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Are all submanifolds locally flat? |
The statement of the Calderon-Zygmund decomposition as given in Duoandikoetxea's book is the following.
For $f$ integrable on $\mathbb{R}^n$ and non-negative and given $\lambda>0$, there exist a sequence $\{Q_j\}$ of disjoint dyadic cubes such that
1) $f(x) \leq \lambda$ for almost every $x \notin \cup Q_j$
2)... |
Given an expander graph family (an injective sequence of graphs with uniformly bounded vertex degree and a Cheeger constant/Laplacian spectral gap uniformly bounded away from zero).
Can the corresponding clique complexes (flag complexes, i.e., all cliques are taken as simplices) all be simply connected? If not, can ... |
Clique complex of expander graphs simply connected? |
I wish to know the exact value of
$$\sum_{n=1}^\infty\frac1{n^n}.$$
I've found on the internet some mentions of the equality (the [Sophomore's Dream][1], as I've learned)
$$\sum_{n=1}^\infty\frac1{n^n} = \int_0^1\frac1{x^x}dx.$$
Wolfram Alpha's computations returns the value of $1.29128599706266\dots$ to the R... |
Why is none trying to find the value of $\displaystyle\sum_{n=1}^\infty\frac1{n^n}$? |
A ball is dropped from rest at a height $h_0$ and bounces from a surface such that the height of the $n$th bounce, $h_n$ is given by $h_n=αh_{n-1}$, where $h_{n-1}$ is the height of the previous, $(n-1)$th bounce. The factor $α$ has value between 0 and 1.
How long does the ball take to cover the distance it travels ... |
I'm currently learning exponential generating functions and am working on a problem that needs me to find the corresponding closed form expression to a given sequence.
I've been given the sequence $1,0,1,0,...$ and decided to use the EGF $e^x$ to find my solution by substituting $0x$ in every other term in the expre... |
In Serre's "Abelian $\ell$-adic representations and Elliptic curves", in order to define the set $S_{\mathfrak{m}}$ he defines of $U_{v,\mathfrak{m}}$ as follows:
$$U_{v,\mathfrak{m}}=\left\{\begin{array}{ll}
\text{The connected component of } K_v^{\times} & \text{ if } v \in \Sigma^{\infty}\\
U_v & \text { if... |
The question is Exercise 3.89 from Advanced Modern Algebra, Rotman.
Stochastic group $\sum (2,\mathbb{F}_4) < \text{GL}_2(\mathbb{F}_4)$ is defined as following:
$\sum (2,\mathbb{F}_4) = \left\{ \begin{pmatrix} a & b \\ 1-a & 1-b \end{pmatrix} \bigg| a,b \in \mathbb{F}_4 \right\}$
$A_4$ is the alternating subg... |
Are all smooth submanifolds locally flat? |
This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$":
Let $F$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique unramified extension of degree $n$. We can denote by $F_{\infty}$ the composite of all these fields that is clearly the maximal unramif... |
I'm currently learning [exponential generating functions](https://en.wikipedia.org/wiki/Generating_function#Exponential_generating_function_(EGF)) and am working on a problem that needs me to find the corresponding closed form expression to a given sequence.
I've been given the sequence $1,0,1,0,...$ and decided to ... |
Given a gravity vector $g$ in a 3D coordinate frame $F$ we can find pitch $p$ and roll $r$ (Euler angles) of $F$ relative to $g$. Assume we apply a rigid transformation to $F$, sense a new gravity vector $g'$, and compute new Euler angles $p'$ and $r'$. Can we predict the yaw angles $y$ or $y'$ given $(p, r, p', r')$? ... |
Given a gravity vector $g$ in a 3D coordinate frame $F$ we can find pitch $p$ and roll $r$ (Euler angles) of $F$ relative to $g$. Assume we apply a rigid transformation to $F$, sense a new gravity vector $g'$, and compute new Euler angles $p'$ and $r'$. Can we predict the yaw angles $y$ or $y'$ given $(p, r, p', r')$? ... |
**Example 8.40** (Introduction to smooth manifolds. Lee)
If we consider $\mathbb{R}^n$ as a Lie group under addition, left translation by an element $b\in\mathbb{R}^n$ is given by the affine map $L_b(x)=b+x$, whose differential $d(L_b)$ is represented by the identity matrix in standard coordinates. Thus a vector fie... |
I want to show that $\partial A= \overline{A}-\text{Int}(A)$, so for the first inlcusion:
Let $x \in \partial A$, then for all $\varepsilon > 0$ we have that
$$B_{\varepsilon}(x) \cap A \neq \emptyset \quad \text{and} \quad B_{\varepsilon}(x) \cap A^c \neq \emptyset$$
From the second afirmation i can say that ... |
I wish to know the exact value of
$$\sum_{n=1}^\infty\frac1{n^n}.$$
I've found on the internet some mentions of the equality (the [Sophomore's Dream][1], as I've learned)
$$\sum_{n=1}^\infty\frac1{n^n} = \int_0^1\frac1{x^x}dx.$$
Wolfram Alpha's computations returns the value of $1.29128599706266\dots$ to the R... |
What is a good reference for this result? |
This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$":
Let $F$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique unramified extension of degree $n$. We can denote by $F_{\infty}$ the composite of all these fields that is clearly the maximal unramif... |
There is nothing invalid with your approach.
The difference between a) and b) is not one of right or wrong, validity versus invalidity.
The difference is one of *logical formality* versus *mathematical strategy*.
What you wrote is logically correct, one must...
> *...show that for any arbitrary given $\ep... |
I need to find the arc length from the function $$ y^{2}= - 2.6\times x$$
Will the result change if I replace the function with? $$x = \frac{y^{2}}{-2.6}$$ I also ask you to check my answer, I did it: $$\frac{y\times\sqrt{1+\frac{y^{2}}{1.69}}+\ln\mid\frac{y}{1.3}+\sqrt{1+\frac{y^{2}}{1.69}}\mid}{3}$$
[1]
... |
I need to find the arc length from the function $$ y^{2}= - 2.6\times x$$
Will the result change if I replace the function with? $$x = \frac{y^{2}}{-2.6}$$ I also ask you to check my answer, I did it: $$\frac{y\times\sqrt{1+\frac{y^{2}}{1.69}}+\ln\mid\frac{y}{1.3}+\sqrt{1+\frac{y^{2}}{1.69}}\mid}{3}$$
[Solving]
... |
It seemingly is a fact that a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ acting on hyperbolic space cannot be compact if it contains a parabolic element.
**I was wondering if the following proof works:**
Let $\Gamma$ be our group and $T \in \Gamma$ be parabolic. Then we can conjugate $\Gamma$ in $\mathrm{P... |
There is nothing invalid with your approach.
The difference between a) and b) is not one of right or wrong, validity versus invalidity.
The difference is one of *logical formality* versus *mathematical strategy*.
What you wrote is logically correct, one must...
> *...show that for any arbitrary given $\ep... |
Lets say $S\subseteq \mathbb{R}^m$ is a set, and we want to determine the following:
1. Check/Prove if $S$ is an open set.
To do this, note that:
$S$ is open $\iff$ $$\forall p\in S: p\in \text{Int}(S)$$
$$\forall p\in S: \exists \varepsilon >0 : B(p,\varepsilon)\subseteq S$$
In order to find if $S$ ... |
I wish to know the exact value of
$$\sum_{n=1}^\infty\frac1{n^n}.$$
I've found on the internet some mentions of the equality (the [Sophomore's Dream][1], as I've learned)
$$\sum_{n=1}^\infty\frac1{n^n} = \int_0^1\frac1{x^x}dx.$$
Wolfram Alpha's computations returns the value of $1.29128599706266\dots$ to the R... |
**Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone. Denoting by $∥v∥_g = \sqrt{|g(v, v)|}$ the “norm” of $v$. Then the reverse C-S inequality holds with equality iff v and w are colinear:
$−g(v, w) ≥ ∥... |
Lets say $S\subseteq \mathbb{R}^m$ is a set, and we want to determine the following:
1. Check/Prove if $S$ is an open set.
To do this, note that:
$S$ is open $\iff$ $$\forall p\in S: p\in \text{Int}(S)$$
$$\forall p\in S: \exists \varepsilon >0 : B(p,\varepsilon)\subseteq S$$
In order to find if $S$ ... |
I am trying to solve this problem:
> Prove that for every natural number $m$, there exists a natural number
$n$ such that in the decimal representation of the number $5^n$ there are at least $m$ zeros.
Before that I proved that for $m$ = $2^a$, $(a \ge 3)$ $ $ $Z^*_{m}$ = $<-1>_2$ $*$ $<5>_{2^(a-2)}$
These ... |
Let $\mathcal{C}$ be a category. By *diagram* I mean a covariant functor $F\colon\mathcal{J}\to\mathcal{C}$ for some category $\mathcal{J}$. In [this source][1] it is said that a *commutative diagram* is a diagram $F\colon\mathcal{J}\to\mathcal{C}$ such that there exists a posetal category $\mathcal{P}$ and diagrams $G... |
Is "up to natural isomorphism" crucial? |
I am looking for a solution $X \in \mathbb{R}^{n \times n}$ which minimizes the following quadratic energy:
$$
\textrm{min}_X \| X A X^T - B\|^2 + \mu \|X \Lambda X^T - \Lambda \|^2
$$
where $A, B \in \mathbb{R}^{n \times n}$ are P.D., $\Lambda$ is diagonal with positive entries, $\mu \in \mathbb{R}_{> 0}$, an... |
Lets say $S\subseteq \mathbb{R}^m$ is a set, and we want to determine the following:
1. Check/Prove if $S$ is an open set.
To do this, note that:
$S$ is open $\iff$ $$\forall p\in S: p\in \text{Int}(S)$$
$$\forall p\in S: \exists \varepsilon >0 : B(p,\varepsilon)\subseteq S$$
In order to find if $S$ ... |
It seemingly is a fact that a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ acting on hyperbolic space cannot be compact if it contains a parabolic element.
**I was wondering if the following proof works:**
Let $\Gamma$ be our group and $T \in \Gamma$ be parabolic. Then we can conjugate $\Gamma$ in $\mathrm{P... |
After reading [answers to this question][1] I can't help being confused about axioms 2 and 4 being truly independent, or about their true meaning for that matter.
Recalling them, for a given set **X**:
1. If *N* is a neighbourhood of x ∈ **X** (i.e., *N* ∈ **N(x)**), then x ∈ *N*.
2. If *N* ∈ **N(x)** and *N* ... |
**Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone. Denoting by $∥v∥_g = \sqrt{|g(v, v)|}$ the “norm” of $v$. Then the reverse C-S inequality holds with equality iff v and w are colinear:
$−g(v, w) ≥ ∥... |
I have the following task: Let $Y$ be a binomial random variable. Find the asymptotic distribution of the ML estimate and find the asymptotic distribution of the estimator $(p(1-p))^{\frac{1}{2}}$ of the standard deviation. What happens if $p=0.5$ ?
For $Y \sim Bin(n,p)$, the MLE of $p$ is $\hat{p}=\frac{Y}{n}$, t... |
If a function $f(x,y)$ is convex in $x \in \mathbb{R}$, and also convex in $y \in \mathbb{R}$ then is $f$ a convex function in $\mathbb{R}^2$? I think no, for example $f(x,y) = xy$ it is convex in $x$, and $y$ but the function is not convex. Am i correct? |
I am trying to solve this problem:
> Prove that for every natural number $m$, there exists a natural number
$n$ such that in the decimal representation of the number $5^n$ there are at least $m$ zeros.
Before that I proved that for $m$ = $2^a$ $(a \ge 3)$, we have $Z^*_{m} = \langle -1\rangle_2*\langle5\rangle... |
It seemingly is a fact that a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ acting on hyperbolic space cannot be compact if it contains a parabolic element.
**I was wondering if the following proof works:**
Let $\Gamma$ be our group and $T \in \Gamma$ be parabolic. Then we can conjugate $\Gamma$ in $\mathrm{P... |
**Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone(ie, both future-directed or past-directed). Denoting by $∥v∥_g = \sqrt{|g(v, v)|}$ the “norm” of $v$. Then the reverse C-S inequality holds with equalit... |
Everything you're saying is valid, but you may be overthinking a bit.
> You may find some $\varepsilon$ for any particular choice $p$, yet how do we find the $\varepsilon (p)$ for some arbitrary $p$?
Let's take your open rectangle $A = (1, 2) \times (-1, 2)$ as an example. Let's take an arbitrary point $x_1 \time... |
Consider a normal operator $N$ on complex Hilbert space $\mathcal{H}$, such that for each $\lambda\in\sigma(N)\setminus\{0\}$, $\lambda$ is isolated in $\sigma(N)$ and $\mathrm{dim}\mathrm{ker}(N-\lambda I)<\infty$. Moreover, let $\rho$ denote the continuous functional calculus on $\sigma(N)$.
How would one verify t... |
I need to use Minkowski’s theorem to show that if α ∈ R and Q ∈ N then there exist q, a ∈ Z such that 1 ≤ q ≤ Q and |qα − a| ≤ 1/Q. I believe that I need to define the lattice in R: Λ = {qα : 0<α<1/Q} but am not sure where to go from there. Is this the correct lattice to use and if so how would I use it in the proof? A... |
**Please note that I am not looking for a solution to the question in the title as it has been asked before.**
I was trying to prove this famous result to a friend:
>If $I$ is an ideal of a ring $R$ and $P_1, P_2$ are prime ideals of $R$ such that $I\subseteq P_1\cup P_2$, then $I\subseteq P_1$ or $I\subseteq P_2... |
On the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq 0}, P)$, let $Z=(Z_t)_{t\geq 0}$ be a continuous-time local martingale (assume $Z_0=0$).
> i.e. there exists a sequence of increasing $\mathbb{R}_+$-valued $(\mathcal{F}_t)_{t\geq 0}$-stopping times $(T_n)_{n\geq 1}$ with $T_n \uparrow \... |
If we discretize a continuous-time local martingale, do we still get a local martingale? |
Let $(a_n)$ be an unbounded and monotone increasing sequence of positive integers and let $(b_n)$ be another sequence of positive integers such that for all $n \in \mathbb{N}$, $a_n \leq b_n$. Evidently, $(b_n)$ is unbounded. Is it the case that $(b_n)$ is also monotone increasing? If not, is there a "$(b_n)$ independe... |
When is a sequence that is bounded below by an unbounded and monotone increasing also monotone increasing? |
So, let us add the subsets that you are missing. There are only three of them: $$\{6,10,15\},$$ $$\{10,12,15\},$$ $$\{6,10,12,15\}.$$ You missed them because each of them doesn’t have a (non-trivial) common divisor for all its elements.
You found all the other subsets though, but made a mistake while counting: $2^5... |
What is a good reference for an elliptic curve over a local field having at least multiplicative reduction over a field extension of degree $\le6$? |
If $a(x,\xi)$ is of $C^{\infty}$ class and positively homogeneous of degree m for $|\xi|\geq1$, i.e.,
$$
a(x, t\xi) = t^{m} a(x, \xi), |\xi| \geq 1, t\geq1,
$$
then $a \in S^{m}_{1,0} = S^{m}$.
We need to prove that $|\partial^{\alpha}_{x}\, \partial^{\beta}_{\xi} a(x,\xi)| \leq C (1+|\xi|)^{m-|\alph... |
$F_V=\iota_Vdv_g$, this equation is valid in 3-dimensional spaces. It basically says that 2-forms are in bijective correspondence with vector fields.
Consider a vector field $V \in \mathfrak{X}(M^3)$ for a 3-dimensional manifold $(M^3,g)$. Take its metric equivalent 1-form denoted by $V^\flat$ and then apply Hodge s... |
I wish to know the exact value of
$$\sum_{n=1}^\infty\frac1{n^n}.$$
I've found on the internet some mentions of the equality (the [Sophomore's Dream][1], as I've learned)
$$\sum_{n=1}^\infty\frac1{n^n} = \int_0^1\frac1{x^x}dx.$$
Wolfram Alpha's computations returns the value of $1.29128599706266\dots$ to the R... |
I've familiarised myself with models of set theory and am beginning to understand the basics, but am still very far away from being a proper model theorist. I currently live under the impression that I understand the idea of the construction of the first and second Fraenkel models.
Hence, unfortunately, I must ask h... |
All of the questions I found are about literally covering "What is the most effective way to cover a circular pot with a square lid etc..."
Here's an example to show what I mean: I have a circular farm of 1000 sq m in area. I have a crop duster that can cover a ten meter wide area (10 meter wingspan) while traveling ... |
What is the most efficient way to cover (not literally cover) a circular area? |
I am looking for a solution $X \in \mathbb{R}^{n \times n}$ which minimizes the following energy:
$$
\textrm{min}_X \| X A X^T - B\|^2 + \mu \|X \Lambda X^T - \Lambda \|^2
$$
where $A, B \in \mathbb{R}^{n \times n}$ are P.D., $\Lambda$ is diagonal with positive entries, $\mu \in \mathbb{R}_{> 0}$, and $ \| \cd... |
Let's say we have a vector space $X$ and a function $p:X\to \mathbb{R}^+$ such that $p(ax)=|a|p(x)$ for all $a\in \mathbb{R}$ and $x\in X$, and such that $p(x)=0$ iff $x=0$.
Then $p$ satisfies the triangle inequality (and is therefore a norm) iff it's a convex function. This is where our connection between the tria... |
$F_V=\iota_Vdv_g$, this equation is valid in 3-dimensional spaces. It basically says that 2-forms are in bijective correspondence with vector fields. By the way, $\iota$ is used for interior product not for "inner product". See this [question][1] to see the relation between them.
Consider a vector field $V \in \mat... |
Show that $\partial x^2 = 2P(x)$ and $\partial \wedge x = 0$ (Geometric algebra)I’m trying to show equations (2-1.32) and (2-1.33) on page 51 of 20 of Hestenes and Sobczyk’s “Clifford Algebra to Geometric Calculus”.
> $\partial |x|^2 = \partial x^2 = 2P(x) = 2x, \tag{1.32}$
> $\partial \wedge x = 0, \tag{1.33}$
... |
It follows from [Dirichlet's approximation theorem][1] that for any irrational $\alpha,\ 0<\left\lvert \frac{p}{q} - \alpha \right\rvert < \frac{1}{q^2} $ for infinitely many pairs of integers $(p,q).$ In fact, [Hurwitz's theorem][2] is a nice result which says that the best we can do with the above is replace $\ \frac... |
We aim to prove that $\tan\left(\frac{|x+y|}{1+|x+y|}\right) \leq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y \in \mathbb{R}$.
Let's assume that $\tan\left(\frac{|x+y|}{1+|x+y|}\right) \geq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y... |
To get something out of the way: not all pairs of parallel epis in $\mathsf{Set}$ have a common section. A simple example is where $E = \{a, b, c, d\}$ maps epimorphically to $\{0, 1\}$ in two ways $f, g$, where $f^{-1}(0) = \{a, b\}$ and $g^{-1}(0) = \{c, d\}$.
The intuition I have for reflexive forks as such is f... |
It seemingly is a fact that a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ acting on hyperbolic space cannot be compact if it contains a parabolic element.
**I was wondering if the following proof works:**
Let $\Gamma$ be our group and $T \in \Gamma$ be parabolic. Then we can conjugate $\Gamma$ in $\mathrm{P... |
We aim to prove that $\tan\left(\frac{|x+y|}{1+|x+y|}\right) \leq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y \in \mathbb{R}$.
Let's assume that $\tan\left(\frac{|x+y|}{1+|x+y|}\right) \geq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y... |
We aim to prove that $\tan\left(\frac{|x+y|}{1+|x+y|}\right) \leq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y \in \mathbb{R}$. Let's assume that $\tan\left(\frac{|x+y|}{1+|x+y|}\right) \geq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y ... |
Hello Math Stack Exchange community,
I'm currently looking to delve deeper into advanced topics in linear algebra, including system of linear equations, eigenvalues and eigenvectors, vector spaces, linear mappings and their applications, and iterative methods. I am seeking advice on the most optimal study methods an... |
how to find the roots for this equation? |
My question is going to focus on quite a counterintuitive thing.
A couple of preliminaries. I understand **geometric space** as a set of points. A **point**, in turn, is an abstract idealization of an exact position in a space. There is a point in each position of geometric space.
While exploring the possible way... |
Can adjacent points exist in geometric space? |
On the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq 0}, P)$, let $Z=(Z_t)_{t\geq 0}$ be a continuous-time local martingale (assume $Z_0=0$).
> i.e. there exists a sequence of increasing $\mathbb{R}_+$-valued $(\mathcal{F}_t)_{t\geq 0}$-stopping times $(T_n)_{n\geq 1}$ with $T_n \uparrow \... |
The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. (Assume that the difference in birth and death rates is 20 million... |
Which logistic differential equation to use? |
What is a good reference for an elliptic curve over a local field having a semi-stable Neron model over a field extension of degree $\le6$? |
Im reading Mazur's Proof of his famous theorem for the torsion of elliptic curves. He points to a result that given an elliptic curve over a given local field, there is a field extension of ramification degree not bigger than 6, over which the given curve has at least multiplicative type reduction. He cites a paper as ... |
After reading [answers to this question][1] I can't help being confused about axioms 2 and 4 being truly independent, or about their true meaning for that matter.
Recalling them, for a given set $X$:
1. If $N$ is a neighbourhood of $x \in X$ (i.e., $N \in \mathcal N(x)$), then $x \in N$.
2. If $N \in \mathcal ... |
I saw [this question][1] on MSE. It's about the general formula of the metric of an N-sphere.
I was wondering, is there a similar general formula for the metric of an N-hyperbolic
(simply connected, one-sheeted) hyperboloid? I found the case for a 2-hyperboloid [(here)][2], but I'm not able to find the general cas... |
What is the formula for the metric of an N-hyperbolic hyperboloid? |
I have a rather basic mis-understanding about Lie groups and Lie algebras. Consider the Lie group $SO(N)$ for $N>3$ of rotations on $\mathbb{R}^N$. On the one hand this Lie group has dimension $N(N-1)/2$, since every $SO(N)$ element can be parametrized as $e^{X}$, where $X$ is an anti-symmetric matrix with $N(N-1)/2$ f... |
Why is the group of rotations in $\mathbb{R}^n$ not $n$-dimensional? |
To understand combinatorics better I formulated a question for myself (not homework!), that I'm trying to solve:
Imagine a sack with 6 red balls, 5 yellow balls and 4 blue balls. Pick six of them without laying them back. What is the probability that at least 2 of the six balls are yellow and at least 1 is red. (The... |
Given a set of points on a hexagonal lattice, is there a simple way to compute a center point $c$ and a minimal integer radius $r$ such that all points are within Manhattan distance $r$ of $c$? |
> Let:
>
>\begin{gather*}
f: \mathbb{R}^n \to \mathbb{R}, \quad A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n, x \in \mathbb{R}^n, c \in \mathbb{R} \\
f(x) = x^T A x + b^T x + c \\
\end{gather*}
>
> If $A$ is symmetric positive definite, show that $f$ is convex.
The domain of $f$ is convex. Algebraic... |
The coin is tossed until the number of heads is exactly equal to twice the number of tails. The coin lands on heads with probability . What is the probability that a coin will be tossed forever?
A similar problem (A coin lands heads with probability . The coin is tossed until the first heads will come up. What is the ... |
I am confused between the transformations between the time-domain and the frequency domain. I have a signal y(t) which is a sum of multiple sinusoids. I band-pass filter this signal to extract one sinusoid and mix this signal with two sinusoids. This is shown in the following figure:
[![Signal filtering and mixing][1]... |
Multiplication of a time-domain sinusoid to a s-domain (Laplace) signal? |
The coin is tossed until the number of heads is exactly equal to twice the number of tails. The coin lands on heads with probability . What is the probability that a coin will be tossed forever?
A similar problem (A coin lands heads with probability . The coin is tossed until the first heads will come up. What is the ... |
# Background
A ***Surface of Constant Width*** is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the surface and paired-planes. As part of a personal project, I want to derive a trio of parametric e... |
Can you translate Section 4.3 of this paper (specifically the functions named $r$) for someone who hasn't gone past Calc 1? |
I am working through a proof outlined in section 4.4 of *Operations Research: Applications and Algorithms*, by Wayne L. Winston. The objective is to show that if a linear program has an optimal solution, then at least one of its basic feasible solutions must be optimal.
Assume:
- $x$ is an optimal solution to the... |
Here:
https://math.stackexchange.com/questions/3146697/proof-that-n-planes-cut-a-solid-torus-into-a-maximum-of-frac16n33n28n
an intuitive idea of understanding the formula of Torus Slicing is given. Nevertheless I wanted to ask if there is any concrete Literature with a formal proof given?
In this book:
Gardner... |
May I mention this:
Gerard Letac (1981) 'Isotropy and sphericity, some characterisations of the normal distribution.' Ann. Statist. 9, 408-417.
**The following part was written by @Amir:**
In the [above paper][1], Prof. Letac presented the following characterization of normal distributions:
[![enter ima... |
I am asked to solve the following problem with the following hint: use Renewal Equation. I get stuck on the hint… it seems a lot easier to solve it without the hint.
Exercise: a machine is working for an exponential time with rate $\lambda_1$ and when it breaks it is being repaired for an exponential time with rate ... |
We aim to prove that $\tan\left(\frac{|x+y|}{1+|x+y|}\right)$ $\leq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y \in \mathbb{R}$. Let's assume that $\tan\left(\frac{|x+y|}{1+|x+y|}\right)$ $\geq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x... |
I found a question in Modern Real Analysis by William P. Ziemer. The question can be found in Section 2.1 Question 1. It goes:
Use the fact that $\mathbb N =$ {$ n: n = 2k$ for some $k \in \mathbb N$} $\cup$ {$ n: n = 2k$ for some $k \in \mathbb N$} to prove $c \cdot c = c$.
**What could they be referring to by c?**... |
What does c mean in the following question on cardinality? |
Can the Residue Integration method be used if the interval of integration does not result in a closed contour, or cannot be leveraged due to symmetry?
For example:
<p><span class="math-container">$$\int_{a}^{b}\frac{1}{k+cosx}\,dx \\$$</span></p>
Letting a=0, b=2$\pi$, then substituting z=$e^{ix}$, then th... |
I am reading the book introduction to functional analysis by Christian Clason. I need help with the problem 4.2 from the book:
Let $T:C[a, b] \rightarrow C[a, b],\ [Tx](t)=tx(t) \quad \forall t \in [a, b]$
1. Show T is a continuous linear operator
2. Calculate $\|T\|$
3. Show or give a counterexample for the clo... |
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