instruction stringlengths 12 30k |
|---|
Since the weights change in discrete periods, I would use a summation and not an integral.
$\sum_\limits{n=0}^{99} (1000 + 60n)(4\text{ meters})(10 \frac {\text {meters}}{\text {seconds}^2})$
$[(10^5) + \frac {(99)(100)(199)}{6}] (40 \text{Nm})$
Nonetheless, the setup for the integral vs. the summation is iden... |
> **Question:** Let $\xi$ be a real vector bundle over a space $B$. Suppose $\xi$ admits stable almost complex structure, i.e., $\xi\oplus
\epsilon^k$ admits almost complex structure for some $k\geq 0$, where
> $\varepsilon$ denotes a trivial real vector bundle. Is it true that
> every integral Stiefel–Whitney clas... |
This is a good start. You've shown that you have the desired inequality when $A \subseteq B$, so next you should ask yourself "is the case $A \nsubseteq B$ any harder?"
Imagine we start with $A \subseteq B$, and then enlarge $A$ by adding elements from $B^c$ to get a set $A'$ with $A \subseteq A' \nsubseteq B$. Beca... |
Let $(a,b,c)$ be the sides of a triangle and $R$ its circumradius. Is it true that
$$
\left(\frac{a}{R}\right)^{\frac{c}{R}} + \left(\frac{b}{R}\right)^{\frac{c}{R}}
- \left(\frac{c}{R}\right)^{\frac{c}{R}} < \pi
$$
**My progress**: In the special case where at least two of the three sides are equal, I have be... |
The result that I want to prove is the following:
Let's associate to each $x\in \mathbb{R}$ a number $r_{x}>0$. Then there is $M\in \mathbb{N}$ and a sequence $ \left( x_{n} \right)_{n\in \mathbb{N}}\subseteq \mathbb{R} $ such that
$$r_{x_{n}}>\frac{1}{M}, \ \ \forall n\in \mathbb{N}.$$
It is clear that the res... |
This semester I'm taking a course in Linear Programming. While the topic is very interesting, all the applications I can find about this topic seem to be outside of mathematics. What are some applications of Linear Programming to math? (specially in number theory / algebra / algebraic geometry) |
Let $X_1,\cdots,X_n$ be $n$ random variables on the same probability space $(\Omega, F, P)$, all with expectation 0. Define $Y_i=X_i - \mathbb{E}[X_i | X_1, \cdots, X_{i-1}]$. Is it true that for any $X = a_1 X_1 + \cdots + a_n X_n$, there exist $b_1,\cdots,b_n$ such that $X = b_1 Y_1 + \cdots + b_n Y_n$, and vice vers... |
Do $(X_i)$ and $(X_i - \mathbb{E}[X_i | X_{<i}])$ represent the same set of random variables? |
Are there any examples of diffusion PDEs with nonlinear complications, that would possess analytical solutions? |
From the book Discrete Mathematics for Computing 2nd Edition in eBook:
[![enter image description here][1]][1]
I know how to perform the Euclidean Algorithm and GCM(a,b). I am however, deeply confused by this:
$$1 = 415 - 69(421 - 1 \times 415)$$
$$ = 70 \times 415 - 69 \times 421$$
Where does the **70**... |
Number Theory, how on earth do they get to this? |
I have looked at similar questions on here but still cannot reach a conclusion for this particular one.
I have gathered that as n increases, 1/n tends to 0, meaning a - 1/n tends to a.
So the set must be bounded below by a - 1, and bounded above by a.
To prove inf(A) = a - 1, I started by assuming that inf(A) = l ... |
From the book Discrete Mathematics for Computing 2nd Edition in eBook:
[![enter image description here][1]][1]
I know how to perform the Euclidean Algorithm and GCM(a,b). I am however, deeply confused by this:
$$1 = 415 - 69(421 - 1 \times 415)$$
$$ = 70 \times 415 - 69 \times 421$$
Where does the **70**... |
The $s$ and $z$ domains are linked by the expression $z = e^{sT}$, where $s$ is the Laplace variable and $T$ is the sampling period. However, I have found no rigorous method that attempts to derive all the well-developed characteristics for the LTI continuous systems such as absolute stability, relative stability, damp... |
Number Theory: how on earth do they get to this? |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
I am working on limits and asymptotics and I would like to know (or where to look for) a correct definition of functions that are asymptotically equivalent (for $x\to \infty$) to shifted versions of themselves. That is, at least for sufficiently small $\delta$,
$$\lim_{x\to \infty}\frac{f(x)}{f(x+\delta)} = 1,$$
suc... |
The $s$ and $z$ domains are linked by the expression $z = e^{sT}$, where $s$ is the Laplace variable and $T$ is the sampling period. However, I have found no rigorous method that attempts to derive all the well-developed characteristics for the LTI continuous systems such as absolute stability, relative stability, damp... |
I have looked at similar questions on here but still cannot reach a conclusion for this particular one.
I have gathered that as n increases, $\frac1n$ tends to $0$, meaning $a - \frac1n$ tends to $a$.
So the set must be bounded below by $a - 1$, and bounded above by $a$.
To prove $\inf(A) = a - 1$, I started by as... |
Consider a non-zero positive element $a$ of a simple C*-algebra $A$. Suppose it has a sub-projection which is Murray-vN equivalent to an infinite projection, i.e. there exists an infinite projection $h$ s.t. $h\precsim a$. Then is $a$ properly infinite?
Here are the steps and I don't see why exactly each one is true... |
I know there are $n$ zeros in the open ball $B_1(1)$ from Rouché's Theorem. However, how do I show that the order of these zeros are all one? I don't know where the zeros are, and this should be true for any arbitrary $n$. One thing I want to try is to show that $f'(z_0)\neq 0$ for any zero $z_0$, so we have
$$
f'(z)... |
Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry i... |
Is the sum of a strictly concave and a log concave functions, log concave? |
The result that I want to prove is the following:
Let's associate to each $x\in \mathbb{R}$ a number $r_{x}>0$. Then there is $M\in \mathbb{N}$ and a sequence $ \left( x_{n} \right)_{n\in \mathbb{N}}\subseteq \mathbb{R} $ of distincts points of $\mathbb{R}$ such that
$$r_{x_{n}}>\frac{1}{M}, \ \ \forall n\in \math... |
*Question*: How does making a spanning tree solution "strongly feasible" for the Network Simplex method assist in cycling prevention?
To start, I was reading *[An Implementation of Network Simplex Method (Big-M) for solving Minimum Cost Flow Problem][1]* (Hassan and Ghalawingy) where they say on page $4$:
>**Stro... |
How does making a spanning tree solution "strongly feasible" for the Network Simplex method assist in cycling prevention? |
Since the weights change in discrete periods, I would use a summation and not an integral.
$\sum_\limits{n=0}^{99} (1000 + 60n)(4\text{ meters})(10 \frac {\text {meters}}{\text {seconds}^2})$
$[(10^5) + (60)(\frac {(99)(100)}{2})] (40 \text{Nm})$
Nonetheless, the setup for the integral vs. the summation is ide... |
*I am not familiar with functional analysis at all. I apologize beforehand if there is any conceptual mistake. The situation is as follows:
Let H be a real Hilbert space, suppose I have a bounded linear operator $T: H \to H$ such that there exists $m > 0$ satisfies $ \lvert \langle Tx, x \rangle \rvert ≥ m\lVert x... |
> Solve the equation:
> $$4x^4 - 36x^3 + 61x^2 + 90x + 25 = 0.$$
This problem was lifted from
[this Youtube Video](https://www.youtube.com/watch?v=M4bKGwrKGxs). I didn't
particularly like the video's approach, so I used my own approach, which I have never
seen discussed anywhere. To understand the approach... |
The goal is to find $m$ swaps $s_1, s_2, \dots, s_m$ such that any $n$-permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \dots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2),\; s_2=(2,3),\; s... |
From the book Discrete Mathematics for Computing 2nd Edition in eBook:
[![enter image description here][1]][1]
I know how to perform the Euclidean Algorithm and GCM(a,b). I am however, deeply confused by this:
$$1 = 415 - 69(421 - 1 \times 415)$$
$$ = 70 \times 415 - 69 \times 421$$
---
**How is this ex... |
> Solve the equation:
> $$4x^4 - 36x^3 + 61x^2 + 90x + 25 = 0.$$
This problem was lifted from
[this Youtube Video](https://www.youtube.com/watch?v=M4bKGwrKGxs). I didn't
particularly like the video's approach, so I used my own approach, which I have never
seen discussed anywhere. To understand the approach... |
Given two path-connected topological spaces $\mathcal{X},\mathcal{Y}$, consider a function $f: \mathcal{X} \to \mathcal{Y}$ with the property, that the preimage $f^{-1}(V)$ is pathconnected for all path-connected $V \subset \mathcal{Y}$.
For example, all homeomorphisms and constant functions should fall into this cl... |
I was wondering if it would be possible (and feasible) to analyse a card game called "declare" which I recently discovered. Suppose there are 9 players playing with 4 decks of 52 cards and starting off with 9 cards in hand. If a person deems the sum of their cards small enough, they can show their cards (they win if it... |
It is known that Ax axiomatized the pseudofinite field F as follows: F is quasifinite [https://en.wikipedia.org/wiki/Quasi-finite_field] and F is PAC [https://en.wikipedia.org/wiki/Pseudo_algebraically_closed_field].
I read Zilber's article about how AСF(algebraically closed field ) is pseudofinite in Zariski langu... |
Is every quasi-finite field algebraically closed? |
The goal is to find $m$ swaps $s_1, s_2, \dots, s_m$ such that any $n$-permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \dots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2),\; s_2=(2,3),\; s... |
I have looked at similar questions on here but still cannot reach a conclusion for this particular one.
I have gathered that as n increases, $\frac1n$ tends to $0$, meaning $a - \frac1n$ tends to $a$.
So the set must be bounded below by $a - 1$, and bounded above by $a$.
To prove $\inf(A) = a - 1$, I started by as... |
Here we deal with the problem of Histogram Equalization.
Let $\Omega$ be the set of all pixels in an image, but you can just treat it as some arbitrary set in this case.\
Let $f: \Omega \rightarrow \{0,...,255\}$ be a map from pixels to pixel intensities.
We then have the pixel intensity distribution given by:... |
Histogram Equalization Derivation and Proof? |
I’m confused by Arthur Besse’s definition of iterated covariant derivative on p. 25 of his book “Einstein Manifolds". He writes:
> **1.17** Let $\nabla$ be a linear connection on a vector bundle $E$ over $M$. For any section $s$ in $\mathcal E$, $\nabla s$ is a section of $T^*M\otimes E$. Now let $D$ be a linear con... |
What’s the definition of iterated covariant derivative? |
Let $R$ be a commutative ring, $I \subset R$ a finitely generated ideal, and $K$ a field containing $R$. Pick a set of generators of $I$, denoted by $\alpha_i$, and define $v = \alpha_1 + ... + \alpha_n \in I$. My question is, does there exist $\gamma \in K - R$ such that $\gamma v \in I$?
I tried to prove that ther... |
Is there an element outside a ring that maps an ideal of the ring to a subset? |
I want to show that for all $x_i > 0$:
$$\sum_{i=1}^{n}\dfrac{x_i}{x_1 + \ldots+x_n}\;\log(x_i) \geq \log\left(\dfrac{x_1 + \ldots + x_n}{n}\right)$$
I thought of Jensen's inequality but since $\log$ is a concave function, this seems to give me the opposite as for $x_i > 0, a_i \geq 0$, it holds: $a_1\log(x_1) + \l... |
I’m confused by Arthur Besse’s definition of iterated covariant derivative on p. 25 of his book “Einstein Manifolds". He writes:
> **1.17** Let $\nabla$ be a linear connection on a vector bundle $E$ over $M$. For any section $s$ in $\mathcal E$, $\nabla s$ is a section of $T^*M\otimes E$. Now let $D$ be a linear con... |
I have to convert the following expression $V(x,y) = x\dfrac{\partial f}{\partial y} - y\dfrac{\partial f}{\partial x}$ to polar coordinates. How do i express the partial derivatives in terms of $r$ and $\theta$? The answer should be something like $V(r,\theta) = \dfrac{\partial F}{\partial \theta}$, where $F(r,\theta)... |
Is there a compactly supported differentiable function that is NOT the Fourier transform of an L1 function? |
It is clear that $\{\log n\pmod 1: n\in\mathbb{N} \}$ [is dense in $[0,1]$ but not uniformly distributed.][1]
> How about $\{n\log n \pmod 1: n\in\mathbb{N} \} ?$ Is it dense in
> $[0,1]?$ If so, is it uniformly distributed?
Definition of uniformly distributed sequence, i.e., equidistributed sequence: https://e... |
Is $\{n\log n \pmod 1: n\in\mathbb{N} \}$ dense in $[0,1]?$ If so, is it uniformly distributed? |
Let $X_1,\cdots,X_n$ be $n$ random variables on the same probability space $(\Omega, F, P)$, all with expectation $0$. Define $Y_i=X_i - \mathbb{E}[X_i | X_1, \cdots, X_{i-1}]$. Is it true that for any $X = a_1 X_1 + \cdots + a_n X_n$, there exist $b_1,\cdots,b_n$ such that $X = b_1 Y_1 + \cdots + b_n Y_n$, and vice ve... |
Let $R$ be a commutative ring, $I \subset R$ a finitely generated ideal, and $K$ a field containing $R$. Pick a set of generators of $I$, denoted by $\alpha_i$, and define $v = \alpha_1 + \cdots + \alpha_n \in I$. My question is, does there exist $\gamma \in K - R$ such that $\gamma v \in I$?
I tried to prove that t... |
What is the expected number of comparisons needed to find a number (if it exists) in an array of [1,2,...,n] using binary search? I understand the limiting time complexity of binary search is $O(\log(n))$, but what is the expected value of the number comparisons? |
Binary search expected number of comparisons? |
I’m confused by Arthur Besse’s definition of iterated covariant derivative on p. 25 of his book “Einstein Manifolds". He writes:
> **1.17** Let $\nabla$ be a linear connection on a vector bundle $E$ over $M$. For any section $s$ in $\mathcal E$, $\nabla s$ is a section of $T^*M\otimes E$. Now let $D$ be a linear con... |
The goal is to find $m$ swaps $s_1, s_2, \dots, s_m$ such that any $n$-permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \dots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2),\; s_2=(2,3),\; s... |
Not related to the body of the post, but rather the title.
One can provide an explicit example to show the Fourier transform is not surjective: https://mathoverflow.net/a/319225/112504 proves that $$g(t) := 1_{[0\leq t\leq e]}\cdot \frac te + 1_{[t>e]}\cdot \frac{1}{\ln t},$$ **extended to be an odd continuous funct... |
I’m confused by Arthur Besse’s definition of iterated covariant derivative on p. 25 of his book “Einstein Manifolds". He writes:
> **1.17** Let $\nabla$ be a linear connection on a vector bundle $E$ over $M$. For any section $s$ in $\mathcal E$, $\nabla s$ is a section of $T^*M\otimes E$. Now let $D$ be a linear con... |
I’m confused by Arthur Besse’s definition of iterated covariant derivative on p. 25 of his book “Einstein Manifolds". He writes:
> **1.17** Let $\nabla$ be a linear connection on a vector bundle $E$ over $M$. For any section $s$ in $\mathcal E$, $\nabla s$ is a section of $T^*M\otimes E$. Now let $D$ be a linear con... |
"I think that after all, an algebraically closed field is pseudofinite."
Certainly not. Let $\varphi$ be the sentence asserting that the definable function $x\mapsto x^2$ is surjective but not injective: $(\forall y\, \exists x\, x^2 = y)\land (\exists z\exists w\, (z\neq w\land z^2 = w^2))$. The sentence $\varphi$ ... |
Your (implicit) claim that $$e_1:=(1,1,1,1),\quad e_2:=(0,1,1,1),\quad e_3:=(0,0,0,1),\quad e_4:=(0,1,0,0)$$ make a basis of $\Bbb R^4$ is correct (how would you prove it?), hence for any choice of four vectors $f_1,f_2,f_3,f_4\in\Bbb R^4$, there is a unique linear transformation $T:\Bbb R^4\to\Bbb R^4$ that maps $e_i$... |
In recent days I am studying semidirect product of groups and I have come up with the following question which has already been answered here (https://math.stackexchange.com/questions/1111239/from-semidirect-to-direct-product-of-groups#comment2266389_1111239), but I can't understand its solution from the comments, can ... |
I’m confused by Arthur Besse’s definition of iterated covariant derivative on p. 25 of his book “Einstein Manifolds". He writes:
> **1.17** Let $\nabla$ be a linear connection on a vector bundle $E$ over $M$. For any section $s$ in $\mathcal E$, $\nabla s$ is a section of $T^*M\otimes E$. Now let $D$ be a linear con... |
I need to do this proof without the use of projections and cartesian equations.
Here's my attempt at proving that if $\vec k$ is non coplanar with $\vec u$ and $\vec v$ then the line supported by $\vec k$ (we call that line $(d)$) is secant to the plane $(P)$ of direction $\vec u$ and $\vec v$.
We work in the euc... |
Proving vector $k$ non coplanar with vectors $u,v$ $\implies (d)$ secant to $(P)$? |
Title: Transforming a Sequence to Obtain a Specific Set
**Question:**
Consider a sequence of integers
( k_1, k_2, k_3, …, k_p ) where ( k_1 >= k_2 >= k_3 >= … >= k_p ). The sum of this sequence is ( k_1 + k_2 + k_3 + … + k_p = n(n+1)/2 ). We are allowed to repeatedly take 1 from each integer in the sequence to ... |
Let $R$ be a commutative ring, $I \subset R$ a finitely generated proper ideal, and $K$ a field containing $R$. Let $\alpha_i$ be an arbitrary set of generators of $I$, and define the column matrix $v = [\alpha_1, \dots, \alpha_n]^\intercal$. My question is, does there exist $\gamma \in K - R$ such that $\gamma v$ has ... |
**Problem**: Let $V$ be a finite-dimensional complex vector space and $\varphi\colon V\to V$ be a linear operator. Suppose there are subspaces $U\subset V$ and $W\subset V$, such that $V=U\oplus W$, $\varphi(U)\subset U$ and $\varphi(W)\subset W$. Prove that there is a basis for $V$, such that the operator $\varphi$ ca... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> **If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.**
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integer... |
Suppose that you initially have a container which has the shape of a paraboloid with equation
$ z = a x^2 + b y^2 $ where $ 0 \le z \le h $
Now you tilt this paraboloid by rotating it about any point (let's say the origin) about a horizontal axis whose unit vector is $v = (\cos \varphi, \sin \varphi, 0)$. Let t... |
I’m confused by Arthur Besse’s definition of iterated covariant derivative on p. 25 of his book “Einstein Manifolds". He writes:
> **1.17** Let $\nabla$ be a linear connection on a vector bundle $E$ over $M$. For any section $s$ in $\mathcal E$, $\nabla s$ is a section of $T^*M\otimes E$. Now let $D$ be a linear con... |
Given $n$ random variables $X_{1},\dots,X_{n}:(\Omega,\mathcal{F}) \to (\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$. We can consider $\sigma(X_{1},\dots,X_{n})$ which is the smallest sigma algebra on $\Omega$ to make $X_{1},\dots,X_{n}$ measurable. I think consider $g:\Omega \to (\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^... |
In recent days I am studying semidirect product of groups and I have come up with the following question which has already been answered here (https://math.stackexchange.com/questions/1111239/from-semidirect-to-direct-product-of-groups#comment2266389_1111239), but I can't understand its solution from the comments, can ... |
Am I thinking about $C^\infty$ rings correctly? |
In Serge Lang's Differential and Riemannian manifold, he defined derivative between two topological vector spaces with the concept tangent to 0:
>Let $\mathbf{E}, \mathbf{F}$ be two topological vector spaces, and $\varphi$ a mapping of a neighborhood of $0$ in $\mathbf{E}$ into $\mathbf{F}$. We say that $\varphi$ is t... |
I was given a question that I can't wrap my head around.
Given a measure space ($X, \Omega,\mu$), with $X$ = \{1,2,3, ... 16 \}, we have
$C_1$ = \{1,2,3,4,5,6,7,8 \}
$C_2$ = \{9,10,11,12,13,14,15,16\}
$C_3$ = \{1,2,5,6,9,10,13,14\}
$C_4$ = \{3,4,7,8,11,12,15,16\}
$C$ denotes the field generated by \{... |
Given a measure of some subsets, how can you determine the number of sets in the sigma algebra? |
Can you efficiently determine the number of possible solutions for an arbitrary starting sudoku configuration? |
I don't know a lot about Riemannian geometry, I only work on complex manifolds but I believe the idea is the same. A connection is a linear map $d^\nabla : \Omega^k(X,E) \rightarrow \Omega^{k + 1}(X,E)$ that verifies a Leibnitz relation. Here, $E$ is a vector bundle (maybe the tangant bundle in your case ?).
Then, w... |
This question in about **GLOBAL** optimization of multivariable polynomials.
Given a multivariable polynomial $q\left(x_1, \ldots, x_n\right) $, one trick to find its global minimum is to construct an auxiliary polynomial $p_\lambda$ by adding a *dominating term* $$\lambda\left(x_1^{2 d}+\ldots+x_n^{2 d}\right)$$ an... |
How to optimize Minkowski dominated polynomials $ \lambda\left(x_1^{2 d}+\ldots+x_n^{2 d}\right)+q\left(x_1, \ldots, x_n\right) $? |
> Solve the equation:
> $$4x^4 - 36x^3 + 61x^2 + 90x + 25 = 0.$$
This problem was lifted from
[this Youtube Video](https://www.youtube.com/watch?v=M4bKGwrKGxs). I didn't
particularly like the video's approach, so I used my own approach, which I have never
seen discussed anywhere. To understand the approach... |
Let $R$ be a commutative ring, $I \subset R$ a finitely generated proper ideal, and $K$ a field containing $R$. Let $\alpha_i$ be an arbitrary set of generators of $I$, and define the column matrix $v = [\alpha_1, \dots, \alpha_n]^\intercal$. My question is, does there exist $\gamma \in K - R$ such that the entries of ... |
I have the following equation $Ax=\lambda x$ where A is a matrix and $\lambda$ a scalar and $x$ a vector.
Then the following should be legal:
$\frac{1}{\lambda}Ax=x$
$\iff \frac{1}{\lambda}Ax-x=0$
$\iff (\frac{1}{\lambda}A-I)x=0$
And yet I this is illegal, because
$Ax-\lambda x=0 \iff (A-\lambda I)x = 0... |
For some reason, you have stopped reading Cunningham's "Theoretical Properties of the Network Simplex Method" *just* as it was about to answer your question. We read on:
> **Lemma 1.** Let $T, T'$ be strongly feasible trees, such that $T'$ is a successor of $T$ with entering edge $e$ and leaving edge $f$, and suppos... |
Long story short: For $f(x)=1$, the ODE does simply not have a solution (with the given decay conditions).
Disclaimer: I am a little bit confused which version of the Fourier transform you are using since the constants are not consistent, I am guessing the unitary angular freuency version? (https://en.wikipedia.org/... |
I have the following equation $Ax=\lambda x$ where A is a matrix and $\lambda$ a scalar and $x$ a vector.
Then the following should be legal:
$\frac{1}{\lambda}Ax=x$
$\iff \frac{1}{\lambda}Ax-x=0$
$\iff (\frac{1}{\lambda}A-I)x=0$
And yet I think this is illegal, because
$Ax-\lambda x=0 \iff (A-\lambda I... |
Suppose we have an underwater robot that can travel at a fixed velocity, v, if there is no current. This robot needs to travel a fixed distance, d. While in the water, the current acts on the robot affecting its velocity. How do we determine how long it takes for the robot to travel d? |
How long to travel fixed distance with variable velocity? |
If $y_{1}(x) = \frac{\sin(x)}{\sqrt(x)}$ is one solution of the differential equation $$x^2y'' +xy' + (x^2-\frac{1}{4})y = 0$$
find the second solution $y_{2}(x)$.
**My effort using Wronskian**
The general form of the homogeneous linear 2nd order differential equation is :
$$ y'' + p(x)y' +q(x)y =0$$
... |
How can I solve a Bessel equation with Reduction of order? |
I have a polynomial $P(x) = Ax^3 + Bx^2 + Cx + D$, with $P(0) = 1$ and $P(1) = 0$. This means that $D=1$ and $A + B + C + D = 0$. Suppose that $P'(0) = d_0 \leq 0$ and $P'(1) = d_1 \leq 0$. Depending on $d_0$ and $d_1$, $P(x)$ may or may not be monotonic. Is there a way to clamp $d_0$ and $d_1$ such that $P(x)$ is mono... |
If $y_{1}(x) = \frac{\sin(x)}{\sqrt(x)}$ is one solution of the differential equation $$x^2y'' +xy' + (x^2-\frac{1}{4})y = 0$$
find the second solution $y_{2}(x)$.
**My effort using Wronskian**
The general form of the homogeneous linear 2nd order differential equation is :
$$ y'' + p(x)y' +q(x)y =0$$
... |
**Short answer**: respecting the parallel postulate the internal angle sum of a triangle is $180°$ -- see [Euclid's Elements, Book I, Proposition 32][1]. In contrast, when disobeying it, in *non-Euclidean geometries*, the sum of a triangle internal angles is seldom $180°$.
**Why?** -- the parallel postulate implies ... |
$\newcommand{\C}{\mathscr{C}}$
$\newcommand{\R}{\mathbb R}$
Let $\C$ be $C^\infty$, that is a set with operations:
$$\Phi_f:\C^n\longrightarrow \C$$
for every smooth function $f$ such that if $g:\mathbb R^n\rightarrow \mathbb R$, and $f_i:\R^m\rightarrow \R$ are smooth functions for $1\leq i\leq n$ then:
\begin{al... |
Given $n$ random variables $X_{1},\dots,X_{n}:(\Omega,\mathcal{F}) \to (\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$. We can consider $\sigma(X_{1},\dots,X_{n})$ which is the smallest sigma algebra on $\Omega$ to make $X_{1},\dots,X_{n}$ measurable. I think consider $g:\Omega \to (\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^... |
Suppose we have an underwater robot that can travel at a fixed velocity, v, if there is no current. This robot needs to travel a fixed distance, d. While in the water, the current acts on the robot affecting its velocity. The current varies over time. How do we determine how long it takes for the robot to travel d? |
"I think that after all, an algebraically closed field is pseudofinite."
Certainly not. Let $\varphi$ be the sentence asserting that the definable function $x\mapsto x^2$ is surjective but not injective: $(\forall y\, \exists x\, x^2 = y)\land (\exists z\exists w\, (z\neq w\land z^2 = w^2))$. The sentence $\varphi$ ... |
This semester I'm taking a course in Linear Programming. While the topic is very interesting, all the applications I can find about this topic seem to be outside of mathematics. What are some applications of Linear Programming to pure math? (specially in number theory / algebra / algebraic geometry) |
I was given a question that I can't wrap my head around.
Given a measure space ($X, \Omega,\mu$), with $X$ = \{1,2,3, ... 16 \}, we have
$C_1$ = \{1,2,3,4,5,6,7,8 \}
$C_2$ = \{9,10,11,12,13,14,15,16\}
$C_3$ = \{1,2,5,6,9,10,13,14\}
$C_4$ = \{3,4,7,8,11,12,15,16\}
$C$ denotes the field generated by \{... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.