instruction stringlengths 12 30k |
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How many binary representation exists that each has five "00", three "01", three "10" and three "11" in it ?! |
Under which minimal conditions on the function $f$ on $\mathbb{R}^+$ and the measure $\mu$ also on $\mathbb{R}^+$ we have:
$$ \int_A f d \mu \leq \mu(A) \int_A f d \lambda$$
for $\lambda$ Lebesgue measure and $A$ non empty measurable set? Is it enough to ask $f$ to be nonnegative and $\mu$ a nonnegative finite me... |
As I wrote in a comment, the adjunction that you propose does hold in an appropriate $\infty$-categorical version of the $S$-construction. I will quickly explain what stable $\infty$-categories are (and how they are related to Waldhausen categories), and how we obtain the functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty... |
Using Lagrange multipliers, I can see that the *maximum* of $f(x, y, z)$ subject to the constraint is $\sqrt{3}$. However, the answer key in my textbook says that the *minimum* is $1$. Can someone explain why the minimum is 1, and how to understand that? |
The law of tangents still has its uses! But most modern school syllabuses seem to have ditched it, solving non-right-angled triangles using the law of sines and law of cosines instead, so your question is a fair one. There's a misconception here:
> I was thinking about it and anything I could think about using the l... |
As I wrote in a comment, the adjunction that you propose does hold in an appropriate $\infty$-categorical version of the $S$-construction. I will quickly explain what stable $\infty$-categories are (and how they are related to Waldhausen categories), and how we obtain the functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty... |
This is the definition of a restriction of a vector bundle given in John Lee's Introduction to Smooth Manifolds.
Let $E,M$ be smooth manifolds with or without boundary and $\pi:E \to M$ a rank-$k$ smooth vector bundle and $S$ be an embedded submanifold with or without boundary of $M$. Then we have the restriction of... |
As I wrote in a comment, the adjunction that you propose does hold in an appropriate $\infty$-categorical version of the $S$-construction. I will quickly explain what stable $\infty$-categories are (and how they are related to Waldhausen categories), and how we obtain the functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty... |
$\sigma:F\to\overline{K_2}$ can be extended to a homomorphism of the polynomial rings $F[x]\to\overline{K_2}[x]$ by $\sigma(\sum\limits_{i=0}^n a_ix^i)=\sum\limits_{i=0}^n\sigma(a_i)x^i$, i.e we simply apply $\sigma$ to the coefficients of the polynomial.
Now let $m\in F[x]$ be the minimal polynomial of $\alpha$ ov... |
Using Lagrange multipliers, I can see that the *maximum* of $f(x, y, z)$ subject to the constraint is $\sqrt{3}$.
To see this, here is what I did:
- Show the constraint in the form $g(x,y,z) = k$, which gives us $g(x, y, z) = x^4 + y^4 + z^4 - 1$.
- To satisfy $\nabla{f(x, y, z)} = \lambda \nabla{g(x, y, z)}... |
I am currently studying the following nonlinear PDE model for cells migrating under the influence of diffusion and chemotaxis:
\begin{align}
\hspace{2cm} \frac{\partial m}{\partial t} & = \frac{\partial^2 m}{\partial x^2} - \chi \frac{\partial}{\partial x} \left(m \frac{\partial c}{\partial x} \right) && \... |
How to "linearize" a term of the form $\frac{\partial}{\partial x} \left( m \frac{\partial c}{\partial x} \right)$ in a nonlinear PDE? |
As I wrote in a comment, the adjunction that you propose does hold in an appropriate $\infty$-categorical version of the $S$-construction. I will quickly explain what stable $\infty$-categories are (and how they are related to Waldhausen categories), and how we obtain the functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty... |
Using Lagrange multipliers, I can see that the *maximum* of $f(x, y, z)$ subject to the constraint is $\sqrt{3}$.
To see this, here is what I did:
- Show the constraint in the form $g(x,y,z) = k$, which gives us $g(x, y, z) = x^4 + y^4 + z^4 - 1$.
- To satisfy $\nabla{f(x, y, z)} = \lambda \nabla{g(x, y, z)}... |
I am trying to use the least squares regression to fit a curve to a table of values representing a sine wave. Similar to the question:
https://math.stackexchange.com/questions/3926007/least-squares-regression-of-sine-wave
Except I want to fit a more general formula of: $a * \sin(b*x + c) + d$
So given a function... |
I am currently studying the following nonlinear PDE model for cells migrating under the influence of diffusion and chemotaxis:
\begin{align}
\hspace{2cm} \frac{\partial m}{\partial t} & = \frac{\partial^2 m}{\partial x^2} - \chi \frac{\partial}{\partial x} \left(m \frac{\partial c}{\partial x} \right) && \... |
I have been asked to show that a smooth plane quartic is never hyperelliptic. I know that
1. The genus of any such curve is 3
2. The statements of Riemann-Roch and Riemann-Hurwitz
3. A curve is hyperelliptic if there's a degree 2 map from it to $\Bbb P^1$.
Can I have a hint about what to do please? Cheers. |
$x = 2\lambda x^3$
Either $x = 0$ or $\lambda =\frac {1}{2x}$.
You have investigated what happens if $\lambda = 2x$ but what about $x = 0?$
|
i am trying to find the coefficients to a general solution for non-linear, autonomous second order ordinary differential equation, i have ended up with a general solution that has 4 integration constants. does this mean that i need 4 initial conditions? and if so which? i know that i can already use the initial conditi... |
Hope I made no mistake here:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
$ \int_V \nabla \mathbf{F} dV = \oint_S \mathbf{F}\cdot\mathbf{n} dS $
So if you have a symmetric problem then this helps you find a solution quickly. But your bo... |
>Prove that if $F(x_1, x_2, \ldots, x_n) = 0$, where $F$ is a polynomial with integral coefficients, has a solution in integers, then the congruence $F(x_1, x_2, \ldots, x_n) \equiv 0 \pmod{m}$ is solvable for any value of modulus $m$.
I am not sure how to prove this.
**My attempt:**
Let $F(x_1, x_2, \ldots, x_... |
Let $\mathbb \S^n$ denote the $n$-sphere, which is the smooth manifold consisting of all points in $\mathbb R^{n+1}$ with Euclidean norm one. Recall that the tangent bundle $\mathbb S^n$, denoted $T \mathbb S^n$, is a $2n$-dimensional smooth manifold. Let $\Delta \subseteq \mathbb S^n \times \mathbb S^n$ denote the dia... |
I'm currently going over the proof of the uniqueness of the infinite open cluster for the percolation model in Grimmet (Probability on Graphs, http://www.statslab.cam.ac.uk/~grg/books/USpgs-rev3.pdf, p. 92).
There he defines the quantities $S= \{x \in \mathbb{Z}^d: dist(0,x) \leq n\}$ and $N_S(0),N_S(1)$, which are... |
At school we were given the following differential equation to solve:
a) $yy'+x=\frac{1}{2}(\frac{x^2+y^2}{x^2})^2$ and
b) $y'=(\frac{3x+y^3-1}{y})^2$.
What I know now are:
1. DE with separable variables
2. Homogeneous DE of the 1st order
3. Linear DE
4. Bernoulli DE
5. Riccati's DE
6. Lagrange's DE
7. ... |
How can I solve $yy'+x=\frac{1}{2}(\frac{x^2+y^2}{x^2})^2$ and $y'=(\frac{3x+y^3-1}{y})^2$? |
At school we were given the following differential equation to solve:
a) $yy'+x=\frac{1}{2}(\frac{x^2+y^2}{x})^2$ and
b) $y'=(\frac{3x+y^3-1}{y})^2$.
What I know now are:
1. DE with separable variables
2. Homogeneous DE of the 1st order
3. Linear DE
4. Bernoulli DE
5. Riccati's DE
6. Lagrange's DE
7. Cl... |
How can I solve $yy'+x=\frac{1}{2}(\frac{x^2+y^2}{x})^2$ and $y'=(\frac{3x+y^3-1}{y})^2$? |
I'm working on a potential flow problem. I have a box, centered at the origin (i.e. $V=[-x_b,x_b]\times[-y_b,y_b]\times[-z_b,z_b]$) that has inside of it a uniform distribution of source strength. We can assume this distribution has unit strength. The potential induced by the distribution of source strength in the box ... |
How can I solve 1. $yy'+x=\frac{1}{2}(\frac{x^2+y^2}{x})^2$ and 2. $y'=(\frac{3x+y^3-1}{y})^2$? |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
Hi I have a specific problem, that is given a 1-d semi-infinite nonhomogeneous heat equation
$$\frac{\partial^2 T}{\partial x^2}+\frac{1}{k}g(x,t)=\frac{1}{\alpha}\frac{\partial T}{\partial t}, \quad 0<x<\infty, t>0$$
for any given $t$, is there a way to get the location of maximum i.e. $x_{max}$, for ${T_{max}(... |
I was studying coercive functions and I came upon three definitions:
1 - it's the usual, the one I saw most.
$$\lim_{||x||\to+\infty} f(x) = +\infty$$
2 - Given $M>0$, and $r>0$. $f(x)>M$ when $||x||> r$
3 - Someone said it to me, so I dont know if its correct. it's like the second one, but without the const... |
Hope I made no mistake here:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_theorem) )
$ \int_V \nabla \mathbf{F} dV = \oint_S \mathbf{F}\cdot\mathbf{n} d... |
> Let $K_{1}$ and $K_{2}$ be two isomorphic fields. Prove that an isomorphism $f \colon K_{1} \xrightarrow{\sim} K_{2}$ can be extended to an isomorphism $\sigma \colon \overline{K_{1}} \xrightarrow{\sim} \overline{K_{2}}$ between their respective closures.
I intended to do this using Zorn's Lemma (akin to the proo... |
Let $q : W \rightarrow V$ be a regular epimorphism in a category $\mathcal{C}$, and consider the pullback functor $q^* : \mathcal{C}/V \rightarrow \mathcal{C}/W$. Is $q^*$ necessarily essentially injective? That is, does $q^*C \cong q^*D$ imply $C \cong D$?
Context: I'm trying to get a better understanding of Schulm... |
Is the pullback functor $q^*$ essentially injective when $q$ is a regular epimorphism? |
Let $f(n)$ denote $n$th number in the Fibonacci sequence. Find a two digit prime such that $p$ divides $f(n) + f(n+100)$ for all $n$.
Well to be honest I have no idea how to solve this problem so I don't really have anything to show for my attempts.
Well I did try to play around with the recursion a bit but it ende... |
I have some data on a game I'm playing and I'm trying to interpret my results.
Let's say I have a soccer team. Whenever that soccer team had scored 5 goals, they won 100% of their games. Whenever they had scored 4 goals, they also won 100% of their games. 3 goals won 80% of games. 2 goals won 78% of games. 1 goal wo... |
Let $\mathbb \S^n$ denote the $n$-sphere, which is the smooth manifold consisting of all points in $\mathbb R^{n+1}$ with Euclidean norm one. Recall that the tangent bundle of $\mathbb S^n$, denoted $T \mathbb S^n$, is a $2n$-dimensional smooth manifold. Let $\Delta \subseteq \mathbb S^n \times \mathbb S^n$ denote the ... |
i am trying to find the coefficients to a general solution for non-linear, autonomous second order ordinary differential equation, i have ended up with a general solution that has 4 integration constants. does this mean that i need 4 initial conditions? and if so which? i know that i can already use the initial conditi... |
Can't you just say $F(x_1,\ldots, x_m) = 0$ $\implies$ $F(x_1, \ldots, x_m) \equiv_m 0$ $\forall m \in \mathbb{N}$, as $0 \equiv_m 0$ for all such $m$? |
No, that's not the right interpretation.
In a loose way, one could even say it's quite the opposite, since the sub-lattice of $L$ generated by $\{a,x,y\}$ is the smallest sub-lattice of $L$ which contains $a,x,y$, in the sense that if $K \leq L$ and $a,x,y \in L$, then the sub-lattice of $L$ generated by $\{a,x,y\}... |
Let $f: X \to \mathbb{R}$ be some function defined on a convex set $X \subset \mathbb{R}^n$. Now assume that for all $x \in X$ and $y \in \mathbb{R}^n$ such that $x+ty \in X$ for all $t \in [0,1]$, the function $f(x+ty)$ is a convex function on $[0,1]$. Does that mean $f$ is convex? To me the standard definition of con... |
I was studying coercive functions and I came upon three definitions:
$$f: R^n \rightarrow R$$
$$M \in R, r \in R$$
1 - it's the usual, the one I saw most.
$$\lim_{||x||\to+\infty} f(x) = +\infty$$
2 - Given $M>0$, and $r>0$. $f(x)>M$ when $||x||> r$.
3 - Someone said it to me, so I dont know if its correct.... |
Can't you just say $F(x_1,\ldots, x_n) = 0$ $\implies$ $F(x_1, \ldots, x_n) \equiv_m 0$ $\forall m \in \mathbb{N}$, as $0 \equiv_m 0$ for all such $m$? |
What are the conditions for a function $f\colon X \to Y$ to be regular when $X$ and $Y$ are prevarieties? |
I want to know if there exist infinitely many irreducible elements in an infinite commutative domain with unit. In other words, is the following statement true?
> Let $R$ be an infinite commutative domain with unit.
Then, $\# \{ a \in R \mid a$ is an irreducible element of $R \} = \infty$
This question is not an... |
Preface
=======
Consider a plane curve defined by parametric equations (for $t_1\le t \le t_2$):
$$x=x\left(t\right)$$
$$y=y\left(t\right)$$
In addition, there is a scalar function $f\left(x,y\right)$ defined for every value of $x$ and $y$.
Giving an example to illustrate (in this example, $t$ is the cent... |
Prove that the division rings $\mathbb{C}((z;\sigma))$ and $\mathbb{H}((x))$ are not isomorphic. The following statement is needed:
\begin{equation}
\text{any ring automorphism $\phi: \mathbb{R}((x)) \rightarrow \mathbb{R}((x))$ has the form:}
\end{equation}
\begin{equation}
\forall a \in \mathbb{R}, \phi(a)=a;
\... |
I was studying coercive functions and I came upon three definitions:
$$f: R^n \rightarrow R$$
$$M \in R, r \in R$$
1 - it's the usual, the one I saw most.
$$\lim_{||x||\to+\infty} f(x) = +\infty$$
2 - For all $M>0$, exist an $r>0$ such that $f(x)>M$ when $||x||> r$.
3 - Someone said it to me, so I dont know... |
I was studying coercive functions and I came upon three definitions:
$$f: R^n \rightarrow R$$
$$M \in R, r \in R$$
1 - it's the usual, the one I saw most.
$$\lim_{||x||\to+\infty} f(x) = +\infty$$
2 - For all $M>0$, exist an $r>0$ such that $f(x)>M$ when $||x||> r$.
3 - Someone said it to me, so I dont know... |
I am solving for a particular situation. A two vehicles approach an intersection and are on a collision course. To avoid the collision, one vehicle needs to apply the brakes such that it slows to allow the other vehicle to cross. There are two variables that control whether the application of brakes will be sufficient ... |
I was studying coercive functions and I came upon three definitions:
$$f: \mathbb{R}^n \rightarrow \mathbb{R}$$
$$M \in \mathbb{R}, r \in \mathbb{R}$$
1 - it's the usual, the one I saw most.
$$\lim_{||x||\to+\infty} f(x) = +\infty$$
2 - For all $M>0$, exist an $r>0$ such that $f(x)>M$ when $||x||> r$.
3 - S... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
Hi I have a specific problem, that is given a 1-d semi-infinite nonhomogeneous heat equation
$$\frac{\partial^2 T}{\partial x^2}+\frac{1}{k}g(x,t)=\frac{1}{\alpha}\frac{\partial T}{\partial t}, \quad 0<x<\infty, t>0$$
for any given $t$, is there a way to get the location of maximum i.e. $x_{max}$, for ${T_{max}(... |
I am trying to evaluate the trigonometric Fourier series expansions of three quasi-square waves, which are essentially the same quasi-square waveform, phase-shifted with respect to each other by multiples of $\frac{2\pi}{3}$. The three waveforms are as shown:
[Phase-shifted quasi-square waveforms][1]
I have evalu... |
$(-.001)^{-.001} \approx 1.007 - .0031634i$, $(-.000001)^{-.000001} \approx 1.000014 - .00000314164$, and $(-.000000001)^{-.000000001} \approx 1.0000000207 - .00000000314159271i$. Notice that as we approach 0 from the negative direction for $x^x$ by magnitudes of ten, the imaginary component more closely approximates ... |
Why does Im($(-1/10^n)^{-1/10^n}$) turn into the digits of pi as integer n gets larger? |
Recently I was confronted with another tournament design problem:
Suppose we have a tournament with $2n$ teams ($n\in \mathbb{N}$). We have $n$ different types of games (say at $n$ distinct locations), to be played by a pair of teams at a time. The tournament will be played in $2n$ rounds. In the first $n$ rounds as... |
Hope I made no mistake here:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_theorem) )
$ \int_V \nabla \mathbf{F} dV = \oint_S \mathbf{F}\cdot\mathbf{n} d... |
Let us consider Fibonacci sequence modulo $41$. One can look up [the Pisano period](https://en.wikipedia.org/wiki/Pisano_period) of $41$, it is equal to $40$, but actually (the $0$-th and then) every $20$-th Fibonacci number is divisible by $41$. Here are Fibonacci numbers modulo $41$:
$$0, 1,1,2,3,5,8,13,21,34,14,7... |
I am guessing this can be argued by the monotonicity of the $f_n$, but is there a reasonable way to do it directly? That is, given $\epsilon > 0$ to find $N$ such that if $n>N$ then $|f_n(x)| < \epsilon$ for all $x \in [0,1]$.
I tried using $f_n(x) = x(x^{\frac{-1}{n}}-1) \leq (x^{\frac{-1}{n}}-1)$ but then the con... |
I understand that the probability of absorbing eventually is one in a simple random walk with a single absorbing boundary.
However, if we depart from this simple random and consider biased random walks with unequal step sizes, the absorbing probability may not always be one. Specifically, let $S_n = X_1 + ... + X_n$... |
While creating an answer key for my Calc 2 students and trying to save a bit of time, I plugged a trigonometric integral into Mathematica and was a little confused about how it approached the problem. The integral was
$$
\int\sin^5x\cos^7x \,dx
$$
The typical way to evaluate this integral would be to use the Pyth... |
Consider a 2-dimensional rectangular plane sheet of paper $S \subset \mathbb{R}^2$, then push both small sides towards the middle to form a Gaussian-like surface, that we call $D$, see the figure below for only one wave. Unless I am mistaken, this is a developable surface since its Gaussian curvature is everywhere zero... |
Hope I made no mistake here:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_theorem) )
$ \int_V \nabla \mathbf{F} dV = \oint_S \mathbf{F}\cdot\mathbf{n} d... |
Let $I=[a,b]$ be an intervall and $x_o \in I$ fixed. I'm trying to show that for any $\epsilon>0$ exists a continously differentiable function $f$ such that $|f'(x_0)| \geq \epsilon \||f\||_{\infty}$.
Can someone give me a hint on how to construct such function? I'm obviously missing something here.
|
While creating an answer key for my Calc 2 students and trying to save a bit of time, I plugged a trigonometric integral into Mathematica and was a little confused about how it approached the problem. The integral was
$$
\int\sin^5x\cos^7x \,dx
$$
The typical way to evaluate this integral would be to use the Pyth... |
Let $\mathbb S^n$ denote the $n$-sphere, which is the smooth manifold consisting of all points in $\mathbb R^{n+1}$ with Euclidean norm one. Recall that the tangent bundle of $\mathbb S^n$, denoted $T \mathbb S^n$, is a $2n$-dimensional smooth manifold. Let $\Delta \subseteq \mathbb S^n \times \mathbb S^n$ denote the d... |
Lately I have been reading about the cut elimination theorem, I think I get the idea however I have been struggling with some technical details concerning quantifiers.
Consider the following rule:
[![enter image description here][1]][1]
Now in the proof of the cut elimination theorem we are using mixes inste... |
I am fairly new to proof writing so I would appreciate if you could help me out!
> Suppose that $(X^*, Y^*)$ and $(X^0, Y^0)$are saddle points of a zero-sum game with payoff matrix $A$. Show that $(X^*, Y^0) and (X^0, Y^*)$ are also saddle points.
This is what I know: If $(X^*, Y^*)$ is a saddle point for a zero... |
Pick **any** absolutely integrable function $f(x)$ such that $\int dx f(x)=1 $. [Then we have an approximation to the identity](https://en.wikipedia.org/wiki/Dirac_delta_function?useskin=vector#nascent_delta_function)
$$
\frac{1}{\epsilon}f\left(\frac{x}{\epsilon}\right) \to \delta (x) \qquad ,\qquad \epsilon \to ... |
Does anyone know how it is possible to lose the n element in the index of the summation, while remaining equal ?
$$
\sum_{k=0}^{n} \binom{n}{k}q^kp^{n-k}(n-k)^1 = np\sum_{k=0}^{n-1} \binom{n-1}{k}q^kp^{n-k-1}
$$
I understand the following steps
$$
\sum_{k=0}^{n} \frac{n!}{k!(n-k)!}q^kp^{n-k+1-1}(n-k)^1
$$
... |
I would like to calculate the following expectations of the output of a Neural network. I make the hypothesis that the $E[\beta_i] = 0$ and also I am assuming that the activation $h_j$ is independent of the weights $W_{ij}$. I could write the expectation of the output as follows:
$$E[(\beta_i + \sum_{j=1}^{D_h} W_{i... |
Let $I=[a,b]$ be an interval and $x_0 \in I$ fixed. I'm trying to show that for any $\epsilon>0$ exists a continuously differentiable function $f$ such that $|f'(x_0)| \geq \epsilon \| f \|_{\infty}$.
Can someone give me a hint on how to construct such function? I'm obviously missing something here.
|
I have to look for a formula which gives the number of i-dimensional singular ubspaces of a polar space of order (s,t) and rank n. My idea was to prove this by induction on i. So I wanted to start with calculating the number of points. Here my idea was to use a double counting on the pairs (point, line through that poi... |
This is a follow up to https://math.stackexchange.com/questions/2978806/a-martingale-converging-in-distribution-but-not-a-s-or-in-probability?noredirect=1&lq=1. Suppose we have a sequence of integer-valued RVs that satisfies
(1) $P(X_n=a~i.o.)=1,a=-1,0,1$
(2) $\sup_n|X_n(w)|<\infty$ a.s.
(3) For some $p\in(0,1... |
Why does these three conditions imply divergence in probability? |
What exactly is the torus $(\mathbb{Z}/n\mathbb{Z})^d$?
I know that $\mathbb{Z}/n\mathbb{Z}$ is the set of equivalence classes: $\mathbb{Z}, \mathbb{Z}+1, \mathbb{Z}+2, \ldots, \mathbb{Z}+(n-1)$. I also know that a simple torus is $S^{1} X S^{1}}$ that is the cartesian product of 2 circles equipped with the product ... |
What exactly is the torus (Z/nZ)^d? |
What exactly is the torus $(\mathbb{Z}/n\mathbb{Z})^d$?
I know that $\mathbb{Z}/n\mathbb{Z}$ is the set of equivalence classes: $\mathbb{Z}, \mathbb{Z}+1, \mathbb{Z}+2, \ldots, \mathbb{Z}+(n-1)$. I also know that a simple torus is $S^{1} X S^{1}$ that is the cartesian product of 2 circles equipped with the product t... |
Hope I made no mistake here:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_theorem) )
$ \int_V \nabla \mathbf{F} dV = \oint_S \mathbf{F}\cdot\mathbf{n} d... |
**Is there a set $S$ that is definable in ZFC, but for which we know neither the aleph nor beth number?**
For example, we do not know the aleph number of $|\mathbb R|$, but we know its beth number, $|\mathbb R| = \beth_1$. Conversely, we trivially know the aleph number of $\aleph_1$, but we do not know its beth num... |
Are there sets whose aleph and beth numbers are both unknown? |
I am interested in analyzing the stability of the periodic orbits resulting from the Van der Pol system periodically perturbed by a time-dependent external forcing. Mathematically it would be the following system of nonlinear differential equations
$\begin{equation} \left \{
\begin{aligned}
\frac{dx}{dt} & = \mu... |
How to calculate the monodromy matrix for a system of nonautonomous nonlinear differential equations? |
I was reading about Many-sorted logic and I kept seeing a lot of authors claiming that "When there are only finitely many sorts in a theory, many-sorted first-order logic can be reduced to single-sorted first-order logic".
I get that this is done by introducing, for every sort $a$, a unary predicate symbol $P_a$ and... |
How to interpret functional symbols in a many-sorted language in the corresponding single-sorted language? |
I am trying to prove the following quoted statement.
> **Theorem 1.** Let $r_1,...,r_n$ be *real* algebraic integers over $\Bbb Q$ with minimal polynomials $x^{d_1}-a_1,...,x^{d_n}-a_n$, respectively. Then the equality $$\left[\Bbb Q[r_1,...,r_n]:\Bbb Q\right]=\prod_{i=1}^nd_i$$holds if and only if for all $s_i\in\... |
Given $\mathcal{O}_X$-modules $\mathcal{M}$ and $\mathcal{M}$, does the sheaf $Isom$ of isomorphisms of $\mathcal{O}_X$-modules, defined as
$$U\mapsto \mathrm{Isom}_{\mathcal{O}_{X|U}}(\mathcal{M}|_U, \mathcal{N}|_U)$$
exist? Is is a subsheaf of the sheaf Hom? |
**Is there a set $S$ that is definable in ZFC and known to be infinite, but for which we know neither the aleph nor beth number?**
For example, we do not know the aleph number of $|\mathbb R|$, but we know its beth number, $|\mathbb R| = \beth_1$. Conversely, we trivially know the aleph number of $\aleph_1$, but we... |
Are there infinite sets whose aleph and beth numbers are both unknown? |
A Fixed Point iteration is given by,
\begin{equation}
x_{k+1} = T(x_k) + q
\end{equation}
By Banach Contraction theorem, if $T$ is a contraction, then the above iteration convergent.
In particular, if $T$ is a finite dimensional linear opearator such that $\| T \|< 1$, the iteration converges linearly i.e,
$$ \... |
I'm currently enrolled in Calculus 1, and everything has been pretty smooth up until these last two sections involving the chain rule and implicit derivation.
After watching multiple YouTube videos, and reading different professors' explanations, all of these resources either explain it in a different way, or use dif... |
The following argument is from a column by Peter Winkler, attributed to Ehud Friedgut; it may be equivalent to Graham and Kleitman's argument, but is easier to read. It works in the general case of a graph $G$ with $n$ vertices and $m$ edges, where the edges are labeled with distinct weights.
Put a person on each ve... |
**Exercise.**
Suppose $U$ is a subspace of $V$ such that $V/U$ is finite-dimensional. Prove that $V$ is isomorphic to $U \times (V/U).$
----------
**Outline of my proof.**
We need to construct a linear bijection between $V$ and $U \times (V/U)$, where $U \times (V/U) = \{(u, v+U) : u \in U, v \in V\}$.
I... |
I showed $V$ is isomorphic to $U \times (V/U)$ without using the assumption that $V/U$ is finite-dimensional. What did I do wrong? |
An [Alexandrov topology](https://en.wikipedia.org/wiki/Alexandrov_topology) on a set $X$ is a topology in which arbitrary intersections of open sets are open. Equivalently, every point has a smallest open neighborhood.
Given a partition on a set $X$, one can form the corresponding [partition topology](https://en.wi... |
Is every regular Alexandrov topology a partition topology? |
Here is something I've found on the internet
$$\begin{aligned}
f-\int f&=1\\
\left(1-\int\right)f&=1\\
f&=\left(\frac1{1-\int}\right)1\\
&=\left(1+\int+\int\int+\dots\right)1\\
&=1+\int1+\int\int1+\dots\\
&= 1+x+\frac{x^2}2+\dots\\
&= e^x
\end{aligned}$$
At first I interpreted this as a joke, but on closer in... |
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