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Given matrix A with $A_{i,j} = f(i,j)$, how can I find out if A is positive semidefinite? |
Suppose you have a linear map between two (countable) infinite dimensional vector spaces, a matrix $A$ representing this map and an vector $v$ in the appropriate vector space. Is something like $$\lim_{n \rightarrow \infty} (A^n \cdot v) = (\lim_{n \rightarrow \infty} A^n)\cdot v$$
correct, if I know that both limit... |
Is matrix-vector multiplication continuous in infinite-dimensional vector spaces? |
Transform the set of causes of the given formula are below:
$\overline{x}\lor y.....(1)$
$\overline{y}\lor x.....(2)$
$\overline{z}\lor w.....(3)$
$\overline{w}\lor z.....(4)$
$\overline{s}\lor \overline{t}.....(5)$
$s\lor t.....(6)$
$\overline{x}\lor \overline{z}\lor\overline{s}.....(7)$
$\o... |
Let's first count *ordered* basis.
You can pick the first vector in the basis any way you want, as long as it's not the zero vector. That gives you $q^n-1$ possibilities.
The second vector should not be any of the $q$ multiples of the first vector. So that gives you $q^n-q$ possibilities.
The third vector sho... |
Let $X$ be a Brownian motion. Then a corollary of the iterated logarithm law says that:
$$\limsup_{s\rightarrow +\infty} \frac{X_s}{(2s\log\log s)^{1/2}} = 1 \quad a.s. \tag{3.6}$$
$$\liminf_{s\rightarrow +\infty} \frac{X_s}{(2s\log\log s)^{1/2}} = -1 \quad a.s.\\ \tag{3.7}$$
With help from an earlier question I p... |
>Let be $K$ a non-empty compact subset of $\mathbb{R}^n$, where $n>1$. Show that $S:=\mathbb{R}^n\setminus K$ is connected.
As $K$ is bounded there exists a real number $c\geq 0$ such that $\Vert x\Vert\leq \frac{c}{2}$ for all $x\in K$.
Of course, we can approach this problem by choosing two arbitrary points $a,... |
Hello I'm studying the moderate effect of one continuous variable, family income, on the relationship between gender (dummy variable) and whether attending school (dummy variable).
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/X9VFQ.png
For the first model, I only put the dummy var... |
Another way to solve this problem is as follows. Place $A,B,C,D$ on the $xy$ plane. Let $ a = AB , b = BC , c = CD, d = DA $, and let $V$ be the apex with a $z$ coordinate that is positive. Let $ e= VA, f = VB, g = VC, h = VD $.
Now express the coordinates of all the points using three angles. Angles $\theta$ an... |
A function $X:(\Omega , \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{B}) $ is called a measurable function(or random variable with probability $P$) if
$X^{-1} (B)= \{ \omega \in \Omega : X(\omega) \in B\ \}$ $\in \mathcal{F}$ for all $B \in \mathcal{B}$
Here, $\mathcal{B}$ is a Borel $\sigma$ -algebra and $\ma... |
Suppose we have a system of linear diophantine equations over non-negative integers:
$$
\left\lbrace\begin{aligned}
&Ax=b\\
&x\in \mathbb{Z}^n_{\geq0}
\end{aligned}\right.
$$
where $A$ is a matrix of size $m\times n$, $a_{ij}\in \mathbb{N},~b\in \mathbb{N}$.
Let $S_{b'}$ be the set of solutions of this system... |
Why are the following two equations equivalent? |
So the question is to find a family of functions, f_n, from [0,1] to the complex plane such that there is pointwise convergence to some function f BUT that the integral over [0,1] of f does not equal the limit as n goes to infinity of the integral of f_n. I came up with the function e^i(2*pi
*x^n) as my family of fun... |
>Let be $K$ a non-empty compact, convex subset of $\mathbb{R}^n$, where $n>1$. Show that $S:=\mathbb{R}^n\setminus K$ is connected.
As $K$ is bounded there exists a real number $c\geq 0$ such that $\Vert x\Vert\leq \frac{c}{2}$ for all $x\in K$.
Of course, we can approach this problem by choosing two arbitrary po... |
I'm working on a problem whose solution relies on finding if an arbitrary matrix A is positive semi-definite. A is real valued, square and symmetric, and each of its entries are given by:
$A_{ii} = f(i)$
$A_{ij} = A_{ji} = f(i,j)$
The problem is that this is part of an induction proof, where the size N of the ... |
I'm working on a problem whose solution relies on finding if an arbitrary matrix A is positive semi-definite. A is real valued, square and symmetric, and each of its entries are given by:
$A_{ii} = f(i)$
$A_{ij} = A_{ji} = f(i,j)$
The problem is that this is part of an induction proof, where the size N of the ... |
I'm trying to understand a calculation on the second page of https://sma.epfl.ch/~filipazz/notes/adjunction_and_inversion_of_adjunction.pdf, and I have a couple questions.
Here is the setup. $X$ is the cone of a projectively normal rational curve of degree $n$, $f:Y\to X$ is the resolution of singularities we get by... |
Suppose $x_n$ and $y_n$ are a sequence of real numbers such that
$$\limsup_{n\rightarrow \infty} \frac{x_n}{y_n} = 1 \quad \mathrm{and} \quad \liminf_{n\rightarrow \infty} \frac{x_n}{y_n} = -1.$$
Intuitively this says that as $n \rightarrow \infty$ the quotient $x_n/y_n$ remains within the the interval $(-1, 1)$. ... |
Why is $(x,z)$ not principal in $k[x,y,z]/(xy-z^2)$? |
Fix a vector $\vec{b}$ and a positive definite (but not necessarily symmetric) matrix $A$, can we prove that the fraction $$
\frac{\vec{b}^TA\vec{y}}{\vec{b}^T\vec{y}}
$$
always has the same sign (If $\vec{b}^T\vec{y}\neq 0$)? |
I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are...
Are they full subcategory of algebraic stacks? Are they full subcategory of algebraic spaces?
Edit: I think my question can be misleading, let me clarify it. I know what are ind-object and pro-object. I also know when i... |
Why are the following two random variables equivalent? |
Every elliptic curve over a field $K$ can be mapped to a smooth, projective genus 1 curve (also defined over $K$) with a $K$-rational point, and vice versa. As I understand it, curves *without* said $K$-rational points can *never* be mapped to elliptic curves (since clearly then the elliptic curve group law cannot be e... |
For any curve $f:\mathbb{R}\to\mathbb{R},$ the gradient of $f(x)$ at the point $x=c$ is $f'(c).$ The curve passes through the point $(c,f(c)),$ and so the equation of the tangent to the curve at $x=c$ has equation $y-f(c) = f'(c)(x-c).$ This can be rearranged to: $ g(x): (= y) = f'(c)x + (f(c) -cf'(c)). $
My questio... |
Given concave decreasing function, does $\exists c\in [0,1]$ s.t. $\frac{(f(c)-cf'(c))\left(c-\frac{f(c)}{f'(c)}\right)}{2}\leq2\int_{0}^{1}f(x)dx?$ |
So the question is to find a family of functions, $(f_n)_n$, from $[0,1]$ to the complex plane such that there is pointwise convergence to some function $f$ but that the integral over $[0,1]$ of $f$ does not equal the limit as $n$ goes to infinity of the integral of $f_n$. I came up with the function $e^{i(2 \pi
x^n)... |
I'm reading chapter 15 of Fulton & Harris. On page 212, they let $\phi$ be the automorphism of $\mathbb{C}^n$ mapping $e_i\mapsto e_j$, $e_j\mapsto -e_i$, and $e_k\mapsto e_k$ for $k\neq i,j.$ Then they claim that $\phi$ induces an automorphism $\text{Ad}(\phi)$ of $\mathfrak{sl}_n(\mathbb{C})$ which, among other thing... |
Let us assume we have linear elements $X = \{x_i\}_i^n$ on the d-sphere. Depending on the number of elements, we may find a configuration that is in expectation, orthogonal, subject to the minimization of $\Vert XX^T - I\Vert^2_F$, where $E_{x,y}[x^Ty] = 0$ and $E_{x,y}[(x^Tx)^2]$ is minimal.
In the case where we a... |
For any curve $f:\mathbb{R}\to\mathbb{R},$ the gradient of $f(x)$ at the point $x=c$ is $f'(c).$ The curve passes through the point $(c,f(c)),$ and so the equation of the tangent to the curve at $x=c$ has equation $y-f(c) = f'(c)(x-c).$ This can be rearranged to: $ g(x): (= y) = f'(c)x + (f(c) -cf'(c)). $
My questio... |
I have the following equation
$$a=a_0+Mx,$$
where $a$, $a_0$, and $x$ are $N\times 1$ complex vectors, and $M$ is a (non-invertible) $N\times N$ complex matrix.
For a given $a_0$ and $M$, I want to find a vector $x$ such that the absolute value of each element of $a$ is equal to 1.
Is there a nice algebraic... |
How to deduce that two random variables are equivalent? |
For any curve $f:\mathbb{R}\to\mathbb{R},$ the gradient of $f(x)$ at the point $x=c$ is $f'(c).$ The curve passes through the point $(c,f(c)),$ and so the equation of the tangent to the curve at $x=c$ has equation $y-f(c) = f'(c)(x-c).$ This can be rearranged to: $ g(x): (= y) = f'(c)x + (f(c) -cf'(c)). $
My questio... |
Take for example $x=3$. Then
$$\begin{align}
\prod_{y=0}^9 (1 + \frac{1}{10x+y}) &= (1 + \frac{1}{30})(1+\frac{1}{31}) \cdots(1 + \frac{1}{39})\\
&=\frac{31}{30}\frac{32}{31} \cdots \frac{40}{39}\\
&=\frac{40}{30}\\
&=\frac{4}{3}\\
&=1+\frac{1}{3}
\end{align}
$$
Can you generalize? |
In my answer to a recent
[MSE question](https://math.stackexchange.com/q/4875434/)
I mention the elliptic curve
[LMFDB 1728.n4](https://www.lmfdb.org/EllipticCurve/Q/1728a1)
with an infinite order generator and integer points $(-1,\pm1)$. The
[elliptic divisibility sequence](
https://en.wikipedia.org/wiki/Ellipti... |
Consider some dynamical system $\dot{\textbf{X}}=F(\textbf{X})$ where $\textbf{X}$ is discretized along a 1-dimensional spatial coordinate $\textbf{x}=(x_1,\dots,x_N)^T$. Let $\rho(\textbf{X},t)$ be the probability density function, then the Liouville equation describes the evolution of $\rho(\textbf{X},t)$ given by,
... |
How to properly discretize and solve the Liouville equation? |
let $R$ be a commutative ring with unit, and fix elements $A(X)$ and $B(X)$, *not both zero*, in $R[X]$.
suppose that there exist polynomials $s(X)$ and $t(X)$ in $R[X]$, respectively of degrees *less than* $\deg(B)$ and $\deg(A)$, and not both zero, such that $s(X) \cdot A(X) + t(X) \cdot B(X) = 0$. (the existence ... |
if $\exists$ nontrivial polynomial relation $s(X) \cdot A(X) + t(X) \cdot B(X) = 0$, then do $A$ and $B$ have a common factor? |
Let $A$ be an $n\times n$ real matrix and suppose that $\vec{v}$ is a vector in ${\mathbb R}^n$ such that $\vec{v}$,$A\vec{v}$,$A^2\vec{v}$ are distinct and non-zero. What can be said about $A$ and $\vec{v}$ if
$$
\vec{v},A\vec{v},A^2\vec{v}
$$
are orthogonal to each other? Is $A$ some kind of a rotation matrix? ... |
Reading about the [abc conjecture](https://en.wikipedia.org/wiki/Abc_conjecture) and the interesting definition and properties of [radicals of an integer](https://en.wikipedia.org/wiki/Radical_of_an_integer), I wondered about the properties and bounds of the sum $$S_{\text{rad}}(k)=\sum_{n=1}^k \text{rad}(n)$$
Note ... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
The Boolean function $g$ of the variables $x_1 , . . . , x_5 , y_1 , . . . , y_5$ , is given by the formula:
$$
\bigwedge_{i=1}^{5} (x_i \ \vee \ y_i)
$$
**Mission:** find the dnf($g$). (the minimum number of conjuncts in the DNF of function $g$)
**Some clarifications:** The
smallest number of conjuncts in the ... |
I'm trying to understand a calculation on the second page of https://sma.epfl.ch/~filipazz/notes/adjunction_and_inversion_of_adjunction.pdf, and I have a couple questions.
Here is the setup. $X$ is the cone of a projectively normal rational curve of degree $n$, $f:Y\to X$ is the resolution of singularities we get by... |
I need to prove that if a sequence $\{a_n\}_{n\in\Bbb N}$ is such that
$$
a_n = \frac{a_{n-1}+a_{n+1}}{2} \quad\forall n\in\Bbb N
$$
then the sequence is arithmetic progression.
I transformed that formula to the following one:
$${a_n-a_{n-1}}=a_{n+1}-a_n$$
And at this point it's pretty obvious that we could sa... |
Some time ago I was curious about a question related to my bath towel, which I hang on a rope to have fun (you can use your own towel to do this experiment in bath-o if you want):
'There is this rectangle with sides $a<b$. The rectangle is bent along a line that passes through the center of the rectangle. At which ang... |
The book:
> The notation ka in additive notation does not represent a product of k and a but, rather, a sum
$ka=a+a+a...+a$
with k terms
I confused about the above definition, because we say that $(-1)$ is generator for integers,but we need negative $k$ values to obtain positive integers. If $ka$ is not product... |
I'm currently working on a solution to a projectile motion problem, and I need to figure out the optimal launch angles by solving this system of equations.
$15\cos(\theta)x+2\cos(\theta)=21 \\\\
(15\sin(\theta)-\frac{1}{2}9.81x)x+2\sin(\theta)+3=0$
The first equation models the horizontal position of the project... |
As a partial response, here is a proof of the claim based on a possible *proof* of Hall's theorem. Maybe it will inspire further progress.
The graph we consider is the usual graph for the system-of-representatives view of the theorem: let $X = \{x_1, x_2, \dots, x_m\}$, $Y = \{y_1, y_2, \dots, y_n\}$, and put an edg... |
Here is the definition of a norm given by my textbook; [screenshot of textbook][1]
[1]: https://i.stack.imgur.com/OVldg.png
(This is from Fourier Series and Boundary Value Problems by James Ward Brown and Ruel V. Churchill, Chapter 7)
I'm confused by what authors say after (9). I was under the impression t... |
Here is the definition of a norm given by my textbook;
[![enter image description here][1]][1]
(This is from Fourier Series and Boundary Value Problems by James Ward Brown and Ruel V. Churchill, Chapter 7)
I'm confused by what authors say after (9). I was under the impression that the area between two curves ... |
Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a joint PDF (i.e., $\|f\|_{L^1}=1$) which satisfies $f(x,y)>0$ for all $(x,y)\in \mathbb{R^2}$.
What is a necessary and sufficient condition under which the marginal $$ f^y (y)=\int\limits_{\mathbb{R}} f(x,y)\, dx $$
exist *for all* $y\in \mathbb{R}$?
*Known facts:* becaus... |
Here is the definition of a norm given by my textbook;
[![enter image description here][1]][1]
(This is from Fourier Series and Boundary Value Problems by James Ward Brown and Ruel V. Churchill, Chapter 7)
I'm confused by what authors say after (9). I was under the impression that the area between two curves ... |
## Context
Both the Fulton and Cox/Little/Schenck books on Toric Varieties include an exercise to show that if an inclusion $\tau\subset \sigma$ of cones induces an open embedding $U_\tau \hookrightarrow U_\sigma$, then $\tau$ is a face of $\sigma$.
[This answer](https://math.stackexchange.com/questions/4703972/if-... |
The main problem is that I do not understand what the hell is even being asked in this question. Is anyone familiar with inductive definitions that could help me go through these definitions and understand what the question is about? I think that for the base case of the definition I need to define the variables of typ... |
If A is decidable, is A* decidable? |
The Dirichlet test for convergence of improper intervals strictly regards improper intervals on unbounded intervals. However, I can't seem to find any mention of a similar test for improper intervals of functions that are unbounded. If I follow the proof, I don't see any reason why it wouldn't work for unbounded functi... |
Let $(x_j)_j$ be an ergodic and stationary sequence of random variables such that $E[x_j]=0$ (for all $j$) and:
$$\frac{1}{n}\sum_{j=0}^{n}f(x_j) \to E[f(x)] < \infty, \quad (n \to \infty)$$
for any function. We know that:
$$\log(1 + z) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k+1} z^{k+1} =z + \sum_{k=2}^{\infty... |
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... |
Does anyone have any alternative solutions for this exercise. I'm self studying this book and I don't find many answers online, and don't even know if mine is correct, I find it a bit long. I would like to see how anyone else reasons through this exercise. I find it curious that you can get every function $Y^X$ from $... |
The book:
> The notation ka in additive notation does not represent a product of k and a but, rather, a sum
$ka=a+a+a+\cdots+a$
with k terms
I confused about the above definition, because we say that $(-1)$ is generator for integers,but we need negative $k$ values to obtain positive integers. If $ka$ is not pro... |
For any curve $f:\mathbb{R}\to\mathbb{R},$ the gradient of $f(x)$ at the point $x=c$ is $f'(c).$ The curve passes through the point $(c,f(c)),$ and so the equation of the tangent to the curve at $x=c$ has equation $y-f(c) = f'(c)(x-c).$ This can be rearranged to: $ g(x): (= y) = f'(c)x + (f(c) -cf'(c)). $
My questio... |
Whilst studying linear algebra I came accross an at first sight peculiar conclusion. The double basis representation of the Identity linear transformation equals the Identity matrix of same dimension as the basis in question.
Now my inquiry is if this pattern is generalisable. Say one were to take a set of basis ve... |
Do basis vectors which are represented with respect to their own basis become the standard basis vectors? |
With respect to the usual metric of **C**
this [subset][1] is not closed, however with another [metric][2] this subset is closed? Can some please explain why?
I apologize for not using latex. This is my first post here.
[1]: https://i.stack.imgur.com/vIVm4.png
[2]: https://i.stack.imgur.com/vAOQ7.png |
How does a metric space affect whether a subset is closed or not closed? |
For two $n\times n$ hermitian matrices $A$, $B$, we have the trace inequality
$$\text{tr}(AB)\leq\sum_{i=1}^{n}\lambda_i(A)\lambda_i(B)$$
where the $\lambda_i(X)$ are the eigenvalues of X ordered in decreasing order.
Are there any inequalities similar to this but for unitary matrices? I am aware of the trace i... |
With respect to the usual metric of **C**
this $${z \in C: 1< |z| <= 2}$$ is not closed, however with another metric such as
$$d(z,w)$$ where $$d(z,w) = 0,z=w$$ and $$d(z,w) = |z| +|w|, z\neq w$$
this subset is closed? Can some please explain why?
|
I have stumbled upon an interesting question whilst working on my thesis.
You are given a matrix of pairwise distinct integers $A=(a_{i,j})$ with $1\leq i\leq k$ and $1\leq j \leq r$ and a tuple $(b_1, \dots, b_r)$ such that $\sum_{i=1}^r b_i=k$.
You have to calculate the lexicographic maximal string $(a_{i_1, j_... |
Whilst studying linear algebra I came accross an at first sight peculiar conclusion. The double basis representation of the Identity linear transformation equals the Identity matrix of same dimension as the basis in question.
Now my inquiry is if this pattern is generalisable. Say one were to take a set of basis ve... |
With respect to the usual metric of **C**
this $${z \in C: 1< |z| <= 2}$$ is not closed, however with another metric such as
$$d(z,w)$$ where $$d(z,w) = 0,z=w$$ and $$d(z,w) = |z| +|w|, z\neq w$$
this subset is closed? Can someone please explain why?
|
Let $X$ be a random variable with finite support contained in $(0,\infty)$. By Jensen's inequality we have $\mathbb{E}[1/X] \geq 1/\mathbb{E}[X]$. I am curious if there exists an elementary proof of this fact which avoids the use of Jensen's inequality.
To rephrase the setup in an elementary way:
Let $S$ be a fin... |
Let $M$ be a compact, connected, oriented manifold. Let $p\in C^\infty(T^*M)$ be a real and positive principal symbol of some degree $2d\in 2\mathbb{N}$ elliptic differential operator $P\in \mathrm{Diff}^{2d}(M)$. That is, $p$ is smooth, $p(x,\alpha\cdot \xi) = \alpha^{2d} p(x,\xi)$ for all $(x,\xi) \in T^*M$ and $\alp... |
Let $(X_n)_{\geq 0}$ be a symmetric random walk on $\mathbf{Z}$. Let $T := \min (n\geq 0 : X_n = 0 \text{ or } X_n = N)$, with $N\in\mathbf{N}$, and $f(n) := \mathbf{P}(X_T = N | X_0 = n)$ for $0 < n < N$. I'm trying to prove that $f(n) = (f(n + 1) + f(n - 1))/2$. The intuition is that we arrive at $n - 1$ or $n + 1$ w... |
>Show $(a, b)$ has the same cardinality as $\Bbb R$ for any interval $(a, b)$.
From the previous chapters of the book (*Understanding Analysis* by Stephen Abbott), I know that $(-1,1)$ has the same cardinality as $\Bbb R$. I am willing to know is there any systematic way to show this or I have to guess some functio... |
Let $X$ be a random variable with finite support contained in $(0,\infty)$. By Jensen's inequality we have $\mathbb{E}[1/X] \geq 1/\mathbb{E}[X]$. I am curious if there exists an elementary proof of this fact which avoids the use of Jensen's inequality.
To rephrase the setup in an elementary way:
Let $S$ be a fin... |
**Context**
I understand from [1] that to define the contour integral, let $f:\mathbb {C} \to \mathbb {C} $ be a continuous function on the directed smooth curve $\gamma$. Let $z:[a,b]\to \mathbb {C} $ be any parametrization of $\gamma$ that is consistent with its order (direction). Then the integral along $\gamma$... |
**Context**
I understand from [1] that to define the contour integral, let $f:\mathbb {C} \to \mathbb {C} $ be a continuous function on the directed smooth curve $\gamma$. Then the integral along $\gamma$ is denoted
$$\int _{\gamma }f(z)\,dz.$$
For a research project that I am working on in pseudo-Riemannia... |
Let $M$ be a compact, connected, oriented manifold. Let $p\in C^\infty(T^*M)$ be a real and positive principal symbol of some degree $2d\in 2\mathbb{N}$ elliptic differential operator $P\in \mathrm{Diff}^{2d}(M)$. That is, $p$ is smooth, $p(x,\alpha\cdot \xi) = \alpha^{2d} p(x,\xi)$ for all $(x,\xi) \in T^*M$ and $\alp... |
To answer why that specific set is closed in that specific metric, hint: consider a $z \neq 0$ and let $0 <r <|z|$. What is the open ball of radius $r$ around $z$? From this you should conclude that many sets are open w.r.t. this metric, including the complement of yours.
|
How to make sense of $\int _{\gamma }f(z)\,\left|dz\right|$? |
$\int_{-1}^{1}\int_{-1}^{1}\frac{dxdy}{\sqrt{(x-.3)^2+y^2}}$ has the following physical meaning:
it is the potential of the uniform surface source distributed on a square $|x|,|y|<1$ observed at a point $O=(0.3,0)$. Obviously the point $O$ is a singular point.
I could evaluate the following integrals using WOLFARM ... |
Let $X$ be a random variable with finite support contained in $(0,\infty)$. By Jensen's inequality we have $\mathbb{E}[1/X] \geq 1/\mathbb{E}[X]$. I am curious if there exists an elementary proof of this fact which avoids the use of Jensen's inequality.
To rephrase the setup in an elementary way:
Let $S$ be a fin... |
Motivated by [this question][1], I ask the question below.
Let $S\subset M$ be a closed in $M$ but not necessary compact, embedded submanifold of a complete Riemannian manifold $M.$ Assume that for each $p \in M$, there exist a **unique** closest point on $S$, i.e a unique point $\tilde s(p) \in S$ such that $d(p,\t... |
Let $M$ be a compact, connected, oriented manifold. Let $p\in C^\infty(T^*M)$ be a real and positive principal symbol of some degree $2d\in 2\mathbb{N}$ elliptic differential operator $P\in \mathrm{Diff}^{2d}(M)$. That is, $p$ is smooth, $p(x,\alpha\cdot \xi) = \alpha^{2d} p(x,\xi)$ for all $(x,\xi) \in T^*M$ and $\alp... |
Take for example $x=3$. Then
$$\begin{align}
\prod_{y=0}^9 \left(1 + \frac{1}{10x+y}\right) &= \left(1 + \frac{1}{30}\right)\left(1+\frac{1}{31}\right) \cdots\left(1 + \frac{1}{39}\right)\\
&=\frac{31}{30} \, \frac{32}{31} \cdots \frac{39}{38} \frac{40}{39}\\
&=\frac{40}{30}\\
&=\frac{4}{3}\\
&=1+\frac{1}{3}
\... |
A friend of mine came up with this problem:
$$
a+b+c = 3(ab-c^2)
$$
Prove no solutions exist for positive integers $a, b$.
They say it's based on an IMO shortlist problem, but claims it's a bit harder.
I gave it a try myself, after isolating $a$ you get this: $a = (3c^2+c+b)/(3b-1)$.
This "simplifies" to t... |
**Background/context:**
If $S \subset \mathbb{CP}^1 \times \mathbb{CP}^1$ is a smooth curve with bidegree $(d_1, d_2)$, we know that its genus is $(d_1 - 1)(d_2 - 1)$ for example by the adjunction formula.
Alternatively, we can attempt to compute the genus via the Riemann-Hurwitz formula as follows:
1. Write ... |
How do repeated roots of the discriminant correspond to the order of repeated roots in the original polynomial? |
A friend of mine came up with this problem:
$$
a+b+c = 3(ab-c^2)
$$
Prove no solutions exist for positive integers $a, b$.
They say it's based on an IMO shortlist problem, but claims it's a bit harder.
I gave it a try myself, after isolating $a$ you get this: $a = (3c^2+c+b)/(3b-1)$.
This "simplifies" to t... |
Consider the primal problem
$$
\min_x \{f(x):G(x)\in Y\}\tag{$P$}
$$
rewritten as
$$
\nu(P^\prime) = \min_x \{f(x):y\in Y,\;y=G(x)\}\tag{$P^\prime$}
$$
where $G:\mathbb{R}^n\to \mathbb{R}^m$ with $G(x)=\{g_1(x),\ldots,g_m(x)\}$ where $Y\subseteq\mathbb{R}^m$.
For simplicity, assume that $g_i$, $f$, and $Y$ are... |
I had a query related to Farkas's lemma. As i understand as per the lemma the following two statements are equivalent:
For a matrix $A \in \mathbb{R}^{m \times n}$,and vector $c \in \mathbb{R}^{n}$ the following two claims are equivalent:
i. The implication $Ax\le 0 \implies c^Tx\le0$ holds true
ii. There exi... |
Present value involving deferred annuities - is this a typo? |
Let $p$ be an positive integer. Show that $\displaystyle\sum_{i=1}^{n}{i^p}$ can be written in the form $$\frac{n^{p+1}}{p+1} + An^p + Bn^{p-1}+Cn^{p-2}+\cdots$$
I tried prove it using induction on $p$.
Notice that the proposition works for:
$$\displaystyle\sum_{i=1}^{n}{i}=\frac{n^2}{2}+\frac{n}{2}$$
$$\displayst... |
I caught myself up in a seemingly simple question. Suppose $f: \mathbb C^n \rightarrow \mathbb C^n$ is a $\mathbb C$-linear map given by a matrix $A \in M_n(\mathbb C)$. I am trying to figure out how the differential of $A$ acts on the standard basis in the complexified tangent space $(T {\mathbb{C}}^n)_{\mathbb{C}}$ -... |
How does the differential of a complex endomorphism of $\mathbb C^n$ act on $\frac{d}{dz^i}$ and $\frac{d}{d\bar{z}^i}$? |
As I am reading wikipedia and some material:
> A basis for cycles of a network is a minimal collection of cycles such that any cycle in the network can be written as a sum of cycles in the basis.
Some other material also point out the cycle basis need to be linear independent and span the graph.
Given a gr... |
Check the linear independency of a set of cycles in a graph? |
Can we prove that $\frac{1}{n}\sum_{k=1}^n a_k \rightarrow \frac{1}{2}$? |
Suppose I have two continuous functions $f(y_I)$ and $g(y_I,m)$, and I am interested in calculating the conditional expectation that $y_I \in [0,m]$:
$$\int_{0}^m f(y_I) g(y_I,m)dy_I.$$
What conditions imply that the conditional expectation is decreasing in $m$? I know that $f(y_I)$ is strictly decreasing in $y_I$ a... |
Suppose that $(X,\mathcal{A})$ is a measurable space and let $\mathcal{A}^0$ be an algebra generating $\mathcal{A}$. For each $\epsilon>0$ and $A\in \mathcal{A}$, [one can find][1] $A^0\in\mathcal{A}^0$ such that $\mu(A\Delta A^0)<\epsilon$, where $\mu$ is a finite measure on $(X,\mathcal{A})$. Now, if we're given anot... |
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