of numbers in a specific range
The question is :
What is the arithmetic mean of all multiples of 10 from 10 to 190 inclusive.
Now I know how many nos there are by using $\frac{190-10}{10}+1 = 19$ but how do I get their sum ?
AI: 180+10 = 190;
170+20 = 190;
160+30 = 190;
...
how many numbers are there? 1... |
H: Is $\mathbb{Z}[\sqrt{2},\sqrt{3}]$ flat over $\mathbb{Z}[\sqrt{2}]$?
Is $\mathbb{Z}[\sqrt{2},\sqrt{3}]$ flat over $\mathbb{Z}[\sqrt{2}]$? The definitions doesn't seem to help. An idea of how to look at such problems would be helpful.
AI: $\mathbb Z[\sqrt{2},\sqrt{3}]$ is freely generated as a $\mathbb Z[\sqrt{2}]$-... |
H: Application of Cauchy's integral theorem
On page 97 of John B. Conway's Functions of one complex variable, the author states that:
"Suppose $G$ is a region (open connected subset) and let $f$ be analytic in $G$ with zeros at $a_1,a_2,...,a_m$. So we can write $f(z)=z(z-a_1)(z-a_2)...(z-a_m)g(z)$ where $g$ is analy... |
H: Continuous mappings pull back closed sets to closed sets
George F Simmons, Topology and Modern Analysis pg.79 Problem 4
Let $X$ and $Y$ be metric spaces. Show that an into mapping $f:X \rightarrow Y$ is continuous $\iff$ $f^{-1}\left(G\right)$ is closed in $X$ whenever $G$ is closed in $Y$.
I can prove the problem... |
H: Given that $5n$ is a square and $75np$ is a cube, why is the smallest possible value of $n+p$ equal to $14$?
I can't solve this problem:
Suppose $n$ and $p$ are integers greater than $1$, $5n$ is the square of a number, and $75np$ is the cube of a number. What is the smallest value for $n+p$?
(Answer given is $14$... |
H: Question about the $\mathrm{Tor}$ functor
Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring.
We take a projective resolution of $M$:
$$ \dots \to P_1 \to P_0 \to M \to 0$$
Now does it matter whether we apply $-\otimes N$ or $N \otimes -$ to this? It sh... |
H: Definition of 'closed relative to'
I know what's the definition of 'open relative'.
I googled 'close relative', but i couldn't find a definition of it.
How come every metric space $X$ is close relative to $X$?
If $p$ is a limit point of $X$, there exists a neighborhood $N_r(p)$ and $q\in X$ such that $q\in N_r(p)$ ... |
H: Going from binomial distribution to Poisson distribution
Why does the Poisson distribution
$$\!f(k; \lambda)= \Pr(X=k)= \frac{\lambda^k \exp{(-\lambda})}{k!}$$
contain the exponential function $\exp$, while its relation to the binomial distribution would suggest it's all about powers of $2$?
AI: The Binomial $(n,p... |
H: Aftermath of the incompletness theorem proof
This is somewhat of a minor point about the incompletness theorem, but I'm always a little unsure:
So one proves that there is a formula which is unprovable in the theory of consideration. Okay, at this point one is done.
Then, as that unproven sentence contains the cl... |
H: Norm of compactly supported functions with disjoint support .
If $u_n \in C_c^\infty (\mathbb R)$ with $u_n=u(x+n)$ , $n\in \mathbb N$ , $u$ is not identically zero.
How do i prove that $||u_{n+k}-u_n||_{L^q}^q =2||u||_{L^q}^q$.
What my doubt is that even if we take $u_{n+k}$ and $u_n$ to have disjoint support, i... |
H: Applying the symmetric difference to multiple sets
I was working on set theory and I came across a rule:
$$(A\cup B) - (A\cap B) = (A\cap B)'$$
I have 4 sets $A$, $B$, $C$, $D$. How can I apply the rule above to all set at same time? is the below mentioned result mathematically correct?
$$(A\cup B\cup C\cup D) - (... |
H: Prove that $\mathop {\lim }\limits_{(x,y,z) \to (0,0,0)} \left( {\frac{{{x^2}y - x{z^2}}}{{yz - {z^2}}}} \right)=0$
Prove that:
For all $\epsilon>0$ exist $\delta>0$ which depends on $\epsilon$, such that:
$$\left| {\frac{{2{x^2}y - x{z^2}}}{{yz - {z^2}}}}-0 \right|<\epsilon$$ ever that
$$0 < \sqrt {{x^2} + {y^2} ... |
H: Counterexample of Compactness
Let $X$ be a metric space and $E\subset X$.
Let {$G_i$} be an open cover of $E$
For every open cover {$G_i$}, there exists a finite subcover {$G_{i_n}$} of $E$ such that $G_{i_n} \in${$G_i$}.
For every open cover {$G_i$}, there exists {$M_n$}, a finite family of open sets, such that $... |
H: Good problem book on Abstract Algebra
I am currently self-studying abstract algebra from Artin. In that background, I am looking for a problem book in a spirit somewhat similar to Problems in Mathematical Analysis by AMS so that I have a lot of problems to solve.
AI: I don't know if I would call it a "problem book"... |
H: How do I work out a new constrained scale height based on a new width?
This is super simple I guess but my mind has gone blank.
I have these dimensions below, and I am trying to scale this rectangle to 610 wide
Width 560
Height 315
From these figures, how do I work out the percentage to scale the height so the rect... |
H: Existence statement and Axiom of Choice
Let $I$ be an infinite set.
Suppose that for every $i\in I$, there exists a set $S_i$ satisfies a statement $\psi(S_i)$.
Here, is constructing a family of such $S_i$ (i.e. $\{S_i\}$ for $i\in I$) using Axiom of Choice??
For example, $\{V_i\}$ is given, suppose for every $i\in... |
H: Quotient of a free $\mathbb{Z}$-module
I'm trying to find the quotient of a free $\mathbb{Z}$-module, but somehow I don't really find the right procedure on how to get the right quotient (nor have I found any sources). I've read What does it mean here to describe the structure of this quotient module? already, but ... |
H: For Maths Major, advice for perfect book to learn Algorithms and Date Structures
Purpose: Self-Learning, NOT for course or exam
Prerequisite: Done a course in basic data structures and algorithms, but too basic, not many things.
Major: Bachelor, Mathematics
My Opinion: Prefer more compact, mathematical, rigorous bo... |
H: Finding Nash Equilibria with Calculus
The problem is summarized as:
There are two players. Player 1's strategy is h. Player 2's strategy is w. Both of their strategy sets are within the range [0,500].
Player 1's payoff function is:
$
P_h(h, w) = 50h + 2hw-\frac{1}{2}(h)^2
$
Player 2's payoff function is:
$
P_w(h,... |
H: Check if below limits exist $\lim_{(x,y,z) \to (0,0,0)} \left( {\frac{{2{x^2}y - x{z^2}}}{{y^2 - xz}}} \right)$?
Check if below limits exist $$\lim_{(x,y,z) \to (0,0,0)} \left( {\frac{{2{x^2}y - x{z^2}}}{{y^2 - xz}}} \right).$$
Is there any succession to prove that this limit is not zero?
AI: Taking the sequence $x... |
H: Open relative and choice
I'm using ZF as my axiom system.
Let $X$ be a metric space and $K\subset Y \subset X$. Suppose $K$ is compact relative to $X$.
Let $\{V_a\mid a\in I\}$ be a family of open sets relative to $Y$ such that $K\subset \bigcup V_a$. Then for every $a\in I$, there exists $G_a$, an open subset of $... |
H: How high will the water rise
I need to know where I am going wrong since I am getting the wrong answer
12 litres of water are poured into an aquarium of $50$ cm length , $30$ cm breadth and $40$ cm Height.How high in cm will the water rise. (Ans 8 cm)
Edit:
So I was making some very illogical assumptions howeve... |
H: Doubling the Sides of the Cube
For the following question
A cubical block of metal weighs 6 pounds.How much will another cube of the same metal weigh if its sides are twice as long ? Ans=48
Here is how I am solving it but I am getting 12 as the answer
For the current cubical block $12$ edges of cube weigh = 6 p... |
H: Is it true that if $\alpha \in \operatorname{Frac}(A)$ and $s\alpha \in A$, then $\alpha \in S^{-1}A$?
In the proof of Proposition 1.9 in Chapter VII of Algebra by Serge Lang, it seems to me that the following property is used.
Let $A$ be a commutative entire ring, $S$ a multiplicative subset of $A$, $0 \not \in S$... |
H: Generating random numbers with the distribution of the primes
I would like to generate random numbers whose distribution mimics that of the primes.
So the number of generated random numbers less than $n$ should grow like $n / \log n$,
most intervals $[n,n+n^\epsilon]$ should contain approximately $n^\epsilon / \log... |
H: Weyl's unitarity trick
Weyl's unitarity trick creates from an irreducible representation of a compact group a unitary representation by averaging with a Haar measure.
Does anyone know a reference to the paper (or book, with page number) where Weyl introduced his unitarity trick?
AI: The first instance I'm aware of ... |
H: Solving linear congruences by hand: modular fractions and inverses
When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:
"Notice" that adding $13$ on the right and subtracting $13x$ on the lef... |
H: If n balls are thrown into k bins, what is the probability that every bin gets at least one ball?
If $n$ balls are thrown into $k$ bins (uniformly at random and independently), what is the probability that every bin gets at least one ball? i.e. If we write $X$ for the number of empty bins, what is $P(X=0)$?
I was ... |
H: Decomposition of $l$ in a subfield of a cyclotomic number field of an odd prime order $l$
Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$.
Let $K = \mathbb{Q}(\zeta)$.
Let $A$ be the ring of algebraic integers in $K$.
Let $G$ be the Galois group of $\mathbb{Q}(\zeta)/... |
H: Largest positive integer $k$ such that $\mu(n+r)=0$ for all $1\leq r\leq k$
Find the largest positive integer $k$, such that $\mu(n+r)=0$ for all $1\leq r\leq k$ where $r,n$ are positive integers.
As far as I could make out, we need to find out the maximum range(if nay) of numbers where each has a square divisor... |
H: Does anyone recognize this function?
I am looking for a function $f(n)$ that satisfies the following two conditions at the same time $$ \frac{f(n-1)}{f(n)}=(-1)^n\quad ,\quad \frac{f(n+1)}{f(n)}=(+1)^n\equiv 1,\quad \forall n\in\mathbb{N}\ . $$
As I am not even sure if such a function exists, I'd appreciate any he... |
H: non-residually finite group
Let $G$ be the subgroup of $\text{Bij}(\mathbb{Z})$ generated by $\sigma : n \mapsto n+1$ and $\tau$ which switches $0$ and $1$.
How can we prove that $G$ is not residually finite? Is it hopfian?
AI: $G$ is not residually finite. Specifically, any map $f:G\to H$ with $H$ finite sends $g... |
H: Anticommutative operation on a set with more than one element is not commutative and has no identity element?
An anticommutative operation on a set X to be a function $\cdot:X\times X\rightarrow X$ satisfying two properties:
(i) Existence of right identity: $\exists r\in X:x\cdot r=x$ for all $x\in X$
(ii) $x\cdot ... |
H: Convex function on Banach space
Let $(Y,\|\cdot\|)$ a Banach space and $b\colon Y\to \mathbb{R}$ a nonnegative convex function such that, for some $\mathcal{E}>0$, the set $\{y\in Y\,:\, b(y)<\mathcal{E}\}$ is nonempty and bounded.
I need prove that exist $z\in Y$, $c>0$ and $C>0$ such that $b(y)-b(z)\geq c\|y-z\|$... |
H: Harmonic coordinates for Ricci flow
It is customary to use DeTurck's argument (or Hamilton's original one involving the Nash-Moser iteration) for proving local existence of the Ricci flow. I am wondering why one cannot use harmonic coordinates for this purpose, as can be done for the Einstein equations.
AI: There a... |
H: Is the axiom of choice needed to show that $a^2=a$?
A comment on this answer states that choice is needed for the statement that $a^2=a$ for all infinite cardinals $a$. In Thomas Jech's Set Theory (3rd edition), his theorem 3.5 proves this statement when $a = \aleph_b$ for some $b$.
It's not clear to me in the proo... |
H: let $f$ be any meromorphic function then $\operatorname{ord}_p(f)=0$
let $f$ be any meromorphic function at $p$ whose laurent series is $\sum_n c_n(z-z_0)^n$, define the $order$ of $f$ at $p$ $$\operatorname{ord}_p (f)= \min\{n:c_n\neq 0\}$$
let $f=\frac{p}{q}$ be a rational function, considered as meromorphic func... |
H: 5 digit no combination problem
How many 5-digit numbers can be created based on the following conditions:
Starting from the left, the first digit is even and nonzero, the second is odd, the third is an odd prime and the fourth and fifth are two random digits not used before in the number?
Note: First digit cannot b... |
H: Show $A$ is "real-equivalent" to its transpose
The problem statement:
Let $A$ be a real $9\times 9$ matrix with transpose $B$. Prove that the matrices are real equivalent in the following sense: There exists a real invertible $9\times 9$ matrix $H$ such that $AH=HB$.
Unfortunately, I don't have much of an attempt a... |
H: Expressing ratio of functions as a constant number
Suppose that we have a function $f$ where $Q(\tau)$ is modified by multiplying $Q(\tau)$ by a real number. Let $f'$ be the modified function, and let $k \in \mathbb{R}$ or $k \in \mathbb{C}$. Take the ratio:
$\frac{f}{{f'}} = k$
$f = \exp \left( { - {{\int\limits_... |
H: Elementary Geometry Nomenclature: why so bad?
A long-ish wall of text, and I apologize.
Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A student piped up and said, "that's a rhombus." The professor stopped mid-st... |
H: What's polynomial composition useful for?
I've made this question here. But I got no answer then I'll make a question with it:
I remember of studying:
Sum of two polynomials;
Difference of two polynomials;
Product of a constant and a polynomial;
Product of two polynomials;
And they kinda make sense for me, but I ... |
H: Are there larger than countable chains in the join-semilattice of Turing degrees?
Recall that Turing degrees are equivalence classes of subsets of $\mathbb{N}$ under Turing equivalence (mutual Turing reducibility). They are partially ordered by Turing reducibility and form a join-semilattice of cardinality $2^{\ale... |
H: The sum of powers of two and two's complement – is there a deeper meaning behind this?
Probably everyone has once come across the following "theorem" with corresponding "proof":
$$\sum_{n=0}^\infty 2^n = -1$$
Proof: $\sum_{n=0}^\infty q^n = 1/(1-q)$. Insert $q=2$ to get the result.
Of course the "proof" neglects th... |
H: How to solve this system of fractional equations?
I have to complete a summer packet of 90 Algebra 2 questions. I have completed 89 of them, the only one I could not get was this. I know the answer is $y = \frac {47}2$, $\frac 17$ according to WolframAlpha, but I have no idea how to reach that answer, my Algebra 2 ... |
H: If $\phi \in C^1_c(\mathbb R)$ then $ \lim_n \int_\mathbb R \frac{\sin(nx)}{x}\phi(x)\,dx = \pi\phi(0)$.
Let $\phi \in C^1_c(\mathbb R)$. Prove that $$
\lim_{n \to +\infty} \int_\mathbb R \frac{\sin(nx)}{x}\phi(x) \, dx = \pi\phi(0).
$$
Unfortunately, I didn't manage to give a complete proof. First of all, I fixe... |
H: Help with radical equation
Please, help me to solve this equation. No advanced math should be needed.
$$
3x^2 - 4x + \sqrt{3x^2 - 4x - 6} = 18
$$
I'm clueless. It should be simple.
AI: Hint : set $u:= \sqrt{3 x^2-4x-6}$ then your equation becomes $u^2+6+u=18$ or $$u^2+u-12=0$$ |
H: A characteristic of intersection with cartesian product
Fix some binary relation $f$.
Does there necessarily exist a set $C$ such that $(x\times x)\cap f\ne \varnothing \Leftrightarrow x\cap C\ne \varnothing$ for all sets $x$?
AI: Not necessarily.
Say $f=\{(a,b)\}$, with $a\neq b$, and assume such a set $C$ exists.... |
H: Does this qualify as a proof of $\sum _{k=0}^{l}\binom{n}{k}\binom{m}{l-k} = \binom{n+m}{l}$? (Spivak's 'Calculus')
I'm working through Spivak's 'Calculus' at the moment, and a question about series confused me a bit. I think I have the solution, but I'm not sure if my "proof" holds.
The question is:
Prove that $\... |
H: A trigonometric identity
If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind,
$$\ln(2) + \sum _ {n=1} ^{\infty} \frac{\cos(n\theta)}{n} = - \ln \left\vert \sin\left(\frac{\theta}{2}\right)\right\vert $$
Is the ab... |
H: Discontinuous functions that are continuous on every line in ${\bf R}^2$
This is exercise 7 from chapter 4 of Walter Rudin Principles of Mathematical Analysis, 3rd edition. (Page 99)
Define $f$ and $g$ on ${\bf R}^2$ by: $$f(x,y) = \cases {0,&if $(x,y)=(0,0)$\\
xy^2/(x^2+y^4) &otherwise}$$
$$g(x,y) = \cases {0,&if ... |
H: Are all $p$-adic number systems the same?
After just having learned about $p$-adic numbers I've now got another question which I can't figure out from the Wikipedia page.
As far as I understand, the $p$-adic numbers are basically completing the rational numbers in the same way the real numbers do, except with a dif... |
H: Find numbers in a set whose sum equals x
Given $S$, a set (unsorted) of $n$ real numbers;
and a real number $x$;
what is the fastest algorithm to determine if $S$ contains two numbers whose sum exactly equals $x$?
We could add each element of $S$ with each other element to determine if a pair whose sum is $x$ exist... |
H: Discreteness of Preimages of points, map between compact riemann surfaces
Let $f:X\rightarrow Y$ be a non-constant holomorphic map between compcat riemann surfaces, we need to show $f^{-1}(y),\forall y\in Y$ is finite and discrete subset of $X$.
What if $X$ and $Y$ are non compact?
well, $f$ is onto clearly, and I ... |
H: A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$
So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling with some algebraic gymnastics for a while now, but I can't ... |
H: Intersection of 2 $p$-simplices is a finite union of some $p$-simplices
I'm looking for a non-painful proof of this assertion.
A p-simplex is defined as the set of all sums $\sum_{i=0}^p t_i x_i$ with $0\leq t_i\leq 1$, $\sum_{i=0}^p t_i=1$ for a geometrically independent set of points $x_i\in \mathbb{R}^{n}$ for $... |
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