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In the course, we do a lot of exemples about optimisation using Taylor method, but I notice that if we have function like:
f(x) = A*x^2 + B*y^2 + C*xy + F*x + G*y + L
without determine the hessian i can ensure that taylor formula will be:
f(a+h)-f(a)= A*x^2 + B*y^2 + C*xy with a a critic point.
So it is always tr... |
What is the relation sense of Taylor Formula in optimisation? |
So I read from various sources that a function can be defined as a binary relation. Then is it valid to say, for example, $f = \{ (1, 2), (2, 3 \}$?
And suppose I have another function $g = \{ (4, 5) \}$. Does it then make sense to write $(f \cup g)(2) = 3$? |
Can a function be defined as the union of two other functions? |
First of all, I was reading a text and stuck with this part:
[![enter image description here][1]][1]
Now regarding the above quote, I want to give two scenarios I made up to make my question clear:
**Scenario 1:**
Imagine we take N measurements from same sensor, and each measurement consists of 10 individual ... |
So I read from various sources that a function can be defined as a binary relation. Then is it valid to say, for example, $f = \{ (1, 2), (2, 3) \}$?
And suppose I have another function $g = \{ (4, 5) \}$. Does it then make sense to write $(f \cup g)(2) = 3$? |
Rouché's Theorem: How many roots does $\lambda-z=\frac{1}{3}e^{z^2}$ have in the strip Re$(z)\in [-1,1]$? |
I got this limit in one of my math exams, but I got stuck because I couldn't decide which term was of a greater significance, ```(x^2+y^2)``` or ```e^(-x-y)```... Maybe I'm not even on the right path. How to solve this limit?
```
lim(,y)→(∞,∞)((x^2+y^2)*e^(-x-y))
``` |
In the course, we do a lot of examples about optimization using Taylor method, but I notice that if we have function like:
f(x) = A*x^2 + B*y^2 + C*xy + F*x + G*y + L
without determine the Hessian, I can ensure that Taylor formula will be:
f(a+h)-f(a)= A*x^2 + B*y^2 + C*xy with a a critic point.
So it is always t... |
What is the relation sense of Taylor formula in optimization? |
Long ago, I learned about the Cauchy completion of metric spaces via the explicit construction via quotienting the Cauchy sequences. For a metric space $X$, let $\hat X$ denote this Cauchy completion of $X$. We have a canonical isometric embedding $\iota\colon X\to\hat X$ ($x\mapsto(x, x, x, \ldots)$). It's easily seen... |
I have recently been trying to learn some basic symplectic geometry, and I have come across two sesquilinear forms which are closely related to the symplectic form.
Fix $\mathbb K$ to be a field, and $V$ to be a $\mathbb{K}$-vector space equipped with a symplectic form $\omega(-,=):V^*\oplus V\to \mathbb{K}$.
Whe... |
I got this limit in one of my math exams, but I got stuck because I couldn't decide which term was of a greater significance, $(x^2+y^2)$ or $e^{(-x-y)}$.
Maybe I'm not even on the right path.
How to solve this limit?
$$\lim_{(,y)→(\infty,\infty)}(x^2+y^2)\cdot e^{(-x-y)}$$
|
In the course, we do a lot of examples about optimization using Taylor method, but I notice that if we have function like:
$f(x) = \alpha \times x^2 + \beta\times y^2 + \sigma \times xy + \gamma \times x + \phi \times y + \xi$
without determine the Hessian, I can ensure that Taylor formula will be:
$f(a+h)-f(a)= \... |
I have recently been trying to learn some basic symplectic geometry, and I have come across two sesquilinear forms which are closely related to the symplectic form.
Fix $\mathbb K$ to be a field, and $V$ to be a $\mathbb{K}$-vector space equipped with a symplectic form $\omega(-,=):V^*\oplus V\to \mathbb{K}$.
Whe... |
I am confused about two things, one of my problem sets for a course on Abelian varieties, I was given a morphism $\mathbb A^1_k\rightarrow \mathbb A^2_k$ "in coordinates" as:
$$f(t)=(t^2-1,t^3-t)$$
Now, since I have never taken a course in classical algebraic geometry, the only way I know how to think about varieties... |
I would like to know more about co-Heyting algebras, particularly from the perspective of their logics (like paraconsistent logics).
What books are available out there on the topic?
It might be that it's too specialised a topic. I get that.
I have looked at the references on the [ncatlab page](https://ncatlab.... |
Assume that the distinct integers 1,...,*N* are in random order and need to be
sorted using selection sort. Here I am interested in the average number of
swaps required, rather than the number of comparisons. Self-swaps are not
counted. Running the selection sort over every possible permutation for a
given *N* p... |
Not a full answer but way to much for a comment :
Let $n$ be a natural number and $f_n$ defined as:
$$
f_n\left(x\right) = e^{-x}\sum_{k=0}^{n}\frac{x^k}{k!}
$$
Now clearly a truncated taylor series for $\exp(x)$ is less than $\exp(x)$ for $x>0$.
So
$$f_n(x) < 1$$
Taylor's theorem with error term giv... |
I do not understand one line in the proof of the following theorem. Within the proof of d) it's stated that, "if $y \in \mathfrak{h}$, then y commutes with x and hence also with s and n." I do not get why the fact that y commutes with x=s+n already implies that y commutes with both s and n. Does anybody know how the to... |
I need to prove the following equality: $\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$.
I know that proving this equality using the properties of $\nabla$ and $(\cdot)$ is easy, what the exercise asks me to do is to use the symbol of Levi-Civita, I'm new to working with turnbuckles and that's why I'm a... |
Does one use the ratio test? If so, how? |
How do you show the convergence of sum_(n=1)^∞ 2^ntan(1/(n!))? |
Below is proof outline. Several places switch the order of expectation $f(E(x))=f(E(x))$ which was confirmed to hold in numeric simulations, but I'm curious how to show it more rigorously:
Given $x\sim \mathcal{N}(0, 1)$, define $y=Ax$ for invertible $d\times d$ matrix $A$ and consider 2 solutions of this system:
... |
How do you show the convergence of $\sum_{n=1}^∞ 2^n\tan(1/n!)$? |
How do you show the convergence of $\sum_{n=1}^∞ 2^n\tan(1/n!)$?
Does one use the ratio test? If so, how? |
I need to prove the following equality: $\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$.
I know that proving this equality using the properties of $\nabla$ and $(\cdot)$ is easy, what the exercise asks me to do is to use the symbol of Levi-Civita, I'm new to working with tensors and that's why I'm a bit... |
Perhaps Green's identity? It says
$$ \int_V \left(\psi \: \nabla \cdot \mathbf{F} + \mathbf{F} \cdot \nabla \psi \right)dV = \int_{\partial V} \psi \: (\mathbf{F} \cdot \mathbf{n})dS $$
for any differentiable $\psi$.
The RHS is zero, the first term of the LHS is zero, and thus it reduces to
$$\int_V \mathbf{F} \c... |
> Give a bijection from the set of strictly increasing sequences of integers from $\{1, \dots, m + d - 1\}$ with length $m$ to the set of non-decreasing sequences of integers from $\{1, \dots, d\}$ also with length m.
I'm sure there is a simple way of looking at this but I have not been able think of a suitable bij... |
How to sketch this set in the complex plane?
$$S=\{ z\in C: Re[(4-i)z] > Re[(-5+7i)(4+6i)] \}$$
Where:
- S - set
- Re[a+bi] - real part of the complex number
- C - complex numbers
- z - complex number |
This is a follow-up on my last question about distributions, https://math.stackexchange.com/q/4818687/1104384, again to potentially help with figuring out the solutions for the post https://math.stackexchange.com/q/4808230/1104384.
To have this be self-contained: last time I managed, thanks to a kind commenter who g... |
I am aware of the well-known Paley-Wiener theorem stating that the Fourier transform is a bijection from the set of compactly supported distributions to the space of exponentially bounded entire functions. I am looking for a reference if there is a similarly nice characterisation of compactly supported Radon measures o... |
How to sketch this set in the complex plane?
$$S=\{ z\in C: Re[(4-i)z] > Re[(-5+7i)(4+6i)] \}$$
Where:
- S - set
- Re[x+yi] - real part of the complex number
- C - complex numbers
- z - complex number
----------
I solved the Real parts inside the brackets and here what I`ve got:
$$Re[(4-i)z] =... |
I'm trying to answer the following problem:
- For $a,b>0$: If $a^2<b^2$, then $a<b$.
I found a different answer but I'd like to understand The answer/hints in the back:
>
The corollary 1.3 is the following:
> i} d x \\
& =\Re\int e^{\cos x+i \sin x} \cdot e^{2 x i} d x \\
&= \Re \int e^{e^{x i}} e^{2 x i} d x\\& \stackrel{y=xi}{=} \Re \left(
\underbrace{\frac{1}{i} \int e^{e^y} e^{2 y} d y}_{J} \right... |
I have encountered many types of sigma algebra generated by a random variable. I would like to clarify my understanding of them.
>The simplest one is generated by a random variable. I skip it here.
>Then I have sigma algebra generated by any family of random variables. i.e. Suppose $X_i$s are rvs from $(\Omega, ... |
I'm trying to answer the following problem:
- **(a)** For $a,b>0$: If $a^2<b^2$, then $a<b$.
I found a different answer but I'd like to understand The answer/hints in the back:
>
The corollary 1.3 is the following:
>![enter image descriptio... |
I have encountered many types of sigma algebra generated by a random variable. I would like to clarify my understanding of them.
>The simplest one is generated by a random variable. I skip it here.
>Then I have sigma algebra generated by any family of random variables. i.e. Suppose $X_i$s are rvs from $(\Omega, ... |
First of all, to clarify what I think the problem is:
We have a curve $\mathcal{C}$ in the $xy$ plane, which encloses a region $\mathcal{R}$. Now consider a function $f:\mathcal{R}\rightarrow\mathbb{R}_{\geq0}$, defining a surface $S=\{(x,y,f(x,y)\,:\,(x,y)\in\mathcal{R}\}$ lying "above" $\mathcal{R}$, and suppose t... |
Given an n-digit integer sequence, e.g. $n=5$ `[1, 1, 5, 6, 4]` where each digit is independent and uniformly random in the range 1-6, what is the probability of having _at least_ k same digits of a certain value (e.g. 1)?
Specifically I'm interested in $k = n/2$ (rounded up) and n in the range 1 - 40.
So ... `f... |
Probability of *at least* k same specific digits in an n-digit sequence? |
Suppose I have an optimization model (P1) which its decision variable is a symmetric matrix $W$ (but not necessarily negative/positive semidefinite). In my case, this model can be converted to a vector-based model using half vectorizing $W$; let this new equivalent model be (P2). I want to **only** work with (P2) and a... |
How to capture low-rankness of a symmetric matrix using its components (not eigenvalues)? |
Consider the function
$$ f(x)=\lim_{k \to \infty}\bigg(\int_0^x \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)\bigg( \int_0^1 \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)^{-1} $$
I want to find a closed form for
$$ f\bigg( e^{-\sqrt{\log n}} \bigg). $$
I simplified this to
$$f\bigg( e^{-\sqrt{\lo... |
For which $n\geq 2$, is it possible to express the identity permutation as the product of all $\binom{n}{2}$ distinct transpositions in $S_n$? Clearly, we require $\binom{n}{2}$ to be even, which requires $n$ to be $0$ or $1 \pmod{4}$.
We have for $n=4$ that (in cycle notation)
$$ I = (34)(14)(23)(13)(24)(12).... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
Let $f: X \to Y$ be a proper birational morphism of Noetherian quas-separated schemes. Let $a: D(QCoh(Y))\to D(QCoh(X))$ be the right-adjoint of the derived pushforward functor $Rf_*: D(QCoh(X))\to D(QCoh(Y))$ (see https://stacks.math.columbia.edu/tag/0A9E) . I have two questions:
(1) Is $a(\mathcal O_Y)\cong \math... |
[**Updated on Nov.10, Dec.05.2023 and Feb.18.2024**] Applying a similar approach followed in these MO questions, for [Catalan's][1] and [Apery's][2] Constants, i.e. Wilf-Zeilberger proofs starting from Dougall's sum, working together with nonlinear $_2F_1$ and $_3F_2$ hypergeometric transformations, I have found four c... |
When searching for derivations of Poisson distribution online, all the articles were making the following two assumptions:
1. Probability of a single event occurring in the interval $[t, t+\mathrm{d}t]$ being equal to $\mu \mathrm{d}t$ for small $\mathrm{d}t$.
2. Probability of two events occurring in the suffici... |
I have the same question as [this][1] post here. To summarise, the poster introduces an apparent counter-example. Consider the manifold $\mathbb{R}$ and a spanning atlas $(\mathbb{R}, \psi)$ where $\psi = x^{1/3}$. Clearly $\psi$ is homeomorphic, but it is not diffeomorphic. I believe the easy counter example is to see... |
I am trying to evaluate the reverse Mellin's transfor for the pochhammer $\frac{1}{2\pi \cdot i} \int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\Gamma(s)}{\Gamma(s+k+1)} y^{-s}ds$.
$\sigma>0$ and $k$ can be any positive integer. Could anyone help me with this evaluation? |
If I had a finite amount of red balls in a infinite sea of blue balls, what is the probability of grabbing a red ball? |
Assume that $X,Y,Z$ are three hypercuboid with the same dimension $k$. Specifically, we have $X=\{x_1,x_2,...,x_k:0\leq x_i\leq c_i, \forall\,i\}$ and $Y=\{y_1,y_2,...,y_k:0\leq y_i\leq b_i, \forall\,i\}$ and $Z=\{z_1,z_2,...,z_k:0\leq z_i\leq a_i, \forall\,i\}$ where each $c_i$ and $b_i$ and $a_i$ are constant positiv... |
If I had a finite amount of red balls in a infinite sea of blue balls, what is the probability of grabbing a red ball?
is it 0 because $\frac{n}{\infty} = 0$? But I can't accept that because if every one of those balls where grabbed by one of infinite people somebody would have needed to grab a red ball in order for a... |
Let $X$ be a complex smooth manifold. The data of a rank-$n$ local system (a sheaf on $X$ that is locally the constant sheaf $\underline{\mathbb{C}^n}$) is the same thing as an open cover $\{U_\alpha\}$ of $X$ and a choice of elements $g_{\alpha \beta} \in GL_n(\mathbb{C})$ telling you how to identify the stalks on ove... |
I am trying to evaluate the reverse Mellin's transform for the pochhammer symbol.
I got to this equation:
$\frac{1}{2\pi \cdot i} \int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\Gamma(s)}{\Gamma(s+k+1)} y^{-s}ds$.
$\sigma>0$ and $k$ can be any positive integer.
I don't know how to proceed. Could anyone help me wi... |
> If $A$ is positive definite relative to a subspace $L \subset \mathbb{R}^n$, i.e.
> $$\forall x \in L, x \ne 0, \quad \langle x, Ax \rangle > 0 $$
> then can we find a positive definite matrix $B$ (on the entire $\mathbb{R}^n$) such that
>$$\forall x \in L, \quad \langle x, Bx \rangle = \langle x, Ax \rangle? $$
... |
I am trying to evaluate the reverse Mellin's transform for the pochhammer symbol.
I got to this equation:
$\frac{1}{2\pi \cdot i} \int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\Gamma(s)} and {\Gamma(s+k+1)} y^{-s}ds$.
$\sigma>0$ and $k$ can be any positive integer.
I don't know how to proceed. Could anyone help ... |
Determine a polynomial of degree $7$ whose Galois group is cyclic of order $7$?
I found the answer [here](https://math.stackexchange.com/a/836420/557708)
It is written that
Let $\zeta = \exp(2\pi i/29)$ be a primitive 29th root of unity.
Then the field $\mathbb Q(\zeta)/\mathbb Q$ has Galois group... |
why value of $f_2(\zeta)=\zeta^{12}, f_3(\zeta)= \zeta^{28}$ and $f_4(\zeta)=\zeta^{17}?$ |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time heads appears.
- The first time tails appears, the game ends and the play... |
Does the St Petersburg Paradox happen for Finite Length Games? |
No tricks requires, just standard techniques.
Lemma: Apply Jensen's to show that
> If $ \frac{ x+y+z}{3} = A $, then $\sum \frac{ x^5 } { x^2 + 1 } \geq 3 \frac{ A^5}{A^2 + 1 }$.
Proof: Work through the [tedious verification that $f''(x) = \frac{ 2x^3 (3x^4 + 9x^2 + 10 } { (x^2+1)^3 } \geq 0$](htt... |
I am trying to evaluate the reverse Mellin's transform for the pochhammer symbol.
I got to this equation:
$\frac{1}{2\pi \cdot i} \int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\Gamma(s)} and {\Gamma(s+k+1)} y^{-s}ds$.
$\sigma>0$ and $k$ can be any positive integer.
I don't know how to proceed. Could anyone help ... |
Determine a polynomial of degree $7$ whose Galois group is cyclic of order $7$?
I found the answer [here](https://math.stackexchange.com/a/836420/557708)
It is written that
>Let $\zeta = \exp(2\pi i/29)$ be a primitive 29th root of unity.
Then the field $\mathbb Q(\zeta)/\mathbb Q$ has Galois grou... |
Assume that the distinct integers 1,...,*N* are in random order and need to be
sorted using selection sort. Here I am interested in the average number of
swaps required, rather than the number of comparisons. Self-swaps are not
counted. Running the selection sort over every possible permutation for a
given *N* p... |
If $\mathbb{K}$ is a field, and we consider the module $\mathbb{K}[x,y]$ over itself, I've seen others give the example that the ideal generated by $\langle x, y \rangle$ is not a free module to demonstrate that a submodule of a free module need not be free. However, I don't understand why.
From the definition of a ... |
Why is the submodule of K[x,y] generated by x and y not a free module? |
Can anyone explain why to compromise between the variances 1/n_in and 1/n_out its 2/(n_in+n_out) its not an average or a harmonic mean it seems random to me?[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/w335I.png
The full paper is here:https://proceedings.mlr.press/v9/glorot10a.htm... |
I want to show that the aplication
$$\phi: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R} \quad \text{given by} \quad \phi(u, v)=u \times v$$
It is bilinear
In first place, note that
\begin{align*}
\phi(\alpha u+ \beta v), w &= (\alpha u+ \beta v) \times w \\
&=\begin{vmatrix}
i & j & k\\
\alp... |
I want to show that the application
$$\phi: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R} \quad \text{given by} \quad \phi(u, v)=u \times v$$
It is bilinear
In first place, note that
\begin{align*}
\phi(\alpha u+ \beta v), w &= (\alpha u+ \beta v) \times w \\
&=\begin{vmatrix}
i & j & k\\
\a... |
Determine a polynomial of degree $7$ whose Galois group is cyclic of order $7$?
I found the answer [here](https://math.stackexchange.com/a/836420/557708)
It is written that
>Let $\zeta = \exp(2\pi i/29)$ be a primitive 29th root of unity.
Then the field $\mathbb Q(\zeta)/\mathbb Q$ has Galois grou... |
How to find $\mathbb Q(\zeta)^H?$ |
Determine if the serie converges or diverges
$\sum_{n=2}^{\infty} \left(1 - \frac{\ln(n)}{n}\right)^{n}$
I tried the divergence theorem, the nth root test and the integral test. Unfortunately, I got no answer and don't know what to do. Is there even an analytic answer?
I would appreciate an orientation or tip |
Does the serie converges or diverges? |
$Q1:$ Let $a_1,a_2,…,a_n$ be non-zero real numbers and $b_1,b_2,…,b_n$ be real numbers. Find the discriminant of the quadratic equation
$$(a_1x-b_1)^2 + (a_2x-b_2)^2 +…+(a_nx-b_n)^2 = 0$$
What can you say about the discriminant?\
$S1: \text{Discriminant},\Delta= 4\left[\left(\sum a_ib_i \right)^2 - \left(\sum a_i^2... |
I recently encountered a question
Let $n \in \mathbb{Z} ^+$, Let $U$ be a non-empty set containing $n$ elements. Let $\mathcal{S} \subseteq P(U)$. Let $m \in \mathbb{Z}^+, m \leq n$.
Prove that
$$
|\mathcal{S}| > \sum_{i=0}^{m - 1} \binom{n}{i} \Rightarrow \exists S' \in P(U). (|S'|=m) \wedge \left(\{S \cap S... |
I recently encountered a question
Let $n \in \mathbb{Z} ^+$, Let $U$ be a set containing $n$ elements. Let $\mathcal{S} \subseteq P(U)$. Let $m \in \mathbb{Z}^+, m \leq n$.
Prove that
$$
|\mathcal{S}| > \sum_{i=0}^{m - 1} \binom{n}{i} \Rightarrow \exists S' \in P(U). (|S'|=m) \wedge \left(\{S \cap S':S \in \m... |
It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/link by changing the variable $V_L(t) \to V_L(t^{-1})$. More generally, the HOMFLY-PT polynomial of the mirror image can be obtained by switching from $P_L(l,m) \to P_L(l^{-1},m... |
The definition of A is a Null set is that $A\subseteq \mathbb{R}$,forall $\varepsilon>0$, exists open cover $\{I_n\}$,so that
$\displaystyle \sum_{n=1}^\infty |I_n|\leq \varepsilon.$
Could I change open intervals set to close intervals set?
I think the essence of the question is how to define the length of in... |
It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/link by changing the variable $V_L(t) \to V_L(t^{-1})$. More generally, the HOMFLY-PT polynomial of the mirror image can be obtained by switching from $P_L(l,m) \to P_L(l^{-1},m... |
I am trying to solve the following:
$\sec(\arctan(\frac{4}{3}))$
The problem tells me to use a relevant right triangle but I am curious as to if I need to create a right triangle for the inside ($\arctan(\frac{4}{3}))$ or the outside ($\sec$)?
Thank you. |
I'm trying to understand the calculation of the following equation 3.25 in the textbook "Numerical Optimization by Nocedal and Wright":
[![Nocedal 3.25][1]][1]
Here is my work:
\begin{gather*}
f: \mathbb{R}^n \to \mathbb{R}, \quad x,b \in \mathbb{R}^n, \quad Q \in \mathbb{R}^{n \times n} \\
f(x) = \frac{... |
If $L$ is regular, is $L' = \{xz \mid \exists y, y \in \Sigma^* \text{ such that } |x|=|y|=|z|\text{ and }xyz \in L\}$ regular? |
I'm in the process of of again learning algbra 1. But I ran across this problem the other day.
The sum of two numbers is 10 and the sum of their reciprocals is 5/12. Find the numbers.
Instinctively, I thought okay x+x=10
And when worked out x=5. Looked up the answer and quite different to mine. Did some research a... |
Assume $X$ is a compact set (not necessarily convex). If the following equality holds true?
$$\text{Conv}(X)=\text{Conv}(\text{extr(X)})$$
where $\text{extr}$ means the extreme point of $X$. The definition of the extreme point is that a point $a$ is said to be an extreme point of the set $X$ if the point $a$ cannot ... |
Convex hull of a set is equivalent to the convex hull of the extreme point of the set? |
For each nonegative integer $x$ and each $Y\subset\{0,1\}$ let the event $R_{x,Y}$ be:
for each $y\in \{0,1\}$ we have $y\in Y$ iff there exists an abscissa-nondecreasing path from $(0,0)$ to $(x,y)$. Then $p(R_{0,\varnothing})=0$, $p(R_{0,\{0\}})=\frac 12$, $p(R_{0,\{1\}})=0$, and $p(R_{0,\{0,1\}})=\frac 12$. Moreove... |
I'm in the process of of again learning algebra 1. But I ran across this problem the other day.
The sum of two numbers is $10$ and the sum of their reciprocals is $5/12$. Find the numbers.
Instinctively, I thought okay $x+x=10$
And when worked out $x=5$. Looked up the answer and quite different to mine. Did some r... |
You tried $5$, but the sum of the reciprocals $1/5+1/5 \ne 5/12$.
Hint: Did you try 3, 4? What are the factors of 12? These will help you figure out what $x$ and $y$ must be.
Answer:
>! If you choose $x=4$ and $y=6$ then $x+y = 4+6=10$ and $1/4+1/6=5/12.$ |
I am trying to solve the following:
$$\sec\left(\arctan\left(\frac{4}{3}\right)\right)$$
The problem tells me to use a relevant right triangle, but I am curious as to if I need to create a right triangle for the inside ($\arctan(\frac{4}{3}))$ or the outside ($\sec$)?
Thank you. |
A textbook I'm reading asks to prove that the special case of a two value softmax is equivalent to a sigmoid/logistic function. It defines the softmax as:
$$
\frac{e^{H_t(a)}}{\sum_{b=1}^ke^{H_t(b)}}
$$
I made the first logical leap to:
$$
\frac{e^x}{e^x + e^y}
$$
Wolfram Alpha tells me that this is equi... |
I found an identity
$$\int_{-\infty}^{\infty}\binom{n}{x}dx=\sum_{i=0}^{n} \binom{n}{i}$$
where LHS can be calculated by the [Reflection relation](https://mathworld.wolfram.com/ReflectionRelation.html) and
[Dirichlet integral](https://en.wikipedia.org/wiki/Dirichlet_integral).
The result is $2^n$, which is appa... |
This is getting almost unbearably tedious. Let's go step by step.
Do you know how to show the algebraic quadratic equation relating $X$ an $Y$? Just get $x'$ and $y'$ from the two equations you showed in the post using $X$ and $Y$, and plug them into the condition
$$
x'^2 + 2\gamma x' y' + y'^2=1.
$$
Now the... |
I'm trying to show that
$$e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu} + \left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2},$$
for an absolute constant $C > 0$, where $\delta \in (0,1]$ and $\mu > 0$.
Using the idea in [this answer][1], I wrote
$$\log(1+\delta) = 0 + \... |
I am trying to evaluate the reverse Mellin's transform for the pochhammer symbol.
I got to this equation:
$\frac{1}{2\pi \cdot i} \int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\Gamma(s)} {\Gamma(s+k+1)} y^{-s}ds$.
$\sigma>0$ and $k$ can be any positive integer.
I don't know how to proceed. Could anyone help me w... |
Is there a perfect group in which not every element is a commutator? |
Is there a perfect group in which not every element is a commutator?
By a well-known fact, it must have order at least $96.$
|
Is there a perfect group in which not every element is a commutator?
By a well-known fact, it must have order at least $96.$
By Ore's conjecture (now a theorem), it must be infinite or non-simple.
|
Does this improper integral converge? How? |
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