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I found a question in the very end of [this][1] video, which essentially is the following :
[1]: https://www.youtube.com/watch?v=Sv7VseMsOQc&t=5s
> Suppose there is a function $f : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that it satisfies the following 2 properties :
>- $f(A\cdot B) = f(A)\cdot f(B)... |
$\newcommand{\R}{\mathbb R}$ $\newcommand{\bd}{\text{Bd}}$
>**Question.**
Let $f:S^n\to \R^n$ and $g:S^n\to S^n$ be continuous functions.
Is it necessarily true that there is an $x\in S^n$ such that $f(x) = f(g(x))$?
The conclusion is true if $n=1$ and an easy proof can be given.
Also, the Borsuk-Ulam theorem ... |
If $N\subset M$ is an inclusion of type II$_1$ factors, Jones define the index $[M:N]$.
If $N$ is the type II subfactor of the type III factor $M$. How to define $[M:N]$? |
For which $n$ the second and third and fourth term's coefficients of binomial expansion's $(x+y)^n$ makes arithmetic sequence? |
Let $G$ be a graph with $n$ vertices and $m$ edges, and $A_k(G)$ be the number of $k$-coloring of $G$. If there is a matching $M$ of $G$ with $|M|=t$, then it is clear that $A_k(G)\leq (k^2-k)^{t}k^{n-2t}$. So it is possible to give a upper bound of $A_k(G)$ about $n,m$. Are there some results about this? |
Are there some researches about upper bound of number of k-coloring of a graph with n vertices and m edges? |
Let Y1, Y2, . . . , Yn be independent and identically distributed random variables having the same population distribution with density:
f(y; θ) = ( θ(3^θ)/y^(θ+1) , y ⩾ 3; 0, elsewhere.) where θ is a positive unknown parameter.
So my question is, how can I use the Rao–Blackwell theorem to find the MVUE of θ? I wa... |
How do I apply the Rao-Blackwell Theorem to find MVUE of parameter theta? |
Let $\mathcal S, \mathcal T$ be triangulated categories and $R: \mathcal S \to \mathcal T$ be an additive functor that commutes with shift. If $L:\mathcal T \to \mathcal S$ is a left adjoint to $R$, then must $L$ also commute with shift?
(Note that $L$ must be an additive functor https://math.stackexchange.com/... |
This is more like a coding question, but I thought I'll post it here because of its mathematical foundation.
Two ODEs first degree can describe a Hamiltonian System. The connection between the ODEs and the Hamiltonian System is given by the canonical equations which are:
$
\dot{x}=\frac{\partial H}{\partial x}
... |
I found a question in the very end of [this][1] video, which essentially is the following:
[1]: https://www.youtube.com/watch?v=Sv7VseMsOQc&t=5s
> Suppose there is a function $f : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that it satisfies the following 2 properties:
>- $f(A\cdot B) = f(A)\cdot f(B),\... |
Given $f$ which is defined in the environment of $x_0$, and given that for every sequence $a_n$ which approaches $x_0$ $f(a_n)$ converges, prove that $\lim_{x \to x_0}{f(x)}=f(x_0)$
Apart from the proof, I would also like to ask, is this not a version of Heine's definition for limits which requires less knowledge ab... |
I found a question in the very end of [this][1] video, which essentially is the following:
[1]: https://www.youtube.com/watch?v=Sv7VseMsOQc&t=5s
> Suppose there is a function $f : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that it satisfies the following 2 properties:
>- $f(A\cdot B) = f(A)\cdot f(B),\... |
Verify whether the function $f$ from $[0,2]$ to $\mathbb{R}$ defined by
$$f(x) =\begin{cases} x+x^2, & x\in\mathbb{Q} \\
x^2 + x^3, & x\in\mathbb{R-Q}
\end{cases}$$
is not Riemann integrable.
I'll explain the solution given in my textbook.
It is clear that for the interval $[0,1]$, $x+x^2$ is the suprem... |
Given $f$ which is defined in some neighbourhood of $x_0$, and given that for every sequence $a_n$ which approaches $x_0$ $f(a_n)$ converges, prove that $\lim_{x \to x_0}{f(x)}=f(x_0)$
Apart from the proof, I would also like to ask, is this not a version of Heine's definition for limits which requires less knowledge... |
Assume that $G$ is a finite group, $p$ a prime number, and $H_1,H_2\le G$ such that $[G:H_1]=p=[G:H_2]$. Assume further that $[G:H_1\cap H_2]<p^2$. Does it follow that $H_2=gH_1g^{-1}$ for some $g\in G$?
As an alternative goal: if the main "guess" fails, will it still follow that $H_1$ and $H_2$ share the same core ... |
Are two subgroups of prime index with a large intersection necessarily conjugate? |
The following is the graph of the $\tan(x)$ function:
[![enter image description here][1]][1]
We can clearly see there how it’s undefined at $\frac \pi 2$.
Now, what if we wanted to find the area between the tangent function and the x-axis in the interval $[0, 2]$?
The following happens:
$$\int \tan(x) ... |
**Large $n$ asymptotics**
Let $S$ be the position of a random walk with steps $X_j$ which are uniformly distributed unit vectors :
$$\tag{1}
S=\sum\limits_{j=1}^nX_j
$$
Then using the multivariate CLT the distribution of $S$ as $n\to\infty$ is a multivariate Gaussian
$$\tag{2}
f_S(s)=\frac{\exp\left(-sM^... |
**Large $n$ asymptotics**
Let $S$ be the position of a random walk with steps $X_j$ which are uniformly distributed unit vectors :
$$\tag{1}
S=\sum\limits_{j=1}^nX_j
$$
Then using the multivariate CLT the distribution of $S$ as $n\to\infty$ is a multivariate Gaussian
$$\tag{2}
f_S(s)=\frac{\exp\left(-sM^... |
Verify whether the function $f$ from $[0,2]$ to $\mathbb{R}$ defined by
$$f(x) =\begin{cases} x+x^2, & x\in[0,2]\cap\mathbb{Q} \\
x^2 + x^3, & x\in[0,2]\setminus\mathbb Q
\end{cases}$$
is not Riemann integrable.
I'll explain the solution given in my textbook.
It is clear that for the interval $[0,1]$, $... |
The mapping function for transforming the exterior of the unit circle with unequal cracks on both sides (in the $\zeta$-plane) to the exterior of the unit circle (in the $z$-plane) is as follows:
$${\omega}(\zeta ) = M(\zeta + \frac{1}{\zeta }) + N + Q(\zeta )$$
where
$$\left\{ \begin{array}{l}
Q(\zeta ) = \ze... |
Assume that $G$ is a finite group, $p$ a prime number, and $H_1,H_2\le G$ such that $[G:H_1]=p=[G:H_2]$. Assume further that $[G:H_1\cap H_2]<p^2$. Does it follow that $H_2=gH_1g^{-1}$ for some $g\in G$?
As an alternative goal: if the main "guess" fails, will it still follow that $H_1$ and $H_2$ share the same core ... |
The crux of the issue lies in the phrase "for the interval $[0,1]$, $x+x^2$ is the supremum of the given function". You said that you understand this, but taken literally, this statement is complete nonsense. How, then, should we interpret this statement?
I would argue that the only meaningful interpretation is th... |
Let $X$ be an object of $\mathcal{T}$. There is a sequence of isomorphisms of functors
\begin{align}
\operatorname{Hom}_{\mathcal{T}}(L(X[n]),?)
&\cong\operatorname{Hom}_{\mathcal{S}}(X[n],R?)\\
&\cong\operatorname{Hom}_{\mathcal{S}}(X,(R?)[-n])\\
&\cong\operatorname{Hom}_{\mathcal{S}}(X,R(?[-n])\\
&\cong\operato... |
How does $\left(p+\frac{2-p^2}{r}\right)^2<2$ simplify to $2pr+2-p^2<r^2$? |
Where did I go wrong trying to simplify this down to $(x \times y)$ boolean expression? |
The crux of the issue lies in the phrase "for the interval $[0,1]$, $x+x^2$ is the supremum of the given function". You said that you understand this, but taken literally, this statement is complete nonsense. How, then, should we interpret this statement?
I would argue that the only meaningful interpretation is th... |
I'm not using the language of schemes. I define projective varieties to be an algebraic set in $\mathbb P^n$ with the homogeneous ideal to be a prime ideal, and a projective curves to be $1$-dimension projective varieties.
Is there any proof of any morphisms between curves being either constant or surjective that do... |
**Large $n$ asymptotics**
Let $S$ be the position of a random walk with steps $X_j$ which are uniformly distributed unit vectors :
$$\tag{1}
S=\sum\limits_{j=1}^nX_j
$$
Then using the multivariate CLT the distribution of $S$ as $n\to\infty$ is a multivariate Gaussian
$$\tag{2}
f_S(s)=\frac{\exp\left(-sM^... |
> Given $E(X)=100,\sigma(X)=10$ and $P(X\le 10k)\ge 4/5$, how do we find a lower bound on $k$ using the Chebyshev's inequality
Let $X$ be any r.v. with finite mean, $μ$, and finite variance. Then $∀a > 0,$
$$
P\bigg(|X-\mu|\ge a\bigg)\le\frac{\sigma^2(X)}{a^2}
$$
which is the Chebyshev’s inequality.
$$
P(X... |
**Large $n$ asymptotics**
Let $S$ be the position of a random walk with steps $X_j$ which are uniformly distributed unit vectors :
$$\tag{1}
S=\sum\limits_{j=1}^nX_j
$$
Then using the multivariate CLT the distribution of $S$ as $n\to\infty$ is a multivariate Gaussian
$$\tag{2}
f_S(s)=\frac{\exp\left(-sM^... |
I have a dataset consisting of N measured parameters along with their covariance matrix (stemming from experimental nuclear physics). These parameters are inherently positive. I use this set to perform simulations and use the covariance matrix to propagate the uncertainty derived from the data. Until now, I have been e... |
How to Rigorously Tackle Uncertainty Propagation for Inherently Positive Data with Covariance? |
I am now just beginning my study in field theory and I am trying to find all embeddings from $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to $\bar{\mathbb{Q}}$ (an algebraic closure of $\mathbb{Q}$).
Here, an embedding from a field $K$ to another field $F$ refers to an injective homomorphism from $K$ to $F$.
I asked my instru... |
The crux of the issue lies in the phrase "for the interval $[0,1]$, $x+x^2$ is the supremum of the given function". You said that you understand this, but taken literally, this statement is complete nonsense. How, then, should we interpret this statement?
I would argue that the only meaningful interpretation is th... |
This is regarding the complexity of an algorithm for counting triangles in an undirected graph which was suggested in a document I came across.
(Link - https://www.cs.cmu.edu/~15750/notes/lec1.pdf)
Suppose the graph has $m$ edges and $n$ vertices. The algorithm goes as follows: sort the vertices in increasin... |
I have a dataset consisting of N measured parameters along with their covariance matrix (stemming from experimental nuclear physics). These parameters are inherently positive. I use this set to perform simulations and use the covariance matrix to propagate the uncertainty derived from the data. Until now, I have been e... |
How to rigorously tackle uncertainty propagation for inherently positive data with given mean values and covariance matrix? |
This answer is partial. It concerns mainly natural solutions of the equation
The brief search showed several related papers, which I listed below as the references. Unfortunately, I had no time to study them all, so my survey below is incomplete.
It seems that all of these references concern natural solutions of ... |
I was wondering whether the following statement is true:
If a $C^*$algebra A is a primitive, and it contains a simple ideal F, then F is contained in every non-zero closed ideal of A.
We say a $C^*$ algebra is primitive if it has a faithful irreducible representation on a Hilbert space.
We say a (closed) ideal... |
Is it true if a C*algebra A is a primitive, and it contains a simple ideal F, then F is contained in every non zero closed ideal of A? |
I found a question in the very end of [this][1] video, which essentially is the following:
[1]: https://www.youtube.com/watch?v=Sv7VseMsOQc&t=5s
> Let $n$ be some natural number. Suppose there is a function $f : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that it satisfies the following 2 properties:
>-... |
Since Gödel's numbering maps formulas into numbers, how would a number inside a number that represents a formula be decoded? Let's say I have the following formula of a formal system called TNT (from the book "Gödel, Escher, Bach" by Douglas Hofstadter):
> ¬∃a:∃b:<PROOF-PAIR(a,b)∧ARITHMOQUINE(c,b)>
It has some Gö... |
Let $A$ be a non-zero orthogonal matrix, that is, a real matrix such that $A^tA=I_3$. Can the trace of $A$ be 0 ? |
I am wondering about the existence of projections from a multivariate polynomial $p\in\mathbb{R}[x_1,...x_n]$ to a polynomial $q\in\mathbb{R}[t]$ in the sense that there is a map $\phi:\mathbb{R}\to\mathbb{R}^n$ such that for all $t\in\mathbb{R}$ we have
$$q(t)=p(\phi(t)).$$
Basically, this amounts to finding a path ... |
Let $A$ be an orthogonal matrix, that is, a real matrix such that $A^tA=I_3$. Can the trace of $A$ be 0 ? |
There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow the same exponential distributions say $a_1,a_2,a_3\to Exp(\lambda)$... |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. The solution of [this question][1] unexpectedly showed that the simple triangle inequality $a+b > c$ is equivalent to the famous [Basel problem][2].
$$
\zet... |
If $(a,b,c)$ are the sides of a triangle, is it true that probability that $a+b > c^{\frac{3}{c}}$ is $\zeta(2)-1$? |
Let $M$ be a smooth manifold. We know that if $X:\ M\to TM$ is a smooth vector field, then for any local chart $x=(x_1,\dots,x_n):\ U\subset M\to\mathbb R^n$, there exists smooth functions $X_1,\dots,X_n:\ U\to\mathbb R $ such that
$$X_p\,=\,\sum X_i(p)\frac{\partial }{\partial x_i}\bigg|_p\ \ \forall p\in U $$
I w... |
Is the component function of a smooth vector field unique? |
**Context summary :**
We perform measurements of a set of parameters, calculate the mean of the population, and the covariance matrix between parameters. These parameters are combined to construct the answer $\epsilon$. The question is, how can we propagate the uncertainty stored in the covariance matrix to the answer... |
There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow the same exponential distributions say $a_1,a_2,a_3\to Exp(\lambda)$... |
There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow the same exponential distributions say $a_1,a_2,a_3\to Exp(\lambda)$... |
Hey I want to check my solutions for this exercise:
Let $K$ be a field with $Char(K) = p > 0$ and let $L/K$ be an extension whose degree $[L : K] = n$ cannot be divided by $p$. Show that $L/K$ is separable.
So my idea was:
Let $f$ be a irreducible polynomial in $L$
We have for $f(x):=x^n+a_{n-1}x^{n-1}+\dots+... |
Try a cyclic matrix
$$
A=\begin{pmatrix} 0 & 0 & 1 \cr
1 & 0 & 0 \cr 0 & 1 & 0 \end{pmatrix}.
$$
Is it an orthogonal matrix? |
A (real-valued) kernel $k$ on $X$ is a positive semi-definite symmetric map $X \times X \to \mathbb{R}$. Specifically, for any $n \in \mathbb{N}$, $x_1,...,x_n \in X$ and $\alpha_1,...,\alpha_n \in \mathbb{R}$ we have that
$$\sum_{i,j=1}^n \alpha_i \alpha_j k(x_i,x_j)\geq 0.$$
Now, it's easy to see that
$$\sum_{i,... |
I have a rectangle with a given length L and width B. There is a point in one of quadrant of this rectangle whose location x value can vary from 0 to B/2 and y can vary from 0 to L/2. Depending on the location of this point a shaded region is formed within the bounds of this rectangle. The shaded region is such that it... |
In [this paper][1] (lemma 3) it is claimed that any column of a unitary matrix that has two entries at zero can be approximated to $\mathbb{Z}[i,1/\sqrt2]$ by solving the Diophantine equation $a^2 + b^2 + c^2 + d^2 = n$ (the original unitary is decomposed in several reflections, each one of them corresponding to a colu... |
I found a question in the very end of [this][1] video, which essentially is the following:
[1]: https://www.youtube.com/watch?v=Sv7VseMsOQc&t=5s
> Let $n$ be some natural number. Suppose there is a function $f : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that it satisfies the following 2 properties:
>-... |
This is probably a very simple question with a negative answer, but I somehow cannot find a counterexample.
Let $X$ be a smooth algebraic variety over an algebraically closed field $k$. Assume that $X$ contains an embedded curve $C$ isomorphic to $\mathbb{A}^1_*$, such that for every $\varphi\in \operatorname{Aut}(X... |
Is an extension of an algebraic group by a multiplicative group a semidirect product? |
I am trying to understand tensors and one particular question have caused me a great deal of confusion. The particular example with the metric tensor below is an attempt to highlight where my confusion is.
In my understanding, one can view the metric tensor $g$ as a bilinear map $g:V \times V \rightarrow \mathbb{R}$... |
Tensor confusion: scalar $\otimes$ vector = OK?... |
I have a rectangle with a given length $L$ and width $B$. There is a point in one of quadrant of this rectangle whose location $x$ value can vary from $0$ to $B/2$, and $y$ can vary from $0$ to $L/2$. Depending on the location of this point, a shaded region is formed within the bounds of this rectangle. The shaded regi... |
Given the tower of fields $F_1\subset F_2\subset K$, and $\alpha\in K$ algebraic over $F_1$, can we deduce that (this is not an exercise, and thus more or less a two-fold question: 1. is the statement correct and 2. is the proof correct)
$$
[F_2(\alpha):F_2]\leq [F_1(\alpha):F_1]
$$?
My attempted proof uses the ... |
This is probably a very simple question with a negative answer, but I somehow cannot find a counterexample.
Let $X$ be a smooth algebraic variety over an algebraically closed field $k$. Assume that $X$ contains an embedded curve $C$ isomorphic to $\mathbb{A}^1_*$, such that for every $\varphi\in \operatorname{Aut}(X... |
We shall bring in probability to simplify the problem.
For the moment, consider Mary and Jane as blue balls, John as a yellow ball, and the other $3$ as red balls.
P(blue balls together) $=P(A)= \frac{5}{\binom62} = \frac13$
P(yellow ball) at ends $= P(B) = \frac26 = \frac13$
$P(A\cap B) = \frac13\fra... |
I am trying to understand tensors and one particular question have caused me a great deal of confusion. The particular example with the metric tensor below is an attempt to highlight where my confusion is.
In my understanding, one can view the metric tensor $g$ as a bilinear map $g:V \times V \rightarrow \mathbb{R}$... |
Let $V,W$ be two finite dimensional Hilbert spaces over $\mathbb{R}$. Suppose that $\{v_1,\cdots,v_m\}$ and $\{w_1,\cdots,w_n\}$ are orthonormal subsets of $V$ and $W$, respectively. Define $T:V\to W$ by $T(v):=\Sigma _{i=1}^ma_i\langle v,v_i\rangle w_i$ with $a_1,\cdots,a_m\in \mathbb{R}$. **My question is:** are the... |
Can the following function
$$
F(y,p) = \int\limits_1^y\ x\frac{(x-1)^{p-1}(y-x)^{p-1}}{(x+1)^{p+1}(y+x)^{p+1}}dx,\quad
y>1, p>0
$$
be expressed in terms of known special and elementary functions? |
I am working on this problem
Consider a qubit $\scr H =\Bbb C^2$ and pure states. Prove the no-clonning theorem
( hint: Use the linearity of the channel to arrive to a contradiction)
I wonder if the following proof of the non-cloning theorem for pure states I came up with is valid:
**My try:**
I have to... |
*Rem:* I have mostly a question on my part "3-" but any general remark on my answer will be greatly appreciated.
### Question:
We have the sequence of r.v. $ (Y_n= X_{n+1} \cdot X_n ) , n \in \mathbb{N}$ with $X_n$ a i.i.d. r.v. defined as follow $\forall n \in \mathbb{N}, X_n \in \{0; n \} $ and $P(X_n=0)=1-... |
We shall bring in probability to simplify the problem.
For the moment, consider Mary and Jane as blue balls, John as a yellow ball, and the other $3$ as red balls.
P(blue balls together) $=P(A)= \frac{5}{\binom62} = \frac13$
P(yellow ball) at ends $= P(B) = \frac26 = \frac13$
$P(A\cap B) = \frac13\fra... |
Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$
>I need asymptotic formula of $I_n$ as $n\to\infty$
Firstly we see that since $x(1-x)\leq \frac{1}{4}$, so the defini... |
**Large $n$ asymptotics**
Let $S$ be the position of a random walk with steps $X_j$ which are uniformly distributed unit vectors
$$\tag{1}
S=\sum\limits_{j=1}^nX_j
$$
Then using the multivariate CLT the distribution of $S$ as $n\to\infty$ is a multivariate Gaussian
$$\tag{2}
f_S(s)=\frac{\exp\left(-sM^{-... |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular. But the definition of function
$$\alpha(\xi):= \inf\{t\in I': \xi\notin E_t[x]\}$$
in the proof seems contradictory. Because th... |
*Rem:* I have mostly a question on my part "3-" but any general remark on my answer will be greatly appreciated.
### Question:
We have the sequence of r.v. $ (Y_n= X_{n+1} \cdot X_n ) , n \in \mathbb{N}$ with $X_n$ a i.i.d. r.v. defined as follow $\forall n \in \mathbb{N}, X_n \in \{0; n \} $ and $P(X_n=0)=1-... |
Does the range of a one-one function need necessarily to be equal to its co-domain? |
I am trying to generalize the triangle inequality in probabilistic terms. Assume that the vertices are uniformly distributed around the circumference of a fixed circle. [In this question][1] it was proved that if $(a,b,c)$ are the sides of a triangle than the probability that geometric mean of any two sides is greater ... |
I am working on this problem
Consider a qubit $\scr H =\Bbb C^2$ and pure states. Prove the no-clonning theorem
( hint: Use the linearity of the channel to arrive to a contradiction)
I wonder if the following proof of the non-cloning theorem for pure states I came up with is valid:
**My try:**
I have to... |
Consider a regular hexagonal lattice, like so
[![enter image description here][1]][1]
Given a circle centred at a vertex in this lattice, with radius $r\in\mathbb{R}^+$, is the number of lattice points inside this circle known?
**Note:** The purpose of this question is simply to inquire whether this is a known... |
Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$
>I need asymptotic formula of $I_n$ as $n\to\infty$
Firstly we see that since $x(1-x)\leq \frac{1}{4}$, so the defini... |
*Rem:* I have mostly a question on my part "3-" but any general remark on my answer will be greatly appreciated.
### Question:
We have the sequence of r.v. $ (Y_n= X_{n+1} \cdot X_n ) , n \in \mathbb{N}$ with $X_n$ a i.i.d. r.v. defined as follow $\forall n \in \mathbb{N}, X_n \in \{0; n \} $ and $P(X_n=0)=1-... |
I have never developed sufficient knowledge in algebraic geometry but I ran into an apparently easy problem, so I apologise in advance if my question sounds naive.
Suppose to have a degree $2$ homogeneous (non-trivial) polynomial $p(\mathbf{x})=\sum\limits_{j=1}^n\sum\limits_{i=1}^ja_{ij}x_ix_j\in\mathbb{F}_2[\mathbf{... |
I was reading D.Knuth's book "Concrete mathematics. Foundation of computer science".
In chapter 8.5 Hashing was given:
$$B_n(z) = \sum_{n_1,n_2,...,n_m \geqslant \ 0 \\ n_1+n_2+...n_m = \ n }
\begin{pmatrix} n \\ n_1,n_2, ..., n_m \end{pmatrix}
z^{\begin{pmatrix} n_1 \\2 \end{pmatrix} + \begin{pmatrix} n_2... |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. The solution of [this question][1] unexpectedly showed that the simple triangle inequality $a+b \ge c$ is equivalent to the famous [Basel problem][2].
$$
\z... |
Let $S$ be some subset of $\mathbb{R}^3$ and $\text{Conv}(S)$ be its convex hull. Say $S$ has property $\mathcal{P}$ if each point $x \notin S$ lies on some line contained in $\mathbb{R}^3 \setminus S$ such that there is a ruled surface in $\mathbb{R}^3 \setminus S$ giving a homotopy to a line in $\mathbb{R}^3 \setminu... |
Is there a name for shapes that can by cut out on a bandsaw? |
*Rem:* I have mostly a question on my part "3-" but any general remark on my answer will be greatly appreciated.
### Question:
We have the sequence of r.v. $ (Y_n= X_{n+1} \cdot X_n ) , n \in \mathbb{N}$ with $X_n$ a i.i.d. r.v. defined as follow $\forall n \in \mathbb{N}, X_n \in \{0; n \} $ and $P(X_n=0)=1-... |
*Rem:* I have mostly doubt on my part "3-" but any general remark on my answer will be greatly appreciated.
### Question:
We have the sequence of r.v. $ (Y_n= X_{n+1} \cdot X_n ) , n \in \mathbb{N}$ with $X_n$ a i.i.d. r.v. defined as follow $\forall n \in \mathbb{N}, X_n \in \{0; n \} $ and $P(X_n=0)=1- \fra... |
I am currently reading the book of Erich Heeke "Lectures on the Theory of Algebraic Numbers", and it is not clear for me to understand his notation of theta series associated to quadratic form over totally real field.
The first question appears in Page 205, formula (175) (I use the English version GTM 77, for Germ... |
What is the transformation formula of Theta series associated the quadratic form over totally real field? |
In *Probability and Stochastics*, (Cinlar), isomorphic measure spaces are defined as follows:
[![][1]][1]
[1]: https://i.stack.imgur.com/WdbAR.jpg
So we have a bijection $f$. But then it's emphasized that we have a *functional inverse* $\hat{f}$. Why is this distinction necessary? Is $\hat{f}$ not also ... |
Is there a name for shapes that can be cut out on a bandsaw? |
I'm developing RPG(Role-Playing Game) game and I'm planning make a formula for 'How much damage is being converted'.
I'm not previliged to put embedded images, but I hope you **check out the image here**(It will make my question clear) >>
[image about my situation][1]
- each '<---->' line is the *duration* of da... |
Is it possible to make formula for 'ratio of conversion in RPG game'? |
I saw a problem where it asked: What question can you ask a knight (who always tells the truth) such that he cannot respond? For this, I came up with the question "Is the answer to this question No?" -- the knight can't say Yes (because he would be lying) and he also can't say no. Similarly for a knave, "Is the answer ... |
I am currently reading the book of Erich Hecke "Lectures on the Theory of Algebraic Numbers", and it is not clear for me to understand his notation of theta series associated to quadratic form over totally real field.
The first question appears in Page 205, formula (175) (I use the English version GTM 77, for Germ... |
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