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Is there any idea of evaluating integer topological invariant that is given by the integral form, say the Gauss-Bonnet theorem, by “not doing the integral”?
In other words, is there any way to evaluate the global quantity by only looking at local quantities? |
Can one find. the Chern number without integral? |
I was trying to find best 'u' value for this approximation:
> $\log_2(x+1)=x+u$
And I did think I can calculate error with this function.
> $f(u)=\int_{0}^{1}|\log_2(x+1)-(x+u)|dx$
Then I wanted to find when the slope becomes zero. (means local min in this situation)
> $\frac{d}{du}f(u)=0$\
$\frac{d}{du... |
Define a 3D axis in space, at (0, 0, 0) with an X, Y and Z axis visually, and a point at (1, 1, 1) is in one of eight octant sections the axis form. Let’s trade up 1 for +# as being any positive value, and –# as any negative value. Starting with (+#, +#, +#), if we clockwise rotate all values 45 degrees, we and up at (... |
I was trying to find best 'u' value for this approximation:
> $\log_2(x+1)≈x+u$
And I did think I can calculate error with this function.
> $f(u)=\int_{0}^{1}|\log_2(x+1)-(x+u)|dx$
Then I wanted to find when the slope becomes zero. (means local min in this situation)
> $\frac{d}{du}f(u)=0$\
$\frac{d}{du... |
Can one find the Euler characeristic of two dimensional manifold without integrating Gaussian curvature? |
Helly's theorem is stated as:
**Helly's theorem**:
Let $C_1, \cdots, C_n \subseteq \mathbb{R}^d$ be convex sets such that any $d+1$ of them intersects. Then all $C_i$ have a common point.
Nerve theorem is stated as:
**Nerve theorem**:
If $U$ is a good cover ($U$ is a good cover if $U$ is a finite family of ... |
This is a follow-up question to https://math.stackexchange.com/a/4872343
Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. Let $A(u)$ be the collection of all finite sets of simple tensors $\{x_1\otimes y_1,\dots ,x_n\otimes y_n\}$ such that:
1. $u=\sum_{i=1}^n x_i\otimes y_i,$
2. ... |
I am reading the book *Measure and Integration: An introduction to Real Analysis* by Richard L.Wheeden, Antoni Zygmund, second edition. In chapter 5, the authors give a definition of a distribution function (p. 97):
>It turns out that there is a remarkably simple and useful representation of Lebesgue integrals over su... |
**Is there an integer $n$ such that $2n^2-1$ has two distinct positive divisors congruent modulo $2n$?**
**We prove that the answer to this question is "NO".**
Suppose such an integer $n$ exists: $\exists p, t \in \mathbb{N}^*$ such that $p$ and $p+2tn$ are divisors of $2n^2-1$.
Let $d = p \wedge (p+2tn)$.
We... |
I was trying to find best 'u' value for this approximation:
> $\log_2(x+1)≈x+u$
And I did think I can calculate error with this function.\
NOTE: for the x values between 0 and 1 i need becouse of IEEE 754 float point uses a scientific notation that (1 + x) * 2^n that 0 <= x < 1
> $f(u)=\int_{0}^{1}|\log_2(x+1... |
A representation of a group $G$ is said to be [trivial][1] (over some vector space V) if $1_G:G\to B(V)$ maps every element to the identity element in $B(V)$.
Furthermore two representations $\rho :G\to B(V)$ and $\pi:G\to B(W)$ are said to be equivalent if there exists an *isomorphism* $\alpha: V\to W$ such that
... |
Let $$f(a,b,c)=a(a+b)^5+b(b+c)^5+c(c+a)^5- \frac{32}{243}(a+b+c)^6$$and the SOS tool gives the following result
$$\begin{aligned} f(a,b,c) & = \frac{10}{818181}\sum \left(- 156 a^{3} - 544 a^{2} b + 143 a^{2} c - 403 a b^{2} + 395 a c^{2} + 149 b^{2} c + 260 b c^{2} + 156 c^{3}\right)^{2} \\ & + \frac{1}{4332331332}... |
I was trying to find best 'u' value for this approximation:
> $\log_2(x+1)≈x+u$
And I did think I can calculate error with this function.\
NOTE: for the x values between 0 and 1 i need becouse of IEEE 754 float point uses a scientific notation that (1 + x) * 2^n that 0 <= x < 1
> $f(u)=\int_{0}^{1}|\log_2(x+1... |
The exercise3 on page915 in D&F's Abstract Algebra states that:
> Show that the map $\mathsf{Ring}$ to $\mathsf{Grp}$ by mapping a ring to its group of units (i.e., $R\mapsto R^{\times}$) defines a functor.
But for a non-trivial ring $R$ without $1$ , how to define $R^{\times}\in\mathsf{Grp}$ , I think it is an e... |
Suppose there is a set of arrays $S= [A_1, A_2, ..., A_k] $, where $A_i$ is a finite subset of $\mathbb{Z}$. Given a positive integer $g$, I want to build $g$ sets $G_1,G_2,\dots,G_g$ such that $\forall i\in[k], \exists j\in[g], A_i \subseteq G_j$ and I want the maximum size of the groups minimized, i.e.,
$$\min \max... |
Is there any idea of evaluating integer topological invariant that is given by the integral form, say the Gauss-Bonnet theorem, by “not doing the integral”?
In other words, is there any way to evaluate the global quantity by only looking at local quantities?
---------
I want to calculate the Chern number of phys... |
Some says covectors are linear map that maps $ V \mapsto R $ (which means it's just a row vector considering vectors are $ n $ x $ 1 $ matrix and mapping is matrix multiplication), while some say it's a certain value whose components follow $ v'_i = \frac{\partial x^j}{\partial x'^i} v_j $ under coordinate transformati... |
I'm trying to formalize some of my code within Category Theory, and I ended up having to prove that a certain construction $(T,\lambda,\rho)$ was a monad. Here is the setup.
Let $(D,\eta,\mu)$ be a monad over $\mathbf{Set}$. Now,I want to define a monad $(T,\lambda, \rho)$ over the slice category $\mathbf{Set}/P$.
... |
Let $(B_t)_{t \geq 0}$ be a Brownian motion and $T \geq 0$. Find a constant $z_T \in \mathbb{R}$ and $(s,\omega) \in \mathcal{V}([0,T])$ such that $F_T(\omega) = z_T + \int_{0}^{T} \phi_T(s,\omega)dB_s(\omega)$ when $F_T(\omega) = \int_0^T B^3_s(\omega) ds$.
The idea is to rewrite $\int_0^T B^3_s ds$ using Itô's lem... |
Suppose we are given a filtered probability $(\mathcal{F},\{\mathcal{F}\}_t,\mathbb{P})$ space fulfilling the "usual conditions". Let $\tau$ be a stopping time with respect to this space and $a$ be a positive real number. Is it true that for any $\epsilon>0$ and $\delta>0$ there exists a stopping time $\sigma$ such tha... |
Let the function $f:\mathbb R^n \rightarrow \mathbb R^n$ be Lipschitz with constant $L$. In a paper, they say the condition that $f:\mathbb R^n \rightarrow \mathbb R^n$ is Lipschitz is equivalent to the condition that there exists a $B < \infty$ such that
\begin{align}
(i)&: \quad \Vert f(x) \Vert \leq B(1+\Vert x\... |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
A representation of a group $G$ is said to be [trivial][1] (over some vector space V) if $1_G:G\to GL(V)$ maps every element to the identity element in $GL(V)$.
Furthermore two representations $\rho :G\to GL(V)$ and $\pi:GL\to B(W)$ are said to be equivalent if there exists an *isomorphism* $\alpha: V\to W$ such th... |
This [Wikipedia article](https://en.wikipedia.org/wiki/Without_loss_of_generality) and [AoPS page](https://artofproblemsolving.com/wiki/index.php/Without_loss_of_generality) have the same proof that is an example of *WLOG*. These and other *WLOG* proofs seem to be connected with logical statements. Am I wrong? However,... |
In Introduction to Set Theory by Hrbacek and Jech, the Recursion Theorem is stated as:
For any set $A$, any $a\in A$, and any function $g: A \times \mathbb{N} \rightarrow A$, there exists a unique infinite sequence $f: \mathbb{N} \rightarrow A$ such that
(a)$f(0)=a$;
(b)$f(n+1) = g(f(n) , n)$ for all $n\in... |
If $f \in \mathcal{H}^\infty(U,F)$ is a bounded holomorphic mapping from an open subset of a complex Banach space $U$ into a complex Banach space $F$, then I know that $T \circ f = 0$, for all bounded linear operator $T \in \mathcal{L}(F,G)$, implies $f=0$.
Nevertheless, is this still true if we change the ideal of ... |
I was given a *positive* polynomial function $f(t,x)=\sum_{k=0}^{2n} a_k(x)t^{2n-k}$, where the coefficients $a_j(x)$ are infinitely differentiable funtion of $x\in \mathbb{R}^n$, and $t \geq 0$. My problems are listed below.
1. The dimention (or Hausdorff dimention) of the zero set of $f$, denoted by $Z(f):=\{(t,x... |
>We will find the formula for Motzkin numbers and show that they enumerate chord diagrams. Let $\mathcal{M}$ be the following class of combinatorial configurations. A configuration of weight $n$ is a circle with $n$ points, with some collection of chords between the vertices under the condition that no two chords are c... |
Consider $\mathbb{R}^2$ as a real Lie algebra. How can we prove that $\mathcal{H}^2 (\mathbb{R}^2)\cong \mathbb{R}$, the second cohomology space of $\mathbb{R}^2$?
I really appreciate if someone could help me about it. Thanks in advance. |
A representation of a group $G$ is said to be [trivial][1] (over some vector space V) if $1_G:G\to GL(V)$ maps every element to the identity element in $GL(V)$.
Furthermore two representations $\rho :G\to GL(V)$ and $\pi:G\to GL(W)$ are said to be equivalent if there exists an *isomorphism* $\alpha: V\to W$ such th... |
Why $\mathbb{CP}^1 \cong S^2$? |
Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function, and it is known that for any constant $u$ it holds that $\Psi(x,x^{1/u})$ tends to $x \cdot u^{-u}$ as $x$ tends to infinity. There is also a large literature on computin... |
I'm trying to formalize some of my code within Category Theory, and I ended up having to prove that a certain construction $(T,\lambda,\rho)$ was a monad. Here is the setup.
Let $(D,\eta,\mu)$ be a monad over $\mathbf{Set}$. Now,I want to define a monad $(T,\lambda, \rho)$ over the slice category $\mathbf{Set}/D P$.
... |
How do I prove a limit of multivariable functions using epsilon-delta definitions? |
When does a distribution and an integral commute? |
I have a question about the computation of a certain integral. Let $\chi_{[0,B]}(\vert k-l\vert)$ be the characteristic function on the interval $[0,B]$. In order to estimate the following integral in terms of the variable $F$ I computed
$$ \int_{\vert k \vert \leq F} d^d k \int_{\vert l \vert \geq F} d^d l \frac{\chi... |
Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function, and it is known that for any constant $u$ it holds that $\Psi(x,x^{1/u})$ tends to $x \cdot u^{-u}$ as $x$ tends to infinity. There is also a large literature on computin... |
Two bijections from a finite set to itself commute with each other when they are both powers of one bijection. They also commute with each other when their cycles do not intersect each other.
But some other pairs commute: $(ab)(cd), (ac)(bd), (ad)(bc)$ commute with each other.
Can anything of interest be said abo... |
REPLY TO COMMENT:
A vector field with respect to coordinates $(x^1, \dots, x^n)$ is of the form $$ V = v^i\frac{\partial}{\partial x^i}. $$ Therefore, if we do a change of coordinates, where we view the old coordinates $(x^1, \dots, x^n)$ as functions of the new coordinates $(\bar{x}^1, \dots, \bar{x}^n)$, then coor... |
I'm attacking to solve the graph isomorphism problem using the adjacent matrices.
I would appreciate it if someone could show me whether following conjecture works or not. I expect this conjecture to work.
Let $A$ and $B$ be $n \times n$ symmetric matrices. Let there is no permutation matrix $P$ such that $A=P^t... |
What if $f(x)$ is defined by
$$f(x) = \begin{cases} x \quad &\text{if $x <0$}\\
1+x \quad & \text{if $x>0$}\\
1 \quad &\text{if $x=0$}
\end{cases}$$
?
|
>What is $(-1)^{2r-1}$?
$$(-1)^{2r-1}=\dfrac{(-1)^{2r}}{(-1)}=\dfrac{[(-1)^2]^r}{(-1)}=\dfrac{1^r}{-1}=\dfrac{1}{-1}=-1.$$
Would this be correct? It's quite confusing. Also is it mathematically legal to have a negative sign in the denominator only? Is there a difference being in the denominator vs. the numerator?... |
Let $V$ be a finite set consist of vectors $v=(v_1, \cdots, v_n)\in \mathbb{N}^n$, how to find a vector $w$ such that the inner product $<w,v>$ is nonzero? |
How to find a vector such that the inner product is not zero? |
Let $R=\mathbb{C}[[x]]$ be the ring of formal power series and $S=M_2(R)$ be the set of all $2\times 2$ matrices of entries in $R$.
Then characterize all finitely generated left-modules on $S$ and their endomorphism rings.
I know that $R$ is a PID and also Noetherian. Is there some similar results about left-mo... |
$\textbf{Q}$ : There are $6$ children each offered a single scoop of any of $3$ flavors of ice cream. In how many ways can each child choose a flavor for their scoop so that some flavor of ice cream is selected by exactly $3$ children?
- I struggled with this for a bit. I'm counting this as $ 3 \cdot C(6,3) \cdot 2 ... |
I'm trying to formalize some of my code within Category Theory, and I ended up having to prove that a certain construction $(T,\lambda,\rho)$ was a monad. Here is the setup.
Let $(D,\eta,\mu)$ be a monad over $\mathbf{Set}$. Now,I want to define a monad $(T,\lambda, \rho)$ over the slice category $\mathbf{Set}/D P$.
... |
I'm attacking to solve the graph isomorphism problem using the adjacent matrices.
I would appreciate it if someone could show me whether following conjecture works or not. I expect this conjecture to work.
Let $A$ and $B$ be $n \times n$ symmetric matrices. Assume that there is no permutation matrix $P$ such tha... |
How Can I Finish off this Proof on the Set of Convex Sets With a Translated Copy of $K$? |
The other answer focuses on whether or not the simplification is correct, but the question itself asks questions about the representation of rational numbers, which have nothing to do with exponentiation, e.g. "...is it mathematically legal to have a negative sign in the denominator only?"
Remember what a "numerator... |
Let $M$ be a von Neumann algebra and $N$ be the von Neumann subalgebra of $M$.
If $N$ is $*$-isomorphic to the hyperfinite type II$_1$ factor $R$.
Can we conclude that $N=R$? |
Let $G$ be a group and $T:G\to GL(V)$ be a presentation of $G$ in a vector space $V$. I have two question about two notions in this statement:
"A $V$-valued one-cocycle is a smooth map $S:G\to V$ satisfying the propert $$S(fg)=T(f)\cdot S(g)+S(f),\quad \forall f,g\in G.$$ Given a one-cocycle $S$, one defines the ass... |
I am trying to solve a question in probability which goes as follows:
There is a series of light bulbs on the street. Each light bulb may burn out with probability *p* every day and be out of use, independently of the other light bulbs.
a) What is the expected value of the number of days for which the first N bul... |
The conditional operator $\Rightarrow$ can be tricky to motivate in the cases where it is True. An approach taken by Elliott Mendelson in Number Systems and the Foundations of Analysis is to examine the expression $(C\land D)\Rightarrow D$. My understanding is as follows: Because English sentences that can be paraphras... |
Let function $f:[0,1]^{N+1} \mapsto \mathbb{R}$ be continuously differentiable.
Let $Y_1,\cdots,Y_{N}$ be random variables with support on $[0,1]$ with continuously differentiable joint PDF.
$$g(x):= Pr(f(x,Y_1,\cdots,Y_{N})<0).$$
There is $y^* \in (0,1)$ such that for $Y_1\gtrless y^*$,
$$
f(0,Y_1,\cdots,Y... |
In considering volumes created by revolving polynomials $y=\beta x^n$ about the y-axis, if we specify $\beta$ so that the curve includes $(0,0)$ and $(a,2a)$ and consider the volumes swept within the cylinder of radius $a$ and height $2a$ defined between $x=[-a,a]$, I was mildly surprised to realize that several simpl... |
How to prove this vector identity? |
*This question is less open-ended than you might think.*
The Question:
---
>For $g,h\in{\rm SL}(2,q)\setminus\{\pm I\}$ with ${\rm tr}(g)={\rm tr}(h)$, how does $h$ ${\rm SL}(2,q)$-conjugated relate to $g^{\pm 1}?$
Context:
---
According to [a preprint](https://arxiv.org/abs/1808.01815v1):
>[I]f $g,h\i... |
For $g,h\in{\rm SL}(2,q)\setminus\{\pm I\}$ with ${\rm tr}(g)={\rm tr}(h)$, how does $h$ ${\rm SL}(2,q)$-conjugated relate to $g^{\pm 1}?$ |
It a rotation matrix in 3D space a products of three basic rotations? |
I have asked this question once before, but I forgot about it and it's been a long time since then. Also, I have learned new things since then so my approach has changed. Therefore I am creating a new question. It goes something like this:
> Suppose a chess knight is positioned on some arbitrary square on a chess bo... |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
Solve x^x - 5x + 6 = 0 using the concept of LambertW function? |
Let $M$ be a von Neumann algebra and $N$ be the von Neumann subalgebra of $M$.
If $N$ is $*$-isomorphic to the hyperfinite type II$_1$ factor $R$.
Can we conclude that $N$ is also a hyperfinite type II$_1$ factor? |
How do I solve $x^x - 5x + 6 = 0$ using the Lambert W function? |
I am looking to re-express ${}_1F_1(n+\frac12,b,x)$ for $n$ a positive integer. Actually, so is b. I wrote:
${}_1F_1(n + \frac12,b,x) = 1 + \sum_{k=1}^\infty \frac{(2n+1)(2n+3)\ldots(2n+k-1)}{2^kb(b+1)\ldots(b+k-1)} \frac{z^k}{k!} = \frac{(2n+1)(2n+2)\ldots(2n+2k-2)}{2^kb (n+1)(n+2)\ldots (n+k-1)(b+1)\ldots(b+k-1... |
I am looking to re-express ${}_1F_1(n+\frac12,b,x)$ for $n$ a positive integer. Actually, so is b. I wrote:
\begin{equation}
\begin{split}
{}_1F_1(n + \frac12,b,x) & = 1 + \sum_{k=1}^\infty \frac{(2n+1)(2n+3)\ldots(2n+k-1)}{2^k b(b+1)\ldots(b+k-1)} \frac{z^k}{k!} \\
& = 1 + \sum_{k=1}^\infty\frac{2n(2n+1)(2n+2)... |
Let $I$ be a linearly ordered set and $\{A_i\}, \{B_i\}$ be two collection of sets indexed by $I$,
in an order-preserving way (i.e. $A_i\subsetneqq A_j\iff i<j$ and $B_i\subsetneqq B_j\iff i<j$).
If for all $i\in I$, ${\rm card} A_i\le {\rm card}B_i$, do we have
\begin{equation*}
{\rm card}\ \bigcup_{i\in I} A_i \l... |
$\{A_i\}, \{B_i\}$ are chain of sets indexed by the same linearly ordered set. If $|A_i|\le |B_i|$ for all $i$, does $|\cup A_i|\le |\cup B_i|$? |
If I have a subset $A \subseteq \mathbb{R}$, a function $f: A \to \mathbb{R}$, and a point $a_{0} \in A$, then (according to the definitions I came across) for $f$ to be differentiable at $a_0$, then there must exist some open interval $I$ so that $a_{0} \in I$ and $I \subseteq A$.
I tried understanding why this req... |
Why is differentiability defined on open interval? |
In a theorem showing that a power set $\mathcal{P(S)}$ is a sigma-algebra on proofwiki.org, there is the following part:
> Let $S$ be a set, and let $\mathcal{P}(S)$ be its power set.
>
> [...]
>
> Let $\langle A_i \rangle$ be a **countably infinite sequence** of sets in $\mathcal{P}(S)$
>
> [...]
But what ... |
In a theorem showing that a power set $\mathcal{P}(S)$ is a sigma-algebra on proofwiki.org, there is the following part:
> Let $S$ be a set, and let $\mathcal{P}(S)$ be its power set.
>
> [...]
>
> Let $\langle A_i \rangle$ be a **countably infinite sequence** of sets in $\mathcal{P}(S)$
>
> [...]
But what ... |
This type of complete bipartite graph has two vertex sets $(V,U)$ where $V$ has points $v_n$ along a straight line parallel to the set of points $u_m$. How should I go about finding a general formula to count the amount of crossing in this particular type of graph? There are 18 crossings, not including crossings that o... |
Working in Zermelo's set theory (with choice for simplicity) - the construction in Hartogs' theorem shows that starting with a set $X$, there is a set $X'$ in at most $\mathcal{P}^4(X)$ (where $\mathcal P$ denotes power set) and a well-order $<$ in at most $\mathcal{P}^6(X)$ such that the well-order type of $(X',<)$ is... |
Is it consistent with ZC that a well-order of type $\omega_\omega$ does not exist? |
Let $Q$ a polynomial in $\mathbb{C}[X]$ (or in $\mathbb{K}[X]$ with $\mathbb{K}$ a field of characteristic zero).
I wonder what is the commutant of $Q$, i.e. the set of polynomials $P$ such that $P \circ Q = Q \circ P$, where $\circ$ is the composition. More precisely, is there any works, any results on that topic ?... |
what are $O(x)$ and $o(x)$? and where did they came from? |
Let $A=\mathbb{Z}\oplus\mathbb{Z}_3.$ I want to find all subgroups of $A$ as well as all direct summand.
For the subgroup, let $H=H_{free}\oplus H_{tor}$. And consider the inclusion: $\iota_1:H_{free}\to A$ and $\iota_2:H_{tor}\to A$. I conclude that $H_{free}=m\mathbb{Z}$ and $H_{tor}=\mathbb{Z}_3,\{0\}$. Are thes... |
Up until now I have used the Laplace transform to solve all the second order differential equations I have met. However I was recently set this equation as a homework problem and have been unable to solve it.
$ x^2y'' + 5xy' +4y=0 $
I have been made aware that you can also use a Fourier transform to solve these a... |
How do you solve second order differential equations with variables as the coefficients using a Laplace transform? |
Let $(B_t)_{t \geq 0}$ be a Brownian motion and $T \geq 0$. Find a constant $z_T \in \mathbb{R}$ and $(s,\omega) \in \mathcal{V}([0,T])$ such that $F_T(\omega) = z_T + \int_{0}^{T} \phi_T(s,\omega)dB_s(\omega)$ when $F_T(\omega) = \int_0^T B^3_s(\omega) ds$.
The idea is to rewrite $\int_0^T B^3_s ds$ using Itô's lem... |
To prove: $T:l^1 \to c_0^*, \langle Tx, y\rangle = \sum x_ky_k$ is an isometric isomorphism
Is this proof correct? Particularly, I dont know if I proved injectivity correctly
Proof:
(i) linearity: let $x, y \in l^{1}(\mathbb{K}), \eta \in c_0(\mathbb{K})$
$T(\alpha x+y)(\eta)=\sum(\alpha x+y)_k \eta_k=\sum \a... |
I was studying polynomial rings over commutative rings from the book Topics in Algebra by I.N Herstein.
From there what I understood was, that:
If $R$ be a commutative ring with a unit element then $R[x]$ is defined to be the set of all symbols $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ where, $a_0,a_1,...,a_n\in R$ and... |
What is the indeterminate in the set of all symbols in $R[x]$ and what does the elements in $R[x_1,x_2]$ looks like? |
For every line in the 2d plane, we can construct an equation (often a circle) such that the graph of that equation is cut by the line into two symmetrical parts.
Does this property of lines extend to all curves?
My idea is if you "zoom in" on a curve, if you get small enough, it might become a line segment, then... |
This is related to a question I posted before: https://math.stackexchange.com/q/4872424/1295158.
Since that question has not received a reply, here I post a simplified version with the $N$ random variables reduced to one random variable. This may help me or other people to think about that question.
Let function $f... |
Define the topology $\tau$ on $\Bbb{Z}/p_n\#$ to be that generated by the basis $B = \{ X_d : d \mid p_n\#\}$ and $X_d = \{ x \in \Bbb{Z}/p_n\#: x^2 \neq 1 \pmod q, \forall \text{ primes } q \mid d\}$. Then clearly $X_c \cap X_d = X_{\text{lcm}(c,d)}$.
The full space $\Bbb{Z}/p_n\#$ is given vacuously by $X_1 = \{... |
Let $U \subset \mathbb R^2$ be an open, regular (meaning $U$ is the interior of its closure), bounded and simply connected set. Assume there exists a simple curve $\gamma : [0,1] \rightarrow \partial U$, $\gamma(0) \neq \gamma(1)$.
Does $\partial U \setminus \gamma((0,1))$ stays connected ?
I think this should be... |
Removing curve from simply connected domain's boundary : does it stay connected? |
To say "R[x] is defined to be the set of all *symbols*..." is not clear. What is a *symbol*?
More formally, a polynomial is a sequence $$(a_0,a_1, a_2,...)$$ where the $a_i \in R$ are almost all zero, which means that there exists an $n$ such that $\forall m>n, a_m=0$.
Then, very classically, you define an addit... |
I am constructing a bijection from open interval $(0,1)$ to open interval $(a,b)$ for $a,b\in\mathbb{R}$ and $a<b$.
I tested the line through points $(0,1)$ and $(a,b)$, namely $f(x)=1+\frac{b-1}{a-0}(x-0)$.
But say $(a,b)=(3,4)$ giving $f(x)=1+\frac{4-1}{3-0}(x-0)$ or $f(x)=1+x$, then we see that $0<x<1$ gives $... |
To say "R[x] is defined to be the set of all *symbols*..." is not clear. What is a *symbol*?
More formally, a polynomial is a sequence $$(a_0,a_1, a_2,...)$$ where the $a_i \in R$ are almost all zero, which means that there exists an $n$ such that $\forall m>n, a_m=0$.
Then, very classically, you define an addit... |
For every line in the 2d plane, we can construct a space-filling curve (often a circle) such that the space-filling curve is cut by the line into two symmetrical parts.
Does this property of lines extend to all curves?
My idea is if you "zoom in" on a curve, if you get small enough, it might become a line segmen... |
Suppose we are given a filtered probability $(\mathcal{F},\{\mathcal{F}\}_t,\mathbb{P})$ space fulfilling the "usual conditions" (.i.e. the filtration is right-continuous and complete). Let $\tau$ be a stopping time with respect to this space and $a$ be a positive real number. Is it true that for any $\epsilon>0$ and $... |
Let $Q$ a polynomial in $\mathbb{C}[X]$ (or in $\mathbb{K}[X]$ with $\mathbb{K}$ a field of characteristic zero).
I wonder what is the commutant of $Q$, i.e. the set of polynomials $P$ such that $P \circ Q = Q \circ P$, where $\circ$ is the composition. More precisely, is there any works, any results on that topic ?... |
To say "R[x] is defined to be the set of all *symbols*..." is not clear. What is a *symbol*?
More formally, a polynomial is a sequence $$(a_0,a_1, a_2,...)$$ where the $a_i \in R$ are almost all zero, which means that there exists an $n$ such that $\forall m>n, a_m=0$.
Then, very classically, you define an addit... |
How do I solve $x^x - 5x + 6 = 0$ using the Lambert W function?
**EDIT:**
I solved equations $2^x - 5x + 6 = 0$ and $3^x - 4x - 15 = 0$ using Lambert W function, but not able to solve $x^x - 5x + 6 = 0$ equation. How and which method can be used to solve this equation?
How to solve this equation without graphi... |
If $R = GF(3)[x]$, that is the set of all polynomials with coefficients in $\{0,1,2\}$
and $I = R{\langle x^3+1\rangle}$, the ideal of $R$ generated by $x^3+1$, then show that the factor ring $R/I$ is a field.
I have found that $R/I = \{a+bx + cx^2 + I\ | \ a,b,c \in R\}$, how can I show that this is a field?
... |
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