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Why the derivative of a vector field along a curve is not defined for a generic manifold? |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
Is my proof correct? Infinite rational numbers |
The questions asks: Let $X$ and $Y$ be independent random variables taking values in the positive integers and having the same mass function $f(x)=2^{-x}$ for $x=1,2,...$ find:
$P(X>=kY)$ for a given positive integer k
$P(X>=kY)$
$= \sum_{y=1}^\infty (X>=kY, Y=y)$
$= \sum_{y=1}^\infty (X>=kY)P(Y=y)$ due to... |
Let $K$ be a field, and let $C$ be a projective curve over $K$. Consider a morphism $\varphi: \mathbb{P}^2 \to C$ over $Spec(K)$.
Is it true that $\varphi$ must be constant?
What I know so far: $\varphi$ is proper (since $\mathbb{P}^2$ is projective), hence closed, which means that the image is either a point of... |
I was just introduced the concept that if $(f_n)$ converges in $L^2$ topology to $g(x)\in L^2([0,2\pi])$ then $\lim_{n\to\infty}\int^{2\pi}_0|f_n(x)-g(x)|^2dx=0$. I would appreciate any hint to how to start proving for example $sin(nx)$ diverges in $L^2$. I know that in the normal sense it doesn't even have a pointwise... |
Does $sin(nx)$ converge in L^2? |
Is there alternatives proofs that if $m^4 + 4^n$ is prime number then $m$ is odd and $n $ is even? |
One property of the characteristic function $\phi(t)=\int_{-\infty}^\infty e^{itx}\mu(dx)$ is that for measures $\mu_1$ and $\mu_2$ with respective $\phi_1(t)$ and $\phi_2(t)$, the convolved measure $\mu_1\ast\mu_2$ has $\phi_1(t)\phi_2(t)$. It suffices to observe that $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^... |
Let T be a positive compact operator on a Hilbert space then we know that an operator equation $X^2 = T$ have a solution.
Im confused since we have a result which says that a compact operator $T \geq 0$ have an unique operator $X$ which is compact, positive and $X^2 = T$. How there can be a compact operator $T\geq ... |
Find modal matrix? |
The questions asks: Let $X$ and $Y$ be independent random variables taking values in the positive integers and having the same mass function $f(x)=2^{-x}$ for $x=1,2,...$ find:
$P(X>=kY)$ for a given positive integer $k$
$$\begin{align*}
P(X>=kY) &= \sum_{y=1}^\infty (X>=kY, Y=y) \\\\
&= \sum_{y=1}^\infty (X>=k... |
Let $G$ be a simple group and $H_1$ and $H_2$ be abelian subgroups of $G$ such that $H_1\cong H_2$ and $H_1,H_2\subseteq Cl_G(e_G)\cup Cl_G(x)$ for some $x\in G$, where $Cl_G(\cdot)$ denotes the conjugacy class on $G$.
Is it true that $H_1$ and $H_2$ are conjugate subgroups in $G?$
That is, is there is an element $... |
I was just introduced the concept that if $(f_n)$ converges in $L^2$ topology to $g(x)\in L^2([0,2\pi])$ then $\lim_{n\to\infty}\int^{2\pi}_0|f_n(x)-g(x)|^2dx=0$. I would appreciate any hint to how to start proving for example $sin(nx)$ diverges in $L^2$. I know that in the normal sense it doesn't even have a pointwise... |
Does $\sin (nx)$ converge in $L^2$? |
Let T be a positive compact operator on a Hilbert space then we know that an operator equation $X^2 = T$ have a solution.
Im confused since we have a result which says that a compact operator $T \geq 0$ have an unique operator $X$ which is compact, positive and $X^2 = T$. How there can be a compact operator $T\geq ... |
Consider a set S of positive integers, and define $q(S) = \sum_{s \in S}{1/s^2}$. Letting $\rho = \pi^2/6$, we have $q(S)$ in the ranges $[0, \rho - 1), [1, \rho)$. I conjecture that for every rational $r$ within those ranges, $\exists S: q(S) = r$. Can anyone prove or disprove this?
Note that for plain unit fractio... |
I failed this question at an exam and would like help in explaining how to solve it: Historical data show that approximately 70 000 vehicles cross Älvsborgsbron every day. What is the probability that more than 7 500 000 vehicles cross Älvsborgsbron in 100 days? For full points you must motivate and explain all assumpt... |
it is easy to show that the derivative of the trace $Tr(A)$ with respect to $A$ if A is a N x N matrix, is
$\frac{\partial Tr(A)}{\partial A} = I_{N \times N}$
However, if I now assume A to be antisymmetric $a_{ij} = -a_{ji}$ the trace $Tr(A) = 0$ but
with above result the derivative would still be $I_{N \time... |
Let $U$ be the universe of a multiple choice questionnaire. We can say $|U| = 1$. We have $k \geq 3$ sets of answers in $U$ such that for all $k$, we have $ S= \sum_{u_i \in U} |u_i| > 1$. The answers are in percentages.
We are interested in the quality of a certain set, say $u_c'$, such that we want to know the per... |
Let $(V,\|-\|)$ be a normed function space, and suppose that $(f_i)_i$ is a sequence of elements of $V$ so that $\sum \|f_i\| < \infty$. Further suppose that this data implies that $f = \sum f_i$, where $\sum_{i=1}^n f_i \to f$ pointwise, is an element of $V$. Then I would think that we can show immediately that $\sum_... |
If $\sum \|f_i\|$ converges and $\sum_{i=1}^\infty f_i$ exists in a normed function space, do we get for free that $\|\sum_{i=1}^n f_i - f\| \to 0$? |
If $G$ is PSD, then there exists a matrix $B$ such that $G = BB^\top$. Then
$$AGA^\top = (AB) (AB)^\top.$$
The latter is not necessarily faster (you would also have to compute $B$) but at least it is more compact. Depending on your purposes with the matrix $AGA^\top$ it might offer some improvements. For example,
$$... |
It seems to me it should look more like this:
$$\operatorname{FCN}_\sigma(x) = \forall X (0\in X \land \forall c.( c\in X \to \operatorname{Suc}(c)\in X))\to x\in X$$
Here I've defined $\operatorname{FCN}(x)$ as a one-place predicate which is true if and only if $x$ is a finite Church numeral. The clause $$
\f... |
It seems to me it should look more like this:
$$\operatorname{FCN}_\sigma(x) = \forall X (0\in X \land \forall c.( c\in X \to \operatorname{Suc}(c)\in X))\to x\in X$$
Here I've defined $\operatorname{FCN}(x)$ as a one-place predicate which is true if and only if $x$ is a finite Church numeral. The clause $$
\f... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
I have the following, possibly trivial question, regarding checking for the essential spectrum:
Given a self-adjoint bounded operator $T\in \mathcal{L}(\mathcal{H})$, does $\dim \Big(\text{Im}\big(\chi_{[a-\epsilon,a+\epsilon]}(T) \big) \Big)=\infty$ imply that $\sigma_{ess}(T)\cap [a-\epsilon,a+\epsilon] \neq \empt... |
If degree of one polynomial (say $f(x)$) is greater than degree of the other polynomial (say $g(x)$), then is it ever possible for the two polynomials to be equal for $x > 1$?
I was struck with this question when trying to work on a conjecture involving powers. |
This question has been taken from **III GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN**:
> A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
I have tried solving it by cuttin... |
I'm trying to find the following limit (without L'Hopital, not there yet):
$$\lim_{x \rightarrow 0} \frac{x -\sin x}{x (1 - \cos x)}$$
Tanking into account that:
$$\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$$
and
$$\lim_{x \rightarrow 0} \frac{1 - \cos x}{\frac{x^2}{2}} = 1$$
If I divide by x I get:
... |
While working on a project, I came across a paper that includes this definition: $g ∈ $, where $$ is the "tetrahedral group" ("a discrete subgroup of isometries $ ⊂ O(3)$"). The paper then uses $g(z)$ as a function with a complex input.
***Is it possible to define an algebraic expression for $g(z)$ in a way that is ... |
Can you explain how to convert this notation to a function to someone unfamiliar with differential geometry? |
Consider a quantity, $a$, which could represent various measures such as the area under a curve. Initially, we estimate the value of $a$. As we refine our approximation, we find that we can make it arbitrarily close to a number $\ell$. This suggests that $a$ equals $\ell$. But why is this the case?
Here's the ration... |
Consider a quantity, $a$, which could represent various measures such as the area under a curve. Initially, we estimate the value of $a$. As we refine our approximation, we find that we can make it arbitrarily close to a number $\ell$. This suggests that $a$ equals $\ell$. But why is this the case?
Here's the ration... |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
Given an infinite poset $P$, does it always contain a proper subposet $Q \subsetneq P$ such that $P$ and $Q$ are isomorphic as posets?
What motivated this question is the following. It can be seen that for the natural number poset given by the divisibility relation, there are (uncountably) infinitely many ways of ch... |
Consider a quantity, $a$, which could represent various measures such as the area under a curve. Initially, we estimate the value of $a$. As we refine our approximation, we find that we can make it arbitrarily close to a number $\ell$. This suggests that $a$ equals $\ell$. But why is this the case?
Here's the ration... |
It seems to me it should look more like this:
$$\operatorname{FCN}_\sigma(x) = \forall X (0\in X \land \forall c.( c\in X \to \operatorname{Suc}(c)\in X))\to x\in X$$
Here I've defined $\operatorname{FCN}(x)$ as a one-place predicate which is true if and only if $x$ is a finite Church numeral. The clause $$
\f... |
Given a symplectic matrix $S \in \text{Sp}(2n,\mathbb Z)$ whereby $S^T\Omega S=\Omega$ with
$$\Omega=\left(\begin{matrix}0&I_n\\-I_n&0\end{matrix}\right)$$
what known decompositions exist such that the matrix $S$ is decomposed into $S=ABC$ whereby $A,B,C \in \text{Sp}(2n,\mathbb Z)$?
When $S'\in\text{Sp}(2n,\math... |
Suppose we have a graph on the vertex set $[n]^m$, i.e, each vertex is of the form $(v_1, ..., v_m)$, where each $v_i \in \{1, ..., n\}$.
We draw an edge between two vertices $v = (v_1, ..., v_m)$ and $u = (u_1, ..., u_m)$ iff $u_i = v_i$ for some $i \in [m]$. My question is, how many edges are in our graph?
My a... |
Let $(V,\|-\|)$ be a normed function space, and suppose that $(f_i)_i$ is a sequence of elements of $V$ so that $\sum \|f_i\| < \infty$. Further suppose that this data implies that $f = \sum f_i$, where $\sum_{i=1}^n f_i \to f$ **pointwise**, is an element of $V$. Then I would think that we can show immediately that $\... |
A cube can fit in a sphere through its vertices in several ways. I mean, we know that every cube has 8 vertices. Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides. One way is that There is no vertex on the sphere and the whole cube is inside the sphere. Another case is that the cube has only one... |
I am a little lost about why subsequences necessarily converge to the same limit as their respective sequence. Consider a sequence where S:N$\rightarrow$Q$\in$[0,3], where the naturals are mapped to every rational in order in the interval. By the definition of convergence, S$\rightarrow$3.
Now consider the sequence wh... |
If degree of one polynomial (say $f(x)$) is greater than degree of the other polynomial (say $g(x)$), then is it ever possible for the two polynomials to be equal for $x > 1$?
I was struck with this question when trying to work on a conjecture involving powers. I think I should share some specific values that I got:... |
AM-GM tells us that $x^2y^3 \le \frac{3}{16}$. It also has an equality case, when all of the terms in the AM-GM are equal—in your case, that would be when $\frac{3x}{2} = \frac{4y}{3}$. Combining this with the assumption $3x + 4y = 5$, this gives us only one pair where equality holds: $(x, y) = (\tfrac{2}{3},\tfrac{3}{... |
I've been doing this question but I'm a little stuck on the second part. The first part is as follows:
The probability of a randomly selected person in a population having a particular genetic trait is 0.00001. A test for this trait successfully detects it, if present, 99.9% of the time, and only returns a false pos... |
This question came to me while reading "Model Categories and their Localization", specifically the definition of path object in a general model category.
Given a topological space $X$, we have a homotopy equivalence $c:X\rightarrow X^I$, where $X^I$ is the space of maps $I=[0,1]\rightarrow X$, sending a point $x$ t... |
The questions asks: Let $X$ and $Y$ be independent random variables taking values in the positive integers and having the same mass function $f(x)=2^{-x}$ for $x=1,2,...$ find:
$P(X\geq kY)$ for a given positive integer $k$
$$\begin{align*}
P(X\geq kY) &= \sum_{y=1}^\infty (X\geq kY, Y=y) \\\\
&= \sum_{y=1}^\in... |
Problem: Prove that $\aleph_0^c=2^c$
My attempt: $\aleph_0^c=\aleph_0^{2^{\aleph_0}}=\aleph_0^{\aleph_1}=\aleph_2$
Then $2^{2^{\aleph_0}}=2^{\aleph_1}=\aleph_2$, so $\aleph_0^c=2^c$.
But my intuition is telling me I may have made a mistake or an assumption that I can't make. Did I mess up? Or is this a valid p... |
I want to prove that $3^{100} > 5\cdot10^{47}$ without using calculator or any approximation of the logarithm.
I tought about finding an integral representation of $(3^{100} - 5\cdot10^{47})$ with the integrand non negative in the chosen integration interval (something like [this][1]), but I don't know how to chose t... |
Consider a finite set S of positive integers, and define $q(S) = \sum_{s \in S}{1/s^2}$. Letting $\rho = \pi^2/6$, we have $q(S)$ in the ranges $[0, \rho - 1), [1, \rho)$. I conjecture that for every rational $r$ within those ranges, $\exists S: q(S) = r$. Can anyone prove or disprove this?
Note that for plain unit ... |
Let $G$ be a finite group, and $V$ a (say complex) finite dimensional representation of $G$.
Let me view $V$ as a $G$-module in the obvious way. Is it true that $$H^n(G;V)=0$$ for $n\geq 1$? I suspect that the answer is no, but I haven't been able to find a counterexample. If the answer (surprisingly) turns out to b... |
In Diestel Theorem 4.3.1 (ii) it says "Every combinatorial isomorphism between two 2-connected plane graphs is topological."
I don't understand, where in the proof do we use the fact that the graphs are 2-connected?
[![1]][1]
[![2]][2]
[![3]][3]
[1]: https://i.stack.imgur.com/9YR4U.png
[2]: https://i.... |
Note that $\mathbb R[x]/(x^2+1)$ is isomorphic to the complex numbers by sending $x\mapsto i$. With that in mind, consider $(y-x)(y+x) + I$. What happens if you add $1$ and subtract $1$ to this expression? |
$\frac{π}{2}$ is a trascendental number, given by the integral:
$$\frac{π}{2}=\int_{-1}^1\sqrt{1-x^2}dx$$
The integrand is also the curve of a semiball of the form:
$$||(x,y)||_2=1$$
Induced by the 2-norm $||\cdot||_2:\mathbb{R^2\rightarrow R_+}$ where $y$ has been parametrized in terms of $x$.
Similarly... |
Are the semiballs induced by even p-norms the integrands of integral representations of trascendental numbers? |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
Let us have a Riemmanian manifold $(M,g)$. Is there some “natural“/canonical way to lift the metric to the cotangent bundle $T^* M$?
I ask because I have read some articles on orthogonal separation of variables (on configuration space) heavily using Killing tensors and other notions from Riemannian geometry and I am... |
Let $ \gamma\colon [a,b[\to \mathbb R^n $ be a curve (i.e. a continuous function). Studying differential geometry I came across two ways in which $ \gamma $ can be called "differentiable".
1. $ \gamma $ is differentiable if it is differentiable in the classical sense in $ ]a,b[ $, and if
$$
\gamma^\prime(a) := \l... |
I have problems visualizing intuitively and understanding the proof of theorem 9.16 in John Lee's "Introduction to Smooth Manifolds", 2nd ed. The proof starts
>"Suppose V is a compactly supported vector field on a smooth manifold M, and let $K= supp \, V$. For each $p \in K$, there is a neighborhood $U_p$ of p and a... |
Consider a sufficiently fast decaying and smooth function $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$ such that
$$
\int_{\mathbb{R}} f(x)dx=0.
$$
Now consider the Schrodinger semigroup $e^{it\partial_x^2}$, that is, the operator such that $$
\mathcal{F}\big[e^{it\partial_x^2}f\big](\xi)=e^{-it\xi^2}\mathcal{F}[f](\... |
From Bosch‘s Algebra (p. 368):
> Determine the galois group of $$X^7 - 8X^5 - 4X^4 + 2X^3 - 4X^2 + 2 \in \mathbb Q[X]$$ and decide if it‘s solvable or not.
How does one find the galois group of that polynomial?
My usual way is to find the roots $\alpha_1, \ldots, \alpha_n$, find a splitting field $L = L(\alph... |
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$
by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(n+1)(2n+1)}{12}-1} \prod_{k=1}^n (y^p+1)^p dy$$
now lets define $a(n,k)$ as the coefficient of $x^k$ in the expression ... |
The formula is correct; there are indeed $\frac12 n^m(n^m - (n-1)^m - 1)$ edges in this graph.
The asymptotics are a bit trickier. When there are two variables involved, our asymptotic analysis has to be done with an understanding of which variable is going to infinity - or if both of them are, how quickly relative ... |
W.Dörfler and D.A.Waller's paper ["A category-theoretical approach to hypergraphs"](http://rd.springer.com/article/10.1007%2FBF01224952#page-1) gives the following definitions:
1. A _hypergraph_ is a triple $H = (V,E,f)$ where $V$ is the set of vertices, $E$ is the set of (hyper)edges and $f \colon E \to \mathcal{P... |
Let $X$ be a Banach space. Assume $(z_{n,m})_{n,m}$ is a sequence in $X$ and $x\in X$ such that
$z_{n,m}\rightarrow^{m\rightarrow \infty} z_n'$ and $z_{n'}\rightarrow z'$ as $n\rightarrow \infty$.
Then, why is it that $(z_{n,m})_{n,m}$ is bounded?
Argument: As $m\rightarrow \infty$, $z_{n,m}$ goes to $z_n'$... |
**Hint**
1. Use [Eisenstein's Criterion][1] to show that $f$ is irreducible.
2. Show that $f$ has exactly $5$ real roots. (Since $f$ has negative discriminant, $f$ has either exactly $1$ or exactly $5$ real roots, hence it suffices to show that $f$ has at least $2$ real roots.)
What kind of permutations do (1), ... |
Problem :
Show by hand without any computer assistance :
>$$e^{e^2}>1000\phi$$
where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$.
I come across this limit showing :
$$\lim_{x\to 0}x!^{\frac{x!!^{\frac{2}{x!!-1}}}{x!-1}}=e^{e^2}$$
I cannot show it without knowing some decima... |
Take a family $F$ of measurable functions on $\mathbb{R}$, then $F$ is uniformly integrable if:
**Definition 1 (Royden):** For each $\epsilon >0$, there is a $\delta>0$ such that for each $f\in F$, if a Borel set $I$ is such that $\lambda(I)<\delta$ (Lebesgue measure), then $\int_I |f|<\epsilon$.
**Definition 2 (... |
A cube can fit in a sphere through its vertices in several ways. I mean, we know that every cube has 8 vertices.
Now imagine a sphere with a fixed radius. Cubes can have arbitrary sides. One way is that There is no vertex on the sphere and the whole cube is inside the sphere. Another case is that the cube has only o... |
I'm faced with the situation as presented on this
[picture][1].
[1]: https://i.stack.imgur.com/Epllj.png
I know the coordinates of $A$ and $B$. And I know that the vectors $AC$ and $CB$ lay in a $45^{\circ}$ angle relative to the plane (as shown on the picture).
I know that if not for the $45^{\circ}$ condi... |
How many ways can a cube be placed in the sphere through its vertices? |
While working on a project, I came across a paper that includes this definition: $g ∈ $, where $$ is the "tetrahedral group" ("a discrete subgroup of isometries $ ⊂ O(3)$"). The paper then uses $g(z)$ as a function with a complex input, where $z$ is "the local complex coordinate on the unit 2-sphere in $^3$ obtained by... |
$\infty$ is not the answer I'm looking for.
Let's say I had a stupidly big number in my head and I wanted to write it down on a sheet of paper. I could start plotting down the digits, but at some point I would run out of space to write. So I could use more sheets of paper, but again, at some point I will run out of ... |
> Let $R$ and $S$ be rings and ${}_{S}A_{R}$, ${}_{R}B$, $C_{R}$, ${}_{R}D_{S}$ (bi)modules as indicated.
>
>(i) $A\otimes_R B$ is a left $S$-module such thats $s(a\otimes b)= sa\otimes b$ for all $s\in S$, $a\in A$, $b\in B$.
>
>**Sketch of proof:** (i) For each $s\in S$ the map $A\times B\to A\otimes_R B$ given b... |
I'm not actually sure if 4704 is the correct number of combinations. If my reasoning (in the next paragraph is correct), I need help showing why it makes sense.
*Scenario:*
Draw 5 cards from a regular deck of 52 cards. How many ways can we draw 5 cards that contains exactly 3 Ace's? No replacement.
*Reasoning... |
This question has been taken from **III GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN**:
> A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
I have tried solving it by cuttin... |
This question has been taken from **III GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN**:
> A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
I have tried solving it by cuttin... |
>Find the value of $\sqrt[5]{0.00000165}$ given $\log165=2.2174839$ and $\log697424=5.8434968$
$\log x=\log\sqrt[5]{0.00000165}$
$\Rightarrow \log x =\dfrac{1}{5}\log0.00000165=\dfrac{1}{5}(\overline{6}.2174839$)
$\Rightarrow \dfrac{1}{5}(\overline{10}+4.2174839) = \overline{2}.8434968$
I don't understand h... |
( x y 1 ) is not a subspace of R^3 .
I understand that is true since it does not contain the origin.
I went on the explore this concept further.
Since a span is by definition a subspace, then for a subspace V, there must be a set S containing R^3 vectors which spans V.
Or,
span(S) = V.
Conversely... |
I am new to this topic of homotopy and the fundamental group.
I have just read the proof that the fundamental group of $S_1$ is isomorphic to $\mathbb{Z}$.
There's something that has been bugging since the beginning.
Can't I define an homotopy between the constant loop staying at $(1,0)$ and the loop going around... |
Let $G$ be a finite abelian group, $f:G\to \mathbb C$ not identically zero. Donoho-Stark uncertainty principle say that $|\textrm{supp} f||\textrm{supp} \hat{f}|\geq |G|$. The prove I have goes as follows:
$$\sup_{\gamma\in \widehat{G}}|\hat{f}(\gamma)|=|\hat{f}(\gamma_0)|=\left|\frac{1}{|G|}\sum_{b\in G}f(b)\overli... |
You want to show for any multiplicative function $f$,
\begin{equation}\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).
\end{equation}
Your question about convergence is a good one. As stated, the equality is *purely formal*, in that it doesn't consider larg... |
I am new to this topic of homotopy and the fundamental group.
I have just read the proof that the fundamental group of $S_1$ is isomorphic to $\mathbb{Z}$.
There's something that has been bugging since the beginning.
Can't I define an homotopy between the constant loop staying at $(1,0)$ and the loop going around... |
For ease of notation, below when I write $|x| \mod N$, I am referring to the absolute value of the integer with smallest absolute value in the same residue class as $x$. For example $|6| \mod 7$ is $1$.
A natural corollary of the simultaneous version of the Dirichlet's approximation theorem is the following statemen... |
I have a poset with the following property:
> For any infinite descending chain $x_1 > x_2 > \dots$ and any $y$ that is a lower bound for the chain ($y < x_i$ for all indices $i$), there exists $y' > y$ that is also a lower bound for the chain.
Does this property have a standard name, or is it equivalent or relat... |
Does this poset property have a name? |
> $\begin{pmatrix} x & y & 1 \end{pmatrix}$ is not a subspace of $\bf{R}^3$.
I understand that is true since it does not contain the origin.
I went on the explore this concept further.
Since a span is by definition a subspace, then for a subspace $V$, there must be a set $S$ containing $\bf{R}^3$ vectors wh... |
This Question asked on [math over flow][1]
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$
by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(n+1)(2n+1)}{12}-1} \prod_{k=1}^n (y^p+1)^p dy$$
now lets define $a(n,k)$ ... |
This should be a piece of cake but I really have no idea about how to do it.
I have to calculate $f^{-1}(2)$ where $f(x) = \ln(x) + 2x^5$.
I proved the function is invertible (that is: injective and surjective), but I am blocked about he request of $f^{-1}(2)$.
I cannot find a way to calculate $f^{-1}$. any h... |
Question from an old exam:
>In a full binary tree of height $n$, each vertex is randomly colored either white or black. A path is considered good if it goes from any vertex to a leaf and all vertices on this path are of the same color. Let $X$ denote the number of good paths. For example, for a tree of height $n = 3$ ... |
Let $X$ be a Banach space. Assume $(z_{n,m})_{n,m}$ is a sequence in $X$ and $x\in X$ such that
$z_{n,m}\rightarrow^{m\rightarrow \infty} z_n'$ and $z_{n'}\rightarrow z'$ as $n\rightarrow \infty$.
Then, why is it that $(z_{n,m})_{n,m}$ is bounded up to subsequence?
Argument: As $m\rightarrow \infty$, $z_{n,... |
Why is it bad to translate from Lp norm to priors? |
This question has been taken from **III GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN**:
> A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral.
> Determine the angles of the original triangle.
My attempts:
I have tried s... |
Theoretically, we know that the range and domain of $g(x)=\cot^{-1}x$ is $(-\infty,\infty)$ and $(0,\pi)$, respectively. Its plot is a smooth continuous function.
Strangely, I find that WulForm Mathematica plots it as an odd
discontinuous (at $x=0$) function in $(-\pi/2,\pi/2)$!
**Question 1:** Have your encoun... |
I'm studying the construction of the tensor product. To do this, I opted for the quotient space's approach, defining first the free vector space over the cartesian $V\times W$ of two vector spaces. The following steps is at usual, take the quotient of the free vector space and the subspace spanned by especifics formal... |
Theoretically, we know that the range and domain of $f(x)=\cot^{-1}x$ is $(-\infty,\infty)$ and $(0,\pi)$, respectively. Its plot is a smooth continuous function.
Strangely, I find that Wolfram Mathematica plots it as an odd
discontinuous (at $x=0$) function in $(-\pi/2,\pi/2)$!
**Question 1:** Have your encoun... |
My question is for formulation or notation of numerators and denominators of variable-containing algebraic irrational terms (<a href="https://en.wikipedia.org/wiki/Solution_in_radicals">radicals</a>) in general.
I'm looking for formulations to talk about variable-containing algebraic irrational terms
- whose numera... |
Let $f:N\rightarrow N$ and for all $i$ we have $f(i)\in \Theta(\sqrt i)$. For all $i, j$ such that $1\le i\le j\le n$ we define
$$g(i, j)=\begin{cases}f(i)& i=j\\\max_{i<k\le j}\{g(i,k-1)+g(k,j)\}& i<j\end{cases}$$
What's the time complexity of $g(1,n)$? |
What's the time complexity of $g(1, n)$? |
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