instruction stringlengths 12 30k |
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Is geodesic distance locally approximated by euclidean distance? |
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power series will have a closed form in terms of well established functions, but also I have no way to know if that is the c... |
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass theorem (BWT)**.
I've been trying to prove the IVT **without the lub property, nor the Archimedean property**, using the BWT as an axiom. The sketch of my pr... |
[enter image description here][1]
[1]: https://i.stack.imgur.com/SmXkC.png
May I ask how to approach this kind of question ? |
A *semialgebra* $\mathcal{S}$ is a nonempty family of a set $X$ satisfying the properties
$1.$ $E,F\in\mathcal{S}\implies E \cap F\in \mathcal{S}$
$2.$ If $E\in\mathcal{S}$ then there exists pairwise disjoint sets $F_1,\dots, F_n\in\mathcal{S}$ such that $$X\setminus E =\bigcup_{k=1}^n F_k$$
Now, let $A,B\in\m... |
In an art museum, there are $n$ paintings, $n \ge 33$, for which there are
used a total of $15$ different colors so that any two paintings have at least one common color and there are no two paintings that have exactly the same colors. Determine all possible values of $n \ge 33 $ so that anyway we color the painting... |
To understand the conversion without standard form I read: https://math.stackexchange.com/questions/249413/directly-from-primal-to-dual-when-primal-not-in-standard-form
The posted LP there is:
$$
Primal =\begin{Bmatrix}
max \ \ \ \ 2x_1 - 2x_2 +x_3 + 4x_4 \\
s.t. \ \ \ \ x_1 - x_2 - x_3... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng). We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying the condit... |
We throw a dice, if we throw a 6, then we throw again (any number of times). Let $X$ be the sum of all thrown numbers. Find $\mathbb{E} (X)$.
I know that if we were just throwing without repeating throws then it would be:
$\mathbb{E} (Y) = 1*P(1) + 2*P(2) + 3*P(3) + ... + 6*P(6) = \frac{7}{2}$
Now if we would ... |
Let $\langle x_n\rangle$ be a sequence of real numbers. Let $\langle \phi_n\rangle$ be an enumeration of the Turing machines (in ZFC?). Consider the following two "definitions" of a computable sequence:
> **Correct definition**: We call $\langle x_n\rangle$ computable if there exists a Turing machine that given $n$ ... |
Why do Lagrange's multipliers not catch cthe solution to $\min x^2+y^2$ subject to $(x-1)^3-y^2=0$? |
I'm having a hard time understanding principal bundles. It seems there are a lot of definitions around which is making it even more confusing for me. Is this a valid definition of a principal bundle:
____
## Fiber Bundle
A fiber bundle is a tuple $(E, B, \pi, F)$ where $E, B, F$ are topological spaces and $\pi: ... |
Is this is valid/standard definition of a principal bundle? |
> Let $w$ be a root of the Complex equation $z^8=1$. Find the value(s) of $$(w^{-3}+w^{-2}+w^{-1}+1+w^1+w^2+w^3+w^4)^\frac{1}{2}$$
May I ask how to approach this kind of question ? |
as title, is there a way to find the minimum value of $x^4-x$ without using calculus?
By calculus it's easy as $(x^4-x)'=4x^3-1$, so we got $x=\frac1{\sqrt[3]4}$, then we get the min value. But without calculus, is it possible?
I tried to use AM-GM but to no avail... |
My question comes from section 1.6 of *Complex Analysis* (4th edition) by Serge Lang: <br/>
Let $U$ be an open subset of $\mathbb{C}$ and let
\begin{align*}
f(x + iy) = u(x,y) + i v(x,y), \qquad x + iy \in U
\end{align*}
be a complex-valued function on $U$. Lang argues that if $f$ is holomorphic at some po... |
Assumptions
---
Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species.
Conjecture $2$: All members of a species influence each other and each influence and member is unique
Conjecture $3$: There is only one dominant i... |
Probability in the logistics equation? |
My question comes from section 1.6 of *Complex Analysis* (4th edition) by Serge Lang: <br/>
Let $U$ be an open subset of $\mathbb{C}$ and let
\begin{align*}
f(x + iy) = u(x,y) + i v(x,y), \qquad x + iy \in U
\end{align*}
be a complex-valued function on $U$. Lang argues that if $f$ is holomorphic at some po... |
During a class assignment, I was presented with the following question:
Provide an integer program that has an exponential number of branches...(expunged excess)
There was more to the question, but I'm primarily concerned and curious about finding more integer programs than I the ones I found that fit this cr... |
Does there exist other integer models that contain an exponential number of branches thats not knapsack for the branch-and-bound method? |
Is "If ⊨□φ then ⊨φ" true? |
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power series will have a closed form in terms of well established functions, but also I have no way to know if that is the c... |
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power series will have a closed form in terms of well established functions, but also I have no way to know if that is the c... |
During a class assignment, I was presented with the following question:
Provide an integer program that has an exponential number of branches...(expunged excess)
There was more to the question, but I'm primarily concerned and curious about finding more integer programs than I the ones I found that fit this cr... |
Assumptions for reaching the [logistic equation][1]
---
Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species.
Conjecture $2$: All members of a species influence each other and each influence and member is unique
Conj... |
I'm studying general topology and a question has come to my mind.
I am referring to the class of theorems that in algebra go by the name of "homomorphism theorems". In my topology course, we have seen some results alike, but it lacks the general result on when, given two continuous maps with the same domain, there e... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass theorem (BWT)**.
I've been trying to prove the IVT **without the lub property, nor the Archimedean property**, using the BWT as an axiom. The sketch of my pr... |
During a class assignment, I was presented with the following question:
Provide an integer program that has an exponential number of branches...(expunged excess)
There was more to the question, but I'm primarily concerned and curious about finding more integer programs than I the ones I found that fit this cr... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass theorem (BWT)**.
I've been trying to prove the IVT **without the lub property, nor the Archimedean property**, using the BWT as an axiom. The sketch of my pr... |
You might also try [Hussemoller's book "Fiber Bundles"](https://books.google.com/books/about/Fibre_Bundles.html?id=DPr_BSH89cAC&source=kp_book_description), Chapter 2, Corollary 4.8.
Usually one proves, in the same breath, that if $f,g : X \to Y$ are two homotopic maps and if $B$ is a vector bundle over $Y$ then $f... |
The question is to compute the Fourier Series of the function
$$h(x) = sin(\frac{x}{4})$$ on $[-\pi, \pi]$
My question is I know normally, we can use the following formula to compute $a_0, a_k, b_k$
$$a_0 = \frac{1}{L} \int_{-L}^{L}f(x)dx$$
$$a_k = \frac{1}{L} \int_{-L}^{L}f(x)cos(\frac{k\pi x}{L}) dx$$
$$b_... |
Fourier series? |
## The Problem
The [gradient descent](https://en.wikipedia.org/wiki/Gradient_descent) algorithm finds a minimum of a convex function, but it does not guarantee that the found minimum is the global one. I don't know if it's possible to find a minimum of a function by its derivative (assuming no, then what would be th... |
How to find the global minimum of a convex function? |
Going through a fluid mechanics book, I encountered this expression: $$\nabla \cdot (uu^T) = u \cdot \nabla u, $$ whereby $u$ is supposed to be the velocity vector field. I can not make sense of this. Considering $u$ as a vector, $uu^T$ will be a 3 by 3 matrix. For the left-hand side of the equation above I took the di... |
The question is to compute the Fourier Series of the function
$$h(x) = \sin\left(\frac{x}{4}\right)$$ on $[-\pi, \pi]$
My question is I know normally, we can use the following formula to compute $a_0, a_k, b_k$
$$a_0 = \frac{1}{L} \int_{-L}^{L}f(x)dx$$
$$a_k = \frac{1}{L} \int_{-L}^{L}f(x)\cos\left(\frac{k\pi... |
Is "If ⊨□φ then ⊨φ" true in modal logic? |
Have a look at Fig. 1 modelizing the position of the extremity of a string rolled/unrolled along a convex polygon (here the particular case of a regular octagon) generating a kind of spiral (modelizing rather well a half-egg-shape) ?
[![enter image description here][1]][1]
Fig. 1.
But one can obtain much g... |
I'm studying general topology and a question has come to my mind.
I am referring to the class of theorems that in algebra go by the name of "homomorphism theorems". In my topology course, we have seen some results alike, but it lacks the general result on when, given two continuous maps with the same domain, there e... |
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power series will have a closed form in terms of well established functions, but also I have no way to know if that is the c... |
How can I make the images that appear in Morse theory?. Type https://math.stackexchange.com/questions/3980682/beginners-question-about-homotopy-type-in-milnors-morse-theory, i try in geogebra but i cannot. |
Note that
\begin{aligned}
\sin(z-w) &= \frac{e^{i(z-w)}-e^{-i(z-w)}}{2i} \\
&= \frac{e^{iz}e^{-iw}-e^{-iz}e^{iw}+e^{iz}e^{iw}-e^{-iz}e^{-iw}}{2i} \\
&= \frac{e^{iw}(e^{iz}-e^{-iz})}{2i}-\frac{e^{iz}(e^{iw}-e^{-iw})}{2i} \\
&= e^{iw}\sin(z)-e^{iz}\sin(w) \\
&= 2(\cos(w) \sin(z)-\cos(z) \sin(w))+e^{-iz}\sin(w)-e^{i... |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be Lipschitz continuous and $g \in C^\infty_c(\mathbb{R})$ (i.e. smooth with compact support). Is the composition $g \circ f$ Lipschitz?
I have tried proving this myself but I'm not sure how to relate $g(f(x))$ to $g(f(y))$ to show that $$|g(f(x)) - g(f(y))| \leq C |x-y|.$... |
If $f$ is Lipschitz and $g \in C^\infty_c(\mathbb{R})$, is $g \circ f$ Lipschitz? |
Going through a fluid mechanics book, I encountered this expression: $$\nabla \cdot (uu^T) = u \cdot \nabla u, $$ whereby $u$ is supposed to be the velocity vector field. I can not make sense of this. Considering $u$ as a vector, $uu^T$ will be a 3 by 3 matrix. For the left-hand side of the equation above I took the di... |
I'm studying general topology and a question has come to my mind.
I am referring to the class of theorems that in algebra go by the name of "homomorphism theorems". In my topology course, we have seen some results alike, but it lacks the general result on when, given two continuous maps with the same domain, there e... |
**Background**
I am trying to understand different notions of aperiodicity in graphs, and how these concepts relate to <a href="https://en.wikipedia.org/wiki/Aperiodic_tiling">aperiodic tilings</a>. There is a different description of aperiodicity in graphs <a href="https://en.wikipedia.org/wiki/Aperiodic_tiling">o... |
Compute $\int_C (y+z,z+x,x+y) d\vec{r}$, where $C$ is the intersection of the cylinder $x^2 +y^2 = 2y$ and the plane $y = z$.
Is is true that all I can do is apply Stokes Theorem: since $C$ is a closed curve, I need to choose a surface $S$ bounded by $C$ such that orientations of $C$ and $S$ are compatible? Since th... |
>Investigate whether the polynomial $q(x) = 2x^5 - 78x^3 + 39x + 21$ is irreducible in $\mathbb{F}_{13}[x]$.
**Solution**: In $\mathbb{F}_{13}[x]$, $q(x) = 2x^5 + 8 = 2(x^5 + 4)$. This polynomial has a root in $\mathbb{F}_{13}:
p(a) = 0 \in \mathbb{F}_{13} $ if and only if $a^5 = -4 \in \mathbb{F}_{13}$.
Since $... |
this is character function in probability theory
$$\phi(u)=\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}ux}f(x)\mathrm{d}x$$
Let an asset price $S_t$ (e.g. a stock) be modeled with a Geometric Brownian motion:
$$\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t$$
where $W_t$ is a Wiener process, $r$ the... |
For a cylinder, the volume formula is given by $V = \pi r^2 H $. To find the differential \( dV \), we differentiate \( V \) with respect to \( r \) and \( H \), yielding $\left( dV = 2 \pi r H dr + \pi r^2 dH \right)$. Upon integrating \( dV \) again, we obtain $( 2 \pi r^2 H )$. However, I'm confused about where the ... |
Consider for the short exact sequence (s.e.s)
\begin{align*}
0 \to \mathbf{Z} \xrightarrow{\cdot 2} \mathbf{Z} \xrightarrow{\pi} \mathbf{Z}/2\mathbf{Z} \to 0
\end{align*}
the covariant left exact functor $\text{Hom}(\mathbf{Z}/2\mathbf{Z}, -): \text{Ab} \to \text{Ab}$. I want to exhibit the long exact sequence asso... |
I must find the Laurent series for $f(z) = \frac{e^z}{z^2}$ in powers of $z$ for the annulus $ |z| > 0$.
I wrote $f(z) = \frac{1}{z^2} \sum_{n=0}^{\infty} \frac{z^n}{n!} = \sum_{n=0}^{\infty} \frac{z^{n-2}}{n!} = \sum_{n=-2}^{\infty} \frac{z^{n}}{(n+2)!} $
Is this correct? I didn't apply the annulus being $z > 0... |
Let $A = \mathbb{R}^\mathbb{N}$ with the product topology. Let $B = \{ (x_n)_n \in A \mid \forall n : |x_n| \leq 1\} = [-1, 1]^\mathbb{N}$ be the $\infty$-dimensional cube. Let $C = \{ (x_n)_n \in A \mid \sum_{n=1}^\infty |x_n| \leq 1 \}$ be the $\infty$-dimensional cross-polytope. Let $\operatorname{d}B = \{(x_n)_n \i... |
let $\mathbb{Z}[M]$ be 'the' monoid ring of $M$ over $\mathbb{Z}$; that is to say (if my understanding is right) the set of finite linear combinations of elements of $M$, with product given by $\sum_n i_n\mathbf{a}_n\times\sum_m j_m\mathbf{b}_m$ $= \sum_{m, n}i_nj_m(\mathbf{a}_n\times\mathbf{b}_m)$. I'm somewhat famili... |
The 'union of factors' comultiplication in a monoid ring? |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying... |
Let's call an *algebraic invariant* of a tensor of order N to an algebraic combination of the components of that tensor. In other words, a homogeneous polynomial in the components of the tensor.
* For $N = 2$ algebraic invariants are well known: given a second-order tensor, defined over vector spaces of dimension $n$... |
I have an assignment to find the number of operations (only multiplications and divisions) needed to solve $N \times N$ linear equation while using Jordan Elimination method. I thought about it in two different ways but I'm not sure if my work is right.
Suppose that that the augumented matrix where I need to apply t... |
There are other good answers here, I wanted to offer a perspective that I think might appeal to a computer scientist: in a proof by induction, the inductive step is often presented as a way to build up $P(n+1)$ from $P(n)$, but it is often better to think of the inductive step as breaking $P(n+1)$ down into smaller, mo... |
Good people!
So I'm trying to work my way through Ernst Kummer's *De Numeris Complexis*, and I've reached a point where I keep stumbling over something that should be very, very simple. After almost an hour of playing around with this, I have been unable to solve it, and so appeal to the community for help.
The b... |
Beware of not confusing a theorem with a metatheorem:
$$\Box\psi\rightarrow\psi$$
is not a theorem of $\mathbf{K}$, 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 noticing that $\Box\psi$ sa... |
The claim is that it decomposes into two smaller polynomials neither of which is a unit, as a factor $x - a$ is not a unit in $\mathbf{F}_{13}[x]$. To answer your second question, think about what happens when we take $k$-th powers of elements in $\mathbf{F}_{13}^*$ with $\text{gcd}(k, 12) = 1$. Hint: when do wo have t... |
Note that
\begin{aligned}
\sin(z-w) &= \frac{e^{i(z-w)}-e^{-i(z-w)}}{2i} \\
&= \frac{e^{iz}e^{-iw}-e^{-iz}e^{iw}+e^{iz}e^{iw}-e^{-iz}e^{-iw}}{2i} \\
&= \frac{e^{iw}(e^{iz}-e^{-iz})}{2i}-\frac{e^{iz}(e^{iw}-e^{-iw})}{2i} \\
&= e^{iw}\sin(z)-e^{iz}\sin(w) \\
&= 2(\cos(w) \sin(z)-\cos(z) \sin(w))+e^{-iz}\sin(w)-e^{i... |
I understand the notion of Linearity typically applied to define Linear Programs (here I will capitalize "Linear" when and only when I use it in this sense). In contrast, this question is focused on the notion of linearity applied in math more broadly, with an eye toward scaling and adding together multiple Mathematic... |
Beware of not confusing a theorem with a metatheorem:
$$\Box\psi\rightarrow\psi$$
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 $\mat... |
So I'm trying to work my way through Ernst Kummer's *De Numeris Complexis*, and I've reached a point where I keep stumbling over something that should be very, very simple. After almost an hour of playing around with this, I have been unable to solve it, and so appeal to the community for help.
The basic proposition... |
So I'm trying to work my way through Ernst Kummer's *De Numeris Complexis*, and I've reached a point where I keep stumbling over something that should be very, very simple. After almost an hour of playing around with this, I have been unable to solve it, and so appeal to the community for help.
The basic proposition... |
$
\DeclareMathOperator{\init}{in}
\DeclareMathOperator{\gr}{gr}
\newcommand{\calO}{\mathcal{O}}
$Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\gr_{\mathfrak{m}}(R)=\bigoplus_{i=0}^\infty\mathfrak{m}^i/\mathfrak{m}^{i+1}$ be the associated graded ring. For $f\in R$, the initial form of $f$ is $\init(f)=f+\... |
Note that
\begin{aligned}
\sin(z-w) &= \frac{e^{i(z-w)}-e^{-i(z-w)}}{2i} \\
&= \frac{e^{iz}e^{-iw}-e^{-iz}e^{iw}+e^{iz}e^{iw}-e^{iz}e^{iw}}{2i} \\
&= \frac{e^{iw}(e^{iz}-e^{-iz})}{2i}-\frac{e^{iz}(e^{iw}-e^{-iw})}{2i} \\
&= e^{iw}\sin(z)-e^{iz}\sin(w) \\
&= 2(\cos(w) \sin(z)-\cos(z) \sin(w))+e^{-iz}\sin(w)-e^{iw}... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass theorem (BWT)**.
I've been trying to prove the IVT **without the lub property, nor the Archimedean property**, using the BWT as an axiom. The sketch of my pr... |
Are Fenchel and Lagrangian duals equivalent? |
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... |
I'm having a hard time understanding principal bundles. It seems there are a lot of definitions around which is making it even more confusing for me. Is this a valid definition of a principal bundle:
____
## Fiber Bundle
A fiber bundle is a tuple $(E, B, \pi, F)$ where $E, B, F$ are topological spaces and $\pi: ... |
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... |
In a textbook that I am reading the author defines a differential equation for values of $t>0$ and an initial condition for $t=0$. As an example consider
$$
y'(t) = 0 \enspace\text{for all } t\in\left]0,\infty\right[, \quad y(0)=1.
$$
I don't fully understand how *solutions* to differential equations are even defin... |
Let $f:\mathbb{R}^{m\times n}\rightarrow \mathbb {R}$ be a function that takes an $m\times n$ matrix $X$ and maps it to the real line. Suppose that the derivative of $f$ with respect to one element $X_{ij}$ is
$$
\frac{df}{dX_{ij}}=a^\top XE1_jb_i
$$
where $a$ is an $n\times 1$ vector, $E$ is an $m\times m$ positi... |
Assumptions for reaching the [logistic map equation][1]
---
Conjecture $1$: Assumption for a species genetically/mimetically/etc there "life" prefers the state found in the maximum population for a species.
Conjecture $2$: All members of a species influence each other and each influence and member is unique
... |
$$\frac{dB}{dS} = -\tau N = \tau(-N)$$
$$\frac{dT}{dS} = \kappa N$$
where $\tau$ is the torsion and $\kappa$ is the curvature.
$\frac{dT}{dS}$, by convention, is defined to be in the direction of $N$, and hence $\kappa$ is always positive. This also means that $\kappa$ is the magnitude of $\frac{dT}{dS}$.
But... |
Torsion for a 3D curve can be positive or negative, while curvature is always taken to be positive. Why? |
Let $f:\mathbb{R}^{m\times n}\rightarrow \mathbb {R}$ be a function that takes an $m\times n$ matrix $X$ and maps it to the real line. Suppose that the derivative of $f$ with respect to one element $X_{ij}$ is
$$
\frac{df}{dX_{ij}}=a^\top XE1_jb_i
$$
where $a$ is an $n\times 1$ vector, $E$ is an $m\times m$ positi... |
Let $\delta$ be an interval in $\mathbb{R}$. Recall that a function $f$ is called Lipschitz continuous on $\delta$ with Lipschitz constant $L$ if there holds
$|f(x) - f(y)| \leq L|x-y|$ for all $x,y$ in $\delta$.
(a) Show that the composition of Lipschitz continuous functions is again Lipschitz continuous.
(b) ... |
I'm trying to solve the 2D heat equation in cartesian coordinates with initial condition of the form $1/r$. The problem is as follows:
$$
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right), & x \in \mathbb{R}, ... |
Let $f:\mathbb{R}^{n\times m}\rightarrow \mathbb {R}$ be a function that takes an $n\times m$ matrix $X$ and maps it to the real line. Suppose that the derivative of $f$ with respect to one element $X_{ij}$ is
$$
\frac{df}{dX_{ij}}=a^\top XE1_jb_i
$$
where $a$ is an $n\times 1$ vector, $E$ is an $m\times m$ positi... |
Consider $\mathbb{R}^{1+2}$ with coordinates $\{t, x, y\}$ and Minkowski metric $g = diag(-1,1,1)$. Suppose we have a hyperboloid $x^2 + y^2 - t^2 = 1$ inside the previous Minkowski spacetime. My question is the following: is it always true that two arbitrary points on the hyperboloid are spacelike separated in $\mathb... |
Is causality preserved between two points on an immersed hyperboloid in Minkowski $R^3$? |
How to prove that $ {\displaystyle \sup_{x \in \mathbb R^n, \, r > 0} r^{-\lambda} \int_{B(x,r)} |f_\alpha(y)|^p \, dy < \infty}$? |
As part of a proof of the admissibility of the complex Morelt wavelet, I am trying to show that the following integral is positive and finite
$$
0<\int_0^\infty{\frac{(e^{\sigma \omega}-1)^2e^{-\omega^2}}{\omega}}d\omega<\infty
$$
Where $\sigma > 0$. Is anyone willing to give any guidance, or provide an explicit ... |
As part of a proof of the admissibility of the complex Morelt wavelet, I am trying to show that the following integral is positive and finite
$$
0<\int_0^\infty{\frac{(e^{\sigma \omega}-1)^2e^{-\omega^2}}{\omega}}d\omega<\infty
$$
Where $\sigma > 0$. Is anyone willing to give any guidance, or provide an explicit ... |
As part of a proof of the admissibility of the complex Morelt wavelet, I am trying to show that the following integral is positive and finite
$$
0<\int_0^\infty{\frac{(e^{\sigma \omega}-1)^2e^{-\omega^2}}{\omega}}d\omega<\infty
$$
Where $\sigma > 0$. Is anyone willing to give any guidance, or provide an explicit ... |
What even is a lower and upper triangular matrix in the context of Jacobi method? |
> How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $?
I know this proof has the form of the triangle inequality, but I can't seem to figure it out. This is from Kenneth Ross 17.1 |
How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $? |
The question arose while trying to derive the moment of inertia of solid vs hollow sphere.
In calculating the surface area of sphere, the integral used is
$\int{2\pi r×Rd\theta}$.
In calculating the volume of sphere the integral is $\int{\pi r^2 dx}$.
After reading from some related [questions]:(volume calcu... |
Chapter 6.3 of Bender & Orszag discusses Integration by Parts for Laplace Integrals
$$
I(x) = \int_{a}^{b} f(t) e^{x \phi(t)} \mathrm{d} t \qquad \text{as} \quad x\to \infty
$$
Using integration by parts one finds
\begin{align}
I(x) = \frac{1}{x} \frac{f(b)}{\phi'(b)} e^{x \phi(b)} - \frac{1}{x} \frac{f(a)}{\... |
Algebras, as well as groups, have representations, in a hopefully obvious sense. One way to view Simon's claim is that we are taking a $*$-representation of the $*$-algebra (i.e., algebra with complex-anti-linear involution) $\mathcal A = \mathbb C[A, A^*]/(A^2 = 0, A^* A + A A^* = 1)$—where $A$ and $A^*$ are treated ... |
Algebras, as well as groups, have representations, in a hopefully obvious sense. One way to view Simon's claim is that we are taking a $*$-representation of the $*$-algebra (i.e., algebra with complex-anti-linear involution) $\mathcal A = \mathbb C[A, A^*]/(A^2 = 0, A^* A + A A^* = 1)$—where $A$ and $A^*$ are treated ... |
Take $$ s\frac{\partial^2}{\partial s^2}\phi_s(x)=-x\frac{\partial}{\partial x}\phi_s(x) \tag{1}$$
which is well posed forward in time under the right conditions, $\forall s >0$ and ill posed backward in time where in finite time we get a singularity in the form of some distribution function that can be found by tak... |
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