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$$\forall n\in \mathbb{N},\left.\frac{d^{2n+1}}{dx^{2n+1}}(x^{\ln x})\right|_{x=e^{\frac{n}{2}}}=0$$
How to prove this equation? I find that if $n=0$ or $n=1$, the answer is $0$. But I can't prove that it is true for all the nonegative integer. My idea is, calculate the analytical expression of its $n$-th derivative a... |
In Wikipedia's [proof of Bertrand's Postulate](https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate), [Legendre's Formula](https://en.wikipedia.org/wiki/Legendre%27s_formula) is used to establish an upper bound to the [p-adic valuation](https://en.wikipedia.org/wiki/P-adic_valuation) of ${2n}\choose{n}$
The... |
I've learned about differentiability of several variables.
If $f(x,y)$ is differentiable then we can use chain rule on it. But the inverse of this proposition is also not right. So, is there a function $f(x(t),y(t))$, $x$ and $y$ is differentiable, and with specific initial conditions when dealing with a specific po... |
> Find all holomorphic functions $f$ on $\mathbb{D}$ s.t the following
> condition holds for all $n>1$:$$\left\vert
f\left(\dfrac{1}{n}\right)\right\vert<2^{-n}$$
I found [this][1], but it does not answer this question. I currently have no idea for this problem, except I can show $f(0)=0$. Any help I deeply apprec... |
I am trying to follow the derivation of the adjoint method derived in the paper "A tutorial on the adjoint method for inverse problems" by Dan Givoli, but I am stuck on trying to derive equation 16, in particular it is derived from equation 12 by integration by parts. The relevant part of equation 12 is:
$-\int_\Ome... |
integration by parts not giving the expected paper results? givoli2021 |
Suppose diagonal $D\in \mathbb R^{n\times n}$ with $D\succeq 0, v\in \mathbb R^n,$ and $\alpha>0$ are given. Can we $\textit{exactly}$ identify the smallest eigenvalue of $D+\alpha vv^T$ and its corresponding eigenvector (based on $v$ and given eigenpairs of $D$)?
Even though the method to identify the whole spectru... |
\lim_{(x,y) \to (0,2)} ye^{-\frac{1}{x}}
Does this limit exist? Or does the limit not exist? |
Does the limit of ye^(-1/x) as (x, y) approaches (0, 2) exist? |
I just wanted to clarify my understanding of the sub-multiplicative property of the matrix operator norm.
***
On [this Wikipedia page](https://en.wikipedia.org/wiki/Matrix_norm#Properties) it says for any matrix:
$
\|AB\|_{\alpha, \gamma} \leq \|A\|_{\beta, \gamma} \|B\|_{\alpha,\beta}
$
But then says for... |
> Find all holomorphic functions $f$ on $\mathbb{D}$ s.t the following
> condition holds for all $n>1$:$$\left\vert
f\left(\dfrac{1}{n}\right)\right\vert<2^{-n}$$
I found [this][1], but it does not answer this question. I currently have no idea for this problem, except I can show $f(0)=0$. It seems like $f=0$ is t... |
**Question:** How could we solve this ODE system
$$u'=\alpha(1-x)-(\alpha+\beta)u, \quad u(0)=u_0 $$
$$v'=\gamma x-(\gamma+\delta)v,\quad v(0)=v_0 $$
$$x'=\delta v(1-x)-\beta ux, \quad x(0)=x_0 $$
Here, $\alpha,\beta,\gamma,\delta>0$ and we can assume that $\alpha\ll \beta, \gamma\ll \delta$?
**My attempt**:The ... |
I have the following task and I am a bit confused by the solution of it. In the official solution it's stated that only the first assumption is correct. But in my approach I am always coming to the end that the first and the third assumptions are correct. I would be happy if someone could verify if my solution is corre... |
How to deduce the monotonicity of a function from another function? |
I have a some difficulties understanding the proof in section 7.2 of "Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics" by Edward Saff and Arnold Snider.
Theorem 1 states:
"If $f$ is analytic at $z_0$ and $f'(z_0) \neq 0$, then there exists an open disk $D$ centered at $... |
I have a some difficulties understanding the proof in section 7.2 of "Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics" by Edward Saff and Arnold Snider.
Theorem 1 states:
"If $f$ is analytic at $z_0$ and $f'(z_0) \neq 0$, then there exists an open disk $D$ centered at $... |
In a math paper of MAA(Mathematical Association of America) by Z.Chen, he proved that-
"If, inside $∆ ABC$, there is a circular region $ R$ for which the sum of the
distances from a point $P$ in $R$ to the three sides of the triangle is independent of the
position of $P$, then $∆ABC$ is equilateral."
Why did he m... |
I've learned about differentiability of several variables.
If $f(x,y)$ is differentiable then we can use chain rule on it. But I suspect the converse of this proposition is not right. So, is there a function $f(x,y)$ such that for every functions $x(t)$ and $y(t)$ differentiable at $0$ and such that $x(0)=y(0)=0$, t... |
I am trying to find the general complex solutions to the equation
$$(az-b)^n=(cz-d)^n,$$
assuming $a,b,c,d\in\mathbb{C}\setminus\lbrace 0\rbrace$ and all are distinct, which is naturally satisfied by
$$z_0=\dfrac{b-d}{a-c}$$
with (I think) multiplicity $1$ but as this is equivalent to the zeros of a complex po... |
I came across this problem from a post on instagram.
$$x^4+y^4+z^4=32$$
$$x^5+y^5+z^5=186$$
$$x^6+y^6+z^6=803$$
$$(x+y+z) \;\;\; \epsilon \;\;\; N$$
$$x+y+z=?$$
**My attempt**: I set the following: $$x+y+z=a$$ $$xy+yz+za=b$$
$$xyz=c$$ $$$$Now I tried to express each equations in terms of $a,b,c$ and hence solv... |
I've learned about differentiability of several variables.
If $f(x,y)$ is differentiable then we can use chain rule on it. But I suspect the converse of this proposition is not right. So, is there a function $f(x,y)$, such that the partial derivatives $\frac{\partial f}{\partial x}(0,0),\frac{\partial f}{\partial y}... |
> Find all holomorphic functions $f$ on $\mathbb{D}$ s.t the following
> condition holds for all $n>1$ $$\dfrac{1}{\sqrt{n}}<\left\vert
f\left(\dfrac{1}{n}\right)\right\vert <\dfrac{2}{\sqrt{n}}.$$
My attempt:\
From hypothesis, we have $f(0)=0$ and $\left\vert nf\left(\dfrac{1}{n}\right)\right\vert\geq \sqrt{n}\t... |
I'm looking for a way to define structures such as a topological manifold, a differentiable manifold, a fiber bundle, a topological foliation etc. in a unified way so that I could define a few of my own and play around with them, but without having to use full extent of the theory of sheaves that's quite intimidating t... |
Structures on topological manifolds as colimits? |
It is well known that [if R is a commutative ring in which all the **prime ideals** are finitely generated then R is Noetherian](https://math.stackexchange.com/questions/1130223/if-r-is-a-commutative-ring-in-which-all-the-prime-ideals-are-finitely-generated/1130240#1130240).
Is this also true for maximal ideals instea... |
if R is a commutative ring in which all the maximal ideals are finitely generated then R is Noetherian? |
The problem I'm trying to solve is:
Let $n$ be an odd composite number. Show there are at most $\frac{n-1}{4}$ integers $a$ such that $1 \leq a \leq n$ and $n$ is a strong pseudoprime with respect to $a$.
I know that a composite number is a number with at least two prime factors. But I'm very confused with how I can ... |
$$\lim_{(x,y) \to (0,2)} ye^{-\frac{1}{x}}$$
Does this limit exist? or does the limit not exist? |
Recently I together with a friend solved problem 14 of https://math.mit.edu/~rstan/myputnam.pdf by using an inductive approach. However we got stuck on a variation of the problem where the variation states that:
> $f_0(x)=1, f_{n+1}(x)=xf_n(x)+f'_n(x)$
Thinking of using an inductive approach a few $f_n$s were ... |
I was working on my Linear algebra homework and I felt I needed to prove this statement (which might not be true), but is at least true for $\mathbb{R}^2$ from a right angled triangle.
Given a inner product space $V$ with derived norm $|| \cdot || = \sqrt{\langle\cdot ,\cdot\rangle}$. Let $a,b,c \in V$. Show that
... |
I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want t... |
$$\lim_{(x,y) \to (0,2)} ye^{-\frac{1}{x}}$$
Does this limit exist? Or does the limit not exist? |
Does the limit of $ye^{-1/x}$ as $(x, y)\to (0, 2)$ exist? |
I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want t... |
Consider the heat kernel on the torus
$$
K(t,z-z_1):= \sum_{m \in \mathbb{Z}^2} \exp\Big(-(1-4\pi^2|m|^2)t +2\pi i m \cdot (z-z_1)\Big)
$$
It satisfies the semigroup property
$$
K(t_1+t_1,z_1-z_2)=\int_{\mathbb{T}^2} K(t_1,z_1-z) K(t_2,z_2-z) dz
$$
Let now $\rho_\epsilon$ a mollifier and consider the regula... |
does resularized heat kernel have the semigroup property? |
Consider the heat kernel on the torus
$$
K(t,z-z_1):= \sum_{m \in \mathbb{Z}^2} \exp\Big(-(1-4\pi^2|m|^2)t +2\pi i m \cdot (z-z_1)\Big)
$$
It satisfies the semigroup property
$$
K(t_1+t_2,z_1-z_2)=\int_{\mathbb{T}^2} K(t_1,z_1-z) K(t_2,z_2-z) dz
$$
Let now $\rho_\epsilon$ a mollifier and consider the regula... |
I have a background in theoretical physics and the first time I came across winding numbers was in the context of the vacuum of QCD.
By the way physicists treat this topic, I thought it had little to do with Winding numbers in complex analysis. Now I, came across some results of complex Brownian motion applied to p... |
Winding numbers in QCD and winding numbers in complex analysis. Is there a relation through a differential geometric generalization? |
**Background**
Let $c(t):[0,1]\mapsto\mathbb{R}^2$ be a curve with the following properties
1. $c(t)$ is closed;
2. $c(t)$ is symmetric with respect to the horizontal axis;
3. $c(t)$ is contained in the rectangle $\mathcal{R}\triangleq[0,1]\times[-1,1]$;
4. $c(t)$ interpolates the points $[0\,\,1]',[0\,\,-... |
The Artin-Rees Lemma says that for an ideal $I$ in a Noetherian ring $R$, and for finitely generated $R$-modules $N \subseteq M$, there exists a positive integer $k$ such that for all $n \geq k$, it holds that $I^nM \cap N = I^{n-k}(I^kM \cap N)$.
In [this mathoverflow post][1], the question arises whether there is ... |
It is well know the following resul holds.
>**Theorem**
>
>For any $(M,\bot,e)$ monoid there exists a unique esternal operation $\wedge_\ast$ from $X\times\omega$ into $X$ such that for any $x$ in $M$ the following results hold:
>
> 1. the equality
$$
x\curlywedge_\bot 0=e
$$
> 2. the equality holds
... |
I am trying to follow Griffiths & Harris, "Principles of Algebraic Geometry", page 131, where they explain how from a given divisor one obtains an element of the quotient sheaf $\mathcal M^*/ \mathcal O^*$. In their notation, $M$ is a complex manifold, $\mathcal M^*$ is the sheaf of not-identically-$0$ meromorphic func... |
Why is the quotient of two local defining functions for divisors nowhere vanishing? |
Let $X$ be a Gaussian-mixture random vector with probability density function (PDF) $f_X(x)=\sum_{i=1}^kp_if_i(x)$, where for $i=1,2,\ldots,k$, $f_i$ is a multivariate Gaussian PDF with mean $\mu_i$ and covariance matrix $\Sigma_i$, and $\sum_{i=1}^kp_i=1$. Is $Y=AX$, where $A$ is an arbitrary matrix of appropriate siz... |
Are Gaussian mixtures stable? |
I am trying to follow Griffiths & Harris, "Principles of Algebraic Geometry", page 131, where they explain how from a given divisor one obtains an element of the quotient sheaf $\mathcal M^*/ \mathcal O^*$. In their notation, $M$ is a complex manifold, $\mathcal M^*$ is the sheaf of not-identically-$0$ meromorphic func... |
I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance.
Note: How I can derive $B(X \otimes Y)Q$ with respect to X and then find the closed form of $XA + the derevative of B(X \otimes Y)Q=C$
Note: This problem is part of optimization ... |
I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance.
Note: How I can derive $B(X \otimes Y)Q$ with respect to X and then find the closed form of $XA + the derevative of (B(X \otimes Y)Q)=C$
Note: This problem is part of optimizatio... |
Let $T$ be a $k$-regular tree for some $k>2$ and let $G=\text{Aut}(T)$ the group of automorphisms on the tree. I need to find a closed subgroup $H$ of $G$, so $H$ is a **free** group (and therefore discrete) that acts [cocompactly][1] on $T$ (and therefore by some thereom $H$ is a lattice).
The topology on $G$, if... |
I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance.
Note: How I can derive $B(X \otimes Y)Q$ with respect to X and then find the closed form of $XA + \frac{dX}{d B(X\otimes Y)Q}=C$
Note: This problem is part of optimization problem... |
I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want t... |
I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance.
Note: How I can derive $B(X \otimes Y)Q$ with respect to X and then find the closed form of $XA + \frac{dX}{d B(X\otimes Y)Q} B(X\otimes Y)Q=C$
Note: This problem is part of optim... |
Sure, here's your text translated into English:
---
Hello, I have the following question that starts from this expression:
$$
u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y) .
$$
If $h \in C^2(\bar{B})$ is harmonic and... |
I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want t... |
Consider the set of equivalence classes of measurable functions on the interval [0,1] and the metric $\rho(f,g) = \int_{0}^{1} \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|} \, d\mu$. It must be proved that $\rho(f_n,f) → 0$ if and only if $f_n$ → f as in [0,1].
To prove that convergence by measure implies convergence by metric,... |
Let $C$ be a closed, convex set in a Hilbert space $H$.
$f(x)=\inf_{y \in C} \frac{1}{2}||x-y||^2$
WTS:
$$\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... |
I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want t... |
Let $S(t)$ be the stock proce at time $t\geq 0$ with $S(0)=s_0$ and $\Pi(s)=(K-s)_+$ the payoff of an american put with strike price $K$ How can I calculate the derivate $\frac{d}{d \psi} \mathbb E[e^{-\delta t} \Pi(S(t))\chi_{\{S(t)<\psi\}} | S(0)=s]$ for fixed $t$ and $s$? I dont know how to start here, thanks for a ... |
I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want t... |
there is a debate and the professor isnt available.
True / False:
Given that f is continuous, does it mean that the [enter image description here][1]
exists?
I think it does, because the if f is continuous that means the lim of f as x approaches 0 exists. and the mentioned lim is 0 when n approaches inf (sim... |
Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}, \quad (t,x)\mapsto u(t,x)$ is a solution to $\partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,t)\in \mathbb{R}\times \mathbb{R}$ with initial conditions $u(0,\cdot)=u_0$ and $\partial_t u(0,\cdot)=u_1$. Do we then have... |
Let $m,n$ be positive integers. Consider $X_1, \dots, X_n$ iid uniform discrete random variables taking values over the set of $mn$ integers $\{0,1,2,\dots,mn-1\}$. Let $Y_n=\min\{X_1,\dots, X_n\}$. What's the expected value of $Y_n$? I'm more than OK with asymptotics for large $n$. |
I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(x_i+x_j+x_k)$, $1\leqslant i<j<k\leqslant7$. So altogether (counting each $x_i$ and $s$) I have $64$ expressions. I want t... |
I have the following question that starts from this expression:
$$
u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y) .
$$
If $h \in C^2(\bar{B})$ is harmonic and Green function $G(x, y)=\Gamma(y-x)-h(y)$, the above integral le... |
It turns out that the familiar result that subjective monotones $\mathbb R\to\mathbb R$ are continuous extends to general LOTS (linearly ordered topological spaces):
> **Theorem**. If $X$, $Y$ are LOTS with $Y$ being dense and complete, then any suejective monotone $X\to Y$ is continuous.
(The same proof from $\m... |
I have been asked to verify the local version of the Gauss-Bonnet theorem for a paraboloid, I am confused about how to work out the sum of (π - interior angles) of the vertices at boundary ∂R which is a circle at the point where z=a ( a constant). As it's a circle I assume there is only one vertice, where the start and... |
there is a debate and the professor isnt available.
True / False:
Given that f is continuous, does it mean that the [enter image description here][1]
exists?
I think it does, because the if f is continuous that means the lim of f as x approaches 0 exists. and the mentioned lim is 0 when n approaches inf (sim... |
For arbitrary nonzero vectors $y\in\mathcal{R}^n$, how to prove maximum singular value of matrix $\frac{yy^T}{y^Ty}$ is equal to 1 ? |
I have a smooth function $f : P \rightarrow R$, where $P$ is the positive orthant of $R^3$. I know that $\partial f / \partial x,\; \partial f / \partial y$ are both $> 0$ everywhere on $P$ and that $\partial f / \partial z < 0$, also everywhere on $P$. Finally, I know that all three partials are functions only of $z$... |
I am trying to build an example of a topology on $R^2$ in which addition is discontinuous and multiplication by scalar is continuous. I think that an example of such a topology can be a system of straight lines passing through the origin (0,0) and their intervals. Then, for example, adding points (1,1) and (-1,1) gives... |
Let $C$ be a closed, convex set in a Hilbert space $H$.
$f(x)=\inf_{y \in C} \frac{1}{2}||x-y||^2$
WTS:
$$\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 $$
Thus $\nabla f=\textbf{I}-P_c$
We have this characterization of the... |
An elliptic curve $E$ over $\overline{\mathbb{Q}}$ can be defined over $\mathbb{Q}$ *if and only if* its j-invariant $j(E)$ is a rational number. More precisely, $E$ can be defined over a number field $K\subset \overline{\mathbb{Q}}$ if and only if $j(E)\in K$.
As Mathmo123 points out above, you can find many ellipt... |
I need some help to show this following identity. $$(\nabla\psi\cdot\nabla)\nabla\psi^*+(\nabla\psi^*\cdot\nabla)\nabla\psi=\nabla|\nabla \psi|^2$$
My attempt:
$$
\partial_i\psi\partial_i\partial_k\psi^*+\partial_i\psi^*\partial_i\partial_k\psi\\ \partial_i(\psi\partial_i\partial_k\psi^*+\psi^*\partial_i\partial_k\... |
I have a trouble proving that:
For $k\in \mathbb{N}$, the double coset
\begin{align*} SL_{2}(\mathbb{Z})\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} SL_{2}(\mathbb{Z})\end{align*}
is consisting of the matrices \begin{pmatrix} a & b \\ c & d \end{pmatrix} with integer entries having gcd 1 and of determinant $k$. I al... |
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)... |
Apologies if this post will lack rigour, I'm only a humble physicist. Given a density operator $\hat\rho$, one can represent it as the integral
$$\hat\rho = \frac{1}{(2\pi)^m}\int_{\mathbb C^m} d^m\xi \text{ tr}[\hat\rho \hat D_{\xi}]\hat D_{-\xi} $$
performed over all the complex components $\xi_1,...,\xi_m$ of the... |
Apologies if this post will lack rigour, I'm only a humble physicist. Given a density operator $\hat\rho$ (i.e. a linear, positive operator with unity trace), one can represent it as the integral
$$\hat\rho = \frac{1}{(2\pi)^m}\int_{\mathbb C^m} d^m\xi \text{ tr}[\hat\rho \hat D_{\xi}]\hat D_{-\xi} $$
performed over... |
Apologies if this post will lack rigour, I'm only a humble physicist. Given a density operator $\hat\rho$ (i.e. a linear, positive self-adjoint operator with unity trace), one can represent it as the integral
$$\hat\rho = \frac{1}{(2\pi)^m}\int_{\mathbb C^m} d^m\xi \text{ tr}[\hat\rho \hat D_{\xi}]\hat D_{-\xi} $$
p... |
Apologies if this post will lack rigour, I'm only a humble physicist. Given a density operator $\hat\rho$ (i.e. a linear, positive self-adjoint operator with unity trace), one can represent it as the integral
$$\hat\rho = \frac{1}{(2\pi)^m}\int_{\mathbb C^m} d^m\xi \text{ tr}[\hat\rho \hat D_{\xi}]\hat D_{-\xi} $$
p... |
I tried to solve an olympiad-level problem with complex numbers by myself. Can you please look at my solution and tell me if it is correct and complete, and how many points I would have gotten out of 7? I hope I didn't miss anything.
Prove that for any complex numbers $z_1 , z_2$ the following inequality holds $$|z_... |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right).$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{1}$$
For an... |
I tried to solve an olympiad-level problem with complex numbers by myself. Can you please look at my solution and tell me if it is correct and complete, and how many points I would have gotten out of 7? I hope I didn't miss anything.
Prove that for any complex numbers $z_1 , z_2$ the following inequality holds $$|z_... |
Let $d<0$ be a discriminant and fix an embedding $\mathbb{Q}(\sqrt{d})\to \mathbb{C}$. Let $\tau\in \mathbb{Q}(\sqrt{d})\cap \mathcal{H}$, then $j(\tau)$ is an algebraic number. Moreover, write $R:=\operatorname{End}(\Lambda_\tau)$, then $R$ is isomorphic to an order in $\mathbb{Q}(\sqrt{d})$ and if $R$ is the *maximal... |
Is $j(\tau)$ an algebraic integer even when $\operatorname{End}(\Lambda_\tau)$ is a non-maximal order? |
there is a debate and the professor isnt available.
True / False:
Given that $f$ is continuous, does it mean that
$$\lim_{n \to \infty} f \left( \frac{(-1)^n}{n} \right)$$
exists?
I think it does, because the if $f$ is continuous that means the limit of $f$ as $x$ approaches $0$ exists. and the mentioned... |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(0, 1) \rightarrow \mathbb R$ where $f(x) = \frac {1}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing ... |
Experimental data show that in any triangle with sides $(a,b,c)$ and circumradius $R$, if $x > 2$ then,
$$
\frac{1}{x} - \frac{4}{x^2 - 4} < \frac{1}{a-b+Rx} + \frac{1}{b-c+Rx} - \frac{1}{c-a+Rx}
< \frac{1}{x} + \frac{4}{x^2 - 4} \tag 1
$$
Can this be proved or disproved?
**Corollary**: In the interval $(... |
Does every triangle satisfy $\frac{1}{a-b+Rx} + \frac{1}{b-c+Rx} > \frac{1}{c-a+Rx}$? |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right).$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{1}$$
For an... |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(0, 1) \rightarrow \mathbb R$ where $f(x) = \frac {1}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing ... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integers $c,d... |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let
$$
f(x) = P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)
$$
The plot of $x$ vs $f(x)$ is show below along ... |
With some intermediate derivation results I found the two integrals are exactly the same. But why?
\begin{align}
I=\int_0^\infty\frac{\sin(x)}{x + x^2}\mathrm dx=\int_0^\infty\frac{\pi-2\tan^{-1}(x)}{2 e^x}\mathrm dx
\end{align}
Both are equal to
\begin{align}
I&=\operatorname{Si}(1)\cos(1)-\operatorname{Ci}(1)... |
Is there any relationship between these two integrals which are equal? |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(0, 1) \rightarrow \mathbb R$ where $f(x) = \frac {1}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing ... |
For a complex random variable If amplitude and phase of a random signal are independent,
then does real and imaginary parts of that signal becomes mutually exclusive (or) independent?
|
If amplitude and phase of a random signal are independent, then does real and imaginary parts of that signal are mutually exclusive (or) independent? |
does the regularized heat kernel have the semigroup property? |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right)}.$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{... |
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