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I want to define a continuous function $f(x)$ such that the following properties hold true.
1. $f(0) = 0.5$
2. $f(-1) = 0.5 - 0.25 = 0.25$
3. $f(1) = 0.5 + 0.25 = 0.75$
4. $f(-2) = 0.5 - 0.25 - 0.125 = 0.125$
5. $f(2) = 0.5 + 0.25 + 0.125 = 0.875$
6. And so on, ad infinitum.
So, I know that I'm looking... |
How do I define a function that's first exponential and then logarithmic? |
I want to define a continuous function $f(x)$ such that the following properties hold true.
1. $f(0) = 1$
2. $f(-1) = 1 - 0.5 = 0.5$
3. $f(1) = 1 + 0.5 = 1.5$
4. $f(-2) = 1 - 0.5 - 0.25 = 0.25$
5. $f(2) = 1 + 0.5 + 0.25 = 1.75$
6. $f(-3) = 1 - 0.5 - 0.25 - 0.125 = 0.125$
7. $f(3) = 1 + 0.5 + 0.25 + 0.12... |
Your confusion seems to arise from having a wrong idea about the relation of the elements of the fields $\Bbb{F}_{16}=GF(16)$ and its subfield $\Bbb{F}_4=GF(4)$. It is correct to construct these two fields as quotient fields of the polynomial ring
by moding out the ideal generated by the defining polynomial:
$$
\Bbb... |
What is the number of non negative integral solutions of
$5w+3x+y+z=100$, such that $w≤x≤y≤z$.
I have the answer key to this but I do not understand how to even start. The second inequality condition is what is troubling me the most. Are there specific ways to approach this? |
Completion of $(k[x_1, \dots, x_n]/I)_{(x_1, \dots , x_n)/I}$ is $k[[x_1, \dots , x_n ]] / Ik[[x_1, \dots , x_n ]]$? |
The card you choose to begin with (card A in your example above) becomes immune to any further changes the moment you make that choice. So the probability of win/loss associated with that card also becomes immune to any change. It will continue to be $1/52$, no matter how many cards you remove (or turn to reveal) from ... |
Your self-answer is unsatisfactory.
Also: In OP, what do you mean by "a polynomial $u(t,x)$ in $t,x$ such that
$$
\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}"\quad\dots\quad ?
$$
The correct statement is that any function $u(t,x)$ (regardless if polynomial or not) that satisfies... |
The card you choose to begin with (card A in your example above) becomes immune to any further changes the moment you make that choice. So the probability of win/loss associated with that card also becomes immune to any change. It will continue to be $1/52$, no matter how many cards you remove (or turn to reveal) from ... |
*Definition*: So according to the definition a sequence $(x_n)$ is convergent to some $x$ if:
For any $\varepsilon > 0$ there exists some $K(\epsilon) \in \mathcal{N}$ such that for all $n \in \mathcal{N}$ if $n \ge K(\epsilon)$, we have $|x_n - x| < \varepsilon$.
a) Now my approach of showing if the given sequence... |
I study compound statements, and I encountered this problem in the book:
[The problem][1]
I tried a solution: Let **p** be proposition "The first door leads to freedom" and let **q** be proposition "The warrior tells the truth". The question I am going to ask should help me to understand which door to choose. I ... |
The card you choose to begin with becomes immune to any further changes the moment you make that choice. So the probability of win/loss associated with that card also becomes immune to any change. It will continue to be $1/52$, no matter how many cards you remove (or turn to reveal) from the other group.
It's differ... |
I started watching a course on ODE, In the course lecture first video, the teacher gave some motivations, this example of Non-Linear population model is first among them, the equation is given by $$\frac{dy}{dt}=ay-by^2 (a,b>0)$$
$t_0$ is the intial time and let $y(t_0)=t_0$
and got the solution as $$\frac{1... |
How should I read or interpret this notation? |
Suppose that we are given an arbitrary set $\mathcal X \subseteq \mathbb R^d$, a probability density function of multivariate Gaussian distribution with mean $\mu$ and variance $\sigma^2$, f$(x;\mu,\sigma)$, and a function $g: \mathbb R^d \rightarrow \mathbb R$ that is monotonically increasing. Can we say anything abou... |
Question:
>Let $X$ be a connected space with a topology not necessarily sequential. What is an example where a point in $X$ is not the limit of any not eventually constant sequence?
**Motivation.**
Suppose $X$ is a topological vector space over $\mathbb{R}$ or $\mathbb{C}$. If we define a *sequentially separated*... |
You are given a bucket with apples and oranges. There are two apples, and the rest oranges. Four children uniformly randomly pick a fruit from the bucket. The probability that all four children pick oranges is A and the probability that two of the four children pick the apples is B. After that, you realize at A = B. W... |
Suppose $M^n$ is a smooth projective variety. A line bundle $L$ on $M$ is nef (numerically effective) if on any complete curve $C$ in $M$, $L$ has positive degree, i.e.
$$
L\cdot C=\int_{C}R_h\geq 0.
$$
where $R_h$ is the curvature form of any Hermitian metric $h$ on $L$.
A line bundle $L$ on $M$ is big if there... |
> Let $K$ be a field and let $P\subset K$ its prime field (or minimal field).Then:
1. $char K=p>0 \iff P\simeq\Bbb F_p$ where p is a prime number
2. $char K=0 \iff P\simeq\Bbb Q$
Therefore , barring isomorphisms, the only prime fields are $\Bbb Q$ and the fields $\Bbb F_p$
For the proof we consider the c... |
Find |A intersection B| if |A| = 7 |B| = 8 and |AUB| = 10.
How is this possible?
Please help. |
Find $|A \cap B|$ if $|A| = 7$, $|B| = 8$ and $|A\cup B| = 10$.
How is this possible?
Please help. |
When $X$ is a topological space, any sheaf $S$ on $X$ is a connected object of $\widehat{\mathcal{O}(X)}$ in the sense that
$\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A+B) \cong \mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A)+\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,B)$,
for all $A,B \in \widehat{\mathcal{O}(X)}$... |
For which sites are all sheaves connected? |
**Textbooks on holomorphic functions of several complex variables**<br>
I will first answer your request for a reference text on holomorphic functions of several complex variables, since it is simpler to answer. In my opinion the best textbook to start a journey in this realm is the textbook by Krantz [1a] since it tr... |
Find $|A \cap B|$ if $|A| = 7$, $|B| = 8$ and $|A\cup B| = 10$.
How is this possible?
Please help.
If |A| = 7, |B| = 8, then how is the cardinality of AUB less than 7 + 8.
If you have two sets, than how is the union of it less than the two sets? |
For $d\times d$ matrix $A$, $x\in \mathbb{R}^d$ with IID standard normal entries and $y=Ax$, I'm interested in the the following quantity where $\cos(u,v)$ refers to [cosine similarity](https://en.wikipedia.org/wiki/Cosine_similarity):
$$E_{A,x}[\cos(A^T y,A^{-1} y)]$$
It appears to be $\frac{1}{\sqrt{2}}$, in si... |
**Problem:**
>Show by hand without any computer assistance:
$$e^{e^2}>1000\phi,$$
where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$.
I come across this limit showing:
$$\lim_{x\to 0}x!^{\frac{x!!^{\frac{2}{x!!-1}}}{x!-1}}=e^{e^2}.$$
I cannot show it without knowing some decimals and ... |
Suppose that we are given an arbitrary set $\mathcal X \subseteq \mathbb R^d$, a probability density function of multivariate Gaussian distribution with mean $\mu$ and covariance matrix $\sigma^2I$, f$(x;\mu,\sigma)$, and a function $g: \mathbb R^d \rightarrow \mathbb R$ that is monotonically increasing. Can we say any... |
For which sites are all sheaves connected (as presheaves)? |
Given, simple model,
$$\begin{eqnarray}
\frac{\mathrm{d}S}{\mathrm{d}t} &=& -\beta SI \\ \nonumber
\frac{\mathrm{d}I}{\mathrm{d}t} &=& \beta SI - \gamma I \\ \nonumber
\frac{\mathrm{d}R}{\mathrm{d}t} &=& \gamma I
\end{eqnarray}$$
Is variable $S$ continuous with respect to $\beta$, if one vary $\beta$... |
Is state variable $S$ is continuous as parameter vary for a system of differential equation? |
Given, simple model,
$$\begin{eqnarray}
\frac{\mathrm{d}S}{\mathrm{d}t} &=& -\beta SI \\ \nonumber
\frac{\mathrm{d}I}{\mathrm{d}t} &=& \beta SI - \gamma I \\ \nonumber
\frac{\mathrm{d}R}{\mathrm{d}t} &=& \gamma I
\end{eqnarray}$$
Is variable $S$ continuous with respect to $\beta$, if one vary $\beta$... |
I wish to compute the integral of
$\displaystyle \int_0^{\infty} \dfrac{1}{x^2} \exp{\left(-\dfrac{a}{x}-bx\right)} dx$
I found something similar in Gradshteyn and Ryzhik 7th Ed. p336 (Sec 3.324) in terms of the modified Bessel function of the second kind $K_\nu (x)$ but it didn't include the factor of $\dfrac{1}... |
Let $X=(x_1,x_2,...,x_n)$ where $x_i \in \mathbb{R}^d$ with $d>=2$.
And where each $x_i = \sum_{j=0}^{d}a_j\cdot e_j$ where $(e_1,...,e_d)$ forms a basis in $\mathbb{R}^d$.
We define the first component of the PCA as the $d$-dimensional vector that maximizes the variance of the data projected into it.
Now suppos... |
Is PCA invariant under transofrmations? |
After some helpful feedback on my first question, I will repost my question with some more additional information. <p>
For a programming project, I need to calculate the L-series to a Groessencharacter. In order to use some built-in SAGE functions for certain Zetafunctions, I have the following problem:
Given a field... |
Let $X=(x_1,x_2,...,x_n)$ where $x_i \in \mathbb{R}^d$ with $d>=2$.
And where each $x_i = \sum_{j=0}^{d}a_j\cdot e_j$ where $(e_1,...,e_d)$ forms a basis in $\mathbb{R}^d$.
We define the first component of the PCA as the $d$-dimensional vector that maximizes the variance of the data projected into it.
Now suppos... |
Is PCA invariant up to a change of basis? |
Here is a sample dataset. I have to calculate the Information gain in `Variable`. Am I missing anything as information gain should not be negative?
| Variable | Class |
|----------|----------|
| Val1 | Positive |
| Val1 | Positive |
| Val1 | Positive |
| Val1 | Positive |
| Val1 | Positi... |
Why is the value of information gain negative? |
If $G(x)=x^2+x+1$ and $H(x)=x^3-x+1$ where $r_G$ and $r_H$ are roots of $G(x)$ and $H(x)$ respectively, what do the elements of (I). $\mathbb{F}_2(r_G)$, (II). $\mathbb{F}_2(r_H)$, (III). $\mathbb{F}_3(r_G)$, and (IV). $\mathbb{F}_3(r_H)$ look like? How many elements does each have?
I guess I'm confused, because for... |
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... |
am reading on clean rings this paper [clean general rings by W.K. Nicholson, Y. Zhou][1]
and I am puzzled by the proof of this corollary [corollary 11][2]
and would appreciate any clarification on the following:
what i ring morphism, did the writter mean a ring homomorphism? if so how did they prove that the map... |
am reading on clean rings this paper [clean general rings by W.K. Nicholson, Y. Zhou][1]
and I am puzzled by the proof of this corollary [corollary 11][2]
and would appreciate any clarification on the following:
what is a ring morphism, did the writter mean a ring homomorphism? if so how did they prove that the ... |
I am working with the SIR model with vital dynamics:
\begin{align}
\frac{dS}{dt} &= \mu (K - S)- \beta SI \label{eq3}\\
\frac{dI}{dt} &= I(\beta S - \gamma -\mu) \label{eq4}
\end{align}
where I discarded the last ODE since it does not appear in the first two.
The region for this problem is $D={(S,I) : S>0, I ... |
How to show this point is a global asymptotically stable point? |
Is there a sufficient condition if satisfied by a differential equation at an ordinary point ensures finding the solution using power series method?
In other words, can power series method fails to solve differential equation at ordinary point? |
Can power series method fails to solve differential equation at ordinary point? |
Exercise 2.14 in Villani's "Topics in Optimal Transportation" asks us to show the following:
Let $Q=[0,1]^{n-1}$, and $\mu$ is uniform on $Q\times \{0\}$, $\nu$ uniform on $Q\times \{1\} \cup Q\times \{-1\}$, with the cost function $c(x,y)=|x-y|^2$. Then the only solution to the Kantorovich problem is $\pi=(\delta_... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but we only know it seems to have a recursive formula. However, is there a concise closed-form expression? I attempted to solve it using Mathematica, but its complexity is daunting. I feel that this sequence shows a pattern starting from 6, where... |
Is there a concise closed-form expression for A254828? |
Suppose you have a circle and two points (A and B) outside the circle in the cartesian plane. The question is to find the shortest path from A to B while touching the perimeter of the circle.
1. Is there straightforward formula to find this shortest path value (distance)?
2. How to find the point V where the the ... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with $\nabla g = \|\nabla g\|\hat n$. Now let every other point in the pl... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but we only know it seems to have a recursive formula. However, is there a concise closed-form expression? I attempted to solve it using Mathematica, but its complexity is daunting. I feel that this sequence shows a pattern starting from $n=6$, w... |
I am working with the SIR model with vital dynamics:
\begin{align}
\frac{dS}{dt} &= \mu (K - S)- \beta SI \label{eq3}\\
\frac{dI}{dt} &= I(\beta S - \gamma -\mu) \label{eq4}
\end{align}
where I discarded the last ODE since it does not appear in the first two.
The region for this problem is $D={(S,I) : S>0, I ... |
> what is a ring morphism, did the writter mean a ring homomorphism?
Yes, morphism and homomorphism usually mean the same thing.
> if so how did they prove that the mapping preserved the multiplication?
>
>(i.e.) ϕ(xy)=ϕ(x)⋅ϕ(y)
That's actually quite easy once you introduce $\xi,\eta\in R$ with $x=e\xi$ and ... |
Here's one way to do it:
Suppose $F(x)=\sum_{n=0}^{\infty}a_nx^n=a_0+a_1x+a_2x^2+...$. Then $f(x)=\sum_{n=1}^{\infty}na_nx^{n-1}-x=a_1+(2a_2-1)x+3a_3x^2+...$
If these functions are to be equal, then we must have the following:
$$a_0=a_1$$
$$a_1=2a_2-1$$
$$a_2=3a_3$$
$$a_3=4a_4$$
$$a_4=5a_5$$
And so on. ... |
Rationalizing radicals $x^{d/(d-1)}+y^{d/(d-1)}+z^{d/(d-1)}$
|$d$|(rationalized) polynomial|degree of polynomial|
|---|---|---|
|$2$|$x^2+y^2+z^2$|$2$|
|$3$|\begin{matrix}\hfil\qquad\small(x^{\frac32}+y^{\frac32}+z^{\frac32})(x^{\frac32}+y^{\frac32}-z^{\frac32})(x^{\frac32}-y^{\frac32}+z^{\frac32})(x^{\frac32}-y^... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but we only know it seems to have a recursive formula. However, is there a concise closed-form expression? I attempted to solve it using Mathematica, but its complexity is daunting.
I feel that this sequence shows a pattern starting from $n... |
Let $\mathbb{K}$ be a fixed field. Take a finite-dimensional $\mathbb{K}$-algebra $A$ and look at the category $\text{mod}(A^e)$ of finite-dimensional $A$-bimodules. It has a full subcategory that has a objects those $A$-bimodules which are projective when considered as left and as right $A$-modules. Is that subcategor... |
Consider the minimal polynomial of radicals $x^{d/(d-1)}+y^{d/(d-1)}+z^{d/(d-1)}$
|$d$|minimal polynomial|degree of polynomial|
|---|---|---|
|$2$|$x^2+y^2+z^2$|$2$|
|$3$|\begin{multline}\shoveright\small(x^{\frac32}+y^{\frac32}+z^{\frac32})(x^{\frac32}+y^{\frac32}-z^{\frac32})(x^{\frac32}-y^{\frac32}+z^{\frac32}... |
I'm currently trying to wrap my head around dualizable objects in monoidal categories and I was wondering whether the following claim holds:
Let $A$ and $B$ be dualizable objects in a monoidal category, i.e. $A \dashv B$ with unit $\eta : I \to B \otimes A$ and $B \dashv A$ with counit $\varepsilon' : B \otimes A \t... |
If $A$ and $B$ are dualizable objects in a monoidal category, is the unit of the one duality the inverse of the counit of the other duality? |
I have this function:
$y=\sqrt{\ln|(sin^2(x)-3cos^2(x)|}$
I'm pretty sure its domain is:
$-\pi/4 + k \pi \leq x \leq \pi/4 + k \pi \; \; \cup\; \; x=\pi/2 + k \pi$
I got it by solving $\ln|(sin^2(x)-3cos^2(x)| \geq 0 \Rightarrow |(sin^2(x)-3cos^2(x)| \geq 1$.
To be sure, I "tested" it but every graph ca... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but we only know it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$. However, is there a concise closed-form expression? I attempted to solve it using Mathematica, but its complexity is daunting.
7 6
8 7
... |
I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$. However, is there a concise closed-form expression? I attempted to solve it using Mathematica, but its complexity is daunting.
7 6
8 7
... |
Consider the minimal polynomial of radicals $x^{d/(d-1)}+y^{d/(d-1)}+z^{d/(d-1)}$
|$d$|minimal polynomial|degree of polynomial|
|---|---|---|
|$2$|$x^2+y^2+z^2$|$2$|
|$3$|$x^6+ y^6+ z^6 - 2 x^3 y^3 - 2 x^3 z^3 - 2 y^3 z^3 $|$6$|
|$4$|$(x^4+y^4+z^4)^3-27x^4y^4z^4$|$12$|
Is it true for all $d\ge2$ that the deg... |
In [SHEAVES, COSHEAVES AND APPLICATIONS](https://arxiv.org/abs/1303.3255), the author introduced a different definition for sheaves. Namely,
**Definition**: X is a topological space and $\mathcal{D}$ is a category, e.g. $\mathbf{Sets}$, $\mathbf{Vect}$, e.t.c.. Suppose $\mathcal{F}:\mathbf{Open}(X)^{op}\rightarrow ... |
I have this function:
$y=\sqrt{\ln|\sin^2x-3\cos^2x|}$
I'm pretty sure its domain is:
$-\pi/4 + k \pi \leq x \leq \pi/4 + k \pi \; \; \cup\; \; x=\pi/2 + k \pi$
I got it by solving $\ln|\sin^2x-3\cos^2x|\geq 0 \Rightarrow |\sin^2x-3\cos^2x| \geq 1$.
To be sure, I "tested" it but every graph calculator d... |
The dual curve of $x^d+y^d+z^d=0$ is the minimal polynomial of $x^{d/(d-1)}+y^{d/(d-1)}+z^{d/(d-1)}$
|$d$|minimal polynomial|degree of polynomial|
|---|---|---|
|$2$|$x^2+y^2+z^2$|$2$|
|$3$|$x^6+ y^6+ z^6 - 2 x^3 y^3 - 2 x^3 z^3 - 2 y^3 z^3 $|$6$|
|$4$|$(x^4+y^4+z^4)^3-27x^4y^4z^4$|$12$|
Is it true for all $... |
> Let $\{X_n\}_{n \geq 1}$ be a sequence of i.i.d. random variables with $\mathbb E(X_1^2) \lt \infty.$ Show that $$\frac {2} {n (n + 1)} \sum\limits_{j=1}^{n} j X_j \xrightarrow{p} \mathbb E(X_1)\ \ \text{as}\ n \to \infty.$$
**My Attempt** $:$ First we note that $$\left (\frac {2} {n (n + 1)} \sum\limits_{j=1}^{n}... |
Six edge lengths of a tetrahedron $ (a,b,c,p,q,r) $ are given.
- What relation should be there in order that a tetrahedron can be enclosed among the side lengths?
- What is the volume V of the tetrahedron? Is it like $V= \frac{\sqrt 2}{4}\sqrt{abcpqr}?$. The constant is obtained from a regular tetrahedronal ... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with $\nabla g = \|\nabla g\|\hat n$. Now let every other point in the pl... |
My attempt:\
The dual curve of $x^d+y^d+z^d=0$ is the minimal polynomial of $x^{d/(d-1)}+y^{d/(d-1)}+z^{d/(d-1)}$
|$d$|minimal polynomial|degree of polynomial|
|---|---|---|
|$2$|$x^2+y^2+z^2$|$2$|
|$3$|$x^6+ y^6+ z^6 - 2 x^3 y^3 - 2 x^3 z^3 - 2 y^3 z^3 $|$6$|
|$4$|$(x^4+y^4+z^4)^3-27x^4y^4z^4$|$12$|
Is it ... |
Is there a known description of the set $\text{Ord}_n(a)$ of multiplicative orders of a fixed unit $a$ modulo all divisors of $n$, i.e. of $\text{ord}_d(a)$ with $d\mid n$? It is easy to see that $\text{Ord}_n(a)$ is a subset of the set of all divisors of $\text{ord}_n(a)$. Indeed, if $a^m\equiv1\pmod{n}$ then $a^m\equ... |
**Statement**:
Complex function $f=u+\textbf i v$ is analytic in domain $D\subset \mathbb C$ and everywhere in $D$ $u=\phi(v)$, where $\phi=\phi(t)\in C^{1}(\mathbb R\to\mathbb R)$ and $\phi$ is *strictly monotone*. Prove that $f$ is constant in $D$.
My question is about sufficiency of strict monotonicity conditi... |
> Let $\{X_n\}_{n \geq 1}$ be a sequence of i.i.d. random variables with $\mathbb E(X_1^2) \lt \infty.$ Show that $$\frac {2} {n (n + 1)} \sum\limits_{j=1}^{n} j X_j \xrightarrow{p} \mathbb E(X_1)\ \ \text{as}\ n \to \infty.$$
**My Attempt** $:$ First we note that $$\left (\frac {2} {n (n + 1)} \sum\limits_{j=1}^{n}... |
Six edge lengths of a tetrahedron $ (a,b,c,p,q,r) $ are given.
- What relation should be there in order that a tetrahedron can be enclosed among the edge side lengths?
- What is the volume V of the tetrahedron? Is it like $V= \frac{\sqrt 2}{4}\sqrt{abcpqr}?$ The constant is obtained from a regular tetrahedro... |
> what is a ring morphism, did the writter mean a ring homomorphism?
Yes, morphism and homomorphism usually mean the same thing.
> if so how did they prove that the mapping preserved the multiplication?
>
>(i.e.) ϕ(xy)=ϕ(x)⋅ϕ(y)
That's actually quite easy once you introduce $\xi,\eta\in R$ with $x=e\xi$ and ... |
Are $(x_n-x_{n-1})$ and $(x_1+...+x_n)$ Shauder basis? |
________________________
1.
Take any separable Hilbert space ($\ell^2$, for example) and a maximal orthonormal set (which must be countable, because our space is separable). This orthonormal set is a Schauder basis. Furthermore we notice that the following series is convergent:
$$S=\sum_{j=1}^\infty \frac{... |
I have a question regarding a regression problem. Suppose vector $\mathbf{p}$ is real and has length N. Now suppose $\mathbf{p}$ is the input to the following equation:
$$
y = f(e^{i\mathbf{p}})
$$
Where $f()$ is some function with a complex vector as input and a complex scalar as output and where $e^{i\mathbf{p}}$... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with $\nabla g = \|\nabla g\|\hat n$. Now let every other point in the pl... |
I am reading this paper on clean rings: [*Clean general rings*, by W.K. Nicholson, Y. Zhou](https://www.sciencedirect.com/science/article/pii/S0021869305000591), and I am puzzled by the proof of Corollary 11:
>**Corollary 11.** If $R$ is a ring and $e^2=e \in R$, then $eR$ is a clean general ring if and only if $eRe... |
Let $\mathbb{K}$ be a fixed field. Take a finite-dimensional $\mathbb{K}$-algebra $A$ and look at the category $\text{mod}(A^e)$ of finite-dimensional $A$-bimodules. It has a full subcategory $\text{lrp}(A^e)$ that has a objects those $A$-bimodules which are projective when considered as left and as right $A$-modules.
... |
> what is a ring morphism, did the writter mean a ring homomorphism?
Yes, morphism and homomorphism usually mean the same thing.
> if so how did they prove that the mapping preserved the multiplication?
>
>(i.e.) ϕ(xy)=ϕ(x)⋅ϕ(y)
That's actually quite easy once you introduce $\xi,\eta\in R$ with $x=e\xi$ and ... |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
***
Notice that double sums over rational functions can ... |
So, from Multivariate Calculus and Linear Algebra, I learned that:
1. Given $f(x,y)$, I can find out if it is convex by checking if the eigenvalues of its Hessian are positive or zero (i.e. if the Hessian is positive-semidefinite)
2. The determinant of any matrix is the product of its eigenvalues
Therefore, I... |
Is $f(x,y) = \dfrac{y+2}{x+y+2}$ convex anywhere? This is driving me crazy. |
Six edge lengths of a tetrahedron $ (a,b,c,p,q,r) $ are given.
- What relation should be there in order that a tetrahedron can be enclosed among the edge side lengths?
- What is the volume $V$ of the tetrahedron? Is it like $V= \frac{\sqrt 2}{4}\sqrt{abcpqr}?$ The constant is obtained from a regular tetrahed... |
So, from Multivariate Calculus and Linear Algebra, I learned that:
1. Given $f(x,y)$, I can find out if it is convex by checking if the eigenvalues of its Hessian are positive or zero (i.e. if the Hessian is positive-semidefinite).
2. The determinant of any matrix is the product of its eigenvalues.
Therefore,... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/71Kbw.jpg
Theorem C is from the book Measure Theory by Halmos...from Chapter 3.
My question: how Sigma finite of E is used in the proof?
It will be very much helpful kindly give some hint or explanation. |
$Q1:$ Let $a_1,a_2,…,a_n$ be non-zero real numbers and $b_1,b_2,…,b_n$ be real numbers. Find the discriminant of the quadratic equation
$$(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've been learning about derived functors recently, and I had conceptualized them as fulfilling the following goal:
Suppose that we had a left-exact functor $F:\mathcal{A} \to \mathcal{B}$ between two abelian categories. If we had a short exact sequence $0 \to A \to B \to C \to 0$, then hitting this with $F$ gives a... |
What is the "goal" of derived functors? |
Definition: An operator $T$ on a Hilbert space $H$ is said to be nonnegative definite if it is self-adjoint and $\langle Tx,x \rangle \geq 0$ for all $x \in H$.
Now suppose $T$ is any element of $\mathcal B(H), T^*T$ is nonnegative definite because it is self-adjoint and $\langle T^*Tx,x \rangle = ||Tx||^2$.
**Qu... |
How to see that $T^*T$ is self-adjoint? |
Let's say for example we divide 2m^2 by 1m, the result is 2m. What is the physical interpretation of this? I mean, dividing an area by a distance makes no sense to me. |
What does it mean to divide an area by a distance? |
I recently got an explanation of the Monty Hall problem and I thought I understood it but after giving it more thought, it still looks wrong.
Instead of using goats and doors, the example used a 52-card poker deck. The cards are facing down on a table and you need to pick one card without looking at it. Let's say th... |
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