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Suppose I have two spheres of arbitrary size $S_1$ and $S_2$ and I intersect them so they overlapped some extent.
What kind of shape would their intersection create? In other words, what does $S_1 \cap S_2$ look like?
---------
EDIT:
Someone has suggested this answer which does seem relevant.
https://... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
What shape does the intersection of two spheres yield? How do you find the center and radius of this circle? |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both dense in $Y$.
- Show that $f$ is not a pointwise limit of continuous functions $X \to
Y$.
- Show that in the above s... |
Let $f: X\to Y$ be a continuous surjective map ( possibly open ) between topological spaces. Let $\{ C_i \}_{i}$ be the components of $X$ and $\{ D_j \}_j$ the components of $Y$. Then for each $j$,
$$f^{-1}(D_j) = \bigcup_{f(C_i) \subseteq D_j} C_i $$
? If so, why?
If this question is true, then we can sho... |
For continuous surjective map $f: X\to Y$, $f^{-1}(D_j) = \bigcup_{f(C_i) \subseteq D_j} C_i $ ( Connected components )? |
Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, k=1,...,n-1$, where $0 < c < 2/(n-1)$.
For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized?
Comment: I think solving this may be sufficient to solve an open probl... |
We are now evaluating the following integral:
$$
\int_{0}^{\infty} \frac {\ln \left( 2 {e}^{x} - 1 \right)}{{e}^{x} - 1} \text {d} x = \, ?
$$
My first observation is
$$
2 {e}^{x} - 1 = {e}^{2 x} - {\left( {e}^{x} - 1 \right)}^{2}.
$$
I am not sure if this is relevant. Anyway, it looks like ${e}^{x} \mapsto x$ ... |
Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, k=1,...,n-1$, where $0 < c < 2/(n-1)$.
For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized?
|
The heat kernel on a manifold $M$ is the solution of
$$(\Delta_x-\partial_t)H(x,y,t)=0,\qquad \text{in}~ M\times M\times (0,+\infty)$$
with initial data $\lim_{t\to 0}H(x,y,t)=\delta_x(y)$ in the sense of distribution, where $\delta_x(y)$ is the point mass at $y.$
I already know that the heat kernel on $\mathbb{... |
Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, k=1,...,n-1$, where $0 < c < 2/(n-1)$.
For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized?
I believe this question should be enough to solve an open problem. |
Jacobi and Madden found infinitely many primitive solutions to,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$
using an elliptic curve. We will use a different approach that, like the method for,
$$a^4+b^4+c^4 = d^4$$
discussed in [this post][1], also yields **pairs** of solutions.
---
**I. The system**
Define,... |
I can show that if $\theta$ is a WFF, then $(\theta)$ is not a WFF. Suppose $ (\theta )$ is a WFF. Then by unique readability, we know $(\theta) \equiv \neg\alpha$ or $(\theta) \equiv (\alpha \square \beta)$ for some formulae $\alpha$ and $\beta$ and $\square$ being one of the four binary connectives. And as $\neg$ is... |
So I'm trying to put together a clearer picture of what is going on when we study set theory. I'll describe my current picture which I'd appreciate some feedback on, and I'll ask some specific questions as well.
So from the start: if we take a Platonist perspective (which I was taught as the most pedagogically effec... |
We are now evaluating the following integral:
$$
\int_{0}^{\infty} \frac {\ln \left( 2 {e}^{x} - 1 \right)}{{e}^{x} - 1} \text {d} x = \, ?
$$
My first observation is
$$
2 {e}^{x} - 1 = {e}^{2 x} - {\left( {e}^{x} - 1 \right)}^{2}.
$$
I am not sure if this is relevant. Anyway, it looks like ${e}^{x} \mapsto x$ ... |
Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, k=1,...,n-1$, where $0 < c < 2/(n-1)$.
For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized?
Comment: I think I reduced a bigger problem I've been trying to solve ... |
Let $f: X\to Y$ be a continuous surjective map ( possibly open ) between topological spaces. Let $\{ C_i \}_{i}$ be the components of $X$ and $\{ D_j \}_j$ the components of $Y$. Then for each $j$,
$$f^{-1}(D_j) = \bigcup_{f(C_i) \subseteq D_j} C_i $$
? If so, why?
If this question is true, then we can sho... |
I have just read a [paper][1] about fractional linear functions written by Justin Lanier. I have tried to prove some observations he left in page 13. Here is the statement I want to prove:
>Given function $f(x)=\frac{ax+b}{cx+d}$ ,then
>
>$$f^{\circ n}(x)=x \iff \frac{a+d}{2\sqrt{ad-bc}}=\cos(\frac{k\pi}{n}), 1\le... |
Here is an exercise of Section 1 of Firedman's *Algebraic Surfaces and Holomorphic Vector Bundles*.
>Let $C$ be an irreducible curve whose singular locus is a single ordinary double point $x$, and let $p,q\in\tilde{C}$ be the preimage under the normalization map $\nu$ of $x$. Show that $\nu^*\omega_C=K_{\tilde{C}}\... |
Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, k=1,...,n-1$, where $0 < c < 2/(n-1)$.
For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I think I reduced a bigger problem I've b... |
The heat kernel on a smooth manifold $M$ is defined to be the solution of
$$(\Delta_x-\partial_t)H(x,y,t)=0,\qquad \text{in}~ M\times M\times (0,+\infty)$$
with initial data $\lim_{t\to 0}H(x,y,t)=\delta_x(y)$ in the sense of distribution, where $\delta_x(y)$ is the point mass at $x.$
I have already known that t... |
$\color{brown}{\textbf{What do we prove?}}$
Really, our goal is to prove the inequality
$$\ln(\ln(500\cdot(\sqrt5+1)))<2.$$
We can get the intermediate levels from tables:
$$e^3\approx 20.085\,536\,923\,19,$$
$$e^6\approx 403.428\,793\, 432\,7,$$
$$500\cdot(\sqrt5+1)\approx 1618.033\,988\,75,$$
$$\ln(500... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{x-1}\}$ that are primes and $1$ which are smaller than $n$.
(in which $p_{x-1}$ is the prime directly before n)
1. Can you find at most 2 distinct $a, b \in S$ (... |
Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, k=1,...,n-1$, where $0 < c < 2/(n-1)$.
For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I have spent hours trying to prove the so... |
Introduce non-negative integers $a,b,c$ as auxiliary variables
Put $\;x = w+a,\;y = w+a+b,\; z=w+a+b+c,\;$ to get
${5w + 3(w+a) + (w+a+b) +(w+a+b+c)}$
$\rightarrow 10w + 5a +2b+c =100$
Now you can use generating functions to solve it
The answer will be the coefficient of $x^{100}\;$ in expansion of $\f... |
Assume that $X,Y,Z$ are three hypercubes with the same dimension $k$. Specifically, we have $X=\{x_1,x_2,...,x_k:0\leq x_i\leq c_i, \forall\,i\}$ and $Y=\{y_1,y_2,...,y_k:0\leq y_i\leq b_i, \forall\,i\}$ and $Z=\{z_1,z_2,...,z_k:0\leq z_i\leq a_i, \forall\,i\}$ where each $c_i$ and $b_i$ and $a_i$ are constant positive... |
Convex hull of union of extreme points is equivalent to convex hull of extreme point of a union? |
This is problem `6.3.8` from *Trigonometry: A Complete Introduction*, by Hugh Neill (ISBN 1473678498). It's in the surveying chapter, which is a practical continuation of the sine and cosine rules of the chapter before that. So I'm assuming you don't need much more than those two rules. It's a simple problem, but I'm ... |
Assume that $X,Y,Z$ are three hypercubes with the same dimension $k$. Specifically, we have $X=\{x_1,x_2,...,x_k:0\leq x_i\leq c_i, \forall\,i\}$ and $Y=\{y_1,y_2,...,y_k:0\leq y_i\leq b_i, \forall\,i\}$ and $Z=\{z_1,z_2,...,z_k:0\leq z_i\leq a_i, \forall\,i\}$ where each $c_i$ and $b_i$ and $a_i$ are constant positive... |
Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, k=1,...,n-1$, where $0 < c < 2/(n-1)$.
For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I've spent hours trying to prove the solu... |
I have a vector of four elements. The elements are named `x, y, z, w`:
```cpp
Vec4(x, y, z, w)
```
Now I have another vector of eight elements:
```cpp
Vec8(x, y, z, w, ?, ?, ?, ?)
```
Now how do you name the additional elements? What's the best practice for naming? |
This is problem `6.3.8` from *Trigonometry: A Complete Introduction*, by Hugh Neill (ISBN 1473678498). It's in the surveying chapter, which is a practical continuation of the sine and cosine rules of the chapter before that. So I'm assuming you don't need much more than those two rules. It's a simple problem, but I'm ... |
Suppose that x + y + z = 203, what would be the maximum value of x + (xy)^(1/2) + (xyz)^(1/3)? The solution is to assume that 4/3(x+y+z)≥x+(xy)^(1/2)+(xyz)^(1/3)and prove it with AM-GM inequality. I was just wondering how would one come up with that assumption normally? And if there's a general formula for larger cases... |
Assume that $X,Y,Z$ are three hypercuboid with the same dimension $k$. Specifically, we have $X=\{x_1,x_2,...,x_k:0\leq x_i\leq c_i, \forall\,i\}$ and $Y=\{y_1,y_2,...,y_k:0\leq y_i\leq b_i, \forall\,i\}$ and $Z=\{z_1,z_2,...,z_k:0\leq z_i\leq a_i, \forall\,i\}$ where each $c_i$ and $b_i$ and $a_i$ are constant positiv... |
What is the number of non negative integral solutions of
$5w+3x+y+z=100$, such that $w≤x≤y≤z$.
I have the answer key to this but I do not understand how to even start. The second inequality condition is what is troubling me the most. Are there specific ways to approach this?
Edit:
I have a solution of this but ... |
I have to study the convergence of this integral:
$\int_{0}^{1} \frac{1}{\sqrt x - \sqrt[5]{x}} dx$
I rewrite it as:
$\int_{0}^{a} f (x) dx + \int_{a}^{1} f (x) dx$ (with $0<a<1$)
in order to deal with the problems in 0 and 1 separately.
For the first one, since $ \sqrt x = o(\sqrt [5] x )$ , for $x \... |
This is problem `6.3.8` from *Trigonometry: A Complete Introduction*, by Hugh Neill (ISBN 1473678498). It's in the surveying chapter, which is a practical continuation of the sine and cosine rules of the chapter before that. So I'm assuming you don't need much more than those two rules. It's a simple problem, but I'm ... |
We have a set of coins with values 1, 2, 5, and 10, and banknotes with values 5 and 10. We want to pay a bill of 15 using a combination of these coins and banknotes. How many ways can we pay the bill?
-----
I attempted to formulate the problem as follows:
$$x_1+2x_2+5x_3+10x_4=15,\quad x_i\ge0$$
and then pr... |
Great question! Many paradoxes exploit self-reference to be formulated. A classic example is the liar's paradox: how do we assign a truth value to the sentence "This sentence is not true"? By referring to itself, we get this circular problem by which the sentence must be false if it is true, but true if it is false. Ru... |
Define $$f_n(x_1,x_2,x_3,x_4)=\sum_{s_1=0}^n a_{n,s_1,s_1,s_1,s_1} x_1^{s_1}x_2^{s_1}x_3^{s_1}x_4^{s_1}+\sum_{s_i\neq s_j, \forall i\neq j,s_i\in[0,n]} a_{n,s_1,s_2,s_3,s_4} x_1^{s_1}x_2^{s_2}x_3^{s_3}x_4^{s_4} $$ where $x_i\in (0,1)$ $\forall i$ and $$a_{n,s_1,s_2,s_3,s_4} =(-1)^{s_1+s_2+s_3+s_4}\binom{n}{s_1}\binom{n... |
The following are excerpts from Folland:
Proposition 2.12:
>2.12 Proposition. Let $(X, \mathcal{M}, \mu)$ be a measure space and let $(X, \bar{\mathcal{M}}, \bar{\mu})$ be its completion. If $f$ is an $\mathcal{M}$-measurable function on $X$, there is an $\bar{\mathcal{M}}$-measurable function $g$ such that $f = g$ $... |
Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it **rejects** *any* and *every* non-standard natural number.
Since PA is $ \boldsymbol \omega $-consistent, therefore any su... |
Convex hull of union over extreme points of hypercuboids is equivalent to convex hull of extreme point of a union over hypercuboids? |
I'm not sure if Folland did assume (implicitly or explicitly) that the measure space it complete, but based on the presence of Proposition $2.12$, I'm guessing he did (although he may not have said to explicitly). Other things you're quoting here seem to confirm that. However, let's cover why, regardless of what Follan... |
Given an inner product on a vector space, I can always define a norm and a metric (and a topology using that metric). Is the converse true? That is, given a metric on a vector space, can I define a norm and an inner product? And what about a metric space that is not a vector space? |
I am trying to solve Weibel Exercise 1.4.4 which asks us to show that a split chain complex is homotopy equivalent to its homology.
Setup:
A. Given a chain complex $C$, its homology is the chain complex $H_*(C):=\{H_n(C),0\}$ ie the differentials are all zero.
B. A chain complex being split means there is a... |
When $X$ is a topological space, any sheaf $S$ on $X$ is a connected object of the category of presheaves $\widehat{\mathcal{O}(X)}$ in the sense that
$\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A+B) \cong \mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A)+\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,B)$,
for all $A,B \i... |
I am studying Jordan forms. I see that Hoffman & Kunze define the companion matrix of a polynomial $a_0+a_x+\dots+a_{k-1}x^{k-1}+x^k$ as
$$ \begin{pmatrix} 0 & 0 & 0 & \dots &0 & -a_0\\ 1 & 0 & 0 & \dots & 0 & -a_1\\ 0 & 1 & 0 & \dots & 0 & -a_2 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 1... |
I am aware that in a first countable space (and thus any metric space) is completely determined by its convergent sequences *and* their limits, *i.e.*,
> If $\tau_1$ and $\tau_2$ are two first countable topologies on a set $X$ such that $x_i\to c$ in $\tau_1$ iff $x_i\to c$ in $\tau_2$, then $\tau_1 = \tau_2$.
Ho... |
Let $Q \in \mathbb{C}^{m \times m}$ be a given positive (semi)-definite matrix such that
$Q = CC^H,$
for some known matrix $C \in \mathbb{C}^{m \times r}$ having full column rank and orthogonal columns.
Suppose we decompose $Q$ as $Q = BB^H$, where the matrix $B \in \mathbb{C}^{m\times r}$ is full column rank. Th... |
Why $lim_{h→0+}\frac{ f (x + 2h) − f (x + h)}{h}$ is not a derivative? |
This post directly follows this [question](https://math.stackexchange.com/questions/4862992/using-euler-lagrange-equations-show-that-the-following-eom-can-be-written-as-d) and is very similar in nature:
>Consider the following Lagrangian:
>$$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\... |
I am trying to find out the Eisenstein integers that have norm values of 3,5,7....
I want to see if there is some pattern.
For the norm value of 3, I was able to find six Eisenstein integers.
For the norm value of 7, I was able to find ten Eisenstein integers.
I am unable to find even a single Eisenstein integ... |
I'll take a crack at this.
Any time you make a logical argument, you maybe presume some things that are true (your *axioms*) and always presume some rules about how you can work with true premises to reach true conclusions (your *rules*). Together, they form a *theory*.
Maybe I'm doing some set theory, and I'm wo... |
Why $\lim_{h→0+}\frac{ f (x + 2h) − f (x + h)}{h}$ is not a derivative? |
I'll take a crack at this.
Any time you make a logical argument, you maybe presume some things that are true (your *axioms*) and always presume some rules about how you can work with true premises to reach true conclusions (your *rules*). Together, they form a *theory*.
Maybe I'm doing some set theory, and I'm wo... |
I can't find a proper formula to prove that
$$(p\rightarrow q) \rightarrow ((r ∨ p) \rightarrow (r ∨ q)) $$
Is a tautology; considering it's almost exclusively made up of implications. Can somebody help me?
just I want to know how to simplify it.
I tried [this][1].
[1]: https://i.stack.imgur.com/vUvUn.... |
[OEIS A169945](https://oeis.org/A169945) introduces the concept of _thickness_ of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(x)$ with coefficients $0$ or $1$ and degree at most $n$, it denotes by $T(n,k)$, the numbe... |
Is there some way to calculate the series for arbitrary $p,n\in \mathcal{R}$
$$
\phi(p,n)=\sum_{N=-\infty}^{\infty}\sum_{k=0}^{\infty}(-1)^{2k+N}\frac{1}{k!(k+N)!} \frac{1}{(2k+N-1)!}\left(\frac{n}{4\pi}(p-\frac{4\pi N}{3})\right)^{2k+N-1}\text{sgn}(p-\frac{4\pi N}{3})
$$
I guess it converge because it converges fa... |
The definition of a Gaussian process given by Rasmussen and Williams (RW) in $\textit{Gaussian Processes for Machine Learning}$ is
Definition: $\textit{A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution}$
Comparing this to other sources for exam... |
Is the collection of random variables in a Gaussian Process infinite? |
If I have the stochastic integral
$$
\int_0^T f(t)\,\mathbb{d} W_t
$$
and perform the change of variables $t = \varphi(s)$, how will $\mathbb{d} W_t$ transform (where the integration is taken in the Itô's sense and $W_t$ is a Wiener process)?
PS: for deterministic integrals
$$
\int_{\varphi(a)}^{\varphi(b)} f... |
I am trying to count all nilpotent matrices in $M_n(\mathbb R)$ upto similarity.
I did the same exercise for idempotent matrices and it was quite simple. I realised that rank of an idempotent matrix is a positive integer and two idempotent matrices are similar iff they have the same rank. So I got the answer $n+1$.... |
How many nilpotent matrices are there in $M_n(\mathbb R)$ up to similarity? |
Question Q1 :
A sum of squares is $>0$ unless all the squares are zero, in which case the sum is zero.
It is known that if a quadratic expression remains always $>0$, it cannot have roots, therefore its discriminant is $<0$.
What happens in the exceptional case where all the squares are zero ? (in which case... |
I am trying to count all nilpotent matrices in $M_n(\mathbb R)$ upto similarity.
I did the same exercise for idempotent matrices and it was quite simple. I realised that rank of an idempotent matrix is a positive integer and two idempotent matrices are similar iff they have the same rank. So I got the answer $n+1$.... |
Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it **rejects** *any* and *every* non-standard natural number.
Since PA is $ \boldsymbol \omega $-consistent, therefore any su... |
I found an identity
$\int_{-\infty}^{\infty}\binom{n}{x}dx=\sum_{i=0}^{n} \binom{n}{i}$
where LHS can be calculated by
https://mathworld.wolfram.com/ReflectionRelation.html
and
https://en.wikipedia.org/wiki/Dirichlet_integral
the result is $2^n$, which is apparently equal to RHS.
This identity is in the f... |
I am studying *Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan*. The following is the beginning of Appendix D:
[![enter image description here][1]][1]
I could not understand the last sentence (the one contains D.2.). It's like a theorem that is well known in Fourier Analy... |
I am studying Jordan forms. I see that Hoffman & Kunze define the companion matrix of a polynomial $a_0+a_1x+\dots+a_{k-1}x^{k-1}+x^k$ as
$$ \begin{pmatrix} 0 & 0 & 0 & \dots &0 & -a_0\\ 1 & 0 & 0 & \dots & 0 & -a_1\\ 0 & 1 & 0 & \dots & 0 & -a_2 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \dots & ... |
The following graph consists of 50 vertices, with 25 vertices in each circle, and each vertex having a degree of 3. How many ways can we select 23 vertices from this graph such that they are not adjacent?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/EvVs6.png
Do you have any sugge... |
I need to solve a stationary density distribution of a Fokker Planck equation
$-\nabla \cdot(\mu \nabla V) + \frac{1}{2}\sigma^2 \Delta\mu = 0 $ where $\mu$ is the stationary density of the SDE:
$dX = -\nabla Vdt+\sigma dW_t$
where $\mu \ge0, \int_\Omega\mu dx= 1$
I tried to solve this by Finite element meth... |
I'll take a crack at this.
Any time you make a logical argument, you maybe presume some things that are true (your *axioms*) and always presume some rules about how you can work with true premises to reach true conclusions (your *rules*). Together, they form a *theory*.
Maybe I'm doing some set theory, and I'm wo... |
Let $E = R_2[X]$.\
Let $a \in \mathbb{R}$.\
Let $ev_a : E \rightarrow\mathbb{R}, P \mapsto P(a)$.\
Let $P = a + bX + cX^2$.
Show that $B = (ev_1, ev_2, ev_3)$ is a base of $E^*$.
---
We have:
- $ev_1(P) = P(1) = a + b + c$
- $ev_2(P) = P(2) = a + 2b + 4c$
- $ev_3(P) = P(3) = a + 3b + 9c$
So let $A = \... |
How duality works with polynomials? |
How to test if the differential equation
$$(1+x)y'''+y y''-y'^2+(2+x)y''+y'=0,$$
$$y(0)=1, y'(0)=1, y'(\infty)=0$$
Has an analytic solution? Solution that equals its Taylor series? |
How to test if a certain differential equation has an analytic solution (solution has Taylor expansion)? |
I have to do the following exercise:
> Consider the endomorphism $h\colon\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$ with unique eigenvalues $0,\,2,\,8$. If $h$ is not diagonalizable and we know that dim$S(0)=1$, dim$S(2)\geq2$, dim$S(8)\geq2$, find all the possibilities for the elementary divisors and invariant factor... |
Let $(X,B)$ be a klt pair where $K_X+B$ is $\mathbb{R}$-Cartier. Let $\pi:X\rightarrow U$ be a projective morphism of quasi-projective varieties. Assume either that $B$ is $\pi$-big and $K_X+B$ is $\pi$-pseudo-effective, or $K_X+B$ is $\pi$-big, then the work of Birkar-Cascini-Hacon-McKernan proves that the log canonic... |
I encountered a problem while working on a mathematical analysis exercise. The problem is as follows: we need to determine the initial value condition $y(x_0)=y_0$ for the ordinary differential equation (ODE) $y'=y^4-x^8$ in order to find a global solution, i.e., a solution defined for all $x$ in the set of real number... |
Lets try to use index notation:
$$
y_i = {A^p}_i x_p
$$
So, for example,
$$
\begin{bmatrix}
y_1 \\ y_2
\end{bmatrix}
=
\begin{bmatrix}
{A^1}_1 & {A^2}_1 \\
{A^1}_2 & {A^2}_2
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
$$
Now we write the differential,
$$
dy_i = d{A^p}_i x_p = \delta^p_{p... |
> Let $(W_t)_{t\ge 0}$ be a Wiener process in $\mathbb R^d, d \ge 1$. For which $d$ holds $\lim_{t\to \infty} |W_t|=\infty$? Provide complete proof of this property/lack of this property for those $d$ for which it holds/does not hold.
I have no idea what properties of the Wiener process I could use to solve this pro... |
For which $d$ holds $\lim_{t\to \infty} |W_t|=\infty$? |
It is well-known that a morphism $\varphi : X \to Y$ of affine varieties $X, Y$ is dominant if and only if the induced homomorphism $\varphi^* : A(Y) \to A(X)$ is injective.
But I'd like to show that this is true for quasi-affine case.
>a morphism $\varphi : X \to Y$ of **quasi-affine** varieties $X, Y$ is domi... |
Lets try to use index notation:
$$
y_i = {A^p}_i x_p
$$
So, for example,
$$
\begin{bmatrix}
y_1 \\ y_2
\end{bmatrix}
=
\begin{bmatrix}
{A^1}_1 & {A^2}_1 \\
{A^1}_2 & {A^2}_2
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
$$
Now we write the differential,
$$
dy_i = d{A^p}_i x_p = \delta^p_{p... |
Let $\mathfrak{g}_1\subseteq\mathfrak{g}_2$ be an inclusion of reductive Lie algebras $\mathfrak{g}_1$ and $\mathfrak{g}_2$ over $\mathbb{C}$.
Is it always possible to find Cartan subalgebras $\mathfrak{h}_1\subseteq \mathfrak{g}_1$ and $\mathfrak{h}_2\subseteq \mathfrak{g}_2$ such that $\mathfrak{h}_1\subseteq \mat... |
Let $L(\mathbb F_2)$ be the group von Neumann algebra of the free group on two generators. The interpolated free group factors of Dykema and Radulescu are defined as $L(\mathbb F_r)=L(\mathbb F_2)^{\sqrt{r-1}}$ for $t>1$, where $M^s$ denotes the amplification of the $\mathrm{II}_1$ factor $M$. This notation is consiste... |
Problem: Find all values of n where 2n^2+7n+3 is prime.
What I know:
All prime numbers are only divided by one and themselves.
To reach a solution, I need to factor the polynomial and equal the binomials to 1.
I did it: (2n+1)=1 and (n+3)=1, and the solutions are n=0 and n=-3
Then n=0 is the only viable solutio... |
Problem: Find all values of $n$ where $2n^2+7n+3$ is prime.
What I know:
All prime numbers are only divided by one and themselves.
To reach a solution, I need to factor the polynomial and equal the binomials to 1.
I did it: $2n+1=1$ and $n+3=1$, and the solutions are $n=0$ and $n=-3$
Then $n=0$ is the only viab... |
Given a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, P)$ and with notation (regular conditional probabiltiy assumed): $$P^{X|\mathcal{F}_t}(B)=P(X\in B|\mathcal{F}_t)=E(1_B(X)|\mathcal{F}_t),$$ how can I prove that for a real-valued measurable function $f$ and a $\mathcal{F}_t$-measurable rando... |
Why is $E(f(X,Y)|\mathcal{F}_t)=\int f(x,Y)P^{X|\mathcal{F}_t}(dx)$? |
I have to do the following exercise:
> Consider the endomorphism $h\colon\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$ with unique eigenvalues $0,\,2,\,8$. If $h$ is not diagonalizable and we know that dim$S(0)=1$, dim$S(2)\geq2$, dim$S(8)\geq2$, find all the possibilities for the elementary divisors and invariant factor... |
Introduce non-negative integers $a,b,c$ as auxiliary variables
Put $\;x = w+a,\;y = w+a+b,\; z=w+a+b+c,\;$ to get
${5w + 3(w+a) + (w+a+b) +(w+a+b+c)}$
$\rightarrow 10w + 5a +2b+c =100$
Now you can use generating functions to solve it
The answer will be the coefficient of $x^{100}\;$ in expansion of $\f... |
Let $(X,B)$ be a klt pair where $K_X+B$ is $\mathbb{R}$-Cartier. Let $\pi:X\rightarrow U$ be a projective morphism of quasi-projective varieties. Assume either that $B$ is $\pi$-big and $K_X+B$ is $\pi$-pseudo-effective, or $K_X+B$ is $\pi$-big, then the work of Birkar-Cascini-Hacon-McKernan proves that the log canonic... |
I am currently working on this problem:
Prove that if p is prime, then among 1!,2!,...,(p-1)! one can find at least (p-1)/2 different remainders modulo p. I don't really have any clue how to prove it. Can anyone help me? |
The Lah numbers satisfy the following rising-falling factorials relation
\begin{equation}
x^{\overline{n}} = \sum_{k=0}^{n} L(n, k) x^{\underline{k}},
\end{equation}
where $x^{\overline{n}}$= $x(x+1)\times ... \times (x+n-1)$ and $x^{\underline{n}}$ = $x(x-1) \times ... \times (x-n+1)$.
What would be the ... |
I this proof of the Lah Numbers Polynomial Identity With Induction on $n$ correct? |
I am currently working on this problem:
Prove that if p is prime, then among $1!,2!,\dots,(p-1)!$ one can find at least $\frac{p-1}{2}$ different remainders modulo $p$. I don't really have any clue how to prove it. Can anyone help me? |
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