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This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math.
The associative property is astonishingly fundamental. More than once I've heard musings that it isn't clear if the as... |
Do we ever reason about non-associative algebgras without embedding it in an associative algebra? |
The probability is given by
$$\frac{1}{2}+\mathbb P (b<-\sqrt{4ac})+\mathbb P (b>\sqrt{4ac})=\frac{1}{2}+\frac{2}{\pi}\color{blue}{\int_0^\infty \left (1-\text{erf}\left(2\sqrt{t} \right) \right )K_0(t)\text{d} t}=\\
\frac{1}{2}+\frac{4}{3\pi}\sqrt{2/π} \left (8 \Gamma(5/4)^2 {}_3F_2(3/4, 5/4, 5/4;3/2, 7/4 ;16) \\-... |
The connection to perfect matchings is just this: in a $k$-edge-coloring of a $k$-regular graph (and in particular, in a $3$-edge-coloring of a cubic graph), every color class is a perfect matching. We will take each of those color classes to be $M$, in turn.
To make use of 16.1.7, we want to choose a set $S$ with a... |
I'm trying to see if I understood the integration by parts correctly. By setting $u(t) = a(t)$ and $dv(t) = b(t)dt$, the integral by part gives $\int_0^k a(t) b(t) dt = u(k)v(k) - u(0)v(0) - \int_0^k v(t) du(t)$. Since $v(t) = \int_0^t b(t') dt'$, is this integral become
\begin{align}
\int_0^k a(t) b(t) dt &= a(k)\in... |
The following is presented as an example to motivate my question. It's my paraphrase of the principle of recursion theorem and proof. From [Fundamentals of Mathematics Foundations of Mathematics: The Real Number System and Algebra][1]
>*Let $c$ be a number and let $F$ be a function of two arguments defined
in $\ma... |
Here is a question I got to thinking about while reading speculation about the world being discrete. Suppose I want to solve a second order ODE numerically. Let's say $y''+ay'+by =f(x)$ for concreteness. Usually when we solve this equation numerically we will have a value for y and a value for y' at every point we simu... |
It is said that the number of "information bits" contained in a certain piece of information can be roughly translated as the number of yes/no-questions that would have to be answered in order to transmit the information. But isn't this entirely dependent on the knowledge of the receiver (what questions they ask) and i... |
Do we ever reason about a non-associative algebra without embedding it in an associative algebra? |
This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math.
The associative property is astonishingly fundamental. More than once I've heard musings that it isn't clear if the as... |
The probability is given by
$$\frac{1}{2}+\mathbb P (b<-\sqrt{4ac})+\mathbb P (b>\sqrt{4ac})=\frac{1}{2}+\frac{2}{\pi}\color{blue}{\int_0^\infty \left (1-\text{erf}\left(2\sqrt{t} \right) \right )K_0(t)\text{d} t}=\\
\frac{1}{2}+\frac{4}{3\pi}\sqrt{2/π} \left (8 \Gamma(5/4)^2 {}_3F_2(3/4, 5/4, 5/4;3/2, 7/4 ;16) \\-... |
I've been studying Von Neumann-Barnays-Gödel axiomatic set theory (NBG) from Dugundji's Topology (1966) presented in chapter 8, and the statement 8.3 in the books still doesn't make sense to me, or at least the logical thinking it takes to get there: "the Russell class $\mathcal{R}(p)$ is not a set".
This is affirme... |
Why the Russell class $\mathcal{R}(p)$ is not a set? |
A commutative monoid $M$ has the nice property that given a set $S$, the set of functions $S \to M$ forms a commutative monoid (under pointwise addition). The same statement without any mention of commutativity is true.
Now, consider a cartesian closed category $C$. In other words, the category $C$ has finite pr... |
A commutative monoid $M$ has the nice property that given a set $S$, the set of functions $S \to M$ forms a commutative monoid (under pointwise addition). The same statement without any mention of commutativity is true.
Now, consider a cartesian closed category $C$. In other words, the category $C$ has finite pr... |
Let $V$ be a vector space, $S\subseteq V$ and $v\in V\setminus S$. Show that $S\cup \{v\}$ is linearly dependent if and only if $v\in\langle S\rangle$
So, can you help me evaluate my proof? Any observation will be appreciated. Thanks!
Dem:
$\Leftarrow)$
$v\in \langle S\rangle\implies S\cup \{v\}$ is linearl... |
A commutative monoid $M$ has the nice property that given a set $S$, the set of functions $S \to M$ forms a commutative monoid (under pointwise addition). The same statement without any mention of commutativity is true.
Now, consider a cartesian closed category $C$. In other words, the category $C$ has finite pr... |
I would like to determine the probability that a random quadratic polynomial has positive discriminant, where the 3 coefficients $a, b, c$ are normally distributed and independent.
That is, given $a,b,c \sim N(0,1)$, what is (a numerical approximation to 5 digits of) $\mathrm{Pr}(b^2 - 4 a c \geq 0)$ ?
---------... |
In want to show that $\Bbb R[x] / \langle x^2 + 1 \rangle \simeq \Bbb C$, so note that $\langle x^2 + 1 \rangle$ is irreducible in $\Bbb R$, and maximal in $\Bbb R[x]$, so $\Bbb R[x] / \langle x^2 + 1 \rangle$ It is isomorphic to a field.
To prove that it is isomorphic to $\Bbb C$ consider the aplication $\varphi(a... |
The following is presented as an example to motivate my question. It's my paraphrase of the principle of recursion theorem and proof. From [Fundamentals of Mathematics Foundations of Mathematics: The Real Number System and Algebra][1]
>*Let $c$ be a number and let $F$ be a function of two arguments defined
in $\ma... |
This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning.
Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). Consider $\bigcap_{n>0} I^n = \underset{\longrightarrow}{\textrm{lim}} \, I^n$ as a colimit. I would like to apply the $\t... |
I'm taking a discrete math course in university right now and we're studying permutations and combinations as one of the chapters. The question I'm confused about states that given a 5-letter alphabet, {A, B, C, D, E}, how many 3-letter strings can be made?
Here's my attempt:
Case 1: no duplicates. This is simply a... |
Suppose we have a 2D rotation matrix. If we plot eigenvalues on the complex plane, angle of rotation is connected to eigenvalues -- larger rotations have eigenvalues further away from 1.
<img src="https://i.stack.imgur.com/UOGTh.gif" width="300">
To extend this to higher dimensions, I want to know the following:
... |
can someone help me figure out how this differential equation
\begin{equation}
y' = p(t)(1-y)
\tag{1}
\end{equation}
gives the following solution?
\begin{equation}
y = 1 + \exp\left[-\int_0^tp(s)ds\right]\left(y(0)-1\right)
\tag{2}
\end{equation}
I tried integrating factors, but I end stuck at this step:
\... |
Chapter 20 Problem 23 of Spivak's *Calculus* asks the user to prove the formula for the *Schwarz second derivative* of $f$ at $a$ using a Taylor polynomial. The prompt reads as follows:
>If $f''(a)$ exists, then \begin{align}f''(a)=\lim_{h \to 0}\frac{f(a+h)+f(a-h)-2f(a)}{h^2} \quad \text{Hint: Use a 2nd-degree Tayl... |
Are there additional assumptions required in Chapter 20 Problem 23 of Spivak's Calculus? Using a Taylor polynomial to derive Schwarz second derivative |
$\lim_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$
Obviously L'hopital is inapplicable here. I guess it can be done by saying that $e^{-\infty}$ is almost zero so the limit is zero but what's the formal way of doing this? |
I have the following definition: Suppose that $g \in \mathrm{GL}_n(k)$ is regular and semisimple. Define $T_g := \mathrm{Cent}_{\mathrm{GL}_n(k)}(g)$ to be the centralizer of $g$ in $\mathrm{GL}_n(k)$. My advisor and I managed to proof that we have $T_g \cong k[g]^\times$ as multiplicative subgroups of $\mathrm{GL}_n(k... |
> What relationship must hold between the constants $a, b$ and $c$ to make:
$$
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm e}^{-\left(\, ax^{2}\ +\ 2bxy\ +\ cy^2\,\right)}\phantom{A\,}{\rm d}x\,{\rm d}y = 1
$$
I am absolutely clueless on how to proceed with this question. I found a solution to this questi... |
Lemma 4.7 in Ergodic Theory and Differentiable Dynamics by Racon, makes the statement,
> Let $A$ be in the $\sigma$-algebra generated by $\lim_{n\to\infty}\bigvee_{i=0}^n\mathscr{A}_i$. Then for all $\varepsilon>0$, there exists an $N$, such that for all $n \geq N$, the union of atoms $A_n$ of $\mathscr{A_n}$, we h... |
This is a question about first countable spaces. Topology of such spaces can be defined in terms of convergent sequences, and many topological properties of such spaces can be expressed in terms of sequences by taking general theorems about nets and replacing the term "net" with the term "sequence". For example: A set ... |
Suppose we have a 2D rotation matrix. If we plot eigenvalues on the complex plane, angle of rotation is connected to eigenvalues -- larger rotations have eigenvalues further away from 1.
<img src="https://i.stack.imgur.com/UOGTh.gif" width="300">
To extend this to higher dimensions, I want to know the following:
... |
There is a common argument in a few Digital Control Systems books about the stability of LTI systems in $z$ domain (discrete systems) which is based on the fact that the location of system poles in $z$ domain is connected to the location of the the poles in $s$ domain according to the expression $z = e^{sT}$, where $s$... |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. If $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta$ of such a polynomial, then by definition
$\Bbb E = 2 \cdot \Bbb P(... |
The stability of discrete systems can be understood by partial-fraction expansion of the discrete transfer function and finding the region of conversion for each divided fraction. Yet, there is a common argument in a few Digital Control Systems books about the stability of discrete systems in $z$ domain which is based ... |
Here is an example:
It is easy to see that
$\{[0, \alpha] \setminus F: \alpha < \omega_1, F \subset \omega_1, F \text{ finite}\}$
is a base for a topology on $\omega_1$.
Obviously, this topology is $T_1$.
For $\alpha < \omega_1$,
$\{[0, \alpha] \setminus F: F \subset [0, \alpha) , F \text{ finite}\}$ is a cou... |
I'm interested in compact spaces that are covered by $\mathbb{R}^2$. I know of the torus and the Klein bottle. Is the double-holed torus covered by $\mathbb{R^2}$? Are there any other spaces, or are these the only two compact spaces? Is there a way to show this/is there a complete characterization of all such spaces?
... |
Is there a complete list of all compact spaces covered by $\mathbb{R}^2$? |
is the $$\lim_{x\rightarrow 0}(\frac{1}{x^2})^x$$ can be solved as $$\lim_{x\rightarrow 0}(\frac{1^x}{x^{2x}})$$? It is in the indeterminate form of $0^{\infty}$ so I think I should transform it to the ln form and solve it. But I am not sure if I can solve it as given above. |
Let's denote $N_m$ as the number of solutions of
$$\varphi(n)=m$$
for all $m\geqslant 2$,
where $\varphi$ is Euler totient function, *i.e.*
>$$N_m:=\#\{n\in \mathbb N,\ \varphi(n)=m\}.$$
We can prove easily that $N_m=0$ if $m$ is odd.
We also know that $N_m<\infty$ for all $m$ (thanks to [this][1]).
... |
I need to show that the area of a regular tube of radius $r$ around a curve $\pmb{\alpha}$ is $2\pi r$ times the arc length of $\pmb{\alpha}$. The parametrization of the tube is that $$\pmb{x}(s,v) = \pmb{\alpha}(s) + r(\pmb{n}(s)\cos(v) + \pmb{b}(s)\sin(v))$$, where $\pmb{n}(s)$ is the unit normal vector and $\pmb{b}(... |
is the $$\lim_{x\rightarrow 0}(\frac{1}{x^2})^x$$ can be solved as $$\lim_{x\rightarrow 0}(\frac{1^x}{x^{2x}})$$? It is in the indeterminate form of ${\infty}^0$ so I think I should transform it to the ln form and solve it. But I am not sure if I can solve it as given above. |
I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around.
(1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$. Suppose $\{F_{m}\colon\mathsf{D}\to\mathsf{Set}\mid m\in\mathbf{N}\}$ is an inverse system of functors. Write $\li... |
Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory? |
The top line is $y=x$ and consists of the primes and four because $\rho(p)=p$ if $p$ is a prime number, and $\rho(4)=4$.
The second to top line is $y=\frac{x}{2}+2$ and consists of all $2p$ where $p\in{P}$ and $p>2$, and the number 8. This is because $\rho(2p)=\rho(p)+\rho(2)=p+2$ and $\rho(8)=\rho(4)+\rho(2)=4+2$.
... |
The key is that $y$ does not appear as a partial in the expression. Thus you can imagine plugging in a *fixed* value of $y$, call it $y_0$, into the expression. Define $f(x) = u(x,y_0)$, then the PDE becomes
$$ 2f''(x) + 5y_0 f(x) + 2y_0^2 f(x) = y_0. $$
This is the same differential equation that we started with... |
Consider a right-angled triangle as shown below, where vertex $O$ is the right angle. If we consider the midpoint of the two legs and draw a line passing through it, we have proved that the area of triangle $OCD$ is one-fourth of the area of triangle $OAB$.
[![enter image description here][1]][1]
Similarly, for a... |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. If $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta$ of such a polynomial, then by definition
$\Bbb E = 2 \cdot \Bbb P(... |
recently learned the minimax theorem for zero-sum games, and had a clarification on it. The minimax solution gives the value for the game V, and is also a Nash Equilibria.
So, does that mean that all Nash EQ in zero-sum games have the same value V, and they are all minimax solutions? If so, I'm not able to come up ... |
Can $\displaystyle \lim_{x\rightarrow 0} \left(\frac{1}{x^2}\right)^x$ be solved as $\displaystyle \lim_{x\rightarrow 0}\frac{1^x}{x^{2x}}$? It is in the indeterminate form of ${\infty}^0$ so I think I should transform it to the $\ln$ form and solve it. But I am not sure if I can solve it as given above. |
In KKT, we have some optimal point $x^*$ with associated $\mu^*$ value for inequality constraints. Are these $\mu^*$ values unique for a given $x^*$ (for the primal problem)?
It seems that when these constraints turn active (you go from inequality to equality constraint) the exact magnitude of $\mu^*$ doesn't actual... |
Do Karush Kuhn Tucker $\mu$ conditions have to be unique? |
Suppose we have a 2D rotation matrix. If we plot eigenvalues on the complex plane, angle of rotation is connected to eigenvalues -- larger rotations have eigenvalues further away from 1.
<img src="https://i.stack.imgur.com/UOGTh.gif" width="300">
To extend this to higher dimensions, I want to know the following:
... |
I'm interested in compact spaces that are covered by $\mathbb{R}^2$. I know of the torus and the Klein bottle. Is the double-holed torus covered by $\mathbb{R^2}$? Are there any other spaces, or are these the only two compact spaces? Is there a way to show this/is there a complete characterization of all such spaces?
... |
I have a collection of data points, $(x_0, y_0)...(x_{n-1}, y_{n-1})$ of the function $y = y(x)$ where the values of *x* are in ascending order. I'm working out an algorithm for smoothing splines, following de Boor (A Practical Guide to Splines, 2001), but I'm confused about one aspect of his explanation.
On page 20... |
Recently, I'm trying to learn geodesic currents on free groups through a [paper](https://arxiv.org/abs/0711.3806) by Kapovich and Lustig. Let $\langle, \rangle: \overline{CV}(F_N)\times Curr(F_N)\to\mathbb{R}$ be the intersection form and $I$ be a bipartite graph where vertices are $\overline{CV}(F_N)\cup\mathbb{P}Curr... |
Let $V$ be a vector space, $S\subseteq V$ and $v\in V\setminus S$. Show that $S\cup \{v\}$ is linearly dependent if and only if $v\in\langle S\rangle$
So, can you help me evaluate my proof? Any observation will be appreciated. Thanks!
Dem:
$\Leftarrow)$
$v\in \langle S\rangle\implies S\cup \{v\}$ is linearl... |
Let $X$ be from a normal distribution $N(\theta,1)$.
a) Find a sufficient statistic for $\theta$.
b) Is $S_n^2$ a sufficient statistic for $\theta$?
*My answers*
For part a)
Since the joint p.d.f is $1 \over (2\pi)^{n/2}$, $e^{{-1 \over 2}\sum(x_i-\theta)^2}$, I can say that $\sum X_i$ is a suf... |
The original title of my post was
<h3>Why is it so hard for mathematicians to say <i>exactly one</i>?</h3>
The proof under critique seems to be an elaborate way of saying: given a function $s$ let $a=s\left(x\right)$ and $b=s\left(y\right)$ then $x=y\implies a=b.$ Which is the definition of a function.
The w... |
The following is presented as an example to motivate my question. It's my paraphrase of the principle of recursion theorem and proof. From [Fundamentals of Mathematics Foundations of Mathematics: The Real Number System and Algebra][1]
>*Let $c$ be a number and let $F$ be a function of two arguments defined
in $\ma... |
The probability is given by
$$\frac{1}{2}+\mathbb P (b<-\sqrt{4ac})+\mathbb P (b>\sqrt{4ac})=\frac{1}{2}+\frac{2}{\pi}\color{blue}{\int_0^\infty \left (1-\text{erf}\left(2\sqrt{t} \right) \right )K_0(t)\text{d} t}=\\
\frac{1}{2}+\frac{4}{3\pi}\sqrt{2/π} \left (8 \Gamma(5/4)^2 {}_3F_2(3/4, 5/4, 5/4;3/2, 7/4 ;16) \\-... |
The permutohedron P4 is the 3d truncated octahedron. Its facets consist of 6 hexagons and 8 squares.
The permutohedron P5 is a 4d polytope with 30 3d facets: there are 10 truncated octahedra, and 20 hexagonal prisms (I believe).
The permutohedron P6 is a 5d polytope with 62 4d facets: there are 12 P4 polytopes, 3... |
This question arose from an attempt of using the Equivariant Rank Theorem to prove question 7.1 of Lee's Introduction to Smooth Manifolds.
The Equivariant Rank Theorem is stated:
> Let M and N be smooth manifolds
and let G be a Lie group. Suppose $F:M\rightarrow N$ is a smooth map that is equivariant
with respe... |
I'm wondering if someone can offer an intuition of what is meant when mathematicians talk of recasting first-order logic statements as algebraic expressions. e.g. Halmos, Barnes, or Mack?
Here is a simple example to illustrate this question, adapted from [forall x: Calgary][1].
If we had two variables "lover" an... |
This question arose from an attempt of using the Equivariant Rank Theorem to prove question 7.1 of Lee's Introduction to Smooth Manifolds.
The Equivariant Rank Theorem is stated:
> Let M and N be smooth manifolds
and let G be a Lie group. Suppose $F:M\rightarrow N$ is a smooth map that is equivariant
with respe... |
Define $Q(a)$ as the sum of digits.
> I want to prove that for any two integers $a$ and $b$:
$$
Q(a b) \leq Q(a) Q(b).
$$
For this I was able to prove this identity:
$$
Q(a b) \leq \min \{a Q(b), b Q(a)\}.
$$
Now has anyone an idea of how to show the quasi-multiplicity? Note that $Q(n)$ is not the digital... |
Is lim x approaches 0 of f(x) - f(a) / x - a the same as saying dy/dx at x = 0 equals some number? |
While teaching my calculus class I happened upon a strange identity. Consider the antiderivative $$\int \tan^5 x \sec^4 x dx.$$
You can do that by writing the integrand as $\tan^5 x (1+\tan^2 x) \sec^2x$, then substituting $u=\tan x$. This gives one answer.
But another way to do it is to rewrite the integrand as $... |
The probability is given by
$$\frac{1}{2}+\mathbb P (b<-\sqrt{4ac})+\mathbb P (b>\sqrt{4ac})=\\ \frac{1}{2}+2\int_0^\infty \left (1-\Phi\left(2\sqrt{t} \right) \right )\frac{K_0(t)}{\pi}\text{d} t= \\ \frac{1}{2}+\color{blue}{\frac{2}{\pi}\int_0^\infty \left (\frac{1}{2}-\text{erf}\left(\frac{2\sqrt{t}}{\sqrt{2}} \r... |
While teaching my calculus class I happened upon a strange identity. Consider the antiderivative $$\int \tan^5 x \sec^4 x dx.$$
You can do that by writing the integrand as $\tan^5 x (1+\tan^2 x) \sec^2x$, then substituting $u=\tan x$. This gives one answer.
But another way to do it is to rewrite the integrand as $... |
**Question: Find the equilibrium points of the following ODE and then use Linear Stability analysis to determine the stability. $$\frac{dy}{dt}= cy - \frac{y}{1+y^2}$$**
**My Attempt:**
I have worked out the equilibrium poitns and I believe they are:
$y=0 \ \text{when} \ c=1$
$y= \pm \sqrt{ \frac{1}{c} - ... |
A connected graph is k-connected if either after removing any k−1
vertices the remaining graph is connected.
If I follow the definition,trinangle is 3-connected (k=3) since removing any two vertices (k-1=2) leaves a single vertex, which is a connected graph.
But all the places it is showing ,it is 2-connected.
... |
Triangle is 2-connected or 3? |
I am working with floating point numbers. There is a square matrix that has determinant 1e-14. It is reported as having rank 3, but that might be because I am working with floating point numbers.
I have reason to believe this matrix is an outer product of two vectors. If the assumption is correct, how would I find o... |
If I assume a matrix is an outer product of two vectors; can I determine the vectors somehow? |
[![proof of rational power derivatives**strong text**][1]][1]
[1]: https://i.stack.imgur.com/KwnH5.jpg
I don't understand this step in my calculus textbook (Calculus one and several variables, Salas)
It's supposed to be a proof of the derivative of rational powers, but I don't understand the 3rd line from ... |
The permutohedron P4 is the 3d truncated octahedron. Its facets consist of 6 hexagons and 8 squares.
The permutohedron P5 is a 4d polytope with 30 3d facets: there are 10 truncated octahedra, and 20 hexagonal prisms (I believe).
The permutohedron P6 is a 5d polytope with 62 4d facets: there are 12 P4 polytopes, 3... |
While teaching my calculus class I happened upon a strange identity. Consider the antiderivative $$\int \tan^5 x \sec^4 x dx.$$
You can do that by writing the integrand as $\tan^5 x (1+\tan^2 x) \sec^2x$, then substituting $u=\tan x$. This gives one answer.
But another way to do it is to rewrite the integrand as $... |
If I assume a matrix is an outer product of two vectors; can I determine the vectors? |
If a matrix is an outer product of two vectors; can I determine the vectors? |
I am working with floating point numbers. There is a square matrix that has determinant 1e-14.
I have reason to believe this matrix is an outer product of two vectors. If the assumption is correct, how would I find out which vectors it is? |
A connected graph is $k$-connected if either after removing any $k−1$
vertices the remaining graph is connected.
If I follow the definition,triangle is $3$-connected $(k=3)$ since removing any two vertices $(k-1=2)$ leaves a single vertex, which is a connected graph.
But all the places it is showing, it is $2$-... |
Triangle is $2$-connected or $3$? |
This is a bit of a loose, explorative question. I think it could still have pretty straightforward answers, but I apologize if it's more of a discussion question than a question with a single good answer.
Given the function sequence $f_p(x, y) = \sqrt[p]{\lvert x \rvert^p + \lvert y \rvert^p}$ we get that $\lim\limi... |
Sequence of functions that approach min at the limit? |
I am working with floating point numbers. There is a 3x3 matrix that has determinant 1e-14.
I have reason to believe this matrix is an outer product of two vectors. If the assumption is correct, how would I find out which vectors it is? |
I don't understand a proof step in my calculus textbook (Calculus one and several variables, Salas). It's supposed to be a proof of the derivative of rational powers, but I don't understand the 3rd line from the bottom.
How did they get $y^{1-q}$ to equal $x^{(1-q)/q}\,$?
[![proof of rational power derivative... |
How does $y^{1-q}$ become $x^{(1-q)/q}$ in this proof of the formula for the derivative of rational powers? |
While teaching my calculus class I happened upon a strange identity. Consider the antiderivative $$\int \tan^5 x \sec^4 x dx.$$
You can do that by writing the integrand as $\tan^5 x (1+\tan^2 x) \sec^2x$, then substituting $u=\tan x$. This gives one answer.
But another way to do it is to rewrite the integrand as $... |
I was aked to determine whether the following statement is true or false.
> Let
$$\mathcal{D} \left( (x_0, y_0), r \right) = \left\{ (x, y) \in \mathbb{R}^2 : (x - x_0)^2 + (y - y_0)^2 \leq r^2 \right\} $$
Let $P = \left\{ \emptyset \right\} \cup \left\{ \mathcal{D}\left( (x_0,y_0), r \right) : x_0, y_0 \in \math... |
In the set $P$ of all circles over $\mathbb{R}^2$, ordered by $(P, \subseteq)$, do two circles always have an infimum? |
I have attempted to solve an interesting mechanical problem, but it seems to have two solutions:
[![enter image description here][2]][2]
My first calculation is as below:
[![enter image description here][3]][3]
.
Second solution:
[![enter image description here][1]][1]
.
Both calculated values are o... |
One mechanical question, two possible solutions. Can mathematics help to predict the actual physical event? |
I was aked to determine whether the following statement is true or false.
> Let
$$\mathcal{D} \left( (x_0, y_0), r \right) = \left\{ (x, y) \in \mathbb{R}^2 : (x - x_0)^2 + (y - y_0)^2 \leq r^2 \right\} $$
Let $P = \left\{ \emptyset \right\} \cup \left\{ \mathcal{D}\left( (x_0,y_0), r \right) : x_0, y_0 \in \math... |
Let the vertices of the pyramid be $A,B,C,D,V$, where $ABCD$ is the base and $V$ is the apex. Using coordinate geometry, and letting the given sides of the base be $a = AB, b = BC, c = CD, d = DA $, we can assign $A,B,C,D$ as follows
$ A = (0,0,0) , B= (a, 0, 0), C= (x_1, y_1, 0), D = (x_2, y_2, 0) $
with $y_1, ... |
Let $(\mathbb{R}, \mathcal{F}, \mu)$ be a measure space. Let $F_n$ be measurable subsets of $\mathbb{R}$. Show that $\lim_{n\to \infty} I_{F_n}(x)=0$ a.e. for $\in \mathbb{R}$ if and only if $\mu(\lim \sup F_n)=0$.
---
My proof is as follows.
(1) Assume that $\lim_{n\to \infty}I_{F_n}=0$.
Note that
$$
\... |
I have tried on my kid's textbook question (so to have meaningful discussion with him), but it seems to have two solutions:
[![enter image description here][2]][2]
My first calculation is as below:
[![enter image description here][3]][3]
.
Second solution:
[![enter image description here][1]][1]
.
B... |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. If $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta$ of such a polynomial is positive, then by definition
$$\Bbb E = 2 ... |
Let $X$ denote a vector space. Let $n:X\to\mathbb R$ be a norm over $X$ and let $s:X\to\mathbb R$ be a seminorm over $X$.
I have proved quite easily that the sum of two norms is a norm itself. I would like to understand what happens if I sum a norm with a seminorm.
Does one obtain a norm or a seminorm?
Accord... |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. Then, if $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta$ of such a polynomial is positive, then by definition
$\Bbb E... |
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