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Why is $p$ (or $p + 1$) not the upper bound of the number of solutions of an elliptic curve mod $p$ (finite field)? |
Given $f:\mathbb{R}\rightarrow \mathbb{R}$ , suppose that exists a function $h:\mathbb{R}\rightarrow \mathbb{R}$ such that for every pair of differentiable functions (g(x),f(g(x)), h(g(x)(g'(x))= ($f\circ g$)'(x). Does it follows that f is differentiable and f'=h? |
Fix a $V\in\mathbb{N}$ with $V\ge 4$. Randomly pick $V$ points on a sphere (independently and uniformly with respect to the surface area measure). You may think of the convex hull of these $V$ points. With probability 1, this constitutes a nondegenerate convex polyhedron. In fact, all faces would be triangles (probabil... |
My updated question is
>$A$ is the point $(0,0,0)\in\Bbb R^3$.\
$C$ is the point $(1,0,0)$.\
$S$ is the point $(0,1,\tan(\alpha))$.\
A line through $S$ intersects the planes $z=0$ and $z=\tan(2\alpha)y$ at $B,D$.\
Then $\angle BAC=\angle DAC$.
The question can be rephrased as:
>Two planes intersect in the li... |
I have the following PDE:
Let $G = (a, b)$ and $J = (0, 1)$. Consider the Dirichlet problem with general coefficient functions $\alpha(x)$, $\beta(x)$, $\gamma(x)$
\begin{equation}
\begin{cases}
\partial_t u - \partial_x (\alpha(x) \partial_x u) + \beta(x) \partial_x u + \gamma(x) u = f(t, x), \text{ in $J \times G... |
I'm trying to prove https://imomath.com/index.cgi?page=inversion (Problem 11) by projective geometry:
> If seven vertices of a (quadrilaterally-faced) hexahedron lie on a sphere, then so does the eighth vertex.
Is it suffice to prove: If seven vertices of a (quadrilaterally-faced) hexahedron lie on a [quadric sur... |
Let $G$ be a second countable profinite group. Then $G$ can be written as a denumerable projective limit $\varprojlim_{i}(\cdots \to G_i\to G_{i-1}\to \cdots )$, where the $G_i$'s are finite and the connecting homomorphisms are surjective. Let $H$ be a finite group and let $G\rtimes H$ be a semidirect product of $G$ an... |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
>Let $C(0,0,\sqrt{2})$ and $F(0,\sqrt{2},0)$ be two points in $\Bbb R^3$.\
$AF,BF$ are arbitrary perpendicular lines through $F$ on the plane $y=\sqrt{2}$.\
These lines project to the lines $DF,EF$ under the perspective projection centered at $C$ onto $xy$-plane.\
Then the lines $DF,EF$ perpendicular?
This questi... |
In a linear space, say $\mathbb{R}^{d}$, I know we can define embedded submanifolds with a locally defining function, which is equivalent to the canonical definition via diffeomorphisms. Following is the definition found in [An introduction to optimization on manifolds](https://www.nicolasboumal.net/book/IntroOptimMani... |
Local defining function without constant rank implies not a manifold? |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
I have a real signal $f(t) $that is periodic on $[0,L]$ where $L=1$. I sampled it uniformly $64$ times and computed FFT (in python). I was asked to find the central frequency, namely the frequency with the largest magnitude. I found the index $i$ of the largest value in the transformed signal.
How do I find the freq... |
Given $f:\mathbb{R}\rightarrow \mathbb{R}$ , suppose that exists a function $h:\mathbb{R}\rightarrow \mathbb{R}$ such that for every pair of differentiable functions (g(x),f(g(x)))
, h(g(x)g'(x)= ($f\circ g$)'(x). Does it follows that f is differentiable and f'=h? |
In Serge Lang's Differential and Riemannian manifold, he defined derivative between two topological vector spaces with the concept tangent to $0$:
>Let $\mathbf{E}, \mathbf{F}$ be two topological vector spaces, and $\varphi$ a mapping of a neighborhood of $0$ in $\mathbf{E}$ into $\mathbf{F}$. We say that $\varphi$ is... |
As mentioned here https://math.stackexchange.com/questions/790106/does-finite-expectation-imply-bounded-random-variable, it suffices to show that
$$ \int_0^T E[|X_t|^n]dt < \infty. $$
We will simply use https://math.stackexchange.com/questions/1280256/application-of-the-burkholder-davis-gundy-inequality?noredirec... |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
I tried to let $x = \sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$, and assume for the sake of contradiction that $x$ is rational. Then I squared numerous times to attempt to isolate an irrational on one side, and a supposedly rational expression on another side, since we assumed $x$ is rational, to arrive at a contradict... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/xhFbP.png]
I'm taking an introductory comp-sci course this semester, and my professor pulled up this graphic while explaining mergesort (the graphic shows blocks of size $2^{-n}$). For some reason, the vertical lines that appear in this ph... |
In an exact category $(\mathbf{C}, \mathcal{E})$, we define an object $P \in \mathbf{C}$ to be projective if for every admissible epimorphism $g : A \rightarrow B$
$$
\mathbf{C}(P,A) \rightarrow \mathbf{C}(P,B)
$$
is a surjective map of abelian groups. Let $\mathcal{P}$ be the class of projective objects in $(\... |
Do the projective modules in an exact category determine the admissible epimorphisms? |
I’m interpreting your question as the following; if h satisfies the chain rule equation for f any of its possible parameterizations, is h = f? The answer is no, but up to an additive constant yes.
If h satisfies the desired equation for any such $g$, it in particular satisfies it for $g(x) = x$. Then your equation s... |
I would like to solve the following problem. I have a $N\times 1 $ complex vector, for example with $N=6$:
$$v=(a \; b \; c\; d \; e \;f)^T$$
and its complex conjugate
$$\bar{v}=(\bar{a} \; \bar{b} \; \bar{c}\; \bar{d}\; \bar{e} \;\bar{f})^T$$
I can access any value of the product of a subset of $v$ with t... |
I’m interpreting your question as the following; if h satisfies the chain rule equation for f any of its possible parameterizations, is h = f’? The answer is no, but up to an additive constant yes.
If h satisfies the desired equation for any such $g$, it in particular satisfies it for $g(x) = x$. Then your equation ... |
I’m interpreting your question as the following; if $h$ satisfies the chain rule equation for $f$ any of its possible parameterizations, is $h’ = f’$? The answer is no, but up to an additive constant yes.
If $h’$ satisfies the desired equation for any such $g$, it in particular satisfies it for $g(x) = x$. Then your... |
*Salvador Dali* applied in his 1947–1949 painting "*Leda Atomica*" a concept that relates to the golden ratio or its conjugate. It is well known that Dali used mathematical formulas to elaborate his works (e.g. golden ratio, catastrophe theory, etc.), this said, we are not talking here about a layman.
[![enter image... |
I am struggling dealing with $o$ notation. Here $k, n$ are all natural numbers, why do we have $o\Big(\sum_{k=0}^{n-1}\frac{(k\log\log k)^{1/2}}{k+1}\Big)=o((n\log\log n)^{1/2})$ as $n\rightarrow\infty$? And another one with $O$ notation, why does $\sum_{k=0}^{n-1}\frac{X_k}{k^2}$ converges in $L^2$ and a.s. as $n\righ... |
Sorry to bother you again, hopefully someone will answer again...
I have some questions about optimisation problems, related to KT conditions (but maybe not only).
My questions are rather theoretical, I believe, so no calculations.
Suppose I have to find, if existent, max and min of $f(x, y, z)$ subject to one ... |
Define the sequence $(x_n)_{n \geq 0}$ with $x_0 >0$ and recursive equation $x_{n+1} = \sqrt{2+x_n}$ for all $n \geq 0$. I wish to determine the limit of the sequence $(x_n)_{n \geq 0}$ in terms of the parameter $x_0 \in (0, \infty)$.
If $x_0 < 2$, then $x_n > 0, (\forall)n \geq 0$, $x_n < 2, (\forall)n \geq 0$ and... |
I am struggling dealing with $o$ notation. Here $k, n$ are all natural numbers, why do we have $o\Big(\sum_{k=0}^{n-1}\frac{(k\log\log k)^{1/2}}{k+1}\Big)=o((n\log\log n)^{1/2})$ as $n\rightarrow\infty$? And another one with $O$ notation, why does $\sum_{k=0}^{n-1}\frac{X_k}{k^2}$ converges in $L^2$ and a.s. as $n\righ... |
I am struggling dealing with $o$ notation. Here $k, n$ are all natural numbers, why do we have $o\Big(\sum_{k=0}^{n-1}\frac{(k\log\log k)^{1/2}}{k+1}\Big)=o((n\log\log n)^{1/2})$ as $n\rightarrow\infty$? And another one with convergence, why does $\sum_{k=0}^{n-1}\frac{|X_k|}{k^2}$ converges in $L^2$ and a.s. as $n\rig... |
Let $B = (B_{t})_{t \geq 0}$ be a 1-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (i.e. that is, $B_t: \Omega \to \mathbb{R}$ for each $t \geq 0$), and let $T > 0$ be a real number. For each $n \in \mathbb{N}$, let
$$0 = t_0^{(n)} < t_1^{(n)} < \cdots < t_{m_n}^{(n)} = T$$
... |
I saw this problem:$\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx =\frac{\pi^2}{4}$ and I tried to solve it.
Here is my attempt
$\textbf{Claim: }$ If $f$ is continuous at $[0,1]$ then $\displaystyle \lim_{n \to \infty }\int_{0}^{1} \frac{nf(x)}{1+n^2x^2}dx=\frac{\pi f(0)}{2} $
Proof
Since $f... |
How to complete this proof of $\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx =\frac{\pi^2}{4}$? |
Your "n" should be 10, as is seen on page 17 of [Matila Ghyka's]( https://en.wikipedia.org/wiki/Matila_Ghyka) [_The Geometry of Art and Life_](https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA2&source=kp_read_button&hl=en&newbks=1&newbks_redir=0#v=onepage&q&f=false).
(This by backtracking the references in the W... |
Let $M$ be a connected closed smooth manifold of dimension $n$, and let $X$ be a smooth vector field on $M$.
Since $M$ is compact, $X$ is complete (i.e., its maximal integral curves are defined on all of $\mathbb{R}$).
Given a smooth map $F:N\to M$, a vector field $\tilde{X}$ on $N$ is said to be $F$-**related** t... |
Suppose that $N\sim(0,1)$ and I have two functions $f$ and $g$.
Are there any theorems or techniques for comparing how close are distributions of $f(N)$ and $g(N)$?
For example, it is known that sum of lognormal random variables $ae^{bN}+ce^{dN}$ can be approximated by one lognormal variable $ue^{vN}$. Is it poss... |
Given $f:\mathbb{R}\rightarrow \mathbb{R}$ , suppose that exists a function $h:\mathbb{R}\rightarrow \mathbb{R}$ such that for every pair of differentiable functions (g(x),f(g(x)))
, h(g(x)g'(x)= ($f\circ g$)'(x).
How does this compare to weak derivative concept? |
I was going through the article coping with copulas.
The following situation was considered there.
A die is rolled twice. $X_1$ is the random variable representing the minimum of the two rolls, $X_2$ is the random variable representing the maximum of the two rolls.
The probability distribution of an individual throw... |
It is my understanding that the multiplicity of the smallest eigenvalue (the zero eigenvalue) of the graph laplacian $L=D-A$ equals the number of connected components of a graph, while the second smallest eigenvalue bounds values like the isoperimetric number of the graph (tellings us things about the how "strongly con... |
Do graph laplacian eigenvectors/values beyond the second smallest eigenvalue mean anything? |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ ( $f_1(z)$ ) is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
where $\operatorn... |
I saw this problem:$\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx =\frac{\pi^2}{4}$ and I tried to solve it.
Here is my attempt
$\textbf{Claim: }$ If $f$ is continuous at $[0,1]$ then $\displaystyle \lim_{n \to \infty }\int_{0}^{1} \frac{nf(x)}{1+n^2x^2}dx=\frac{\pi f(0)}{2} $
Proof
Since $f... |
Let $X_1,\cdots,X_n$ be $n$ random variables on the same probability space $(\Omega, F, P)$, all with expectation $0$. Define $Y_i=X_i - \mathbb{E}[X_i | X_1, \cdots, X_{i-1}]$. Is it true that for any $X = a_1 X_1 + \cdots + a_n X_n$, there exist $b_1,\cdots,b_n$ such that $X = b_1 Y_1 + \cdots + b_n Y_n$, and vice ve... |
What's the relationship between $(X_i)$ and $(X_i - \mathbb{E}[X_i | X_{<i}])$? |
Acute triangle $ABC$ has side length $a \ge b \ge c$. We drop the altitude from point $C$ to side $c$, and let its length be h. We construct a square with side length $h$ so that side $c$, points $A$ and $C$ lie on the square's sides and point $B$ lies on its vertex. Then, we construct another square with side $s$ so t... |
Which square is bigger? |
I was going through the article coping with copulas.
The following situation was considered there.
A die is rolled twice. $X_1$ is the random variable representing the minimum of the two rolls, $X_2$ is the random variable representing the maximum of the two rolls.
The probability distribution of an individual throw... |
Prove that $\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$ is irrational
I tried to let $x = \sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$, and assume for the sake of contradiction that $x$ is rational.
Then I squared numerous times to attempt to isolate an irrational on one side, and a supposedly rational expressio... |
I find another charming proof in [*Lectures on the theory of algebraic numbers*](http://books.google.com.tw/books/about/Lectures_on_the_Theory_of_Algebraic_Numb.html?id=QEocn1kbfxAC&redir_esc=y) by **Hecke**.
It is constructive in appearance but is impractical in fact.
Let $p(x)$ and $q(x)$ be two polynomia... |
Let $X_1,\cdots,X_n$ be $n$ random variables on the same probability space $(\Omega, F, P)$, all with expectation $0$. Define $Y_i=X_i - \mathbb{E}[X_i | X_1, \cdots, X_{i-1}]$. Is it true that for any $X = a_1 X_1 + \cdots + a_n X_n$, there exist $b_1,\cdots,b_n$ such that $X = b_1 Y_1 + \cdots + b_n Y_n$, and vice ve... |
Suppose $h_n$ is a sequence of non-negative functions in $L^2([0,1])$ converging weakly to $h$ (i.e., for every $g\in L^2([0,1])$ it holds $\int g\cdot h_n \,d\lambda \to \int g\cdot h\, d\lambda$).
Let $T_n\subseteq [0,1]$ be a measurable set for each $n$ such that $h_n$ vanishes almost everywhere on $T_n$.
Suppos... |
Does the weak limit of a sequence in $L^2([0,1])$ vanish on the limit set of vanishing sets? |
Let $X_1,\cdots,X_n$ be $n$ random variables on the same probability space $(\Omega, F, P)$, all with expectation $0$. Define $Y_i=X_i - \mathbb{E}[X_i | X_1, \cdots, X_{i-1}]$. Is it true that for any $X = a_1 X_1 + \cdots + a_n X_n$, there exist $b_1,\cdots,b_n$ such that $X = b_1 Y_1 + \cdots + b_n Y_n$, and vice ve... |
Let $B = (B_{t})_{t \geq 0}$ be a 1-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (i.e. that is, $B_t: \Omega \to \mathbb{R}$ for each $t \geq 0$), and let $T > 0$ be a real number. For each $n \in \mathbb{N}$, let
$$0 = t_0^{(n)} < t_1^{(n)} < \cdots < t_{m_n}^{(n)} = T$$
... |
You've made a mistake somewhere. For $x_{n+1} = \sqrt{2 + x_n}$, and assuming that all $x_n > 0$,
- $x_{n+1} > x_n$ for $x_n < 2$, and
- $x_{n+1} < x_n$ for $x_n > 2$.
So in either case $x_{n+1}$ is closer to $2$ than $x_n$ (either increasing from below or decreasing from above).
Why? Let's look at the $... |
How to find a separating hyperplane? |
I met the definition of exponential tilting and am struggling about this definition:
for every $\theta\in\mathbb R$, we define a random variable $X_1^\theta$ with law given by $$\mathbb E[f(X_1^\theta)]:=\mathbb E[f(X_1)\frac{e^{\theta X_1}}{M(\theta)}]$$ for $f\geq 0$ measurable.
What does this exponential tilt... |
What does exponential tilting actually do? |
>Also, is there a better way to show that the right angle at the center of the square is preserved by the perspective projection and $xw$-rotation when $\text{distance}=\sqrt{2}$?
When $\text{distance}=\sqrt{2}$, the perspective center will lie on the bisector plane of the original plane $P_2P_3P_{10}P_{11}$ and the... |
Let $C$ be the convex subset of $\mathbb{R}^2$ defined by:
\begin{align}
C = \{(x,y):~~ \log(x) + \log(y) \geq 1;~~ x\geq 0;~~ y\geq 0\}
\end{align}
According to the [hyperplane separation theorem][1], if a point $(x_0,y_0)$ is not in $C$, then there is a hyperplane separating $(x_0,y_0)$ from $C$.
MY QUESTION... |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous and nowhere differentiable, and $g: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable everywhere and non constant. Can $f \circ g$ ever be differentiable?
My intuition is no, but I'm not sure. Perhaps $g$ is differentiable but not continuously so and its de... |
In the build-up to the enunciation of the Birch, Swinnerton Dyer conjecture (BSD), the following back-of-the-envelop idea comes up:
Because half of the $1$ to $n-1$ elements mod $p$ are quadratic residues (squares), for the expression $y^2= x^3 +\cdots$ mod a certain prime $p,$ **on average** $1/2$ of the input $x$ ... |
Can a composition of a continuous nowhere differentiable and a nonconstant differentiable function be differentiable? |
Computing the Adams spectral sequence is hard. Even determining the $E_2$-page can be difficult, but it can be simplified in some cases. In particular, when $H^*(X; \mathbb{F}_2) = A//A(n)$, for $A$ the mod 2 Steenrod algebra, we can use the change of rings isomorphism:
$$\text{Ext}_A(A//A(n), \mathbb{F}_2) = \text{Ex... |
Orthogonal projection method:
when $K = L$ and matrix $A$ is symmetric positive definite (SPD), show that $\tilde{x}$ minimizes $(A(x−x^*),(x−x^*))≡ \|x - x^* \| ^{2} _A = ( A ( x − x^* ) , ( x − x^* ) ) $ over all vectors $x$ in the affine space $x_0 +$ where $x^*$ is the exact solution (i.e., $Ax^* = b$).
He... |
Prove the optimal property of an orthogonal projection method -- how to show that the approximate solution minimizes the *A-norm* of the error? |
Given an orthogonal projection method (when $K = L$) and a symmetric positive definite (SPD) matrix $A$, show that $\tilde{x}$ minimizes $(A(x−x^*),(x−x^*))≡ \|x - x^* \| ^{2} _A = ( A ( x − x^* ) , ( x − x^* ) ) $ over all vectors $x$ in the affine space $x_0 +$ where $x^*$ is the exact solution (i.e., $Ax^* = b$).... |
I hope/think that this is an appropriate soft-question.
I am a student of mathematics and have been increasingly feeling doubtful about whether or not I really understand the theory. What I mean by this is highly attributed to the style of textbooks in presenting information.
Authors simply write definitions and ... |
How to motivate and deal with speculation? |
I (and probably most people here) can empathize with you. I believe this is a symptom of how papers are meant to be written - most of the dirty groundwork that motivated everything gets swept under the rug, and can become really mysterious.
That said, some books and some survey papers actually attempt to give some m... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
A circular hoop of radius $r$ is stood vertically on a flat surface and held in place. A smooth wire is tightly stretched between the upmost point of the hoop and a point on the hoop at height $h$. A bead of mass $m$ is held at rest at the top of this wire, upon which it is threaded. When the bead is released, how long... |
Fractional linear transformation is a map from extended complex plane to itself, defined by:
\begin{equation}
z\to \frac{az+b}{cz+d}
\end{equation}
with $ad-bc\ne0$.
Wikipedia says that *"a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator ar... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
Let $F$ be a number field and $S$ a finite set of places including all infinity places. Let $f\in \mathcal{A}(F_\infty)$ be a rapidly decreasing function (in the usual sense), and $K\subseteq F_{S\setminus\infty}$ be a compact open subgroup. I would like to study the bound of the series
$$\sum_{x\in\mathcal{O}_F^S\cap... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
Let $F$ be a number field and $S$ a finite set of places including all infinity places. Let $f\in \mathcal{A}(F_\infty)$ be a rapidly decreasing function (in the usual sense), and $K\subseteq F_{S\setminus\infty}$ be a compact open subgroup. I would like to study the bound of the series
$$\sum_{x\in\mathcal{O}_F^S\cap... |
For a $k$-cycle $(1, 2, 3, 4, 5)$ and a transposition $(1, 2)$.Calculation of "$(1, 2, 3, 4, 5) \cdot (1, 2)$". We all know that groups satisfy the associative property, i.e. $(a \cdot b) \cdot c=a \cdot (b \cdot c)$.
I will get $ \left \{ 1, 2, 3, 4, 5 \right \} \longmapsto \left \{ 2, 3, 4, 5, 1 \right \} $, wh... |
>Also, is there a better way to show that the right angle at the center of the square is preserved by the perspective projection and $xw$-rotation when $\text{distance}=\sqrt{2}$?
In the post the "view space" is $w=1$. We change this to $w=0$ (will just rescale the projection) then the point $(0,1,-1,0)$ lies on the... |
Let $C$ be a closed, convex set in a Hilbert space $H$.
$f(x)=\inf_{y \in C} \frac{1}{2}||x-y||^2$
WTS:
$$\lim_{\alpha \to 0^{+}} \frac{||x+\alpha y-P_c(x+\alpha y)||^2-||x-P_c(x)||^2}{2\alpha}= \langle y | \textbf{I}(x)-P_c(x) \rangle $$
Thus $\nabla f=\textbf{I}-P_c$
We have this characterization of the... |
There is a well known fact, that if we have non-decreasing weight $w(x)$ in the finite interval $[a,b]$, then if $\{ p_n(x) \}$ is the set of corresponding orthogonal polynomials, the functions $|\sqrt{w(x)}p_n(x)|$ attain their maximum in $[a,b]$ for $x=b$. (the proof can be found in Szego book)
My question is: are... |
The contruction was used in two different sources on the web: [a Geogebra resource][1] and [a video using inRm3D][2] so I think it must be documented and proved somewhere, but I didn't find any.
Here is the construction (following the second source):
1. On the xy-plane, draw a conic symmetric about $y$-axis. (Igno... |
>Also, is there a better way to show that the right angle at the center of the square is preserved by the perspective projection and $xw$-rotation when $\text{distance}=\sqrt{2}$?
The center of the square $P_2P_3P_{10}P_{11}$ is $(0,1,-1,0)$.
In the post the "view space" is $w=\text{distance}-1$. We change this t... |
here is a lemma:
> the jointly continuous r.v X, Y are independent. iff $f_{y|x}(y|x) = f_y(y)$ for all $x$ outside a set $E$ with $\int_E f_x(x) dx = x$
However, consider a case, for example with $Y ~ N(0, 1)$ and $X ~ Exponential(1)$. If this makes $f_{y|x}(y|x)$ defined only on $x \ge 0$, wouldn't it fail to b... |
how can the conditional pdf remain a pdf when the negligible set of the conditioning random variable overlaps? |
I met the definition of exponential tilting when reading about Cramer's theorem, and am struggling about this definition:
for every $\theta\in\mathbb R$, we define a random variable $X_1^\theta$ with law given by $$\mathbb E[f(X_1^\theta)]:=\mathbb E[f(X_1)\frac{e^{\theta X_1}}{M(\theta)}]$$ for $f\geq 0$ measurable... |
I met the definition of exponential tilting when reading about Cramer's theorem, and am struggling about this definition:
for every $\theta\in\mathbb R$, we define a random variable $X_1^\theta$ with law given by $$\mathbb E[f(X_1^\theta)]:=\mathbb E[f(X_1)\frac{e^{\theta X_1}}{M(\theta)}]$$ for $f\geq 0$ measurable... |
Consider the multivariate polynomial $Q(x,y) = y^2 - f(x)$ over a finite field $\mathbb{F}_q$ where $f(x) \in \mathbb{F}_q[x]$ has $m$ distinct roots in $\mathbb{F}_q$. In his book "*Equations over Finite Fields*", Schmidt claims that if $Q$ is absolutely irreducible then $N$, the number of points on the algebraic vari... |
Let $B = (B_{t})_{t \geq 0}$ be a 1-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (i.e. that is, $B_t: \Omega \to \mathbb{R}$ for each $t \geq 0$), and let $T > 0$ be a real number. For each $n \in \mathbb{N}$, let
$$0 = t_0^{(n)} < t_1^{(n)} < \cdots < t_{m_n}^{(n)} = T$$
... |
For $\mu \in \mathcal P(\mathbb R)$, let $m(\mu)$ be the median of $\mu$, defined as the smallest of all medians of $\mu$ as follows:
$$
m(\mu) = \inf \left\{ x \in \mathbb R \,\middle|\, \mu((-\infty,x]) \ge \frac12 \right\}.
$$
Assume that $\mathcal P(\mathbb R)$ is endowed with the weak convergence topology. T... |
Is the median a measurable function of the probability distribution? |
Let $(X,\vec X,+)$ a real affine space, $a\in X, b\in X, \vec A $ and $\vec B$ two subspaces of $\vec X$.
$A=a+\vec A$ and $B= b+\vec B$ two affine subspaces.
I want to understand what the "affine subspace generated by $A$ and $B$" is, because Jean Frenkel speaks of it in *Géométrie pour l'élève professeur*(Geom... |
What is "affine space generated by two affine subspaces"? |
>Consider $S$ the following recursively defined set; $S$ contains the empty string and, for any strings $a$ and $b$ in $S$, the string $(a b)$ is also in $S$. Here are the first few elements of $S$ :
$$
(),(()),((())),(()()),(((()))),((()))),,((()))),(()(())) \ldots
$$
The weight $w(s)$ of a non-empty string $s$ is... |
I'm new to stochastic processes and have problems understanding martingales, conditional probability, $\sigma$-algebras etc.
I have two proofs that I'm now sure how to handle.
**Problem 1**.
Prove that an integrable stochastic process $\{X(t),\mathcal{F}_t, t\in T\}$ is a martingale if and only if for any bounde... |
I tried to find a *small* natural deduction proof for Meredith's single axiom
| Infix notation | Polish notation |
| -------------- | --------------- |
[((((ψ→φ)→(¬χ→¬ξ))→χ)→τ)→((τ→ψ)→(ξ→ψ))](https://www.umsu.de/trees/#((((a%E2%86%92b)%E2%86%92(%C2%ACc%E2%86%92%C2%ACd))%E2%86%92c)%E2%86%92e)%E2%86%92((e%E2%86%92a)... |
Is there an analytical solution to this ordinary differential equation? |
Consider a complex, separable Hilbert space $H$ and a (densely defined) self-adjoint operator $A: \mathcal D(A)\to H$. Assume that $A$ admits an orthonormal basis of eigenvectors $(\varphi_n)_{n\in \mathbb N}$ with $A\varphi_n=\lambda_n \varphi_n$.
Next, define for $\psi\in \mathcal D(A^2)$ with $\|\psi\|=1$ the foll... |
Let $(X,\vec X,+)$ a real affine space, $a\in X, b\in X, \vec A $ and $\vec B$ two subspaces of $\vec X$.
$A=a+\vec A$ and $B= b+\vec B$ two affine subspaces.
I want to understand what the "affine subspace generated by $A$ and $B$" is, because Jean Frenkel speaks of it in *Géométrie pour l'élève professeur*(Geom... |
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