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We have that Y = $e^x$ and that X is a random variable with $X∼N(μ,σ^2)$. I want to obtain the density function $f_Y(y)$ of $Y$, by finding the cumulative distribution function $F_Y(y)$, and then differentiate. This is my work so far.
Suppose that $y\gt 0$.
$F_Y(y)=\Pr(Y\le y)=\Pr(e^X\le y)=\Pr(X\le \ln y).$
T... |
> Ten chairs are evenly spaced around a round table and numbered
> clockwise from $1$ through $10$. Five married couples are to sit in
> the chairs with men and women alternating, and no one is to sit either
> next to or across from his/her spouse. How many seating arrangements
> are possible?
>
> $\mathrm{(A)}\... |
I was reading [G. B Jeffrey's][1] article with The Royal Society as publisher, and I came across two integrals, for which there were solutions but no procedure how they were obtained. I was trying to do it myself but failed multiple times (I didn't get very far), moreover other authors don't provide the calculation eit... |
I cannot understand how double/nested induction works. I am supposed to prove that the addition operation is both commutative (x + y = y + x) and associative (x+(y+z)) = ((x+y)+z) from the following definitions of the naturals and addition (no other axiom is given).
Naturals: Base: $0 \in \mathbb{N}$ and
Ind. $n \i... |
We have, for $z = e^{\frac{2 \pi i}{17}}$, $$(z + z^2 + z^4 + z^8 + z^{16} + z^{15} + z^{13} + z^9) (z^3 + z^5 + z^6 + z^7 + z^{10} + z^{11} + z^{12} + z^{14}) \in \mathbb Z.$$
This happens because the Galois group $\operatorname{Gal}(\mathbb Q(z)/\mathbb Q)$ acts on the partition of roots by either swapping them or i... |
I am currently reading the paper "Variational Autoencoding Neural Operators", which performs experiments on 1D gaussian random fields (i.e., gaussian processes).
They generate functions according to the following process.
"A zero-mean Gaussian random field on $X = [0,1]$, with zero boundary conditions and covaria... |
We have, for $z = e^{\frac{2 \pi i}{17}}$, $$(z + z^2 + z^4 + z^8 + z^{16} + z^{15} + z^{13} + z^9) (z^3 + z^5 + z^6 + z^7 + z^{10} + z^{11} + z^{12} + z^{14}) \in \mathbb Z.$$
This happens because the Galois group $\operatorname{Gal}(\mathbb Q(z)/\mathbb Q)$ acts on the partition of roots by either swapping them or i... |
I am currently reading Chapter 3.4 of Galois group and Fundamental group by of Szamuely and I am stuck at an important step of lemma 3.4.2.
[![The highlighted part][1]][1]
[1]: https://i.stack.imgur.com/CAU3E.png
I am not sure I understand what they mean by a "direct factor" of the tensor product, and wha... |
I'm looking for clarification as to whether or not the following proof is valid. I am unsure in justifying the last paragraph.
Thank in advance as always,
M
------
**Claim:** Let $R$ be a ring and $(M_{k} , q_{k,j})_{k \in \mathbb{N}}$ be a directed inverse system of finitely-generated $R$-modules with surj... |
We have, for $z = e^{\frac{2 \pi i}{17}}$, $$(z + z^2 + z^4 + z^8 + z^{16} + z^{15} + z^{13} + z^9) (z^3 + z^5 + z^6 + z^7 + z^{10} + z^{11} + z^{12} + z^{14}) \in \mathbb Z.$$
This happens because the Galois group $\operatorname{Gal}(\mathbb Q(z)/\mathbb Q)$ acts on the partition of roots by either swapping them or i... |
Consider the following step function:
$f: \mathbb{N} \rightarrow \mathbb{N}, f(n) = \lfloor \frac{n^{2}}{2023} \rfloor$
Where $n \in [1, 2023]$. How many distinct values does the step function take?
Thanks! |
This seemingly simple question and some variation thereof (‘where does math come from?’ or ‘what is the basis of math?’ or ‘what justifies math?’) is often asked on this site, but (as you probably suspect) it has no simple answer.
Indeed, this question is probably more suited for the Philosophy.SE site, and I would ... |
Let $\sigma\in S_n$. For each $i\in\{1,2,\ldots,n\}$ let $k_i$ be the smallest positive integer such that $\sigma^{k_i}(i)=i$. Suppose now that $k_1,\ldots,k_n$ are all even. Is it true that $n$ must be even?
**My attempt:** Suppose that $\sigma=\sigma_1\cdots \sigma_r$ is the decomposition of $\tau$ into disjoint ... |
I'm looking for clarification as to whether or not the following proof is valid. I am unsure in justifying the last paragraph.
Thank in advance as always,
M
------
**Claim:** Let $R$ be a ring and $(M_{k} , q_{k,j})_{k \in \mathbb{N}}$ be a directed inverse system of finitely-generated $R$-modules with surj... |
I am having trouble understanding the solution to the question, how does when it starts on a black square must end on another black square, and if the king's moves change the colors of the squares how does that constitute making the number of traversed squares odd?
**Question:**
A king Visited all squares of the us... |
A function $f: \mathbb R^d \rightarrow \mathbb R$ is said to have quadratic form iff it can be written as $f(x)=(x-b)^T A(x-b)$ for some $A\in \mathbb R^{d\times d}$ and $b \in \mathbb R^d$. If I have two functions having quadratic forms $f(x)=(x-b_1)^T A_1(x-b_1)$ and $g(x)=(x-b_2)^T A_2(x-b_2)$ can I write $h(x):=f(x... |
This seemingly simple question and some variation thereof (‘where does math come from?’ or ‘what is the basis of math?’ or ‘what justifies math?’) is often asked on this site, but (as you probably suspect) it has no simple answer.
Indeed, this question is probably more suited for the Philosophy.SE site, and I would ... |
Let $\sigma\in S_n$. For each $i\in\{1,2,\ldots,n\}$ let $k_i$ be the smallest positive integer such that $\sigma^{k_i}(i)=i$. Suppose now that $k_1,\ldots,k_n$ are all even. Is it true that $n$ must be even too?
**My attempt:** Suppose that $\sigma=\sigma_1\cdots \sigma_r$ is the decomposition of $\sigma$ into dis... |
There is a challenging question assigned regarding Counting. here is the question:
>we call every path from point A to B a **good** path, if in every step it has moved up, down, or right and not crossed the same line more than once. we write down the number of squares beneath every possible *good* path. what would b... |
Let $G = \mathrm{GL}_n$, let $B$ be a Borel subgroup of $G$.
The set $\mathcal{B}$ of $G$-conjugates of $B$ can be given an algebraic variety structure and is known as the *variety of Borel subgroups* or the *complete flag variety*. It is isomorphic to $G/N_G(B) = G/B$.
A classic result is the following:
**The... |
This seemingly simple question and some variation thereof (‘where does math come from?’ or ‘what is the basis of math?’ or ‘what justifies math?’) is often asked on this site, but (as you probably suspect) it has no simple answer.
Indeed, this question is probably more suited for the Philosophy.SE site, and I would ... |
Say if I have the limit as $z$ approaches $0$ of $\frac{\bar{w}}{w}$, I can set $w = re^{i\theta}$, which would give the limit as $z$ approaches $0$ of $e^{-2i\theta}$. Am I correct in assuming that this limit doesn't exist since as $z$ approaches $0$, $r$ tends to $0$ and $\theta$ can range between $-\pi$ and $\pi$ so... |
I have been trying to solve this problem.
I was able to observe that from the given conditions $x^2 + f^2(x) + (f'(x))^2 = c$, where c is some constant, but it does not help me in any of the options.
Any hints or ideas would be appreciated
Let $f$ be a twice differentiable function that assumes real values satis... |
Let $(X,d)$ be a metric space and let $\mathrm{Iso}(X)$ denote the group of isometries of $(X,d)$, where the group operation is the composition. Equip $\mathrm{Iso}(X)$ with the topology of pointwise convergence. I have not been able to find any source supporting the claim that $\mathrm{Iso}(X)$ is a topological group.... |
*Definition:* A real sequence $\ (x_n)_n\ $ is *convex* if $\ x_n - x_{n+1} \geq x_{n+1} - x_{n+2}\quad \forall\ n\in\mathbb{N}. $
Suppose $\ (x_n)_n\ $ is a positive, decreasing sequence of real numbers such that $\ \displaystyle\sum x_n\ $ converges.
Proposition: There exists a convex sequence $\ (y_n)_n,\ $ su... |
Jan Bohr's answer is already a great answer on how to solve your problem in coordinates, let me just add a bit more on that. In particular, I want to answer your questions:
- why is $\mathcal{A}(L)$ an affine space?
- where does the logarithm come from?
While starting from the standard definition of connectio... |
For Markov Chains, there are well known formulae (e.g. fundamental matrix approach, first step analysis) which show us how to calculate the Expected Time to Absorption (e.g https://en.wikipedia.org/wiki/Absorbing_Markov_chain). That is, given a transition matrix and the current state - how many iterations (i.e. steps... |
Say if I have the limit as $w$ approaches $0$ of $\frac{\bar{w}}{w}$, I can set $w = re^{i\theta}$, which would give the limit as $w$ approaches $0$ of $e^{-2i\theta}$. Am I correct in assuming that this limit doesn't exist since as $z$ approaches $0$, $r$ tends to $0$ and $\theta$ can range between $-\pi$ and $\pi$ so... |
I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", page 344). The article reads:
> We define an _umbral calculus_ $(C, r)$ to consist of a coalgebra $C$ in $\mathcal{C}_{R}$ [the category of coalgebras over a commutative ring ... |
This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning.
Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). Consider $\bigcap_{n>0} I^n = \underset{\longrightarrow}{\textrm{lim}} \, I^n$ as a colimit. I would like to apply the $\t... |
In a topological space we do not talk about closeness in the sense of distance. Distance is a concept defined in metric spaces, not in topological spaces. We talk about abstract closeness, something more related to similarity or likeness than to distance.
In a metric space, when somebody asks “how close are these tw... |
**Problem :** Given $ t \in \mathbb{C}$. Complex numbers $x,y,z$ satisfy that $|x|=|y|=|z|=1$ and $\frac{t}{y}=\frac{1}{x}+\frac{1}{z}$. Calculate $$\left|\frac{2xy+2yz+3zx}{x+y+z}\right|$$
**My attempt :**
$$\left|\frac{2xy+2yz+3zx}{x+y+z}\right|=\left|\frac{2y(x+z)+3zx}{x+y+z}\right|$$
as $y(x+z)=txz$
$$\left... |
Let $ a > 1 $. Find all continuous functions from $ \mathbb{R}^*_+ \to \mathbb{R}^*_+ $ that are solutions to the functional equation $\forall x\in \mathbb{R}^*_+ , f(x) = f(x^a f(x)) $.
It's easy to see that
all positive constant functions are solutions
If $f: \mathbb{R}^*_+ \to \mathbb{R}^*_+ $... |
Let $n$ be a large positive integer and let $X$ be a random element of hypercube $\{0,1\}^n$ such that $\mathbb P(|X|= \ell) \propto \ell^{-\beta}$ for all $\ell \in \{0,1,\ldots,n\}$. Here $\beta \ge 0$ is a constant. For example, when $\beta=0$, then $X$ is uniformly distributed. Now, fix $x \in \{0,1\}$, $r \in (0... |
Say if I have the limit as $w$ approaches $0$ of $\frac{\bar{w}}{w}$, I can set $w = re^{i\theta}$, which would give the limit as $w$ approaches $0$ of $e^{-2i\theta}$. Am I correct in assuming that this limit doesn't exist since as $w$ approaches $0$, $r$ tends to $0$ and $\theta$ can range between $-\pi$ and $\pi$ so... |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and ... |
> Giap secretly gave the judges seven natural distinct numbers whose sum equal to
> $2024$ and enable to show Thin the forth largest one. Find the forth
> largest that can help Thin guess all the numbers of Giap.
It's my City Test for Mathematic. There are many solutions on the Internet, but they have a little bit... |
I having some trouble understanding how mean and variance are be used in sampling rates and temporal rates (per unit of time).
**Example**
If I eg. wanted to evaluate car crashes per hours driven I would like to know how evaluate the expected crashes per hours driven and how dispersed the data is. For simplicity ... |
I having some trouble understanding how mean and variance are be used in sampling rates and temporal rates (per unit of time).
**Example**
If I eg. wanted to evaluate car crashes per hours driven I would like to know how evaluate the expected crashes per hours driven and how dispersed the data is. For simplicity ... |
>A set $E\subseteq \mathbb{R}$ is called $m^*$-measurable if for all $A\subseteq \mathbb{R}$
$$m^*(A)=m^*(A\cap E)+m^*(A\cap E^c)$$
>The set of all measurable sets is called Lebesgue sigma algebra
Does not sigma algebra has properties? this is a way to "build" Lebesgue sigma algebra from "another way" by measur... |
I recently noticed that my go-to reference for stuff related to Hilbert spaces uses the terms "Hilbert space" and "complex Hilbert space" synonymously. Hence the following questions:
- Are there any "basic" theorems that only hold in the complex (or the real) case? By "basic" I perhaps mean that they would be discus... |
I have been trying to solve this problem.
I was able to observe that from the given conditions $x^2 + f^2(x) + (f'(x))^2 = c$, where c is some constant, but it does not help me in any of the options.
Any hints or ideas would be appreciated
Let $f$ be a twice differentiable function that assumes real values satis... |
its my first question here. I'm re-self-studying linear algebra from different sources and one of them is Linear Algebra and Its applications by g.strang 4th ed. . While i have studied a bunch of material i still don't grasp some basics and i really struggle. [Page 4-5][1] gives two approaches for the geometry of linea... |
Let $P = \mathbb{R}^2 \setminus \{(0,0)\}$. We define a relation on $P$ as follows: $(x,y) \sim (z,w)$ if and only if there exist $\lambda \in \mathbb{R}\setminus \{0\}$ such that $x = \lambda z$ and $y = \lambda w$.
Define the function $\phi : \mathbb{R} \to {^{P}\big/_{\sim}}$ by $\phi(t) = [(t,1)]$ for $t \in \ma... |
According to the convention that we our taught in our schools, focus = (a,0) [for a standard parabola] and directrix: y=-a. But is there any way to obtain the equation without knowing these conventions?
I know the basic definition is this: A Parabola is the locus of all points whose distance from a point is equal t... |
What's the correct way to derive the eq. of a parabola without knowing the conventional values of focus and directrix? |
Let $u_0 = \dfrac{b - a}{\| b - a\|}$, and let $u_1 = \dfrac{a + b}{\| a + b \| }$ , and $u_2 = \dfrac{a \times b }{ \| a \times b \| }$, then the axis of rotation is given by
$ U = \cos \alpha u_1 + \sin \alpha u_2 $
vectors $v_1$ and $v_2$ are orthogonal to $U$, where
$ v_1 = \sin \alpha u_1 - \cos \alpha u_... |
According to the convention that we are taught in our schools, focus = (a,0) [for a standard parabola] and directrix: y=-a. But is there any way to obtain the equation without knowing these conventions?
I know the basic definition is this: A Parabola is the locus of all points whose distance from a point is equal t... |
According to the convention that we are taught in our schools, focus = (a,0) [for a standard parabola] and directrix: x=-a. But is there any way to obtain the equation without knowing these conventions?
I know the basic definition is this: A Parabola is the locus of all points whose distance from a point is equal t... |
Confusion can arise by calling the unknowns $x$ and $y$. In the picture on the left, they are actual $x,y$ coordinates of a point with geometrical significance. In the picture on the right, $x,y$ are called $x$ and $y$, but they're not connected to $x$ and $y$ coordinates anymore. I think it's conceptually better to se... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
Let $n,w\in\mathbb{N}$, and consider the matrix ${\bf R}\in\mathbb{Z}^{2n\times w}$ sampled from a gaussian distribution with standard deviation $s>\sqrt{n}$. What I'm curious about is how small the first singular value of $\bf R$ be, i.e. how to minimize $$s_1(\bf R)=\max_{||u||=1}\{||\bf Ru||\}$$ for a certain amount... |
I've been reading the chapter about currents in Simon's Geometric Measure Theory (see e.g. http://web.stanford.edu/class/math285/ts-gmt.pdf). In section 2 he defines the Mass of an $n$-current $T$ by
$$M(t):=\sup_{|w|\leq 1, w\in D^n}T(w)$$
and in Remark 2.6 he mentions that there is some freedom in defining this Mas... |
Does $\sum_{n=2}^{\infty}\frac{\cos(n)}{\ln(n)}$ converge? |
Collapsing either one of the circles in the bouquet of two circles to the basepoint, how can I describe this by a map (in the free product) and how is this related to that $\mathbb{R}^2\backslash\{p,q\}$ is **not** homeomorphic to a bouquet of two circles! However there is a **deformation retraction** of that space to ... |
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... |
Fix a set $X$. For a topology $\tau$ on $X$, let $F_\tau = \{f: X \to X \mid f \text{ continuous w.r.t. } \tau\}$. What would be an example of a set $X$ and two Hausdorff topologies $\tau$ and $\sigma$ on $X$ such that $\tau \neq \sigma$ but $F_\tau = F_\sigma$? |
For any given $j,k\in\mathbb{N}$, with $j\leq k$, how is it possible to prove the following equivalence
$$
\frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!}?
$$
My attempt is:
$$
\frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!}\\
2^j\binom{2k-j}{k-j}k!=\sum_{l=j}^k\binom{2k-l-1... |
[Here][1], the Ramanujan Master Theorem is proven in the following way:
$$ \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} \lambda(n)x^n = \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} E^n\lambda(0)x^n = \int_0^\infty x^{s-1}e^{-Ex}dx \;\lambda(0) = \frac{\Gamma(s)}{E^s}\lambda(0) = \Gamma(s)\lambda(-s).$$
Th... |
In [this](https://github.com/mitmath/18303/blob/master/supp_material/poissonFD.ipynb) notebook from MIT's [Intro to Linear PDEs](https://github.com/mitmath/18303/tree/master) course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations... |
Is anyone familiar with results about properties of the limit set of the local rule for a cellular automaton? I haven't been able to find any good materials on the subject from an initial search, and I thought that someone might help refer me to such results.
I am considering a cellular automaton (CA) $G$ on $\mathc... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
I'm considering the following (stochastic) PDE: $${\rm d}U_t=f(t)\Delta U_t{\rm d}t+\sigma W_t$$ on $[0,1)^2$, where $f:[0,T]\to(0,\infty)$ is linear and $W$ is a cylindrical Wiener process on $H:=L^2([0,1)^2)$. I somehow need to compute/estimate (or at least find a sharp upper bound on) $\operatorname E\left[\|g(U_t)\... |
I recently had a bit of an obsession with the definition of limits, and that made think about this function: $$f: \mathbb{Q} \to \mathbb{R}; \ \ \ \ f(x) = x$$
What is $\ \lim_{x \to 0} f(x)\ $? Intuitively, I think $\ \lim_{x \to 0} f(x) = 0\ $. I then looked at the definition provided by Wikipedia:
[![enter image... |
Suppose a space $Y$ is given along two subspaces $Y_1,Y_2$ s.t. $Y=Y_1^{\circ}\cup Y_2^{\circ}$ and the intersection $Y_0=Y_1\cap Y_2$ is non-empty. If $(Y_1,Y_0)$ is $p$-connected and $(Y_2,Y_0)$ is $q$-connected for some $p,q\ge0$, then the inclusion induces isomorphisms $\pi_n(Y_1,Y_0,\ast)\rightarrow\pi_n(Y,Y_2,\as... |
Is openness crucial in Puppe's proof of the Blakers-Massey Theorem? |
Given a total ordered abstract set $(S,\preceq)$. The set is said to be complete if any non-empty subset that is bounded from above/below has a supremum/infimum.
Given a $(S,\preceq)$ may not be complete, could we always complete it in certain ways?
To be specific, I am considering the set of Hardy's L-functions... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Here, the faces are numbered 1,2,3,4,5,6 and the throws are assumed to be independent of each other.
If we try working backwards from the throw, lets say $ r^{th}$ throw, in which we observed the third distinct number we can find the probability of $X = r$, where X is the number of throws to get three distinct numbers... |
A fair 6-sided dice is rolled repeatedly. What is the expected number of throws to observe three distinct numbers for the first time? |
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... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
I was watching this video https://www.youtube.com/watch?v=cHvYlrud8j8&ab_channel=AndrewMcCrady and in the proof, he states that when $0 \lt \delta = min (1, \epsilon, \frac{\epsilon}{|L| + 1}) $,
$(L + \epsilon)(1 - \delta) - \delta \lt \frac{f(\alpha)}{g(\alpha)} \lt (L+\epsilon) + \delta$
implies that
$ L - 2... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Given Hermitian Matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Given a real number $x>1$ and a natural number $n$, what can we say about the integer part of $x^n$ in terms of $x$ and $n$?
For simplicity, let us assume $x<2$. For the first few values of $n$, $\left\lfloor x^n\right\rfloor =1$. But, once it crosses $1$, it starts growing quite rapidly. Is it possible to express $... |
This is the Leray-Hirsch theorem on Hatcher's Algebraic Topology.
Theorem 4D.1. Let $F \rightarrow E \xrightarrow{p} B$ be a fiber bundle such that, for some commutative coefficient ring $R$ :
(a) $H^n(F ; R)$ is a finitely generated free $R$-module for each $n$.
(b) There exist classes $c_j \in H^{k_j}(E ; R)$ wh... |
Given Hermitian matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
Suppose I have a sequence of i.i.d. random variables $X_1, X_2, \dots$ with positive mean $E[X_1] = \mu$. For $A>0$, is the function
$$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A)
$$
monotonically non-increasing in $n$ (perhaps for $n$ large enough)?
Intuitively, I think this should be the case. By the law of lar... |
The following proof is weaker than the previous answer since it only works for $\sqrt{2}$.
We prove the following result: if $p, q$ are natural numbers, then
$$\left|\frac{p}{q} - \sqrt{2}\right| > \frac{1}{3q^2}.$$
Denote $r := p/q$. If $q=1$,
$$|r - \sqrt{2}| = |p - \sqrt{2}|\geq \sqrt{2} - 1 > \dfrac{1}{3}.... |
I was reading [G. B Jeffrey's][1] article with The Royal Society as publisher, and I came across two integrals, for which there were solutions but no procedure how they were obtained. I was trying to do it myself but failed multiple times (I didn't get very far), moreover other authors don't provide the calculation eit... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
It is a well-known theorem that the axiom of choice (AC) is equivalent to the statement that every vector space over any field has a basis. In this question I am interested in the following somewhat dual concept to that of a basis:
If $V$ is some vector space over a field $K$, we call a subset $S\subseteq V'$ of the... |
What axioms are needed to show that every vector space has a "coordinate system"? |
Let $ a > 1 $. Find all continuous functions from $ ]0,+\infty[ \to ]0,+\infty[ $ that are solutions to the functional equation $\forall x\in ]0,+\infty[ , f(x) = f(x^a f(x)) $.
It's easy to see that
all positive constant functions are solutions
If $f: ]0,+\infty[ \to ]0,+\infty[ $ is invertible... |
Refer to [This book][1] for this discussion.
\
**NOTE**: By Fourier Sine Transform, I am **NOT** referring to complex part of complex Fourier Transform (which is $0$ for even functions), but the single sided sine transform as defined below, which can be non-zero for even functions too.
Definitions from the book... |
Given Hermitian matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
An intuitive explanation of the Poisson equation? |
Most (all?) textbooks on calculus state unconditionally that a function *f* is differentiable at _x_ **if and only if** both of the one-sided derivative $f_+'(x)$ and $f_-'(x)$ exist and are equal. On the other hand, answers to [this question][1] implies that that is not the case. So, are the textbooks or answers to th... |
Is a function differentiable at a point if and only if both of the one-sided derivatives exist and are equal? |
I have a question about the definition of Cauchy sequence in $\sigma$-finite measure, e.g. Lebesgue measure. The reason why I have this question is because I was taught that
for a Cauchy sequence $(f_n)$ we can assume its limit $f$ for all $x\in A$, and zero otherwise where each $f_n$ is in $L^{\infty}$.
My quest... |
[Schanuel's conjecture](https://en.wikipedia.org/wiki/Schanuel%27s_conjecture) says the following about the transcendence of numbers related by the complex exponential function:
> Given any $n$ complex numbers $z_1, ... z_n$ that are linearly independent over the rational numbers $\mathbb{Q}$, the field extension $\ma... |
Is the real-imaginary Schanuel's conjecture equivalent to the full Schanuel's conjecture? |
Let $R$ be a unique factorization domain and $f(x)$ be a non constant polynomial in $R[x].$ We denote the content of the polynomial as $c(f)=d$ and consequently, write $f(x)=df_1(x).$ Prove that $f_1(x)$ is a primitive polynomial.
The proof given is as follows:
Let $f(x)\in R[x]$ be given where $f(x)$ is a nonco... |
In the book (Anders Klarbring and Peter W. Christensen, "An introduction to Structural Optimization"), the *Pareto optimality* has been briefly mentioned which is a method used in the case when we have an optimization problem that involves several objective functions to be minimized simultaneously:
$\text{minimize}... |
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