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Intuitively speaking, how do existential quantifiers appear?
I'm just starting to get familiar with these definitions.
Top and conjunctions appear because of finite products. (Plus, I assume, something that makes them work nicely with the existential quantifiers.)
But my understanding of existential quantifier... |
How are existential quantifiers present in the internal logic of regular categories? |
Let $A, B$ and $C$ be three small categories and let $f: A\to B$ and $g: C\to B$ two functors such that $g$ is conservative. We can take the pullback as in the following picture:
\[\begin{tikzcd}[ampersand replacement=\&]
P \& C \\
A \& {B.}
\arrow["g", from=1-2, to=2-2]
\arrow["f"', from=2-1, to=2-2]
\arrow... |
I know that equivalent metrics do not conserve completeness (e.g $d_1 = |\operatorname{arctan}(x) - \operatorname{arctan}(y)|$ is equivalent to $d_2 = |x - y|$, but $d_1$ does not conserve completeness).
However, equivalent norms do conserve completeness. Does this mean that if two norms are equivalent, their induce... |
Do equivalent induced metrics conserve completeness? |
Let $x$ be a $\mathbb{R}^n$ valued path with finite variation on $[0,T]$ and $A$ be a matrix of $d\times n$. It is easy to see that $Ax$ is a path with finite variation.
Let $S^N(x)_{0,1}$ be the step-N signature of $x$ on $[0,1]$.
What is the relationship between $S^N(x)_{0,1}$ and $S^N(Ax)_{0,1}$?
|
Let $A, B$ and $C$ be three small categories and let $f: A\to B$ and $g: C\to B$ two functors such that $g$ is conservative. We can take the pullback as in the following picture:
$\require{AMScd}$
\begin{CD}
P @>{g^{*}}>> A\\
@VVV @VVV\\
C @>{g}>> B.
\end{CD}
My question is: $g^{*}$ is always conservative?
... |
Suppose that we have an invertible map $T(u,v)=(x,y)$.
The Jacobian of $T$ is given by
$
\text{Jac}(T)=
\left| {\begin{array}{cc}
x_u & x_v \\
y_u & y_v \\
\end{array} } \right|$.
Now if we consider the inverse map of $T$, i.e. $T^{-1}(x,y)=(u,v)$, its Jacobian is given by
$
\text{Ja... |
What's the problem with the fact that $\infty$ is not a constant? For each $n\in\Bbb N$, $a_n\geqslant n$, and therefore $\lim_{n\to\infty}a_n=\infty$. So, $\limsup_{n\to\infty}a_n=\liminf_{n\to\infty}a_n=\infty$.
In general, if $l\in\Bbb R\cup\{\pm\infty\}$, then$$\lim_{n\to\infty}a_n=l\iff\limsup_{n\to\infty}a_n=\... |
In order to calculate the Fourier transform of a formula as follow:
$$g(x, y) = \exp\left(-\frac{(x-x_0)^2 + (y-y_0)^2}{W^2}\right)e^{-il\tan^{-1}(\frac{y}{x})} $$ Where $B$ and $l$ are constants. $$G(f_x, f_y)=\iint g(x, y)\exp\{-2\pi(xf_x +yf_y)\}dxdy$$ Where $F\{\}$ is the Fourier transform.
After transform th... |
I am reading [this paper](https://www.sciencedirect.com/science/article/abs/pii/S0167668799000268) of Lin and Willmot. I dont understand how they come up with formula $2.7$ and why $\tilde{\phi}(s)=\frac{\tilde{H}(s)}{1+\beta-\tilde{g}(s)}$. Can someone help me?
So i want to show that $$\phi(u)=\frac{1}{1+\beta}\int... |
How to determine all valuations of the field $\mathbb{Q}(\sqrt[n]{2})$? |
I am reading [this paper](https://www.sciencedirect.com/science/article/abs/pii/S0167668799000268) of Lin and Willmot. I dont understand how they come up with formula $2.7$ and why $\tilde{\phi}(s)=\frac{\tilde{H}(s)}{1+\beta-\tilde{g}(s)}$. Can someone help me?
So i want to show that $$\phi(u)=\frac{1}{1+\beta}\int... |
I'm currently studying the book "Galois Theory" by "David. A. Cox" in page $103$ he tries to prove the following statement:
> Given $f_1 \in F_1[x]$ and isomorphism of fields $\phi: F_1 \rightarrow F_2$ there exists an isomorphism $\bar{\phi}: L_1 \rightarrow L_2$ such that $\bar{\phi}{\big|}_{F_1}=\phi$
where i... |
Let $f_n \in L^2([0,1])$ be non-zero. Prove there exists a function $g \in L^2([0,1])$ such that
$$\int_0^1 g(x) f_n(x) dx \neq 0.$$
So first I tried contradiction which implies $g(x)f_n(x)$ need be zero almost everywhere which would imply $g$ to be the zero function. But then again I thought maybe $g(x)=1$ on th... |
Suppose $2k + 1 \equiv 3 \mod 4$ in $\Bbb{Z}_{\geq 1}$.
Is the polynomial: $p_k(x) = x^{2k + 1} + x^{2k - 1} \dots + x + 1$ irreducible in $\Bbb{Z}_2[x]$?
I do not know whether it is true or not...
(*Side note:*
I've had a question in a test that is solved immediately if the statement above turns out to be t... |
I know that every square root $\sqrt{n}$ of an integer is contained in some cyclotomic extension $\Bbb Q[\omega_m]$. If we know $\sqrt{n}\in\Bbb Q[\omega_m]$ and that $n$ is square-free, can we conclude that for its prime divisor $p|n$ we have $\sqrt{p}\in\Bbb Q[\omega_m]$ as well? |
In this
>answer to [Is there any valid complex or just real solution to $\sin(x)^{\cos(x)} = 2$?](https://math.stackexchange.com/a/4693242/905886),
one must calculate $$\frac{d^{n-1}}{dw^{n-1}}\left.\frac{4^{-\frac{n}{\sqrt w}}}{\sqrt w}\right|_1=\sum_{m=0}^\infty\frac{\Gamma\left(\frac12-\frac m2\right)(-\ln(4)... |
I am working on deriving the intensity equations for the dynamical diffraction of neutrons following along with a paper by Hartmut Lemmel (Hartmut Lemmel. Dynamical diffraction of neutrons and transition from beam splitter to phase shifter case. Phys. Rev. B, 76:144305, Oct 2007).
The first task is to get the follow... |
Suppose $2k + 1 \equiv 3 \mod 4$ in $\Bbb{Z}_{\geq 1}$.
Is the polynomial: $p_k(x) = x^{2k + 1} + x^{2k - 1} \dots + x + 1$ irreducible in $\Bbb{Z}_2[x]$?
I do not know whether it is true or not...
(*Side note:*
I've had a question in a test that is solved immediately if the statement above turns out to be t... |
At the beginning of *"Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de
vecteurs. C. R. Acad. Sci. Paris S´er. I Math. 305, no. 19, 805–808"*, there is:
$\int_{K}\nabla{f} \cdot w = 0$ for any $f \in C^1\left(K\right)$, where $K$ is a compact subspace of $\mathbb{R}^N$ and $w:K\mapsto K... |
I know that every square root $\sqrt{n}$ of an integer is contained in some cyclotomic extension $\Bbb Q[\omega_m]$. If we know $\sqrt{n}\in\Bbb Q[\omega_m]$ and that $n$ is square-free, can we conclude that for its prime divisor $p|n$ we have $\sqrt{p}\in\Bbb Q[\omega_m]$ as well?
For example, could anyone give me ... |
Let $f_n \in L^2([0,1])$ be non-zero for all $n \in \Bbb{N}$. Prove there exists a function $g \in L^2([0,1])$ such that
$$\int_0^1 g(x) f_n(x) dx \neq 0.$$
So first I tried contradiction which implies $g(x)f_n(x)$ need be zero almost everywhere which would imply $g$ to be the zero function. But then again I thou... |
It's not a typo.
What is a "tuple"? Normally we think of a "tuple" with index set $I$ as a function with domain $I$. So, viewed as a set of ordered pairs, a tuple $(x_i)_{i\in I}$ would be the set
$$\{ (i,x_i)\mid i\in I\}.$$
For index sets that are of the form $I=\{1,2,3,\ldots,n\}$, we often denote this as $(a_... |
You are missing what the Author is trying to convey.
(1) A graphical illustration is **useful** for understanding (by humans) the notions and procedures of linear programming, but as a computational method (by computers) it is **worthless**.
(2) Sometimes it **may** (not "will") even be misleading, since objec... |
Ross and Pekoz's book A Second Course in Probability gives the following question in their chapter on Stein-Chen Method:
Compute a bound on the accuracy of a normal approximation for a Poisson random variable with mean 100.
I believe this post was seeking help on the same question, but never found it https://math... |
Consider a set of probabilities $\{p_1, ..., p_N\}$ where $p_1\leq p_2\leq ... \leq p_N$. Now consider the difference between the largest and smallest probabilities $D =p_N-p_1$. I want to know if one can use the Shannon entropy of the distribution to find a bound on $D$. The Shannon entropy is given by $H = -\sum_i p_... |
For a probability distribution is the difference between the largest and smallest probabilities bounded by the Shannon entropy? |
I am studying topology through the *Topology without Tears* book, and on the first unit the finite-closed topology is presented with the following definition:
> 1.3.1
**Definition**.
Let $X$ be any non-empty set. A topology $\tau$ on $X$ is called the finite-closed topology or the cofinite topology if the closed sub... |
I'm studying general topology and a question has come to my mind.
We have defined a connected topological space to be a space that (viz. such that its underlying set) cannot be written as the union of two non-empty and disjoint open subsets. The notion extends to subsets by means of the subspace topology, viz. a su... |
Disclaimer: I'm still learning, this is my stab at an answer, but correction/improvements are much appreciated.
I think the key here is that *the goal* with a principal bundle is to attach **a $G$-torsor** to each point of the base space, **not** to attach the group $G$ to each point of the base space. There is a bi... |
I have been long time puzzled by the correct application of Fubinis theorem. Is it true that the bounds need to be constant if the integrals can be interchanged? Please correct if not.
Let's consider the following Laplace transform pair.
\begin{align}
\mathcal{L}^{-1}\Bigl[\frac{2(\cos(at)-\cos(bt))}{t}\Bigr]=\ln\... |
What happened to interchange Laplace transform and one definite integral (Fubini's theorem is not satisfied)? |
Given eigenvalues and eigenvector, how to find the matrix there are fewer eigenvectors than its rank? |
In order to calculate the Fourier transform of a formula as follow:
$$g(x, y) = \exp\left(-\frac{(x-x_0)^2 + (y-y_0)^2}{W^2}\right)e^{-il\tan^{-1}(\frac{y}{x})} $$ Where $B$ and $l$ are constants. $$G(f_x, f_y)=\iint g(x, y)\exp\{-2\pi(xf_x +yf_y)\}dxdy$$ Where $F\{\}$ is the Fourier transform.
After transform th... |
Are conservative functors stable under pullback? |
I'm studying general topology and a question has come to my mind.
[Definition of *connected topological space* and *connected subset* of a topological space] We have defined a connected topological space to be a space that (viz. such that its underlying set) cannot be written as the union of two non-empty and disjo... |
I'm studying general topology and a question has come to my mind.
[Definition of *connected topological space* and *connected subset* of a topological space] We have defined a connected topological space to be a space that (viz. such that its underlying set) cannot be written as the union of two non-empty and disjo... |
I'm studying general topology and a question has come to my mind.
[Definition of *connected topological space* and *connected subset* of a topological space] We have defined a connected topological space to be a space that (viz. such that its underlying set) cannot be written as the union of two non-empty and disjo... |
It is well known that hanldes of indices k and (k+1) may be cancelled if the attaching sphere of (k+1)-handle *intersect* the belt sphere of k-handle *at exaclty one point*. Is it also necessary condition for the cancelling? In Rourke and Sanderson book I found that for dimension $n\geq 6$ and for sufficiently small ... |
I'll try to give some intuitions and point the interested people to related material.
**One variable case.**
An indefinite integral $\int f(x) \, dx$ is understood as a function $F$ used to evaluate the definite integral over an interval $[a,b]$ in the following way: given the numbers $a$ and $b$, $$\int_a^b f(x)... |
Let $b,d,m$ be positive integers such that $d|m$ and $0\leq b<d$.
I want to find a minimal integer $h\in\frac{m}{bd}\mathbb{Z}\cap\frac{m}{b^2}\mathbb{Z}\cap\frac{m}{d^2}\mathbb{Z}$.
First, I found minimal integers for each set. They are $\frac{m/d}{(m/d,b)},\frac{m}{(m,b^2)}$, and $\frac{m/d}{(m/d,d)}$, where ( ... |
I have been long time puzzled by the correct application of Fubinis theorem. Is it true that the bounds need to be constant if the integrals can be interchanged? Please correct if not.
Let's consider the following Laplace transform pair.
\begin{align}
\mathcal{L}^{-1}\Bigl[\frac{2(\cos(at)-\cos(bt))}{t}\Bigr] = ... |
I have some problems with this exercise. I don't know if it can be done.
>Consider the polynomial $ x^n - a \in \mathbb{Q}[x]$. Can I compute the Galois group of this over $\mathbb{Q}$?
Maybe having a nice basis. The splitting field is given by $\mathbb{Q}(\zeta_n,\alpha)$, where $\zeta_n$ is a primitive root of u... |
$F$ is a function from $R^2$ to $R^2$. For every line $L$ in $R^2$, $F(L)$ is a line, too. Then, is $F$ a linear transform? |
If image of every line is a line, is that transform linear? |
**Does there exist two binary operations + and ∗ on {0,1} such that + and ∗ satisfy the same equations, but not the same quasi-equations? If not, are there such binary operations on a finite set of higher cardinality, and if so, what is the smallest possible cardinality?**<p>
(Partial) answers:<p>
<ol>
<li> No, ... |
While exploring possible applications for this *[new trick][1]*, I stumbled upon an entire family of integrals that "always" yield $a\pi^n$, where $a$ is an algebraic number and $n$ is a natural number. The following integral captivated me greatly:
>$$\boxed{\int_{-1}^{1} \frac{125}{12}\sqrt[10]{\frac{1 + x}{1 - x}}... |
$F$ is a function from $R^2$ to $R^2$. For every line $L$ in $R^2$, $F(L)$ is a line, too. Also it maps origin to origin. Then, is $F$ a linear transform? |
Note that if $h$ is an integer multiple of $\frac{m}{b^2}$ and $\frac{m}{d^2}$, then it is automatically an integer multiple of $\frac{m}{bd}$. Indeed, we have integers $x$ and $y$ such that $h = \frac{m}{b^2}x$ and $h = \frac{m}{d^2}y$, and multiplying both equations and taking square roots shows that $h = \frac{m}{bd... |
I am trying to solve the following problem and what I need is to split a function on the square of a finite set into tensor of two functions:
>$R(\Gamma)$ is a k-algebra of maps from $\Gamma$ to k. $\Delta$ is a functor acting from $R(\Gamma)$ to $R(\Gamma) \otimes R(\Gamma)$: $(\Delta f)(g,h)=f(gh)$
>If $\Gamma... |
I was supposed to prove:
Theorem: Let $a \neq 0$, the geometric series $\sum_{n=1}^{\infty}(aq^{n-1})$
i) if $|q|< 1$, converges and has the result $S=\frac{a}{1-q}$.
ii) if $|q| \ge 1$, diverges.
I did it as follows:
Let the geometric series $\sum_{n=1}^{\infty}(aq^{n-1})$ and consider $S_n$ the n-th su... |
Given $f:[0,\infty)\to \mathbb{R} $ a uniformly continuous function and $g:[0,\infty)\to \mathbb{R} $ a continuous function.
If $$ \lim_{i=0} g(x) - f(x) = 0$$ is g uniformly continuous over $[0, \infty)$? Thank you for your help! |
Say you have a list of $n$ random integers in the range $[1, m]$. The binomial PMF can find the probability of exactly $k$ occurrences of a number with $p = \frac{1}{m}$
$$ {n \choose k} p^k (1-p)^{n-k} $$
Now say the list is constructed in the following way: the $i$th element in the list is a random number in $[1,... |
At the beginning of *"Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de
vecteurs. C. R. Acad. Sci. Paris S´er. I Math. 305, no. 19, 805–808"*, there is:
$\int_{K}\nabla{f} \cdot \overrightarrow{w} = 0$ for any $f \in C^1\left(K\right)$, where $K$ is a compact subspace of $\mathbb{R}^N$ ... |
I solved the following equation today for fun and I have the exact real result:
$x^{x^{x+1}}=2 \Leftrightarrow x=\frac{W(ln(2))}{W(W(ln(2)))}=\exp ^{W(W(ln(2)))}$
Where $W$ is the Lambert W function.
But how can I now solve $x^{x^{x}}=2$ exactly? I know it's another similar question, but I just love exact res... |
$x^{x^{x}}=2$ How can I solve this exactly? |
$x^{x^{x}}=2$ : How can I solve this exactly? |
> Let $Q_c(x)=x^2+c$. Prove that if $c<\frac14$, there is a unique
> $\mu>1$ such that $Q_c$ is topologically conjugate to
> $F_\mu(x)=\mu(1-x)$ via a map of the form $h(x)=\alpha x+\beta$.
Interpretation: $h(x)$ is a linear map. $Q_c(x)$ and $F_\mu(x)$ are a quadratic and affine map respectively. The claims is th... |
It would not be easy to produce simple formulas. Roughly speaking, the connection between arithmetic and bitwise operations is complicated as seen from computer science. Had there been, math education in elementary school can be significantly simplified, since the formula must be able to re-produce the results for natu... |
Given $f:[0,\infty)\to \mathbb{R} $ a uniformly continuous function and $g:[0,\infty)\to \mathbb{R} $ a continuous function.
If $$ \lim_{x=0} g(x) - f(x) = 0$$ is g uniformly continuous over $[0, \infty)$? Thank you for your help! |
I have recently being interested in the estimation of discontinuities or jumps from noisy signals or densities and spend some time reading "Image Processing and Jump Regression Analysis" by Peihua Qiu. I wonder if there is a similar book that presents more recent results ?
The subject is strongly connected with edge... |
I'm at a loss for ideas on what to do.
I'm given two paraboloids and their equations:
$z_1 = x^2 + 1$ and $z_2 = 5 - y^2$.
I know their intersection is when $z_1 = z_2$.
This gives me $x^2 + y^2 = 4$, the equation of a circle. Which makes sense, because the projection of the intersection on the xy plane... |
Given $f:[0,\infty)\to \mathbb{R} $ a uniformly continuous function and $g:[0,\infty)\to \mathbb{R} $ a continuous function.
If $$ \lim_{x \to \infty} g(x) - f(x) = 0$$ is g uniformly continuous over $[0, \infty)$? Thank you for your help! |
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined:
- I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic order. Take [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution#Mean) as an example. If $d=1$, then $N^{... |
I am trying to solve the following ODE $$\begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned}$$
Using central FD, I can write $$\frac{y_{i+1}-2 y_i+y_{i-1}}{h^2}+2 \frac{y_{i+1}-y_{i-1}}{2 h}+y_i=0$$ which I have simplied to $$(1+h) y_{i+1... |
At the beginning of *"Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de
vecteurs. C. R. Acad. Sci. Paris S´er. I Math. 305, no. 19, 805–808"*, there is:
$\int_{K}\nabla{f} \cdot \overrightarrow{w} = 0$ for any $f \in C^1\left(K\right)$, where $K$ is a compact subspace of $\mathbb{R}^N$ ... |
have a homework that I'm working on and my teacher said that I overcounted. Can you help me?
100 guests buy a drink/snack; 35 of them order wine, 25 order champagne, 20 order sweet snack, 15 order salty snack, 5 don't order anything. You talk to 5 people at random and find out what they bought. Find the probability... |
In the urn we have a white, b red and c black balls. We draw with replacement. Calculate the expected number of white balls, which are drawn until we drew black ball for the k-th time.
My solution:
The expected number of draws till be draw black ball for the k-th time is:
$$\left(\frac{a+b+c}{c} \right)^k$$ That s... |
Assume that $X\in \mathrm{PSD}(n)$ is a symmetric, positive semi-definite matrix with real entries and $C>0$ is a positive real number. If $X$ satisfies $$
\forall \alpha\in\mathbb{Z}^n\setminus\{0\},\qquad \alpha^T X \alpha \geq C,
$$ then is it true that $X$ is positive definite? That is, if $X$ is lower bounded on... |
For starters, let's have an ensemble of $n$ random variables $\mathbf{x}, \mathbf{y}$,
$$
\begin{align}
\mathbf{x} &= (x_1, \cdots, x_n) \\
\mathbf{y} &= (y_1, \cdots, y_n)
\end{align}
$$
and we want to check for independence of $\mathbf{x}$ and $\mathbf{y}$. It's said that the variables are independent if **all... |
I have recently being interested in the estimation of discontinuities or jumps from noisy signals or densities and spend some time reading "Image Processing and Jump Regression Analysis" by Peihua Qiu. I wonder if there is a similar book that presents more recent results ? Also the book focus on 2D image, is there any ... |
While exploring possible applications for this *[new trick][1]*, I stumbled upon an entire family of integrals that "always" yield $a\pi^n$, where $a$ is an algebraic number and $n$ is a natural number. The following integral captivated me greatly:
>$$\boxed{\int_{-1}^{1} \frac{125}{12}\sqrt[10]{\frac{1 + x}{1 - x}}... |
I'm very confused on inequalities. I just came from inequalities with absolute values and tried to solve the above inequality like I did with absolute values. When I compared with my students they seemed to have a different and much more simpler Answer.
\begin{align*}
\text{Given equation:} \quad & \frac{x+1}{x-1... |
How do I find the solution for $(\frac{x+1}(\x-1})>(\frac1{x})$? |
Are $\operatorname{Tr}(AB)$ and $\operatorname{Tr}(AB^T)$ bilinear forms? inner products? |
It would not be easy to produce simple formulas. Roughly speaking, the connection between arithmetic and bitwise operations is complicated as seen from computer science. Had there been, math education in elementary school can be significantly simplified, since the formula must be able to re-produce the results for natu... |
For starters, let's have an ensemble of $n$ random variables $\mathbf{x}, \mathbf{y}$,
$$
\begin{align}
\mathbf{x} &= (x_1, \cdots, x_n) \\
\mathbf{y} &= (y_1, \cdots, y_n)
\end{align}
$$
and we want to check for independence of $\mathbf{x}$ and $\mathbf{y}$. It's said that the variables are independent if **all... |
How do I find the solution for $\frac{x+1}(x-1}>\frac 1{x}$? |
How do I find the solution for $\frac{x+1}(x-1}>\frac{1}{x}$? |
How do I find the solution for $\frac{x+1}{x-1} \gt \frac 1{x}$? |
Suppose we have $A,B \in \operatorname{Mat}(n, \mathbb{R})$, and let $\alpha(A,B) = \text{Tr}(AB)$ and $\beta(A,B) = \text{Tr}(AB^T)$. Then is it possible to show that both $\alpha$ and $\beta$ cannot be bilinear forms? If not then what can be shown as a counter-example?
Furthermore, how can I show that only $\beta$... |
I have a question that may seem trivial.
I am studying Kelley's General Topology. I was reading Chapter 2 when I came across an unusual definition. Tipically, I define a subsequence as follows:
> a subsequence of the sequence $(a_n)_{n \in \mathbf{N}}$ is any
> sequence of the form $(a_{n_k})_{k \in \mathbf{N}... |
I am trying to find all incongruent primitive roots of $14$. I know that there are $\phi{(\phi{(14)})}=2$ incongruent primitives roots, where I have found that both $3$ and $5$ are primitive roots mod $14$.
My question is: why aren't $11$ and $13$ also primitive roots mod 14? Both $11^6 \equiv 13^6 \equiv 1$ mod $1... |
> You are playing a one-player game with two opaque boxes. At each turn,
> you can choose to either "place" or "take". "Place" places $1 from a
> third party into one box randomly. "Take" empties out one box randomly
> and that money is yours. This game consists of 100 turns where you
> must either place or take. A... |
I'm very confused on inequalities. I just came from inequalities with absolute values and tried to solve the above inequality like I did with absolute values. When I compared with my students they seemed to have a different and much more simpler Answer.
\begin{align*}
\text{Given equation:} \quad & \frac{x+1}{x-1... |
On page 78 of Hartshorne's "Algebraic Geometry," we encounter the following correspondence:
Proposition 2.6: Let $k$ be an algebraically closed field. There exists a natural, fully faithful functor $t: \mathfrak{Var}(k) \rightarrow \mathfrak{Sch}(k)$ from the category of varieties over $k$ to schemes over $k$. For a... |
I don't know if there are people who recognize the following pattern in Isosceles triangles and they are new to me, it's difficult to understand and I've been working on it and doesn't seem to get nowhere. Any explanation?
Table of isosceles triangles.
$$
\begin{array}{1|2|3|4|5|6}
\\
\hline
1& 1+1+1=... |
Should there be the table of triangles as in the table of all elements since there are patterns in the laws of triangles? |
### Background
In physics, there is the concept of "[mean free path][1]," which is the expected value for the distance a molecule (for example) can travel before it hits another one.
If they're all the same size, and the neighbor molecules are motionless, this [can be calculated][2] as $1\over{\pi d^2 n_v}$ (wher... |
We notice that
$$\Box\psi\rightarrow\psi$$
is not a theorem of $\mathbf{K}$, hence
$$\not\vdash_{\mathbf{K}}\Box\psi\rightarrow\psi$$
However,
$$\vdash\Box\psi\implies\vdash\psi$$
equivalently,
$$\Vdash\Box\psi\implies\Vdash\psi$$
is a metatheorem of $\mathbf{K}$. We may intuitively see this no... |
### Background
In physics, there is the concept of "[mean free path][1]," which is the expected value for the distance a molecule (for example) can travel before it hits another one.
If they're all the same size, and the neighbor molecules are motionless, this [can be calculated][2] as $1\over{\pi d^2 n_v}$ (wher... |
Consider the following question (I only need help with the last part)
>The dynamics of a directly transmitted microparasite can be modelled by the system
\begin{align*}
\frac{d X}{d t} & =b N-\beta X Y-b X, \\
\frac{d Y}{d t} & =\beta X Y-(b+r) Y, \\
\frac{d Z}{d t} & =r Y-b Z,
\end{align*}
where $b, \beta$ an... |
>Let $G$ be a finite abelian group. Show that there is a Galois extension $K/\Bbb Q$ with $\text{Gal}(K/\Bbb Q) \cong G$.
I have seen a proof using that for a fixed positive integer $n$, there are infinitely many prime numbers $p$ such that $p ≡ 1 \bmod n$.
But is there any easy way to prove this? Please reply |
What is the probability of rolling n consecutive numbers when rolling k dice? |
So, I'm working on an app and a big part of that is tracking traversed distance using an accelerometer, GPS, and sensor fusion on a phone.
I'm stuck because I need to transform the Euler angles derived from a stabilized gyroscope (otherwise I could just use vectors, but the angles need to be stabilized) into a rotat... |
How to calculate a rotation matrix from Euler angles, specifically in pitch-yaw-roll format? |
Consider a probability distribution on the set of all ordered binary strings of length $M$. Each element may then be written as $( s_1, s_2, \cdots, s_M )$ with each $s_k \in \{ 0, 1\}$ and the probability distribution is a non-negative function $p(\{s_k\})$ which sums to $1$.
Just by parameter counting, the probab... |
Say $H$ and $K$ are formally smooth group schemes (or, even better, $p$-divisible groups). Can I deduce that that any extension of $H$ by $K$ is formally smooth? It seems like this should be well-known for algebraic groups or group schemes but I can’t seem to find a reference for any of these cases.
I am interested ... |
The sub-resultant GCD algorithm allows to get the GCD of two polynomials $A$ and $B$ with coefficients in a UFD. Is it possible with this algorithm to get the quotients $A / gcd(A,B)$ and $B / gcd(A, B)$?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/6C0sB.png |
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