instruction stringlengths 12 30k |
|---|
An interesting problem that I've run into during my research work involves sets in $\mathbb{Z}_{2n}$ of size $m$ and their connection to "uniformity" with respect to a counting function
$$\Delta(b) = \sum_{k=0}^{N-1} 1_{S}(k+b)$$
My strong suspicion is that $\Delta(b)$ being sufficiently uniform, i.e. $\vert \Del... |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have a proposition of holomorphic line bundle over $\mathbb P^n$,
> Proposition 2.2.6: The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ be an open covering of... |
I am asked to construct two 3-manifolds $M_1$,$M_2$ both covered by two open sets $U$,$V$ (different for each manifold) s.t. the intersection is diffeomorphic to $\mathbb S^1 \times\mathbb R^2$ but the second de Rham group of $M_1$ and $M_2$ is trivial and not trivial respectively.
My thought so far is to take $M_1=\m... |
>$u(x,y)=e^{x^2-y^2}(e^y \cos(x-2xy)+ e^{-y} \cos(x+2xy))$
Solving the Cauchy-Riemann equations for this is not practical. Alternatively, I could try to express $u(x,y)$ in the form of the real or complex part of a (analytic) function in $z$. In doing that, I either observe it right away, or put in $x=\frac{z+\bar z... |
It is well known that any Ito local martingale of the form $dX_t=Y_sdB_s$ with the initial condition that $X_0=0$ and $Y_s$ being adapted and locally square integrable can be reparametrized as a Brownian motion by defining:
$$\tau_t:= \inf\\{s \geq 0: \int_0^s Y_u^2du = t\\}$$
And we have that the process $X_{\tau_t}... |
I saw this proof in Harvey Friedman [notes][1] titled THE AXIOMATIZATION OF SET THEORY BY EXTENSIONALITY, SEPARATION, AND REDUCIBILITY.
This a quote from page 5.
> LEMMA 1.2. Let $\varphi$ be a formula in $L(\in)$ whose free variables
> are among $x_1,…,x_n , n \geq 0$. let $\varphi^W$ be the result of
> rela... |
Help with understanding this proof about reflection in H. Friedman's axiomatization of set theory by Extensionality, Separation, and Reducibility? |
It is well known that any Ito local martingale of the form $dX_t=Y_sdB_s$ with the initial condition that $X_0=0$ and $Y_s$ being adapted, continuous, and locally square integrable can be reparametrized as a Brownian motion by defining:
$$\tau_t:= \inf\\{s \geq 0: \int_0^s Y_u^2du = t\\}$$
And we have that the proces... |
An interesting problem that I've run into during my research work involves sets in $\mathbb{Z}_{2n}$ ($n \in \mathbb{N}$) of size $m$ and their connection to "uniformity" with respect to a counting function
$$\Delta(b) = \sum_{k=0}^{N-1} 1_{S}(k+b)$$
My strong suspicion is that $\Delta(b)$ being sufficiently unif... |
As you said, we \try to do in reverse way. For simple inequalities, our task should be try to reduce it into a 'nice' form, (get rid of fraction, square roots), so to get some 'always true' stuff.
\begin{align*}\dfrac{a}{2(a+b)\sqrt{b}}<\dfrac{1}{\sqrt{b}}-\dfrac{1}{\sqrt{a+b}}&\iff \dfrac{a}{2(a+b)\sqrt{b}}<\dfrac{\s... |
I think what is confusing you is that you are "starting out" with a graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$ nodes ... |
An interesting problem that I've run into during my research work involves sets in $\mathbb{Z}_{2n}$ ($n \in \mathbb{N}$) of size $m$ and their connection to "uniformity" with respect to a counting function
$$\Delta(b) = \sum_{k=0}^{n-1} 1_{S}(k+b)$$
My strong suspicion is that $\Delta(b)$ being sufficiently unif... |
> In this case, the mean curvature at $p$ is given by $H=\frac{f_{xx}\;\;+f_{yy}}{2}$ (average of second derivatives at $p$) and principal curvatures are $f_{xx},f_{yy}$.\
Is this correct?
I think it is not always true that the principal curvatures are $f_{xx},f_{yy}$.
We need the x-axis and y-axis along the di... |
I think what is confusing you is that you are "starting out" with a graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$ nodes ... |
I want to know, whether the following calculations for a vector of variables $A_{i}$ of size N, drawn independently at random from a complex normal distribution, with mean zero and standard deviation one are correct:
$\sum_{ij} \mathbb{E}(A_i \cdot A_j^*) = \sum_i \mathbb{E} (|A_i|^2) = N$
$\sum_{ijk} \mathbb{E}(... |
> By choosing local coordinates we can identify $U$ as part of the tangent plane of $S$ at $p$, furthermore we can set $f^{-1}(p)=(0,0)$.\
In this case, the mean curvature at $p$ is given by $H=\frac{f_{xx}+f_{yy}}{2}$ (average of second derivatives at $p$) and principal curvatures are $f_{xx},f_{yy}$.\
Is this corre... |
It's well-known that over $\Bbb Q$ the factorization into irreducible cyclotomic polynomials$$x^n-1=\prod_{d|n}\Phi_d(x)$$and over $\Bbb F_p$ it's well-known as well from basic Galois theory that $\Phi_d(x)$ factors into $\displaystyle\frac{\phi(n)}{k}$ factors of degree $k$, where $k$ is the order of $p$ in $(\Bbb Z/d... |
How does $x^n-1$ Factor in an Arbitrary Field? |
If $f,f':X \rightarrow Y$ ($X,Y$ topological spaces) are continuous functions respectively defined on disjoint closed subsets $A,A'\subset X$ . Then their disjoint union; the function $F:A\cup A'\rightarrow Y$, defined as $f$ on $A$ and $f'$ on $A'$ is continuous. Is the same result true for general sheaves?
That is... |
It's well-known that over $\Bbb Q$ the factorization into irreducible cyclotomic polynomials$$x^n-1=\prod_{d|n}\Phi_d(x)$$and over $\Bbb F_p$ it's well-known as well from basic Galois theory that $\Phi_d(x)$ factors into $\displaystyle\frac{\phi(d)}{k}$ factors of degree $k$, where $k$ is the order of $p$ in $(\Bbb Z/d... |
I recently saw a poster claiming free WiFi if you could answer the integral. This is the integral: $$\int_{-2}^2{\left(x^3cos\frac x 2+\frac 1 2\right)\sqrt{4-x^2dx}}$$the only problem is that the dx is under the square root. You could say this is a mistake and integrate and you would get $\pi$. But that answer is not ... |
Can you raise the variable of integration to a power like a second or third derivative? |
Yes, this is true:
Let $C$ be a cofinal chain in $P$ and $Q$ a cofinal subset of $P$.
Obviously, $P$ and $Q$ are upward directed.
W.l.o.g, we may assume that $\emptyset \neq C$ has no maximum, hence $C$ contains a cofinal subset order-isomorphic to a regular cardinal.
Hence, we may assume that $C = \{c_\alpha: ... |
Let $\Omega \subset \mathbb{R}^n$ be an bounded domain with smooth boundary $\partial \Omega$ and consider the functional $I:H_0^{1}(\Omega) \to \mathbb{R}$ defined by
$$I(u)=\frac{1}{2}\lVert u \rVert_{H_0^{1}(\Omega)}^2+\int_{\Omega} u \cos(u)dx-\int_{\Omega} \sin(u)dx, $$
there exist some $u_0$ such that $I(u_... |
Are you looking for this kind of expansion [Laurent series 1/sin(z)][1]?
For $z\ne k\pi\:(k\in\mathbb Z)$, the expansion takes
\begin{align}
\frac1{\sin(z)}&=\sum_{k=-\infty}^\infty\frac{z(-1)^k}{z^2-k^2\pi^2}.
\end{align}
Its proof can be found here [Lobachevsky Integral Formula][2].
[1]: https://www.wolfr... |
What does this sum $\sum_{t=c}^{n-1}(\frac{1}{t^2-1})$ equal?
For context, I am trying to digest the [cardinal payoff variant][1] of the secretary problem. There is an interview process with $n$ candidates each having some value associated with them which comes from i.i.d. interval $[0;1]$, the relative value of can... |
Why is a repelling fixed point also an isolated fixed point? |
Is it possible to find a function $u_0 \in H_0^{1}(\Omega)$ such that $I(u_0)<0$? |
> Suppose that each $X_n$ and $X$ are non-negative random variables and $X_n \stackrel d \to X.$ Assume that $\{X_n \}_{n \geq 1}$ is uniformly integrable. Prove the following decomposition $$\mathbb E(X_n) = \int_{0}^{M} \mathbb P (M \geq X_n \gt t)\ dt + \int_{X_n \gt M} X_n\ d \mathbb P.$$ Hence, show that $\mathbb ... |
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-dimensional Hilbert space. First, some context:
We note that, by concentration of measures for spheres, we have the following: Let $S^{n-1} = \{x \in \mathbb{R}^n: \|x\|_2 = 1\}$ denote the... |
Let $\Omega \subset \mathbb{R}^n$ be an bounded domain with smooth boundary $\partial \Omega$ and consider the functional $I:H_0^{1}(\Omega) \to \mathbb{R}$ defined by
$$I(u)=\frac{1}{2}\lVert u \rVert_{H_0^{1}(\Omega)}^2+\int_{\Omega} u \cos(u)dx-\int_{\Omega} \sin(u)dx, $$
there exist some $u_0$ such that $I(u_... |
> Suppose that each $X_n$ and $X$ are non-negative random variables and $X_n \stackrel d \to X.$ Assume that $\{X_n \}_{n \geq 1}$ is uniformly integrable. Prove the following decomposition $$\mathbb E(X_n) = \int_{0}^{M} \mathbb P (M \geq X_n \gt t)\ dt + \int_{X_n \gt M} X_n\ d \mathbb P.$$ Hence, show that $\mathbb ... |
>$e^{2y}+x^2+2xe^{2y}\frac{dy}{dx}=0 \\ y(-1)=1$
This IVP is exact and:
\begin{align*}
\frac{d(e^{2y}x + \frac{x^3}{3})}{dx}=0 \\ \iff e^{2y}x + \frac{x^3}{3}=k; \ k \in \mathbb{R} \\ \underbrace{\iff}_{y(-1)=1} e^{2y}x + \frac{x^3}{3}=-e^2-\frac{1}{3}
\end{align*}
$$e^{2y}x + \frac{x^3}{3}=-e^2-\frac{1}{3}... |
$\frac{dy}{dx} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} x = 2x$
So $dy= 2x dx$
Upon integration we get,
$y= x^2 + \mathrm{constant \, of \, integration}$
Is this what you are asking? |
I am currently proving a theorem on the expressivity of functions which I have called "autoregressive". Take a finite set $T$ and two natural numbers $n,m\in\mathbb N$. For any function $a:T^n\to T$ we can define a function $\phi[a]:T^n\to T^m$ by applying $a$ $m$ times in an "autoregressive" way, by which I mean for $... |
It's well-known that over $\Bbb Q$ the factorization into irreducible cyclotomic polynomials$$x^n-1=\prod_{d|n}\Phi_d(x)$$and over $\Bbb F_{p^m}$ it's well-known as well from basic Galois theory that $\Phi_d(x)$ factors into $\displaystyle\frac{\phi(d)}{k}$ factors of degree $k$, where $k$ is the order of $p^m$ in $(\B... |
I haven't really found a solution to this confusion I currently have.
I know in certain cases, we can use "without loss of generality" to proceed in a proof.
For example, to prove that "if a|b or a|c -> a|bc", we can assume without loss of generality that a|b and show that a|bc follows.
But let's say I have a t... |
Linear Algebra: Is my proof of equivalence of cosets of $\mathsf W$ containing $v$ okay? |
Hopefully this is not a fail. But, before I explain the math, I think it will be easier if I give a background of what I'm trying to do...
Imagine 2 line segments randomly defined in a $2D$ room. The line segments are 2D... Both are given a speed $v$ and $\omega$ (angular velocity). The point of the math, is to find... |
It is well known that any Itô local martingale of the form $dX_t=Y_sdB_s$ with the initial condition that $X_0=0$ and $Y_s$ being adapted, continuous, and locally square integrable can be reparametrized as a Brownian motion by defining:
$$\tau_t:= \inf\{s \geq 0: \int_0^s Y_u^2du = t\}$$
And we have that the process ... |
I'm trying to grasp proofs of the fact that compact Riemann surfaces are algebraic curves. It seems as though no textbook fully treats this, and I wanted to ask about what was going on. On one hand, I know that one can use Riemann-Roch to furnish meromorphic functions and then an embedding into projective space, whenc... |
The question is basically in the title. I want to evolve an image $u_0$ (i.e. a $n_1\times n_2$ resolution set of discrete values in $[0,1)$, which is a discrete function on $\{0,\ldots,n_1-1\}\times\{0,\ldots,n_2-1\}$) by the PDE $$\partial_tu=\nabla\cdot\kappa(t,\nabla u)\nabla u\tag1$$ with Neumann boundary conditio... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not integrable on unbounded domain but its dirivative is integrable in that unbounded domain.
I only... |
In the context of evaluating the line integral of a vector field $\vec{F}:\vec{x}\in\mathbb{R}^n\mapsto \vec{F}(\vec{x})\in\mathbb{R}^n$ along a regular parameterized curve $\gamma:t\in{I}\subseteq\mathbb{R}\mapsto \vec{x}(t)\in\mathbb{R}^n$, we define $$\int_{\gamma}\vec{F}\cdot d\vec{x} = \int_{I}\vec{F} \left(\vec{x... |
This is a second part of a two-part question.
In the first part I was asked to prove that any group of order $5784 = 2^3 \cdot 3 \cdot 241$ has subgroups of the following *indexes*: 3, 6, 8, 12, and 24.
I proved that part by showing:
- G has subgroups of orders 2, 2^2, 2^3, 3, 241 (by Sylow theorem).
- More... |
**Theorem (of the invariance of domain).** If $U$ is an open set of $\mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ is a continuous and injective application, then $f(U)$ is open.
**Consequence**. If $U$ is a non empty open set in $\mathbb{R}^n$, then an application $f:U\to \mathbb{R}^m$ continuous and injective cannot exi... |
**Question**
A homogeneous subset of ℕ is one where all the elements have the same parity. A student decided to prove that the set all homogeneous subsets of ℕ is uncountable. The student proposed that every homogeneous subset of N can be represented as an infinite binary word with all its 1s either in even position... |
Define:
For $n \geq 0$, on note $X_n=(n+1) \mathbb{1}_{[n+1,+\infty}$, and $\mathcal{F}_n=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+$ $1,+\infty[)$
and $\forall k \in \mathbb{N}^*, \mathbb{P}(\{k\})=\frac{1}{k}-\frac{1}{k+1}$
I am trying to prove that it's a martingale.
We have that $$X_{n+1}=(n+2) 1_{[n+2,+\inf... |
>Is there a simple geometric proof of why two simple harmonic oscillators draw a circle?
No, there is no geometric proof, because it is not always true. If the two oscillators are oscillating with different amplitudes, the resulting figure is an ellipse, not a circle. If the two oscillators are oscillating at to dif... |
**Theorem (of the invariance of domain).** If $U$ is an open set of $\mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ is a continuous and injective application, then $f(U)$ is open.
**Consequence**. If $U$ is a non empty open set in $\mathbb{R}^n$, then an application $f:U\to \mathbb{R}^m$ continuous and injective cannot exi... |
Lately I am thinking about music from a mathematical point of view. Let us consider $\mathbb{H} =\mathcal{L}^2([0,T])$ as the Hilbert space of functions that represents the time-intensity plot of a sound. Quite tautologically, playing together sounds amount to sum them. So, up to scale volumes, the chords one can produ... |
How can I prove that the union of $m+3$ $m$-dimensional convex sets in $\mathbb{R}^n$ such that the intersection of every $m+2$ sets is not empty is contractible? |
I know that we need base e to differentiate but I don't see what makes this formula work.
$$ P = P_0 e^{rt} $$
where the $P_0$ refers to initial population, $r$ the rate, and $t$ the time.
Changing the base changes the curve, so why does base $e$ work? I mean $r$ and $t$ are pretty straightforward numbers so ... |
Lately I am thinking about music from a mathematical point of view. Let us consider $\mathbb{H} =\mathcal{L}^2([0,T])$ as the Hilbert space of functions that represents the time-intensity plot of a sound. Quite tautologically, playing together sounds amount to sum them. So, up to scale volumes, the chords one can produ... |
in "Introductory Quantum Mechanics" by Richard L. Liboff, question 1.21 part o asks the reader to derive the following sum.
$$
\sum_{n=1}^{L} e^{(-i\frac{2\pi n}{L})} = 0
$$
The given hint asks the reader to separate n from the other powers and treat the expression as a geometric sum. Using the geometric sum eq... |
Is euler's identity invalid in this case? |
in "Introductory Quantum Mechanics" by Richard L. Liboff, question 1.21 part o asks the reader to derive the following sum.
$$
\sum_{n=1}^{L} e^{(-i\frac{2\pi n}{L})} = 0
$$
The given hint asks the reader to separate n from the other powers and treat the expression as a geometric sum. Using the geometric sum eq... |
Yes, this is true:
Let $C$ be a cofinal chain in $P$ and $Q$ a cofinal subset of $P$.
Obviously, $P$ and $Q$ are upward directed.
W.l.o.g, we may assume that $\emptyset \neq C$ has no maximum, hence $C$ contains a cofinal subset order-isomorphic to a regular cardinal.
Hence, we may assume that $C = \{c_\alpha: ... |
I am reviewing some commutative algebra notes for an upcoming course, and I found an exercise I wanted to solve, but I found out I absolutely have no idea how.
>Let $K$ be a field (I think $\operatorname{char}K\neq 2$, but the exercise doesn't outright state it), find whether $$R = k[x,y]/(2x^2 + 3y^2 - 11, x^2 - y^... |
If we have a planar graph, why is it true that there always exists a vertex such that $deg(v) \leq 3$ or a face $f$ such that $deg(f) \leq 3$? |
in "Introductory Quantum Mechanics" by Richard L. Liboff, question 1.21 part o asks the reader to derive the following sum.
$$
\sum_{n=1}^{L} e^{-i\frac{2\pi n}{L}} = 0
$$
The given hint asks the reader to separate n from the other powers and treat the expression as a geometric sum. Using the geometric sum equa... |
I'm studying commutative algebra and I need to find, for the course I'm attending, the primary decompositions of the following ideals, with some extra questions:
>In $\mathbb{C}[x,y]$, find, if you can, two primary decompositions of the ideal $(x^2y, xy^2)$, and find all isolated primes. Is the ideal radical? Is the... |
Given a finite dimensional vector space, what is the linear transformation
$$L : V \rightarrow V $$, which has the most fixed points? ($L$ can not be the identity). |
By your reasoning, we could show that all powers are equal to $1$; e.g.,
$$e^z = (e^{2\pi i})^{z/(2\pi i)} = 1^{z/(2\pi i)} = 1.$$
Where is the flaw? Clearly, the second equality is suspect:
$$e^{ab} \overset{?}{=} (e^a)^b.$$
Indeed, this isn't even true in the real numbers:
$$-8 = (-2)^3 = (-2)^{2(3/2)} = ((-2)^... |
polynomials with leading coefficient unit are cancellable? |
Voters arrival is modelled as a Poisson process. How to find the probability of a certain event, conditioned on the number of votes? |
Arrival of voters is modelled as a Poisson process. How to find the probability of a certain event, conditioned on the number of votes? |
Given a finite dimensional vector space over a finite field, what is the linear transformation
$$L : V \rightarrow V $$, which has the most fixed points? ($L$ can not be the identity). |
[Image of the Problem from a Textbook (Calculus by James Stewart pg 812)][1]
[1]: https://i.stack.imgur.com/KGJGO.png
For Part A, I'm getting g'(x) = g(x) * (n/x). How did they get g'(x) = (k*g(x))/(x+1)? |
How to Prove the Binomial Series by Differentiation? |
> (a) Let $g(x) = \sum_{n=0}^\infty \binom{k}{n} x^n$. Differentiate this series to show that $$g'(x) = \frac{kg(x)}{1+x}, \quad -1 < x < 1.$$
> (b) Let $h(x) = (1+x)^{-k} g(x)$ and show that $h'(x) = 0$.
>
> (c) Deduce that $g(x) = (1+x)^k$.
For part (a), I'm getting $g'(x) = g(x) \frac{n}{x}$. How did they get... |
I saw this Corollary on page 21 of [this book](https://u.cs.biu.ac.il/~plotkin/resources/MFAtiyah_IGMacDonald_IntroToCommAlgebra.pdf) *"Introduction to commutative algebra"* by Atiyah and Macdonald.
> Corollary 2.5. Let $M$ be a finitely generated $A$-module and let $a$ be an ideal of $A$ such that $aM=M$. Then th... |
I'm struggle understanding the last statement below from Qing Liu's "Algebraic geometry and arithmetic curves".
>Lemma 2.5.22 (pg. 73): Let $A$ be a finitely generated integral domain over $k$, and let $p$ be a prime ideal of $A$ of height $1$. Then $\dim A/p=\dim A-1.$
Proof. Let $f\in p\backslash \{0\}$. Then $... |
Exactly $\frac{1}{5}$ people that live on earth have a condition
.There are two tests for this condition, the Z1 test and the Z2 test. When a person
goes to a doctor to test for this condition, with probability $\frac{2}{3}$ the doctor conducts Z1 on them and with probability $\frac{1}{3}$ the doctor conducts Z2 on ... |
Let $X,Y,Z$ be real random variables with joint density $f(x,y)$ of $X$ and $Y$ and density $g(z)$ of $Z$. Is it true that
$$E(h(X,Y,Z))=\int\int\int h(x,y,z)f(x,y)g(z)dxdydz$$ or do we need that $(X,Y)$ and $Z$ are independent or something like this? |
I saw this interesting article https://www.mdpi.com/2073-8994/14/4/741 about how the quantization of the eigenvalues of the Legendre equation is a consequence of parity symmetry and imposing that the solutions should be finite in the interval [-1,1]. I wonder if a similar argument could be made for the quantization of ... |
I wish to show the followin: For any positive definite symmetric tridiagonal matrix $A$;
$$A=\left[\begin{array}{ccccc}a_1 & c_1 & & & \\ c_1 & a_2 & c_2 & & \mathbf{0} \\ & \ddots & \ddots & \ddots & \\ \mathbf{0} & & c_{n-2} & a_{n-1} & c_{n-1} \\ & & & c_{n-1} & a_n\end{array}\right]$$
we have that $2D-A$ is posit... |
If $I=[0,1]$, let $f:I\to I$ be an involution. Then $\sim_f:=\Delta_I\cup \text{Gr}(f)\subseteq I^2$ defines an equivalence relation on $I$ where $\Delta_I=\{(t,t):t\in I\}$ is the diagonal of $I$ and $\text{Gr}(f)=\{(t,f(t)):t\in I\}$ is the graph of the function $f$. My question then is
>**Q:** What involutions $f$... |
At [this answer](https://math.stackexchange.com/a/4883740/567304) it is provided an interval of the form $[kn,(k+1)n]$ where each integer is divisible by at least one of the integers of the interval $[2,n]$.
We define a permutation of the integers of the interval $[2,n]$ dividing the integers of the interval $[kn,(k... |
I wish to show the followin: For any positive definite symmetric tridiagonal matrix $A$;
$$A=\left[\begin{array}{ccccc}a_1 & c_1 & & & \\ c_1 & a_2 & c_2 & & \mathbf{0} \\ & \ddots & \ddots & \ddots & \\ \mathbf{0} & & c_{n-2} & a_{n-1} & c_{n-1} \\ & & & c_{n-1} & a_n\end{array}\right]$$
we have that $2D-A$ is posit... |
Let's consider $2024$ variables $x_i$ in $[-2;3]$ such that $x_1+x_2+...+x_{2024}=500$.
What is the greatest value of :
$$x_1^4+x_2^4+...+x_{2024}^4$$
Here is my approach but it is not rigorous. I wanted to choose the $x_i$ with extreme values. I wanted to take $167$ times a $3$ and $2$ times a $-0,5$. So this is ... |
I saw this proof in Harvey Friedman [notes][1] titled THE AXIOMATIZATION OF SET THEORY BY EXTENSIONALITY, SEPARATION, AND REDUCIBILITY.
This a quote from page 5.
> LEMMA 1.2. Let $\varphi$ be a formula in $L(\in)$ whose free variables
> are among $x_1,…,x_n , n \geq 0$. let $\varphi^W$ be the result of
> rela... |
Let $E, F$ be normed spaces and $\gamma > 1$. If $T: E \to F$ is a continuous operator (not necessarily linear) and there is $c > 0$ such
that
$$||T(x)||_F \geq c||x||_E^{\gamma}, \quad \forall \ x \in E,$$
so there is the inverse operator $T^{-1}$? If so, is it possible to conclude that it is continuous?
Note: T... |
I would like to know an example showing that second-order mathematical induction is stronger than first-order mathematical induction.
For a condition $P(x)$ that can be written in the language of Peano arithmetic, we can construct the set $\{x\in\mathbb{N};P(x)\}$.
Which subset of $\mathbb{N}$ cannot be construct... |
I stumbled accross this question and would like to know if my approach makes sense, and if there is a better one:
For which values of $t$ does the following integral converge?
$$\int_1^\infty e^{1/x}-t(\sin(\frac1x)+\cos(\frac1x)) dx$$
I thought of using Taylor's polynomial to the 2nd degree, and using Lagrang... |
In the following proof, (source https://jmp.sh/X5KwZ1Zt)
[![enter image description here][1]][1]
[![enter image description here][2]][2]
Using statement (i) as stated I only get
$X_{F^t}+D\varphi_F^tX_{G^t}\circ (\varphi^t_F)^{-1}=X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}$ but why is this equal to $ X_{F^t +G^... |
Let $X$, $Y$, $Z$ be real random variables with joint density $f(x,y)$ of $X$ and $Y$ and density $g(z)$ of $Z$. Is it true that
$$E(h(X,Y,Z))=\iiint h(x,y,z)f(x,y)g(z)\, \mathrm dx\, \mathrm dy\, \mathrm dz$$ or do we need that $(X,Y)$ and $Z$ are independent or something like this? |
Is it true that $E(h(X,Y,Z)=\iiint h(x,y,z)f(x,y)g(z)\, \mathrm dx\, \mathrm dy\, \mathrm dz$? |
In the following proof, (source https://jmp.sh/X5KwZ1Zt)
[![enter image description here][1]][1]
[![enter image description here][2]][2]
Using statement (i) as stated I only get
$X_{F^t}+D\varphi_F^tX_{G^t}\circ (\varphi^t_F)^{-1}=X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}$ but why is this equal to $ X_{F^t +G^... |
In the following proof, (source https://jmp.sh/X5KwZ1Zt)
[![enter image description here][1]][1]
[![enter image description here][2]][2]
Using statement (i) as stated I only get
$X_{F^t}+D\varphi_F^tX_{G^t}\circ (\varphi^t_F)^{-1}=X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}$ but why is this equal to $ X_{F^t +G^... |
I am trying to prove the following.
**Theorem?** (Seifert-Van Kampen theorem for Lie-groupoids, a partial statement). Let $G_1\substack{\to \\ \to}G_0$ (denoted simpy by $G$) be a Lie-groupoid. Let $x_0\in G_0$. If $G_0$ is the union of open sets $A_\lambda$ each containing $x_0$ and if each restriction $G|_{A_\lamb... |
The inclusion $H\Gamma/\Gamma \hookrightarrow G/\Gamma$ is proper for closed subgroup $H$ and discrete subgroup $\Gamma$? |
Is there a representation for tensors as multidimensional arrays that is best for a general product definition?
If $\bf v$ is a column vector and $\bf w$ is a covector (row vector), then $\bf w v$ is a scalar, but $\bf v w$ is a matrix. Is there an general operation and representation for tensors unifies all instanc... |
The question is simple: how to find an unbounded operator $T:H\to H$ where $H$ is a Hilbert space such that $\text{Sp} T = \mathbb C$? This seems a very basic thing, but I have not found an example in the literature.
In some proofs, we need to consider this case separately. This example should be quite important. |
Unbounded operator whose spectrum is the entire complex plane? |
I need some help with a lemma I need to prove.
First I will provide some background with previous lemmas that I already have been able to prove. Maybe these lemmas are needed to proof the last lemma
>Given $f(x)\in \mathbb{Z}[X]$ and $\Omega =\{f(z)|z\in \mathbb{Z}\}$, if $\Omega$ contains infinitely many prime num... |
So let's list out the weight of each of the $12$ subjects for the three positions (given as percentages).
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Subjects \rightarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline
Position 1 & 3.3 & 3.3 & 3.3 & 3.3 & 3.3 & 3.3 & 15 & 15 & 10 & 5 & 5 & ... |
I do not understand how
P(A|B,C)=P(A|C)
is derived from
P(A∩B|C)=P(A|C)P(B|C)
on this [site][1].
I'm stuck on the statement:
> By conditioning on C, we obtain
> P(A|B,C)=P(A∩B|C)/P(B|C)
Should the left hand side of the equation be P(A|B|C)?
[1]: https://www.probabilitycourse.com/chapter1/1... |
Choose positive integers D and N. Roll a fair D-sided die N times, recording the number of times each of the D outcomes are rolled, say r1, r2, …, rD. What are the asymptotics of the min(r1, r2, …, rD)?
If it makes it easier you can assume N >> D. Of course the leading term is N/D but what’s the second-order term? M... |
How Can I Prove this Version of Minkowski’s Theorem: $vol(C)>k2^d$ with $2k$ Lattice Points? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.