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Suppose that $X$ is a Polish metric space (with a distance $d$). We know that for each $\epsilon>0$, $X$ can be partitioned into a countable number of non-empty Borel subsets, $\{B_n\}$, whose diameters do not exceed $\epsilon$. Given a strictly positive measure $\mu$ on $(X,\mathcal{B}(X))$ (i.e., $\operatorname{supp}... |
Consider the equation $a_1+a_2+a_3+a_4+a_5+a_6 = 30$ where $a_i$ are non-negative integers with a value at most 6. Also, none of the $a_i$ can be $5$.
Following are some hints to find the no. of solutions:
>! Let $m$ denote the no. of $a_i$s which are exactly 6. Is $m=0$ possible?
>! How many solutions for $... |
There’s a bit of multilinear-algebraic folklore knocking around in my head that I’d like to cite and use but that I can’t track down for the life of me:
> Let $n > 2$ be an integer, and let $a,b \in \operatorname{GL}(n,\mathbb{R})$. If $\bigwedge^2 a = \bigwedge^2 b$, then $a=\pm b$.
Note that the condition $\big... |
Consider the equation $a_1+a_2+a_3+a_4+a_5+a_6 = 30$ where $a_i$ are non-negative integers with a value at most 6. Also, none of the $a_i$ can be $5$.
Following are some hints to find the no. of solutions:
>! Let $m$ denote the no. of $a_i$s which are exactly 6. Is $m=0$ possible?
>! How many solutions for $... |
I'm reading *《Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry》* by *Simon Donaldson & Sun Song*. In their Section 2, named *Complex differential geometry: the Hormander technique*, they mention the following elliptic estimate and Sobolev inequality for sections of holomorphic line bundles:
$(U^n,J... |
Suppose that $X$ is a Polish metric space (with a distance $d$). [We know that][1] for each $\epsilon>0$, $X$ can be partitioned into a countable number of non-empty Borel subsets, $\{B_n\}$, whose diameters do not exceed $\epsilon$. Given a strictly positive measure $\mu$ on $(X,\mathcal{B}(X))$ (i.e., $\operatorname{... |
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: *"Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \subset P$ be generic over $M$"*.
Afterwards, the corresponding boolean algebra $B=B(P)$ is constructed (for which $P$ is de... |
Let Y and Z be two random variables on $(\Omega,\mathcal{F},P)$, it is well-known that $\sigma(Z)\subset \sigma(Y)$ is equivalent to that there exists some Borel measurable function such that $Z=f(Y)$.
I am wondering if the analogous is true for stochastic processes. Let ${Y_t}$ and $Z_t$ be two stochastic processes... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
in class i was asked to expand $f(z)=\frac{1}{z^2}$ into a series about the point $z_0 =i$
i solved it in two different ways and got two different answers, i would like to know where my error is
(1) using succesive derivatives:
$f(z) = \frac{1}{z^2}, f^{(n)}(z) = \frac{(-1)^n*{(n+1)}!}{z^{n+2}}$
plug that into ... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
The assignment I'm solving right now is to prove that the computation of a FWHM of a laser line can be expressed as a convolution with some kernel function. And the purpose is fast peak detection. Although in reality we're working with frequency, this is not necessary for the math and we can abstract the variable as "x... |
How can I prove that if x^5-x^3+x=2, with x a real number, than x^6>3, using only middle school algebra? |
Yes, it is. The $\forall x$ isn’t quantifying any free variables, since there are no free variables in $\exists \ x<x$, and the $\forall x$ is therefore called a ‘null quantifier’. However, using the standard semantics for quantifiers we can still evaluate such a statement: we effectively ask: is it true for all object... |
How can I prove that if $x^5-x^3+x=2$, with $x$ a real number, then $x^6>3$, using only middle school algebra?
I have been struggling with this one. And I can't find any solution that doesn't rely on derivatives. |
How can I prove that if $x^5-x^3+x=2$, with $x$ a real number, then $x^6>3$, using only middle school algebra? |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
> How can P1 ⇔ P2 ⇔ P3, ... ⇔ Pn be True when P1 =/= P2 =/= P3 =/= Pn and all P are not false ?
Your ‘P1 =/= P2 =/= P3 =/= Pn’ is a little unclear, but from the title of your post, it is clear that your question is: how can $P1 ⇔ P2 ⇔ P3, ... ⇔ Pn$ be true if it is not the case that all $P$’s are true, nor the c... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
> How can P1 ⇔ P2 ⇔ P3, ... ⇔ Pn be True when P1 =/= P2 =/= P3 =/= Pn and all P are not false ?
Your ‘P1 =/= P2 =/= P3 =/= Pn’ is a little unclear, but from the title of your post, it is clear that your question is: how can $P1 ⇔ P2 ⇔ P3, ... ⇔ Pn$ be true if it is not the case that all $P$’s are true, nor the c... |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
I was watching Survivor (reality show) and they got a following puzzle.
There is 4x4 matrix where each column has a color and each row has a symbol.
By moving around or replacing you should get to the point, where in each row or column there is no repeating symbol or repeating color.
I was able to solve that, bu... |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
You're correct; we should add σ-Locally Finite Base + Weakly Lindelöf => Second Countable to the pi-Base. https://github.com/pi-base/data/issues/584
First, I'll confirm that your claim that every locally finite (or even locally countable) collection in a weakly Lindelöf space is countable. So a σ-locally finite base... |
Consider the function $w = \frac{1}{2}(z+ \frac{1}{z})$
with domain Z = {z ∈ C | Im z > 0, |z| > 1}
and H = {w ∈ C | Im w > 0}
Is the function Zhukovsky function bijective from Z onto H?
I have already demonstrated $w$ to be analytic and maps the set Z to the upper halfplane H
How can I demonstrate bijectiv... |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
Yes, it is. The $\forall x$ isn’t quantifying any free variables, since there are no free variables in $\exists \ x<x$, and the $\forall x$ is therefore called a ‘null quantifier’. However, using the standard semantics for quantifiers we can still evaluate such a statement: we effectively ask: is it true for all object... |
Suppose that $X$ is a Polish metric space (with a distance $d$). [We know that][1] for each $\epsilon>0$, $X$ can be partitioned into a countable number of non-empty Borel subsets, $\{B_n\}$, whose diameters do not exceed $\epsilon$. Given a strictly positive measure $\mu$ on $(X,\mathcal{B}(X))$ (i.e., $\operatorname{... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
The answer is no, and we give two counter-examples below: one in $2$ and one in $3$ and more dimensions. The second example also appears in a slightly different form in [these old lecture notes of M. Wolf][1] (Example 8.1). However, in my opinion, this specific result is rather difficult to find if you're not already f... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
I am reading a proof of the irrationality of $e^2$ and I am stuck on the following inequality:<br>
Let $S := -a\left(\frac{1}{n+1} - \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} \mp ...\right)$ (just because of space issues) with $a \in \mathbb{Z}$, $n \in \mathbb{N}$.
$$
-\frac{a}{n} < S < -a\left(\frac{1}{n+1}... |
If I apply the change of variable method to calculate a double integral, and I pose \( x = u \) and \( y = v-u \), with \( x \) and \( y \) being strictly positive, then we will find that the domain is \( 0 < u < v \). However, I encounter a problem regarding the domain of \( v \). Is it \( v > 0 \) or \( v > u \)? Bec... |
How we do double integration with variable bornes? |
Is it true that if we have $R\subset E$ an irreducible root system then $E$ is an irreducible representation of the Weyl group? |
An elliptic curve $E$ over $\overline{\mathbb{Q}}$ can be defined over $\mathbb{Q}$ *if and only if* its j-invariant $j(E)$ is a rational number. More precisely, $E$ can be defined over a number field $K\subset \overline{\mathbb{Q}}$ if and only if $j(E)\in K$.
As Mathmo123 points out above, you can find many ellipt... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
Are there rings where factorization is unique, but does not necessarily exist? |
In Woit's book chapter 20, he considers character of the group $(\mathbb{R}^d,+)$. Now he claims that irreducible representations of $(\mathbb{R}^d,+)$ are one dimensional and given by $\alpha_{\mathbf{p}}(\mathbf{a}) = e^{i\mathbf{p}\cdot \mathbf{a}} $.
For $\mathbf{a} \in \mathbb{R}^d$, and $\mathbf{p}$ is in anothe... |
I'm required to determine analytically the range of the following function:
[f(x,y) = 2 + sqrt(9-x^2-y^2)][1]
[1]: https://i.stack.imgur.com/Q0igU.png
The domain for this function is {(x,y) ∈ R^2 : x^2 + y^2 ≤ 9}. I've found that the range for a very similar function (which is f(x,y) = sqrt(9-x^2-y^2)) is ... |
Let
$$E:=\overbrace{-2C_0-(n-2)C_1}^A+(4+(n-2)\overbrace{(2C_1+C_2)}^B)\cdot p-(n-2)\overbrace{(C_1+C_2-C_3)}^Cp^2\\$$
I got two questions when I tried to give asymptotics analysis of $E,E^2$. It looks contradiction to the computation obtained in mathematica, which pretty confused me...
(1) Consider $C_0=C_1=... |
>Solve $y+3=3\sqrt{y+7}$
$\Rightarrow (y+3)^2=9(y+7)$
$\Rightarrow y^2+6y+9=9y+63$
$\Rightarrow y^2-3y-54=0$
$\Rightarrow (y-9)(y+6)=0$
$y=9, -6$
But plugging $y=-6$ into $y+3=3\sqrt{y+7}$ yields $-3=3$.
I know the convention is $\sqrt{y}$ is equal to a positive number, but in this case $\sqrt{y+7}... |
I am currently working on analyzing the Flip bifurcation using Jury's condition. I derived the following condition for the bifurcation to not occur:
\begin{equation}
\frac{1}{2(-4+b^2)}\left(-2(b^2(-2+c)+4(2+c))+a(8\alpha+2b^2\alpha+b^3(-1+\alpha)\alpha+4b(1-3\alpha+2\alpha^2))\right) < 0
\end{equation}
Since t... |
I am trying to derive the following result from [this paper](https://journals.aps.org/prl/supplemental/10.1103/PhysRevLett.113.020405/Supplement.pdf) (Equation S3):$$e^{-ik_z\hat{z}}\hat{\rho}e^{+ik_z\hat{z}} - \hat{\rho} \approx k_z^2[\hat{z},[\hat{z}, \hat{\rho}]]$$with operators $\rho, z$.
The authors state the e... |
So I have been strugling with to prove the following:
\[ \left|\sum_{k\geq 0}(-1)^k {N-k-1 \choose N-2k}\right| = \begin{cases} 1, & \text{if $N\equiv 1 \mod 3$} \\ 0, & \text{otherwise} \end{cases} \] \\\\
I was given the hint of considering the coefficient of both sides of the equality:
\[
\frac{1-x^2}{1+x^3}... |
So I have been strugling with to prove the following:
$\left|\sum_{k\geq 0}(-1)^k {N-k-1 \choose N-2k}\right| = \begin{cases} 1, & \text{if $N\equiv 1 \mod 3$} \\ 0, & \text{otherwise} \end{cases} $ \\\\
I was given the hint of considering the coefficient of both sides of the equality:
\[
\frac{1-x^2}{1+x^3}=\f... |
So I have been strugling with to prove the following:
$\left|\sum_{k\geq 0}(-1)^k {N-k-1 \choose N-2k}\right| = \begin{cases} 1, & \text{if $N\equiv 1 \mod 3$} \\ 0, & \text{otherwise} \end{cases} $ \
I was given the hint of considering the coefficient of both sides of the equality:
$
\frac{1-x^2}{1+x^3}=\frac{... |
So I have been strugling with to prove the following:
$\left|\sum_{k\geq 0}(-1)^k {N-k-1 \choose N-2k}\right| = \begin{cases} 1, & \text{if $N\equiv 1 \mod 3$} \\ 0, & \text{otherwise} \end{cases} $ \
I was given the hint of considering the coefficient of both sides of the equality:
$
\frac{1-x^2}{1+x^3}=\frac{... |
So I have been strugling with to prove the following:
>$\left|\sum_{k\geq 0}(-1)^k {N-k-1 \choose N-2k}\right| = \begin{cases} 1, & \text{if $N\equiv 1 \bmod 3$} \\ 0, & \text{otherwise} \end{cases} $
I was given the hint of considering the coefficient of both sides of the equality:
$$
\frac{1-x^2}{1+x^3}=\f... |
I am reading Kawamuro's paper on Morton-Franks-Williams inequality (https://arxiv.org/abs/math/0509169). It says that a knot $K$ with braid index $b$ and maximal/minimal degrees of the variable $v$ being $d_+$ ($d_-$) in its HOMFLY-PT polynomial $P(v, z)$ satisfies inequality
$$1 + \frac{d_+ - d_-}{2} \le b_K$$
and... |
When is Morton-Franks-Williams inequality for knots strict? |
https://imgur.com/a/GnLqVSj
I've been trying to solve this but I just can't find any solved examples on the internet, can someone help me with these? |
I've been trying to solve this but i just can't find any solved exemples on the internet, can someone help me with these? |
What is the degree of extension by the zerors of the polynomial $f((x)=x^{2m}-p^{2n}=0$ over $\mathbb{Q}_p$? |
$$
\sum_{i=1}^3 \sum_{j=1}^4 x_{ij} = 15
$$
$$
\sum_{j=1}^4
k_{ij} = i+1,\qquad i=1,2,3$$
$$
\sum_{i=0}^4 x_{ij}=5, \qquad j=1,2,\ldots,4
$$
I've been trying to solve this but I just can't find any solved examples on the internet, can someone help me with these? |
What is the degree of extension by the zeros of the polynomial $f((x)=x^{2m}-p^{2n}=0$ over $\mathbb{Q}_p$? |
Let
$$E:=\overbrace{-2C_0-(n-2)C_1}^A+(4+(n-2)\overbrace{(2C_1+C_2)}^B)\cdot p-(n-2)\overbrace{(C_1+C_2-C_3)}^Cp^2\\$$
I got a question when I tried to give asymptotics analysis of $E^2$. It looks contradiction to the computation obtained in mathematica, which pretty confused me...
I am analyzing the asymptot... |
What is the degree of extension by the zeros of the polynomial $f((x)=x^{2m}-p^{2n}=0$ over $\mathbb{Q}_p$? |
So I have been strugling with to prove the following:
>$\left|\sum_{k\geq 0}(-1)^k {N-k-1 \choose N-2k}\right| = \begin{cases} 1, & \text{if $N\equiv 1 \bmod 3$} \\ 0, & \text{otherwise} \end{cases} $
I was given the hint of considering the coefficient of x^N on both sides of the equality:
$$
\frac{1-x^2}{1+... |
So I have been strugling with to prove the following:
>$\left|\sum_{k\geq 0}(-1)^k {N-k-1 \choose N-2k}\right| = \begin{cases} 1, & \text{if $N\equiv 1 \bmod 3$} \\ 0, & \text{otherwise} \end{cases} $
I was given the hint of considering the coefficient of $x^N$ on both sides of the equality:
$$
\frac{1-x^2}{... |
Consider the polynomial $f(x)=x^{2m}-p^{2n}=0$ over the $p$-adic field $\mathbb Q_p$. Note that $m,n$ are positive integers and $\gcd(m,n)=1$. I want to study the field extension by the zeros of $f(x)$.
We can rewrite $f(x)=(x^m-p^n)(x^m+p^n)=f_1(x) f_2(x)$, where $f_1(x)$ and $f_2(x)$ are Eisenstein polynomials and... |
Let $x\in R^d$ and $A\in R^{d\times d}$ positive definite. Is the map
$$
B
\mapsto x^T B^{1/2} A B^{1/2} x
$$
always concave? One known result that gives a little hope is the Lieb inequality
(cf. https://arxiv.org/abs/quant-ph/0404126 for instance) on the concavity of $tr[A^T \sqrt B A \sqrt B]$ is concave. |
For a fixed positive definite $A$ anb vector $x$, is $B\mapsto x^T B^{1/2} A B^{1/2} x$ always concave? |
Is there a setup in which is the Woodbury formula actually usefull? |
I have `3x + y = 1100`, the combination x samples out of 500 `C(500, x)` and the combination y samples out of 500 `C(500, y)`.
I want to maximize `C(500, x) * C(500, y)`.
Any one could you solve `x`, `y` using high school math? Thanks a lot!
A intuition would be letting x and y be as close as possible to 250 to ... |
I want to know Why $\sum|n^ka_n|<\infty$ implies $f \in C^k$?
where $a_n$ is the $nth$ Fourier coefficient of $f$
Thank you for your time. |
The usual way to denote RSS is $\sum\limits_{i=1}^n (Y_i-\hat{Y_i})^2$
By writing in the matrix form $Y=X \beta + \epsilon$. Then I know $\hat Y=PY$
Thus, the RSS=$|| (I_n-P)Y ||^2 = || (I_n-P)(Y-X\beta) ||^2= \epsilon^T (I_n-P)\epsilon$
How to see the above last two equal signs? |
I am finding the Mordell-Weil group of the elliptic curve of the form $y^2=x^3+ax+b$ over the rational field. Using Magma, I found that many of them have the form $\mathbb{Z}/2\times \mathbb{Z}/2\times \mathbb{Z}$ or $\mathbb{Z}/2\times \mathbb{Z}/2\times \mathbb{Z}\times \mathbb{Z}$. Can we determine full candidates o... |
Yes, it is. The $\forall x$ isn’t quantifying any free variables, since there are no free variables in $\exists x \ x<x$, and the $\forall x$ is therefore called a ‘null quantifier’. However, using the standard semantics for quantifiers we can still evaluate such a statement: we effectively ask: is it true for all obje... |
If a norm $\lVert \cdot \rVert$ is strictly convex and $\lambda \in (0,1)$, then
$$
\lVert \lambda v + (1-\lambda)w \rVert = \lambda \lVert v \rVert + (1-\lambda)\lVert w \rVert \tag{1}\label{1}
$$
implies $v = w$, what happens if the norm is not strictly convex? For example, if we conser de 1-norm in $\mathbb... |
Let $F$ be a field and $M_n (F)$ the ring of $n\times n$ matrices with entries in $F$. I know that that $F[x]$ is a PID. Trivially, $M_n (F)$ is not commutative and not a field. My question is that is every left ideal of $M_n (F)[x]$ principal? |
Is every left ideal of $M_n (F)[x]$ principal? |
Let $\pi_n : R[x] \rightarrow R[x]_n$ be the map that cuts off all terms of higher than nth degree, i.e., $\pi_n(\sum_{i \geq 0} a_i x^i) = \sum_{i=0}^{n} a_i x^i$.
Let $q \in R[x]_m$ be a fixed polynomial and look at the figure
$M_q : R[x]_n \rightarrow R[x]_n$
$p \rightarrow \pi_n(p q)$
(a) Show that $M_q... |
Suppose I have a sample $\textbf{x} = (x_1, \dots, x_n)$ from not necessarily independent random variables. I want to estimate the scalar parameter $\theta$ of the distribution using MLE, i.e.,
$$ \hat{\theta}_{MLE} = \text{argmax}_{\theta} \;\mathbf{L}(\theta, \textbf{x}) $$
I am interested in the case when I will... |
Suppose I have a sample $\textbf{x} = (x_1, \dots, x_n)$ from stationary and $\beta$-mixing time series. I want to estimate the scalar parameter $\theta$ of the distribution using MLE, i.e.,
$$ \hat{\theta}_{MLE} = \text{argmax}_{\theta} \;\mathbf{L}(\theta, \textbf{x}) $$
I am interested in the case when I will ta... |
> Let $Y_1,Y_2,\dots,Y_n$ be a sequence of i.i.d. random variables. Each of them is integrable.
>
> Let $X_1=(Y_1+Y_2+\cdots+Y_n)/n,X_2=(Y_1+Y_2+\cdots+Y_{n-1})/(n-1),\dots,X_{n-1}=(Y_1+Y_2)/2,X_n=Y_1$
>
> Show that $X_1,X_2,\dots,X_n$ is a martingale relative to natural filtration $\mathcal F_n=\sigma(X_1,X_2,\dot... |
Consider the polynomial $f(x)=x^{2m}-p^{2n}=0$ over the $p$-adic field $\mathbb Q_p$. Note that $m,n$ are positive integers and $\gcd(m,n)=1$. I want to study the field extension by the zeros of $f(x)$.
We can rewrite $f(x)=(x^m-p^n)(x^m+p^n)=f_1(x) f_2(x)$, where $f_1(x)$ and $f_2(x)$ are Eisenstein polynomials and... |
I am trying to integrate this equation:
$$
\frac{dz}{dt} = \alpha \cdot (s(t)-z(t-1))
$$
I came up with this equation to model a "leaky integrator" system where s(t) are some samples from the environment and z(t) is a "decision variable". Thus this system can continuously decide between 2 alternatives based on ... |
Here is an exercise V.1.11.12.1 in J. Kollar's book *Rational curves on algebraic varieties*:
> Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$. Let $\mathscr{L}$ be a line bundle on $X$ such that $\mathscr{L}^{n+1}\cong \omega_X^{-1}$ and $h^0(X,\mathscr{L})\geq n+1$. ... |
IEEE 754 floating point numbers contain the concept of `` (not a number), which "dominates" arithmetical operations ($+,-,⋅,÷$ will return `` if either argument is ``). That is, `` is an [absorbing element](https://en.wikipedia.org/wiki/Absorbing_element). However, from an algebraic point of view, FP numbers are very u... |
If $$z=e^xcos y$$
With
$$x^3+e^x-t^2-t=1$$
$$yt^2+y^2t-t+y=0$$
Find
$$\frac {dz}{dt}$$
When I asked my teacher for any tips he said something about derivating both sides of the 2nd and 3nd equation. But I just can't see how this could help me or get anywhere closer to the solution. How can I find $\frac{dz}{dt}$?... |
If $$z=e^xcos y$$
With
$$x^3+e^x-t^2-t=1$$
$$yt^2+y^2t-t+y=0$$
Find
$$\frac {dz}{dt}$$
When I asked my teacher for any tips he said something about differentiating both sides of the 2nd and 3nd equation. But I just can't see how this could help me or get anywhere closer to the solution. How can I find $\frac{dz}{... |
Consider a vector field $v$ defined in $U \subset \mathbb{R} ^n$ such that $ v \in W^{1, p} (U)$ with $U$ boudned and as musch regular as you wish. I know Sobolev embeddings, and that in general $v$ is continuous (Hölder in fact) if $p>n$, and that there are counterexamples in the limit case $p=n$. However I was wonder... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
Recently, my teacher told me about Socle of a group $G$. So, I know that the socle $Soc(G)$ of a group $G$ is the subgroup generated by all minimal normal subgroups of $G$. I was thinking then that the socle of a simple group $S$ is the group $S$ itself. What I can't find is that examples of non-abelian groups with tri... |
Alternative proof for Dini's theorem. Is it correct? |
Two references that set up the definition, in slightly different ways, in terms of an auto-equivalence rather than an auto-isomorphism are:
<cite authors="Keller, Bernhard; Vossieck, Dieter">_Keller, Bernhard; Vossieck, Dieter_, **Sous les catégories dérivées. (Beneath the derived categories)**, C. R. Acad. Sci., Pa... |
Suppose that waiting in queue at the post office (in hours) has probability density function:
\begin{equation}
f(x) =
\begin{cases}
\frac{3}{8}(2 - \sqrt{x}) & \text{if } x \in [0, 4],\\
0 & \textrm{otherwise}
\end{cases}
\end{equation}
Question:
Given the information that **we are ... |
I did a problem in competitive programming, the problem is split a number into x and y (x <= y), and make a new number by (x - 1) * (y + 1), do it until a new number different zero, and count how many different number can have by this split way.
It's easy for me but, when i draft, a new number always lower than origi... |
How to proof it? |
Except the (rather famous) example $\mathbb F_p(t)= \{ \frac{f(t)}{g(t)}:\ f,g \in \mathbb F_p[t],\ g\neq 0 \}$, which has the inseparable extension containing the one multiple root of $x^p- t$. Do we know other examples of inseparable extensions?
It seems unfortunate if we had a definition of separability only beca... |
Do we know all non-perfect fields? |
> Does there exists a global solution $u\in L^{2}(\mathbb{R}^{3})$ to
> the Laplace equation $\Delta u=0$?
Using the spherical symmetry of the problem, one can consider the usual solution obtained by seperation of variables which yields something like this
$$u(r,\theta,\varphi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}c_... |
I did a problem in competitive programming, the problem is split a number into $x$ and $y$, $(x \le y)$, and make a new number by $(x - 1)(y + 1)$, do it until a new number different zero, and count how many different number can have by this split way.
It's easy for me but, when i draft, a new number always lower than... |
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
Here are some things I will define to make the process easier:
* Let $p$ be a prime number.
* Let $S(n)$ be the digit sum function.
* Let $d_p$ be the number of digits in $p$.
* Let $\mathbb P$ be the set of primes {$n ... |
I am reading Kawamuro's paper on Morton-Franks-Williams inequality (https://arxiv.org/abs/math/0509169). It says that a knot $K$ with braid index $b$ and maximal/minimal degrees of the variable $v$ being $d_+$ ($d_-$) in its HOMFLY-PT polynomial $P(v, z)$ satisfies inequality
$$1 + \frac{d_+ - d_-}{2} \le b_K$$
and... |
> Does there exists a global and non-trivial solution $u\in L^{2}(\mathbb{R}^{3})$ to
> the Laplace equation $\Delta u=0$?
Using the spherical symmetry of the problem, one can consider the usual solution obtained by seperation of variables which yields something like this
$$u(r,\theta,\varphi)=\sum_{l=0}^{\infty}\... |
I have the equation $f(x) = \arccos(x) + (C × (1 - x^2) - x) × \sqrt{1 - x^2}$, where the parameter $C$ is a real number. When $-\frac{2}{3} ≤ C$, the function has one real zero at $x = 1$. When $C$ drops below $-\frac{2}{3}$, a second zero appears, starting at $x = 1$ and decreasing towards $x = -1$ as $C$ tends towar... |
Can the zero of this function be calculated explicitely, or is it transcendental? |
Suppose that we have a symmetric block matrix of the following form:
$$ S =
\begin{pmatrix}
S_{11} & \ldots & S_{1m} \\\\
\vdots & \ldots & \vdots \\\\
S_{m1} & \ldots & S_{mm}
\end{pmatrix} \in \mathbb{R}^{pnm\times pnm},$$ where $$S_{ji}=S_{ij}=\begin{pmatrix}
K_{ij} & 0_{n \times n(p-1)} \\... |
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