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Where will a plane intersecting a cone maximize the distance to inscribed spheres? (V. Arnold) |
Let $G$ be an abelian group.
Let $\widehat{G}$ be a profinite completion of $G$.
Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are **finite index** subgroups of $G$ and $N\subset M$.
Let $G_1$ and $G_2$ be groups and suppose there is an injecti... |
### Setup
Let random variables $X_1,\ldots, X_n$ be i.i.d. $N(0, \sigma^2)$.
Suppose that the observed value $x^2_i$ of $X^2_i$ is given.
### Problem
What is the difference between
$$
\frac{\max_{i=1,\ldots, n}{X^2_i}}{\sum_{i=1}^n X^2_i} \overset{P}{\to} 0
$$
and
$$
\frac{\max_{i=1,\ldots, n}{x^2_i}}{\sum_... |
### Setup
Let random variables $X_1,\ldots, X_n$ be i.i.d. $N(0, \sigma^2)$.
Suppose that the observed value $x^2_i$ of $X^2_i$ is given.
### What I think
If
$$
\frac{\max_{i=1,\ldots, n}{x^2_i}}{\sum_{i=1}^n x^2_i} \to 0,
$$ then
$$
\frac{\sqrt{n}\bar{X}}{\sqrt{\frac{1}{n}\sum_{i=1}^n x^2_i}}= \frac{\sum_{i... |
I am trying to learn about the Laplace Approximation of Integrals (https://en.wikipedia.org/wiki/Laplace%27s_method). Supposedly, this is closely related to "Gaussian Functions".
Here is my current understanding - note that the Laplace Approximation is supposedly only valid for integrating functions of the form $\in... |
Relationship between Laplace Method and Gaussian Distribution? |
### Setup
Let random variables $X_1,\ldots, X_n$ be i.i.d. $N(0, \sigma^2)$.
Suppose that the observed value $x^2_i$ of $X^2_i$ is given.
### What I think
If
$$
\frac{\max_{i=1,\ldots, n}{x^2_i}}{\sum_{i=1}^n x^2_i} \to 0,
$$ then
$$
\frac{\sqrt{n}\bar{X}}{\sqrt{\frac{1}{n}\sum_{i=1}^n x^2_i}}= \frac{\sum_{i... |
I have a number that has odd number of divisors, for example 36, and I have all divisors of this number 1, 2, 3, 4, 6, 9, 12, 18, 36. How many ways do we have to get 36 from divisors if we need to multiply 3 any divisors? (repetition is allowed, order matters) For example 1,6,6 - one way to get 36 |
I have a number that has odd number of divisors, for example 36, and I have all divisors of this number 1, 2, 3, 4, 6, 9, 12, 18, 36. How many ways do we have to get 36 from divisors if we need to multiply k(from 2 to 1000) any divisors? (repetition is allowed, order matters) For example 1,6,6 - one way to get 36 |
I have a number that has odd number of divisors, for example 36, and I have all divisors of this number 1, 2, 3, 4, 6, 9, 12, 18, 36. How many ways do we have to get 36 from divisors if we need to multiply k(from 2 to 1000) any divisors? (repetition is allowed, order matters) For example, if k = 3: 1,6,6 - one way to g... |
Let $A\subseteq \Bbb R^n$, then:
$A^{c\bar{}c}=\mathring A$
Notation: As an example $(\bar{A^c})^c=A^{c\bar{}c}$
Dem:
We know that $A\subseteq \bar A$, now if we take the complement on both sides: $A^{\bar{}c}\subseteq A^c$
Now, knowing that $A^{\bar{}c}$ is an open set, then $A^{\bar{}c}\subseteq \ba... |
Suppose you have a game and you want to interpolate, using linear interpolation, a value's 'target value' from a starting value towards zero and then linearly interpolate the current value towards the target value each frame, in that order.
Let $a(x, t)$ be the 'target' function and $b(x, t)$ be the 'current' funct... |
> Prove that: $$\cos(u+v) = \cos(u)\cos(v) - \sin(u)\sin(v)$$ using the fact that the distance from A to C is the same as the distance from B to D in the picture below:
![picture of the unit circle the problem refers to][1]
I tried using the laws of cosine and sine by looking at the triangles OBD and OAC (where ... |
I have a number that has odd number of divisors, for example 36, and I have all divisors of this number 1, 2, 3, 4, 6, 9, 12, 18, 36. How many ways do we have to get 36 from divisors if we need to multiply k(from 2 to 1000 - for each k there is number of ways) any divisors? (repetition is allowed, order matters) For ex... |
This problem was originally asked here: https://math.stackexchange.com/questions/4870650/lim-x-to-1-sqrtx2-8-3-prove-this-using-epsilon-delta. It got closed and is no longer accepting answers.
-------
Prove this using epsilon-delta
$$\lim_{x\to 1} \sqrt{x^2 + 8} = 3$$
-------------
Definitions: $\lim_{x\to... |
Hi am reading https://www.cs.unm.edu/~saia/classes/362-s08/lec/lec24-2x2.pdf proving that the problem of checking if a graph has a hamiltonian cycle is NP-complete. It uses the fact that the problem of checking if a graph has a vertex cover of a given size is NP-complete, and it reduces the problem of vertex cover to t... |
Let $X^1, X^2, X^3,\ldots $ be $L^2$-integrable martingales. Assume that for each $t\in[0,T]$ there exists an $X_t\in L^2(\Omega)$, such that $X_t^n\rightarrow X_t$ in $L^2(\Omega)$ for $n\rightarrow\infty$.
I want to show, that $(X_t)_{t\in[0,T]}$ is a martingale. This is the first part of *Exercise 21.4.3* taken f... |
I have a circle of fixed radius $r$. I have a target that is $x$ units laterally separated from the center of the circle, and $y$ units vertically. I need to calculate the angle $θ$ which is the declination of the radius that connects that target to the upper tangent on the circle.
[![visual depiction of the above... |
I'm having trouble understanding how to calculate the first order derivative of matrix, and was wondering if anyone could help me.
The function with respect to P is
f(P)=*Tr*($XPAP^TPA^TP^TX^T$)=$||XPAP^T||^2_F$
where *Tr* is the trace, $A$ and $X$ are matrix.
The derivative of *Tr(AP)* is $A^T$, but how to... |
I have an individual selected for a job. He was competing against 10 candidates. All individuals are independently drawn from uniform distribution U(0,1). Him being chosen means he is better than the other. Normally, expected ability would be -
$$ E(a|chosen) = \frac{\int_{b=0}^1 \int_b^1 a ~ da db}{\int_{b=0}^1 \in... |
Let $X^1, X^2, X^3,\ldots $ be $L^2$-integrable martingales. Assume that for each $t\in[0,T]$ there exists an $X_t\in L^2(\Omega)$, such that $X_t^n\rightarrow X_t$ in $L^2(\Omega)$ for $n\rightarrow\infty$.
I want to show, that $(X_t)_{t\in[0,T]}$ is a martingale. This is the first part of *Exercise 21.4.3* taken f... |
This is a repost of a [question](https://stats.stackexchange.com/questions/639693/toad-movement-model-example/639699?noredirect=1#comment1195861_639699) from a different forum that didn't get much attention and I am dying for a further explanation/elaboration.
Before I proceed, this is against community standards, p... |
I am mainly looking for a reference request, if it exists, to save some time looking for it.
Let $E/\mathbb Q$ an elliptic curve in Short Weierstrass form and $\phi_n,\omega_n,\psi_n$ the usual division polynomials ( $nP=( \frac{\phi_n(P)}{\psi_n(P)^2} , \frac{\omega_n(P)}{\psi_n(P)^3})$ ). If $E: y^2=x^3+Ax+B$ and ... |
This is a repost of a [question](https://stats.stackexchange.com/questions/639693/toad-movement-model-example/639699?noredirect=1#comment1195861_639699) from a different forum that didn't get much attention and I am dying for a further explanation/elaboration.
Before I proceed, this is against community standards, p... |
This follows from the Markov property, as mentioned here https://math.stackexchange.com/questions/4656831/the-solution-x-of-the-sde-mathrm-d-x-t-ft-x-t-mathrm-d-t-gt-x-t?noredirect=1&lq=1 one source is in : Schilling; Th.21.23. p. 403
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/kbsq... |
Let $M$ be a smooth manifold. Let $V$ be the canonical vector field on $T M$ (also called the Liouville vector field), which if $(x, y)$ are local coordinates on $T M$ is defined by $V = y^i \frac{\partial }{\partial y^i}$. Also denote by $J : TTM \to TTM$ the canonical endomorphism of $TTM$ (also called a tangent stru... |
I like addition theorems in trigonometry and recently YouTuber Dr Barker posted the video ["My New Favourite Trig Function"](https://youtube.com/watch?v=bn38o3u0lDc) playing around with following:
Define $$ \operatorname{cas}(x) := \cos(x) + \sin(x) $$
Then: $$ \operatorname{cas}(x+y) = \frac{1}{2} ( \operatornam... |
I am following chapter 4 of this [paper (1)](https://pure.manchester.ac.uk/ws/portalfiles/portal/208486475/FULL_TEXT.PDF).
**Background information:** The stochastic differential equations (eq’s 4.2 to 4.5) are given by:
$$
\begin{aligned}
& d S_t=\sqrt{\nu_t} d W_t \\
& d \nu_t=\theta\left(\alpha-\nu_t\right)... |
> Existence of second derivative of characteristic function of r.v. X at 0 implies
> that the second moment of X is finite.
There is a proof of that statement in Chung's "A Course in Probability Theory" (2001 edition, Theorem 6.4.1 - even more general, but it involves doing that for k = 2) which has one transition ... |
Can geodesics on any surface ( smooth, embedded $\mathbb R ^3)$ be mapped as straight lines on the plane? Please provide examples.
Are there some other properties of surfaces that may need to be satisfied for a plane straight line representation ?
Stereographic projection of great circles of a sphere clearly do n... |
I am following chapter 4 of this [paper (1)](https://pure.manchester.ac.uk/ws/portalfiles/portal/208486475/FULL_TEXT.PDF).
**Background information:** The stochastic differential equations (eq’s 4.2 to 4.5) are given by:
$$
\begin{aligned}
& d S_t=\sqrt{\nu_t} d W_t \\
& d \nu_t=\theta\left(\alpha-\nu_t\right)... |
> A cone is cut by a plane, making a closed curve. Two spheres, each inscribed into the cone, are tangent to the plane, at points $A$ and $B$ respectively.
>
> Find the point $C$ on the cut line (i.e. the curve formed by the intersection of the cone and plane) such that $CA + CB$ is at an extrema.
>
> --Vladimir ... |
Can geodesics on any surface ( smooth, embedded $\mathbb R ^3)$ be mapped as straight lines on the plane? Please provide examples.
Are there some other global properties of surfaces that may need to be satisfied for a plane straight line representation ?
Stereographic projection of great circles of a sphere clear... |
Let me formulate everything for you coordinate-independently, so there is no question whether the maps $T\pi_M$, $\operatorname{vl}$, and $S$ are well-defined globally without restriction.
## Global formulas ##
First $\pi_M : TM \to M$ is the structure map of the fiber bundle $TM$. Since $TM$ and $M$ are smooth m... |
Given a bilinear form $b$ on a vector space, one defines the *radical*, which is the set of vectors which are orthogonal (with respect to $b$) to every other vector (so essentially the orthogonal complement of the whole space). My question is: Why is it called radical? Is there an example that motivates this name? |
I was pondering how to solve a limit of the kind $$\lim_{n \to ∞} n^k \left(\dfrac{1}{n}\sum_{k=0}^{n-1} f \left(\dfrac{k}{n} \right) -\int_0^1 f(x)dx \right)$$
where k is chosen such that the order of infinity is just right for the limit to be non-zero but not infinite. For instance, if $f(x)=x^2$, then the limit b... |
Define a category $\textbf{All}$ as follows:
- The objects of $\textbf{All}$ are the pairs $(X, C)$, where $C$ is a category and $X$ is an object of $C$.
- A morphism from $(X, C)$ to $(Y, D)$ in $\textbf{All}$ is an ordered pair $(F:C\to D, f:F(X)\to Y)$, where $F$ is a functor and $f$ is a morphism in $D$.
- Com... |
Has this "everything category" been defined before? |
2. Start with a positive integer, then choose a negative integer. We’ll use these two numbers
to generate a sequence using the following rule: create the next term in the sequence by
adding the previous two. For example, if we started with 6 and −5, we would get the
sequence 6, −5, 1, −4, which is the alternating pa... |
Why is a=43 an exception? |
If $(f_n)$ and $(g_n)$ converge uniformly on $A \subseteq \mathbf{R}$ then $(f_n + g_n)$ converges uniformly, but not necessarily $(f_ng_n)$. Why? |
2. Start with a positive integer, then choose a negative integer. We’ll use these two numbers
to generate a sequence using the following rule: create the next term in the sequence by
adding the previous two. For example, if we started with 6 and −5, we would get the
sequence 6, −5, 1, −4, which is the alternating pa... |
Start with a positive integer, then choose a negative integer. We’ll use these two numbers
to generate a sequence using the following rule: create the next term in the sequence by
adding the previous two. For example, if we started with 6 and −5, we would get the
sequence 6, −5, 1, −4, which is the alternating part,... |
Let $\mathbb{F}$ be a field with char $\mathbb{F} \neq 2$.
Let $A \in M_n(\mathbb{F})$.
Does there exist symmetric matrices $B,C \in M_n(\mathbb{F})$ such that $A=BC$?
The answer is yes when $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. But how about other fields? |
Let $A\subseteq \Bbb R^n$, then:
$A^{c\bar{}c}=\mathring A$
Notation: As an example $(\bar{A^c})^c=A^{c\bar{}c}$
Dem:
We know that $A\subseteq \bar A$, now if we take the complement on both sides: $A^{\bar{}c}\subseteq A^c$
Now, knowing that $A^{\bar{}c}$ is an open set, then $A^{\bar{}c}\subseteq \ba... |
Let $A\subseteq \Bbb R^n$, then:
$A^{c\bar{}c}=\mathring A$
Notation: As an example $(\bar{A^c})^c=A^{c\bar{}c}$
Dem:
We know that $A\subseteq \bar A$, now if we take the complement on both sides: $A^{\bar{}c}\subseteq A^c$
Now, knowing that $A^{\bar{}c}$ is an open set, then $A^{\bar{}c}\subseteq \ba... |
Let $X$ be a norm space and $X=M\oplus N$ where $M,N$ are closed subspace of $X$. Take a closed set $A\subseteq X$.
I know that $(A\cap M)\oplus (A\cap N)\neq A$, for example consider $X=\mathbb{R}^2$, $M=\mathbb{R}\times \{0\}$, and $N=\{0\}\times \mathbb{R}$. Then if $A= \Delta_X=\{(x, x):x\in\mathbb{R}\}$,then $(... |
If $X=M\oplus N$, is it true $(A\cap M)\oplus (A\cap N)\subseteq A$? |
The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
[![enter image description here][1]][1]
In the last step of the proof what exactly "turn the path through an angle" means? I reparametrize $x$ to $xe^{-\frac12 \arg z}$ and I got $$\int_{-\infty}^{\i... |
Given $N$ independent quadratic functions such as
$$x_i(t)=a_it^2 + b_it +c_i$$
is there a linear system of ODEs
$$\dot{x}_i=f(x_i)$$
such that the ODEs depend on $x_i$ and not (directly) on $t$? Of course, one can just take the derivative of $x_i$: $\dot{x}_i=2a_it + b_i$, but this does not yield a system of ODEs.... |
$(sin2t,sin3t),0\leq t\leqq 2\pi$
I've tried almost everything but can't get when the curve crosses itself. Could someone help me? thank you very much. |
# Background #
Hi. I am currently writing my undergraduate thesis which mainly revolves around the generalized log-Moyal distribution pioneered by Bhati and Ravi (see [here][1]). In the aforementioned article, Bhati and Ravi addressed a proposition regarding the generalized log-Moyal distribution, one of which is the ... |
In the last step of the proof of [Theorem 10.1][2] in the book *Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan* I couldn't understand what exactly "turn the path through an angle" means. I reparametrize $x$ to $xe^{-\frac12 \arg z}$ and I got $$\int_{-\infty}^{\infty} e^{- \pi... |
Evaluate the following double summation over all acceptable value of $m$ and $k$:
$$S=\sum_{0 \leq k < m \leq n+1} \binom {n+1}{m}\binom {n}{k}$$
This sum arose in a probability question a friend of mine came across in her homework, in which two people A and B were tossing $(n+1)$ and $n$ coins respectively and $... |
I understand that there are multiple ways to calculate the knot determinant, one is through the Alexander polynomial, the other is by creating another matrix which uses the linesections and crossings, the last method is explained in [this pdf][1].
My problem lies in the fact that I don't understand how both can gi... |
$$\frac{1}{1-z+qp^kz^{k+1}}, \space (p+q = 1 \space\space p, q > 0)$$
To find a pole, other than 1/p,
I got to know the pole is between 1 and 1/q.
Then, by using a fixed point iteration, $z_{n+1} = 1+qp^kz^{k+1}_n $, starting $z_0 = 1$(lowerbound)
I have two difficulties with that.
Q1) How to find t... |
$$\frac{1}{1-z+qp^kz^{k+1}}, \space (p+q = 1 \space\space p, q > 0, \space k : positive \space integer)$$
To find a pole, other than 1/p,
I got to know the pole is between 1 and 1/q.
Then, by using a fixed point iteration, $z_{n+1} = 1+qp^kz^{k+1}_n $, starting $z_0 = 1$(lowerbound)
I have two difficul... |
Are the $x_i$ scalars? Then $N \le 3$. And if $N = 3$ you have
$$ \pmatrix{x_1\cr x_2\cr x_3\cr} = A \pmatrix{1\cr t \cr t^2\cr}$$
for some invertible $3 \times 3$ matrix $A$. Then
$$\pmatrix{\dot{x_1}\cr \dot{x_2}\cr \dot{x_3}} = A \pmatrix{0 \cr 1 \cr 2 t\cr} = A \pmatrix{0 & 0 & 0\cr 1 & 0 & 0\cr 0 & 2 & 0\cr... |
$$\frac{1}{1-z+qp^kz^{k+1}}, \space (p+q = 1 \space\space p, q > 0, \space k : positive \space integer \ge 2)$$
To find a pole, other than 1/p,
I got to know the pole is between 1 and 1/q.
Then, by using a fixed point iteration, $z_{n+1} = 1+qp^kz^{k+1}_n $, starting $z_0 = 1$(lowerbound)
I have two di... |
$$\frac{1}{1-z+qp^kz^{k+1}}, \space (p+q = 1 \space\space p, q > 0, \space k : positive \space integer \ge 2)$$
To find a pole, other than 1/p,
I got to know the pole is between 1 and 1/q.
Then, by using a fixed point iteration, $z_{n+1} = 1+qp^kz^{k+1}_n $, starting $z_0 = 1$(lowerbound)
I have two di... |
**I don't have much guidence in real life unfortunately and career counsillors are not available in my country. Also,I was badly harassed by my classmates in my home country due to some discrimination and by many professors who didn't want many communities in higher education in my masters degree. This is why I was so ... |
Am I chasing an obsession or should I try doing a 2 nd masters 3rd time? |
How to prove the existence of this progression? Struggling with a BDMO problem. |
[![enter image description here][1]][1]
Copied from the Knot Atlas and annotated. As you see, $7_2$ needs $4$ braid strands (any rational knot needs at most 4), but if we cheat and define "closure" differently, we could say "The knot $7_2$ can be mapped to a $2$-tangle, which can be obtained by a Temperley-Lieb clos... |
We want to prove that there exists a Sobolev function that is unbounded on each open set. Let $N \ge 2$, and $p \in [1, N )$. Then:
(i) For $α ∈ (0, ∞)$, consider the function $u : B(0, 1) → R$ defined as
$u(x) := |x|^{−\alpha},$ for $x \neq 0 $
Find for what $p$'s and $\alpha$'s, it holds that $ u \in W^{1,p}(B(0... |
I'm new to complex analysis so I might be overlooking something. I'm thinking I could somehow use Liouville to make $u$ constant by taking $e^f$ where $f=u+iv$, with $v$ being the harmonic conjugate of $u$, but I seem to be stuck here. Another thing I tried was to use the Identity Theorem since taking $u(z)$ over purel... |
Let be a unital C*-algebra. Is there a unital positive self-map :→ which is invertible (i.e. injective and surjective) but neither a ∗-automorphism, nor a *-antiautomorphism (i.e. a *-isomorphism with the opposite algebra)? If yes, how does appear its Gelfand-Naimark-Segal covariant representation associated to a stat... |
Let $f$ be a nonnegative Schwartz function on $n$-dimensional Euclidean space.
Then I wonder if it is possible to find a sequence of complex-valued Schwartz functions $f_n$ such that $\lvert f_n \rvert^2$ converges to $f$ in the Frechet topology of the Schwartz space.
I think it is plausible but cannot figure out... |
Is it possible to approximate any nonnegative Schwartz function by square of Schwartz functions? |
In a categorical poset, the elements of the poset are objects of the category, whereas in a categorical group, the elements are arrows. why does such an apparent asymmetry exist, and how can I understand it?
Thank you in advance for any insights into this possibly naive question. |
Let $R$ be a local ring which is not Cohen-Macaulay, and let $I\subset R$ be an ideal.
If I know dim($R$), depth($R$) and codim($I$), can I know depth($I$)? |
When is the operator associated to a sesquilinear form normal? |
Let $M$ be a smooth manifold. Let $V$ be the canonical vector field on $T M$ (also called the Liouville vector field), which if $(x, y)$ are local coordinates on $T M$ is defined by $V = y^i \frac{\partial }{\partial y^i}$. Also denote by $J : TTM \to TTM$ the canonical endomorphism of $TTM$ (also called an almost tang... |
How can I find sum $\sum_{n=k}^{\infty} \frac{\binom{n}{k}}{2^n}$? I tried differentiating binomial, but it didn't really help :( |
I'm reading **Elements for Physics by Albert Tarantola**, and get stuck on page 16.
Before this page, the author defined a **metric** on a linear space $S$ as a map to its dual $\mathbf{G} : S \to S^*$ that is invertible, linear and symmetric.
So if there is a metric defined over $S$, then we could define the no... |
Let $(S,\mathcal{S})$ be a measurable space, and let $p: S \times \mathcal{S} \to \mathbb{R}$ satisfies the properties: $p(s,\cdot)$ is a measure for each $s \in S$ and $p(\cdot, A)$ is a measurable function on $S$ for each $A \in \mathcal{S}$.
Let $f$ be a measurable function on $S$, assumed to be non-negative or... |
Do Markov chains preserve measurability of functions? |
>The total number of non-negative integers $n$ satisfying the equations $n^2=p+q$ and $n^3=p^2+q^2$, where $p$ and $q$ are integers, is...
Case$1$: If both $p$ and $q$ are non-negative integers then,
using, AM of mth power $\ge$ mth power of AM, we get,
$\frac{p^2+q^2}{2}\ge(\frac{p+q}{2})^2\implies \frac{n^3}... |
I'm reading an article on wikipedia about Cauchy product of series: <https://en.m.wikipedia.org/wiki/Cauchy_product>.
My doubt is in the part of generalized form of the cauchy product:
Let $\sum_{k_1}^{\infty}a_{1,k_1} \cdots \sum_{k_n}^{\infty}a_{n,k_n}$ be $n$ series with complex coefficients, from which all ... |
I'm reading an article on wikipedia about Cauchy product of series: <https://en.m.wikipedia.org/wiki/Cauchy_product>.
My doubt is in the part of generalized form of the cauchy product:
Let $\sum_{k_1}^{\infty}a_{1,k_1} \cdots \sum_{k_n}^{\infty}a_{n,k_n}$ be $n$ series with complex coefficients, from which all e... |
Is there a transformation that produces a vector with sum 0? There are positive and negative values and the transformation does not need to be preserve the weights. |
Hi it's related to the following conjecture https://math.stackexchange.com/questions/3388172/an-inequality-for-polynomials-with-positives-coefficients :
We have the first conjecture :
>>Let $x,y>0$ then we have :
$$(x+y)f\Big(\frac{x^2+y^2}{x+y}\Big)(f(x)+f(y))\geq 2(xf(x)+yf(y))f\Big(\frac{x+y}{2}\Big)$$
Where... |
There is a theorem of Shelah, stated in the following way:
If all $\Sigma_3^1$ sets of reals are measurable, then $\aleph_1$ is an inaccessible cardinal in $L$.
In some textbooks (for example Schindler, 2014: Set Theory, Exploring Independence and Truth) the theorem is stated slightly different. Instead of inacc... |
I am reading the book "Engineering Optimization: Theory and Practice, 5th Edition" on Chapter 7 and page 409 and stumble upon this system of equation
$\begin{array}{*{20}{c}}
{\left\{ {\begin{array}{*{20}{c}}
{{{\left( {{x_1} + 1} \right)}^2} - 2r\left[ {\max \left( {0,1 - {x_1}} \right)} \right] = 0}\\
{1 - 2r\l... |
What is the explicit max-min arithmetic use in this transformation? |
Tao's paper: ["Almost all orbits of the Collatz map attain almost bounded values" (via arXiv.org)][1]
- 1st related question/discussion, I did not understand: [Meaning of "almost all" in Terrence Tao's work on the Collatz map][2]
- 2nd related question/discussion, I did not understand: [Notion of "almost all" usi... |
What does "almost all in the sense of logarithmic density" mean in Tao's Collatz paper? |
In a class I learned that optimization problems can have "dual counterparts"
For example, a problem of the type
$$\min_{x} f(x) + g(x)$$
has a "Fenchel" dual problem:
$$\max_{y} -f^\ast(y) - g^\ast(-y)$$
As another example:
$$\min_{x} f_0(x) \text{ s.t. } f_i(x) \leq 0$$
has a "Lagrange" dual p... |
Are there practical problems in optimization where its dual problem is computationally easier to solve? |
Is there a transformation that produces a vector with sum 0? There are positive and negative values and the transformation does not need to be preserve the weights.
E.g.: f(x1, x2, x3) = (x1', x2', x3') s.t. sum(x1', x2', x3') = 0 |
Is there a transformation that produces a vector with sum 0? There are positive and negative values and the transformation does not need to be preserve the weights.
E.g.: f(x1, x2, x3) = (x1', x2', x3') s.t. sum(x1', x2', x3') = 0 |
Hi it's related to the following conjecture https://math.stackexchange.com/questions/3388172/an-inequality-for-polynomials-with-positives-coefficients :
We have the first conjecture :
>>Let $x,y>0$ then we have :
$$(x+y)f\Big(\frac{x^2+y^2}{x+y}\Big)(f(x)+f(y))\geq 2(xf(x)+yf(y))f\Big(\frac{x+y}{2}\Big)$$
Where... |
Is there a transformation that produces a vector with sum $0$?
There are positive and negative values and the transformation does not need to be preserve the weights.
E.g.: $f(x_1, x_2, x_3) = (x_1', x_2', x_3')$ s.t. $$\sum_{i=1}^3x_1' = 0$$ |
Is there a transformation that produces a vector with sum $0$?
There are positive and negative values and the transformation does not need to be preserve the weights.
E.g.: $f(x_1, x_2, x_3) = (x_1', x_2', x_3')$ s.t. $$\sum_{i=1}^3x_i' = 0$$ |
We want to prove that there exists a Sobolev function that is unbounded on each open set. Let $N \ge 2$, and $p \in [1, N )$. Then:
(i) For $α ∈ (0, ∞)$, consider the function $u : B(0, 1) → R$ defined as
$u(x) := |x|^{−\alpha},$ for $x \neq 0 $
Find for what $p$'s and $\alpha$'s, it holds that $ u \in W^{1,p}(B... |
$(\sin 2t,\sin 3t),0\leq t\leq 2\pi$
I've tried almost everything but can't get when the curve crosses itself. Could someone help me? thank you very much. |
**Plateau's problem for concentric circles**
[Visualisation figure set (will be refered to with fig.1-5)][1]
Given two concentric circles $C_1$ and $C_2$ in $\mathbb{R}^3$ with the center points $(x_1,0,0)$ and $(x_2, 0, 0)$, let $\mathbb{S}$ be the set of all $C^0$-continuous surfaces which boundaries exactly e... |
Is my proof on the topic of the minimal surface of revolution valid? |
We want to prove that there exists a Sobolev function that is unbounded on each open set. Let $N \ge 2$, and $p \in [1, N )$. Then:
(i) For $α ∈ (0, ∞)$, consider the function $u : B(0, 1) → R$ defined as
$u(x) := |x|^{−\alpha},$ for $x \neq 0 $
Find for what $p$'s and $\alpha$'s, it holds that $ u \in W^{1,p}(B... |
I'm reading an article on wikipedia about Cauchy product of series: <https://en.m.wikipedia.org/wiki/Cauchy_product>.
My doubt is in the part of generalized form of the cauchy product:
Let $\sum_{k_1}^{\infty}a_{1,k_1} \cdots \sum_{k_n}^{\infty}a_{n,k_n}$ be $n$ series with complex coefficients, from which all e... |
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