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Is it true that $m_*(E) = m^*(E)? (Lebesgue inner and outer measures) |
If $m$ is the Lebesgue measure on $\mathbb{R}^n$, for any $E \subseteq \mathbb{R}^n$ we can define the inner and outer Lebesgue measure as:
$$ m^*(E) = \inf \{ m(U) : E \subseteq U, \text{ $U$ is open } \} $$
$$ m_*(E) = \sup \{ m(F) : F \subseteq E, \text{ $E$ is closed } \} $$
I know that if $E$ is measurable ... |
Is it true that $m_*(E) = m^*(E)$? (Lebesgue inner and outer measures) |
I'm reading a proof of the statement that each closed subset $F \subset \mathbb{R}^n$ can be written as an intersection of a sequence of open subsets. The corresponding sequence is given by $U_n := \{x \in \mathbb{R}^n | \;||x-y||_2 < \frac{1}{n} \text{ for some } y \in F\}$, i.e. one wants to prove that $F = \bigcap_{... |
Do we need the axiom of choice for choosing a sequence out of countably infinite nonempty sets? |
If $m$ is the Lebesgue measure on $\mathbb{R}^n$, for any $E \subseteq \mathbb{R}^n$ we can define the inner and outer Lebesgue measure as:
$$ m^*(E) = \inf \{ m(U) : E \subseteq U, \text{ $U$ is open } \} $$
$$ m_*(E) = \sup \{ m(F) : F \subseteq E, \text{ $F$ is closed } \} $$
I know that if $E$ is measurable ... |
Is it true that $m_*(E) = m^*(E)$? (Lebesgue inner and outer measures) |
I'm reading a proof of the statement that each closed subset $F \subset \mathbb{R}^n$ can be written as an intersection of a sequence of open subsets. The corresponding sequence is given by $U_n := \{x \in \mathbb{R}^n | \;||x-y||_2 < \frac{1}{n} \text{ for some } y \in F\}$, i.e. one wants to prove that $F = \bigcap_{... |
Suppose there is a sequence of$N$ numbers $x_1, x_2, x_3, ... x_N$.
There are then gaps $|x_i - x_j|$, and the minimum gap:
$\delta (N) = \text{min}_{i \ne j \le N} \{ | x_i -x_j | \}$.
Let the mean gap be normalized to $1/N$.
If the sequence of numbers $x_1, x_2, x_3, ... x_N$ is random, the minimum ga... |
Take an equilateral triangle with side length 1. Then, inscribe the largest possible rectangle in this triangle such that the bottom of the rectangle is a segment of the bottom of the triangle. Then inscribe the largest possible regular pentagon inside this rectangle where the bottom of the pentagon is also a segment o... |
Assume $E_1\subset E_2\subset ...$ is a sequence of Frechet spaces, each one embedded in the next.
Let $E$ be their union, and one puts the LF topology on $E$.
I wished one can say the following: **If $U\subset E$ satisfies that for every $k$, $U\cap E_k$ is open in $E_k$, then $U$ is open in $E$.**
But from... |
> But—and in this mathematics is distinguished from other sciences—these
> extensions of definitions no longer allow scope for arbitrariness; on
> the contrary, they follow with compelling necessity from the earlier
> restricted definitions, provided one applies the following principle:
> Laws which emerge from the... |
Take an equilateral triangle with side length $1$. Then, inscribe the largest possible rectangle in this triangle such that the bottom of the rectangle is a segment of the bottom of the triangle. Then inscribe the largest possible regular pentagon inside this rectangle where the bottom of the pentagon is also a segment... |
Take an equilateral triangle with side length $1$. Then, inscribe the largest possible square in this triangle such that the bottom of the square is a segment of the bottom of the triangle. Then inscribe the largest possible regular pentagon inside this square where the bottom of the pentagon is also a segment of the b... |
Let $X$ be a projective variety over $\mathbb C$, and $X_h$ its associated analytic space. If there is a variety $Y$ that $Y_h$ is a open subset of $X_h$, would $Y$ or some subvariety of $Y$ be a subvariety of $X$? I don't know if GAGA would be helpful, since $Y_h$ is not really open in the projective space. The Euclid... |
Is an "algebraic" subspace of an associated analytic space a subvariety of the original variety? |
In Darij Grinberg's "Hopf algebras in combinatorics", there is a statement about existence of quotient coalgebras:
"Indeed, $J ⊗ C + C ⊗ J$ is contained in the kernel of the canonical map $C ⊗ C → (C/J) ⊗ (C/J)$; therefore, the condition $∆(J) ⊂ J ⊗ C + C ⊗ J$ shows that the map $C → C ⊗ C \twoheadrightarrow (C/J) ⊗ (... |
What does it mean for a map to factor through another map? |
$p\implies q$ does not necessarily indicate that $q\implies p$, which is why there exists a double-sided implication: $\iff$.
$p\iff q$ is the same as saying $(p\implies q)\land(q\implies p)$. However, sometimes when I am illustrating my working out, I use the one-sided $\implies$ symbol even when using the $\iff$ s... |
Very simple question: Suppose $X$ is subgaussian with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$?
Some context: It is not hard to show that bounds like $p_X(x)\le e^{-x^2}$ imply subgaussianity. I am wondering about the converse, or can the density of a subgaussian RV b... |
Very simple question: Suppose $X$ is subgaussian with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$?
Some context: It is not hard to show that bounds like $p_X(x)\le e^{-x^2}$ (for large $x$) imply subgaussianity. I am wondering about the converse, or can the density of a ... |
Very simple question: Suppose $X$ is subgaussian with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$?
Some context: It is not hard to show that bounds like $p_X(x)\le e^{-x^2}$ imply subgaussianity. I am wondering about the converse, or can the density of a subgaussian RV b... |
Very simple question: Suppose $X$ is subgaussian with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$?
Some context: It is not hard to show that bounds like $p_X(x)\le e^{-x^2}$ imply subgaussianity. I am wondering about the converse, or can the density of a subgaussian RV b... |
Take two Banach spaces $X,Y$ and consider their tensor product $W=X\otimes Y$.
Any $w\in W$ can be written, generally not in a unique way, as a linear combination of simple tensors: $$w=\sum_{i=1}^n (x_i\otimes y_i)\ \ \ \ \text{ with }x_i\in X \text{ and } y_i\in Y\ \forall i\quad\quad\quad (1)$$
Under which assu... |
> Prove that a set of lines, each of which intersect the others, are either coplanar or share a single common point.
>
> Source: Hadamard
This is a surprisingly difficult problem, because it requires ping-ponging back and forth between two dissimilar concepts: _lines being coplanar_ and _lines sharing a common poi... |
This is a preliminary in the proof of Cauchy-Goursat Theorem for Triangles in complex analysis,
The solid triangle enclosed by $a,b,c$ in the complex plane is defined as $$\Delta(a,b,c):=\{t_1a+t_2b+t_3c: t_1+t_2+t_3=1,t_i\geq0, i=1,2,3 \}$$
I already proved that this is a convex and compact set, now I need to p... |
8 people total. Seated around a circular table. Hibachi dinner? You can only talk to the 2 people seated directly next to you.
Each person has a connection value with each other person. eg : #1 and #3 are friendly and have a "conversation value" of 7. #1 and #2 are not friends, and have a "conversation value... |
Assuming $y=\sum_{n\geq 0}a_nx^{n+r}$ (Fröbenius method), $xy''+y'-xy=0$ reduces to
$$\sum_{n\geq 0}a_n(n+r)(n+r-1)x^{n+r-1}+\sum_{n\geq 0}a_n(n+r)x^{n+r-1}-\sum_{n\geq 0}a_nx^{n+r+1}=0$$
Reindexing the last sum like $n\to n-2$ and combining the first two sums, we get
$$\sum_{n\geq 0}(n+r)^2a_nx^{n+r-1}-\sum_{n\geq ... |
> The chocolate bar is a rectangle $m*n$ , divided by a recess into
> single squares. Two players play the next game in turn. On each turn,
> it is allowed to pick up one square of chocolate that has not yet been
> taken, as well as squares that have not yet been taken, lying on the
> right and above (including squ... |
Birthday problem: how to show the scaling with $1/N^2$? |
Take two Banach spaces $X,Y$ and consider their tensor product $W=X\otimes Y$.
Any $w\in W$ can be written, generally not in a unique way, as a linear combination of simple tensors: $$w=\sum_{i=1}^n (x_i\otimes y_i)\ \ \ \ \text{ with }x_i\in X \text{ and } y_i\in Y\ \forall i\quad\quad\quad (1)$$
Under which assu... |
$H$ is the space of all real $N \times N$ matrices.
Let $X$ denote the $H$ such that the Frobenius distance is at most $\alpha$ from $R$. Find the closest matrix $A \in X$, for any given $B \in H$.
My attempt:
$A \in X$, then $||A-R|| \leq \alpha$
$$\min_{A \in X}||A-B||_F \forall B \in H $$
So, in terms ... |
> Approximate the large positive root of the equation
> $$e^{e^x} = 10^{10} x^{10}e^{10^{10} x^{10}}$$
>
> (attributed to G.H. Hardy)
This is a great problem which I can find no verifiable source or discussion of online. Is there a source where Hardy actually asks this question? How did Hardy (or others) approa... |
For any $n$, what is the commutator subgroup of the dihedral group $D_n$?
My solution is below. I request verification, feedback, and improvements.
_Note: Other proofs may be available. This question asks to verify or critique **this** proof._
----
**Solution:** The commutator subgroup of $D_n$ is isomorp... |
What is the commutator subgroup of the dihedral group $D_n$? |
In Brezis's *Functional Analysis*, there is a corollary
>**Corollary 3.30.** Let $E$ be a separable Banach space and let $\left(f_{n}\right)$ be a bounded sequence in $E^*$. Then there exists a subsequence $\left(f_{n_{k}}\right)$ that converges in the weak topology $\sigma\left(E^*, E\right)$.
I would like to... |
There exists the converse of this corollary from Brezis? |
I've seen the axiom of choice as $\forall X\left[\varnothing \notin X\implies \exists f\colon X\rightarrow \bigcup_{A\in X}A,~~~ \forall A\in X\,(f(A)\in A)\right]$.
But isn't there a version which doesn't need the definition of a function and of the union? |
Very simple question: Suppose $X$ is a subgaussian RV with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$?
Some context: It is not hard to show that bounds like $p_X(x)\le e^{-x^2}$ imply subgaussianity. I am wondering about the converse, or can the density of a subgaussian... |
$O(n)$ has two connected components, $(\det)^{-1}(1)$ and $(\det)^{-1}(-1)$. While I know it is not true, I am wondering why the above is not enough to say that $O(n)$ is the double cover of $SO(n)$ since $SO(n)$ can be identified with $(\det)^{-1}(-1)$.
Is this because there is no way to associate $(\det)^{-1}(1)$... |
Why is $O(n)$ not the double cover of $SO(n)$? |
For instance, when differentiating four-vectors the result is straightforward:
$$\frac{\partial x^\mu}{\partial x^\nu}=\delta_\nu^\mu$$
as the derivative is only non-zero when the Lorentz indices match. Here $\mu, \nu = 0,1,2,3$.
But when differentiating type $(0,2)$ tensors, for example,
$$\frac{\partial\left(\... |
I wonder, is there any subgroups of $\mathbb{Z}_{2}\oplus\mathbb{Z}_{16}$ isomorphic to $\mathbb{Z}_{4}\oplus\mathbb{Z}_{4}$? It seems like the answer is no since $\mathbb{Z}_{16}$ is not isomorphic to $\mathbb{Z}_{4}\oplus\mathbb{Z}_{4}$, but I want to know more
explicit and accurate proof. Or may be there is some c... |
Groups of order $2^n$ have most non-isomorphic types and are notoriously hard to classify. For example, order $32$ has $51$ non-isomorphic groups, and order $64$ has $267$ ones. In general group theory texts, only groups of order up to $15$ are (completely) classified. Even the order of $16$ is not available for a full... |
How to classify $2^n$ groups up to isomorphism? |
> But I’m a little confused on how you actually know that the right
> triangle you draw isn’t just the one special right triangle with which
> you can actually form such a square.
If I am not mistaken, this is the figure you are referring to.
[![enter image description here][1]][1]
Ask yourself, what is it a... |
**Can I truncate the intermediate values without losing absolute accuracy?**
Short answer: **no**.
Not so short answer: Regarding add (and substraction) *absolute* errors sum up, for multiplication (also dividing) *relative* errors sum up. While only interested in a result of 20 bits you may calculate what trunca... |
Definitions try to precisely formalize intuitive ideas. For example, the concept of mathematical expectation attempts to rigorously formalize the idea of quantifying the average outcome of a random variable, considering all possible values and their respective probabilities. So, given a random variable of the continuou... |
Why is absolute convergence required? |
> But I’m a little confused on how you actually know that the right
> triangle you draw isn’t just the one special right triangle with which
> you can actually form such a square.
If I am not mistaken, this is the figure you are referring to.
[![enter image description here][1]][1]
Ask yourself, what is it a... |
Suppose there is a sequence of$N$ numbers $x_1, x_2, x_3, ... x_N$.
There are then gaps $|x_i - x_j|$, and the minimum gap:
$\delta (N) = \text{min}_{i \ne j \le N} \{ | x_i -x_j | \}$.
Let the mean gap be normalized to $1/N$.
If the sequence of numbers $x_1, x_2, x_3, ... x_N$ is random, the minimum ga... |
I’m trying to show the second equation of the second answer to [this question][1] on Physics StackExchange:
>If $e_1,…,e_n$ is a frame, and $\theta^1,…,\theta^n$ is its dual frame, and the direct frame satisfies the commutation relations $$ [e_a,e_b]=C^c_{\ ab}e_c, $$ then it is easy to show that the dual frame sati... |
I guess the following is very easy to answer for someone knowledgeable in algebra: Consider a real even-dimensional vector space $V$ of (real) dimension $d=2n$. Furthermore, we consider a linear complex structure $J:V\to V$ on $V$. This provides a (complex?) direct sum decomposition
$$V_{\mathbb{C}}=V_{+}\oplus V_{-}$... |
How do I go "backwards" with error propagation? |
**question;how to evaluate this logarithm integral?**
$$
I=\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x=?
$$
## My attemt
$$
\begin{aligned}
& I=\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x=\frac{1}{2} \int_0^1 \log ^2(x) \log (1-x) d\left(\log \left(x^2+1\right)\right) \xrightarrow{I B P} \\
&... |
Let $G_1, G_2, \ldots, G_t$ be finite cyclic groups, and define $G = G_1 \times G_2 \times \ldots \times G_t$. Let $n_j = |G_j|$, such that $|G| = \prod_{j=1}^{t} n_j$.
For part (a) of the problem, we are asked to suppose that $\gcd(n_i, n_j) > 1$ for some $i \neq j$ and show that $G$ is not cyclic.
I und... |
> Prove that a set of lines, each of which intersect the others, are either coplanar or share a single common point.
>
> Source: Hadamard
This is a surprisingly difficult problem, because it requires ping-ponging back and forth between two dissimilar concepts: _lines being coplanar_ and _lines sharing a common 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) $P(X_n=a~i.o.)=1$ for each $a=-1,0,1$
(2) $\sup_n|X_n(w)|<\infty$ a.s.
(3) For som... |
I have to look for a formula which gives the number of i-dimensional singular subspaces 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 t... |
Given circle A (center point and radius $R$ is given), and a line segment (2 points are given: $P_{0}$ and $P_{1}$).
$P_{0}$ is on the circle.
There is another circle with given radius $r$, and it is tangential on the circle and the line segment.
What is the circle's location? (there are 2 possible locations)
... |
Find circle center point that is tangential to line segment and another circle? |
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) $P(X_n=a~i.o.)=1$ for each $a=-1,0,1$
(2) $\sup_n|X_n(w)|<\infty$ a.s.
(3) For som... |
I was struggling a bit with the time integral of an Ito process.
Say I have this:
$$\int^T_t \alpha\circ X_t ds$$
Where $X_t$ is an Ito process, and $\alpha$ is a continuous function. What can we say about the variance and expectation of such a function? Thanks! |
I am interested in the idea of arithmetic performed in binary done without electronics. In particular, I’d like to try calculating the binary logarithm of x (where 10₂ > x >= 1) using the square/division algorithm wherein I square the mantissa, write a 0 if the result is less than 10₂, else write a 1 and divide the man... |
I am interested in the idea of arithmetic performed in binary done without electronics. In particular, I’d like to try calculating the binary logarithm of x (where 10 > x >= 1) using the square/division algorithm wherein I square the mantissa, write a 0 if the result is less than 10, else write a 1 and divide the manti... |
The following proof is weaker than the pervious answer since it only works for $\sqrt{2}$.
We prove the following result: if $p, q$ are natural numbers, then
$$\left|\frac{p}{q} - \sqrt{2}\right| > \frac{1}{3q^2}.$$
Denote $r := p/q$. If $q=1$,
$$|r - \sqrt{2}| = |p - \sqrt{2}|\geq \sqrt{2} - 1 > \dfrac{1}{3}.... |
Let $(W_i)_{0 \leq i \leq n}$ be a simple asymmetric random walk which starts from 0 (that is the probability to go up at each step is equal to $p \neq \frac{1}{2}$). Define $W^*_n = \max\limits_{0 \leq i \leq n} W_i$. Assume that $n$ and $a$ are even positive integers, $b$ is an even integer, $b \leq a \leq n$, and $2... |
Problem:
Given that $y = e^{2x}$ is a solution of
$$ (2x+1)y'' - 4(x+1)y' + 4y = 0 $$
find a linearly independent solution by reducing the order. Write the general solution.
Answer:
Let $y = e^{2x}v$. We have:
\begin{align*}
\dfrac{dy}{dx} &= e^{2x} \dfrac{dv}{dx} + 2 e^{2x} v \\
\dfrac{d^2y}{dx^2} &=
... |
For instance, when differentiating four-vectors the result is straightforward:
$$\frac{\partial x^\mu}{\partial x^\nu}=\delta_\nu^\mu$$
as the derivative is only non-zero when the Lorentz indices match. Here $\mu, \nu = 0,1,2,3$.
But when differentiating type $(0,2)$ tensors, for example,
$$\frac{\partial\left(\... |
I am trying to analyze a bit how we can extend a differential form to the compactification $V\otimes\mathbb{C}=V_{\mathbb{C}}$ of the vector space. Of course, you can do this for a general $k$-form, but I think the simpler the example the better I can grasp the idea.
So let $V=\{<v_1,\ldots,v_m>\}$ be a real vector ... |
It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and write some errata emails.
A standard definition of a limit of a diagram is that it’s a terminal object in the category of... |
Do Wikipedia, nLab and several books give a wrong definition of categorical limits? |
Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ is $1 \frac{1}{3}$?
I know this is true because for $n = 8 + 6i$ where $i$ is an integer $\geq 0$, the difference is $1 \frac{1}{3}$. Plugging other values for $n$ gives a difference that is less than $1 \frac{1}{3}$.
Ho... |
Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ is $1 \frac{1}{3}$? |
I think the step you suspect is illegal is illegal indeed.
Note that the strategy $X^*$ is optimal, when we are sure that the other player plays $Y^*$, but we don't know yet, if it is optimal, when the other player plays any other strategy.
I'd rather approach the problem in the following way. Since $E(X, Y^*) \l... |
I’m trying to show the second equation of the second answer to [this question][1] on Physics StackExchange:
>If $e_1,…,e_n$ is a frame, and $\theta^1,…,\theta^n$ is its dual frame, and the direct frame satisfies the commutation relations $$ [e_a,e_b]=C^c_{\ ab}e_c, $$ then it is easy to show that the dual frame sati... |
In the past, I came across a very elegant definition (below) of a function, which is based on the fundamental concepts of triples, pairs, and sets. However, I find it difficult to search the internet for any citations that I could refer to. One professor once told me that this definition could be over 200 years old, ... |
It is maybe good to answer the question by working explicitly in the given example.
Let $A=1+a=1.3$ and $B=1-b=0.5$.
----------
So we have a discrete process $(X_n)$ with $X_0=1$ on the time line $T=\{0,1,2,3,4,\dots\}$ and the following passage probabilities and values from step $n$ to step $n+1$:
- wi... |
> A group of n friends get together for a cheese tasting where each member of the group brings a cheese taster. Before starting, the friends sit at a round table with n seats and the cheese samples are distributed randomly to everyone. Every member of the group receives one and only one taster. Assume someone has recei... |
I have the following function:
$$f(\omega) = arctan(\frac{-\omega*\frac{R}{L}}{-w^2 + \frac{1}{C*L}})$$
When I try to plot it (`atan(f(w)`), I get the following:
[![enter image description here][1]][1]
This is clearly not a `atan` plot. I investigated a bit and found `atan2`. When I use it instead (`atan2(-... |
I understand that the gradient of a vector field is a rank two tensor. I am having trouble understanding the RHS of the below equation for the stress field generated in a liquid crystal:
$$ \sigma_{\alpha\beta}^{elastic} = - \kappa \partial_{\alpha}P_{\gamma}\partial_{\beta}P_{\gamma} $$
Here, $\kappa$ is a scal... |
Expected number of picks from normal distribution such that the sum exceeds ? Does the value converge? |
I have the following function:
$$f(\omega) = arctan(\frac{-\omega*\frac{R}{L}}{-w^2 + \frac{1}{C*L}})$$
When I try to plot it (`atan(f(w)`), I get the following:
[![enter image description here][1]][1]
This is clearly not a `atan` plot. I investigated a bit and found `atan2`. When I use it instead (`atan2(-... |
Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ where $n$ is an integer is $1 \frac{1}{3}$?
I know this is true because for $n = 8 + 6i$ where $i$ is an integer $\geq 0$, the difference is $1 \frac{1}{3}$. Plugging other values for $n$ gives a difference that is less tha... |
For $n \geq 3$, the dihedral group $D_n =\langle r, s \rangle$, where $r ^n = s^2 = e$.
Within this group, we can distinguish two types of elements:
1. Those of the form $r^i$, where $i$ is any integer
2. Those of the form $sr^i$, where $i$ is any integer
The latter includes not only $sr^i$ but _any_ element ... |
How can we distinguish elements of $D_n$ that include reflections versus those that don't? |
**Base Case $n=1$:** if $ax+b=0$ for all values of $x$, then $a=b=0$
\begin{align}
ax+b=0\\
(0)+b=0\\
b=0\\\\
ax+(0)=0\\
ax=0\\
a(1)=0\\
a=0
\end{align}
**Inductive Step:** if $(a_nx^n+a_{n-1}x^{n-1}+...+a_1x^1+a_0=0)\Rightarrow(a_n=a_{n-1}=...=a_0=0)$ for all values of $x$, then $(a_{n+1}x^{n+1}+a_{n... |
Is This Proof For $a_nx^n+a_{n-1}x^{n-1}+...+a_1x^1+a_0=0\Rightarrow a_n=a_{n-1}=...a_1=a_0=0$, $\forall n$ Okay? |
**Can I truncate the intermediate values without losing absolute accuracy?**
Short answer: **no**.
Not so short answer: Regarding add (and substraction) *absolute* errors sum up, for multiplication (also dividing) *relative* errors sum up. While only interested in a result of 20 bits you may calculate what trunca... |
Let's consider the following equation:
$$
a_2.f(t+2) + a_1.f(t+1) + a_0.f(t+0) = g(t)
$$
where $a_0,a_1,a_2$ are non zero reals and $g(t)$ is a known function.
Although it looks like an ordinary differential equation of order 2, the "changing" parts on the left part of this equation doesn't involve derivatives b... |
I'm just wondering if you can solve the first order PD wave-equation similar to the second order one by separation. For me however it turns out weird.
Having
$$\mathrm{\partial_t u(x,t) = -c\cdot \partial_x u(x,t)}$$
one might separate $\mathbf{u}$ into separate function of space (x) and time (t)
$$\mat... |
There are two ways to answer your question, first through different ideas of what 'foundation' might mean, and second about multiple different foundations of the same kind.
- What does foundation mean? Set theory (and logic) are attempts at foundations of mathematical knowledge about __how we can know__ mathematical... |
In calculus, we have the partial derivatives such as
\begin{align}
\frac{\partial(x^2-y)}{\partial x}(x,y),\ \ \ \frac{\partial(x+e^y)}{\partial y}(x,y),\ \ \ \frac{\partial(x+y)}{\partial(x+y)}(x,y)\ \ ...
\end{align}
But I wonder if these sort of derivatives
\begin{align}
\frac{\partial(ax+by)}{\partial(cx+dy)... |
Let function $u:[0,1]^{N+1} \mapsto \mathbb{R}$ be continuously differentiable.
Let $Y_1,\cdots,Y_{N}$ be random variables with support on $[0,1]$ with the continuously differentiable joint PDF $f$.
$$v(x)\equiv Pr(u(x,Y_1,\cdots,Y_{N})<0).$$
There is $y^* \in (0,1)$ such that for $Y_1\gtrless y^*$,
$$
u(0,... |
Let $r=x$. We can write the ODE as
$$
x\,\frac{d}{dx}\left[x\cdot x\,\frac{d\phi}{dx}\right]
=-K\phi\,x\,\frac{d}{dx}\left[\frac{x}{\phi^2}\cdot x\,\frac{d\phi}{dx}\right]
$$
Substituting $x=e^t,x\,\frac{d}{dx}=\frac{d}{dt}$ results in, after simplification
$$
\phi''+\phi'=-K\left[\frac{\phi'\phi+\phi''\p... |
Problem:
Given that $y = e^{2x}$ is a solution of
$$ (2x+1)y'' - 4(x+1)y' + 4y = 0 $$
find a linearly independent solution by reducing the order. Write the general solution.
Answer:
Let $y = e^{2x}v$. We have:
\begin{align*}
\dfrac{dy}{dx} &= e^{2x} \dfrac{dv}{dx} + 2 e^{2x} v \\
\dfrac{d^2y}{dx^2} &=
... |
I have been reading the paper "Heegner points and derivatives of $L$-series" by Gross and Zagier.
[Link to the paper.](https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01388809/fulltext.pdf)
In section III of the paper, they use intersection theory to express a local height at a non-archimedean place in... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
-------
**Base Case $n=1$:** if $ax+b=0$ for all values of $x$, then $a=b=0$
\begin{align}
ax+b=0\\
(0)+b=0\\
b=0\\\\
ax+(0)=0\\
ax=0\\
a(1)=0\\
a=0
\end{align}
**Inductive Step:** if $(a_nx^n+a_{n-1}x^{n-1}+...+a_1x^1+a_0=0)\Rightarrow(a_n=a_{n-1}=...=a_0=0)$ for all values of $x$, then $(a_{n+1}x^{... |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
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