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> In case of spherical geometry one can imagine everything well as a 2 dimensional sphere in 3 dimensional space and all the major facts on parallel lines make sense.
I would argue that the better analogy is elliptic geometry. That's the same as spherical geometry, except that you treat antipodal points as a unit. L... |
We think about a relation between a function $\psi(x,\epsilon)$ and a divergent power series in $\epsilon$: $f_{0}+\epsilon\,f_{1}+...+\epsilon^{p}f_{p}+...$
The coefficients $f_{0}$, $f_{1}$... can be a function of $x$ only or of $x$ and $\epsilon$.
Now, **let**: $\psi_{p}=f_{0}+\epsilon\,f_{1}+...+\epsilon^{p}f_{... |
### Question:
Let a r.v. $0\leq X \leq 1$. Prove that $V(X) \leq \frac{1}{4}$. When does we have an equality?
### Answer:
1- $0\leq X \leq 1 \Rightarrow X^2 \leq X \Rightarrow \int X(\omega)^2 dP (\omega) = E[X^2] \leq E[X] = \int X(\omega) dP (\omega)$ because the pdf $f(x)$ is strictly positive.
So... |
We think about a relation between a function $\psi(x,\epsilon)$ and a divergent power series in $\epsilon$: $f_{0}+\epsilon\,f_{1}+...+\epsilon^{p}f_{p}+...$
The coefficients $f_{0}$, $f_{1}$... can be a function of $x$ only or of $x$ and $\epsilon$.
Now, **let**: $\psi_{p}=f_{0}+\epsilon\,f_{1}+...+\epsilon^{p}f_{... |
The context might be an old photograph of (say) soldiers in a trench or a picnic in a forest, but without a horizon or any vanishing points.
My conjecture is that 4 known points in world space mapping to 4 points in projected image space are enough, with the constraint that the points form a solid object and a quadr... |
This question arise from Lemma 4.14 of Kwon, H. D., & Palczewski, J. (2022). Exit game with private information. arXiv preprint arXiv:2210.01610.
Let us consider the following optimal stopping problem $u(x)=\sup_\tau\mathbb E_x[\int^\tau_0e^{-rs}D(X_s)ds+e^{-r}\theta]-\theta$ where $X_t$ is an Ito diffusion following ... |
### Question:
Let a r.v. $0\leq X \leq 1$. Prove that $V(X) \leq \frac{1}{4}$. When does we have an equality?
### Answer:
1- $0\leq X \leq 1 \Rightarrow X^2 \leq X \Rightarrow \int X(\omega)^2 dP (\omega) = E[X^2] \leq E[X] = \int X(\omega) dP (\omega)$.
So $V(X) = E[X^2] - E^2[X] \leq E[X] - E^2[X]... |
Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a quadratic form:
$$\frac{1}{\sqrt 2}\frac{x_1^2 x_2 + x_1 x_2^2 + x_1^2 x_3 - 6x_1 x_2 x_3 + x_2^2 x_3 + x_1 x_3^2 + x_... |
I will try to give you an answer, though with a slightly different approach. There are several things to be agreed upon before entering into this discussion.
First, what is "chaotic dynamics"? Although, to my knowledge, there is no a universally accepted mathematical definition of chaos, we do have ways to "classify... |
I have a little question about of the associativity of the tensor products of two modules.
If `M, N, P` are `A`-modules, we know that `(M \otimes N) \ P \cong M \otimes N \otimes Q`. If i want to prove that, i have to find a bilinear-map from `(M \otimes N) \times P \to M \otimes N \otimes P`, but in Atiyah-Macdona... |
Let $p$ be a negative prime number such that $p \equiv 5\pmod 8$.
Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$.
I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ is trivial.
In this context, $p$ and $2$ are unramified in $\mathbb{Q}(\sqrt{p})/\mathbb{Q}$ because one can ver... |
I have a little question about of the associativity of the tensor products of two modules.
If $M, N, P$ are $A$-modules, we know that $(M \otimes N) \ P \cong M \otimes N \otimes P$ If i want to prove that, i have to find a bilinear-map from $(M \otimes N) \times P \to M \otimes N \otimes P$, but in Atiyah-Macdonal... |
Given a (autonomous) dynamical system, one can prove instability of a point via the Lyapunov method, by simply finding a Lyapunov function that increases in a neighbourhood of the point. This ensures a stronger property than instability. It ensures the existence of a small enough neighbourhood, such that all orbits sta... |
I have a game with five die with six faces (A normal dice). The player rolls all the dice and what I am trying to work out is the probability of rolling exactly two dice with the same number. For example, XXYYZ would not be allowed as it has two pairs. I originally had the following equation:
$$\frac{6 \times 1 \tim... |
I'm reading about the process of gluing manifolds and have come across the following example in my studies:
### Text
[![enter image description here][1]][1]
### My question
I understand that we are considering a collection of manifolds and gluing them together using homeomorphisms. However, I am a bit uncle... |
How is different unbounded and bounded? |
I'm reading about the process of gluing manifolds and have come across the following example in my studies:
### Text
[![enter image description here][1]][1]
### My question
I understand that we are considering a collection of manifolds and gluing them together using homeomorphisms. However, I am a bit uncle... |
### Question:
Let a r.v. $0\leq X \leq 1$. Prove that $V(X) \leq \frac{1}{4}$. When do we have an equality?
### Answer:
1- $0\leq X \leq 1 \Rightarrow X^2 \leq X \Rightarrow \int X(\omega)^2 dP (\omega) = E[X^2] \leq E[X] = \int X(\omega) dP (\omega)$.
So $V(X) = E[X^2] - E^2[X] \leq E[X] - E^2[X] =... |
Let a r.v. $0\leq X \leq 1$. Prove that $V(X) \leq \frac{1}{4}$. When do we have an equality? |
Given an orthonormal matrix $R$ and two diagonal matrices $D_1$ and $D_2$, is there an easy way to obtain the eigenvalues of:
$$ D_2 R D_1 R^T D_2 $$
Edit:
I found
https://math.stackexchange.com/questions/67231/singular-value-decomposition-of-product-of-matrices
and
and
https://math.stackexchange.com/ques... |
Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues
of $ρ_A$ and $|e_i⟩$ corresponding orthonormal eigenvectors. If some eigenvalue appears more
than once then this decomposition is not unique.
Show that, nevertheless, any purification $|\psi_{AB}⟩$ of $ρ_A$ has a Schmidt decomposition ... |
Given an orthonormal matrix $R$ and two diagonal matrices $D_1$ and $D_2$, is there an easy way to obtain the eigenvalues of:
$$ D_2 R D_1 R^T D_2 $$
Edit:
I found
https://math.stackexchange.com/questions/67231/singular-value-decomposition-of-product-of-matrices
and
and
https://math.stackexchange.com/ques... |
What is the demonstration of the "sample" standard deviation being somehow more exact than the classic standard deviation, if x/n ≠ x/(n-1)? |
If we have a $\mathbb{k}$-algebra $R$ over a field $\mathbb{k}$ together with an $R$-bimodule $M$ we can consider the center $Z(M)=\{m\in M \vert \ \ rm=mr \ \ \forall r\in R\}.$ This is again an $R$-bimodule. If we consider $R$ as the regular bimodule, then $Z(R)\cong \operatorname{Hom}_{R-R}(R,R)$ as $\mathbb{k}$-alg... |
Why is the demonstration of the "sample" standard deviation somehow more exact than the classic standard deviation, if $\frac{x}{n} ≠ \frac{x}{n-1}$? |
I am not sure if these formulations of the $m$-equivalence of two structures $\mathfrak{A}$ and $\mathfrak{B}$ are correct.
Two structures $\mathfrak{A}$ and $\mathfrak{B}$ are $m$-equivalent iff
- for every substructure $\mathfrak{a} \subseteq \mathfrak{A}$ with $\left|\mathfrak{a}\right| \le m$ there exists a par... |
Are these restatements of m-equivalence correct? |
$$\sum_{n>0} \frac{\mu(n)}{n^s}$$
Sum from $1$ to infinity of The Möbius function$/n^s$, i.e., Möbius function/Riemann-zeta function?
Sorry, I forgot to mention that the way that I am suppose to tackle this question is with the information that: $L(s,f\ast g) = L(s,f)\ast L(s,g)$.
The problem was that I couldn... |
I am new to mathematical logic so forgive me if this is a bad question.
I understand that the Continuum Hypothesis (CH) is independent of ZFC and therefore there exist models of ZFC in which the CH is false. In such models, by the very definition of CH being false, there must exist a set whose cardinality is strictl... |
Having watched the otherwise splendid [Numberphile video with Edward Frenkel explaining the Langlands program][1], two mysteries remained completely open to me:
Given the equation $y^2 + y = x^3 - x^2$ how would one ever come up in the first place with the infinite product
$$q\ \Pi_n(1-q^n)^2(1-q^{11\ n})^2$$
... |
How to come up with the generating function of an elliptic curve? |
My friend gave me the following series about trigonometric functions:
$$ s(x) := \sin \Biggl( \sum_{n=0}^\infty \frac{(-1)^n \, \zeta(4n+2)}{2n+1} \, x^{4n+2} \Biggr) $$
and
$$ c(x) := \cos \Biggl( \sum_{n=0}^\infty \frac{(-1)^n \, \zeta(4n+2)}{2n+1} \, x^{4n+2} \Biggr) $$
where $\zeta(s)$ is the Riemann ze... |
I have been trying to show the following fact:
$$\epsilon -(1+\frac{1}{1!}+ ... + \frac{1}{n!})<\frac{1}{nn!}$$
My steps:
it seems reasonable to me to try to do this problem by induction, but when I tried it I came up with the following:
inductive hypothesis: satisfies for n.
for the case n+1 we will have ... |
Having watched the otherwise splendid [Numberphile video with Edward Frenkel explaining the Langlands program][1], two mysteries remained completely open to me:
Given the equation $y^2 + y = x^3 - x^2$ how would one ever come up in the first place with the infinite product
$$q\ \Pi_n(1-q^n)^2(1-q^{11\ n})^2$$
... |
Let $A$ be a $m\times n(m<n)$ matrix, $\mu>0$, I am interested in showing that $$\lim_{\mu\to 0} (A^TA+\mu I)^{-1}A^T= A^T(AA^T)^{-1},$$
where $I$ is the identity matrix. Could you provide insights or a step-by-step proof for this limit expression?
|
How to show this matrix limit? |
I encountered a mathematical problem while studying mathematical analysis.
The problem is as follows:
Let $n\geq 1$ and consider a polynomial $R$ of degree $2n$ in two variables that is only equal to zero at the point $(0,0)$. Additionally, let $P$ and $Q$ be polynomials of degree at most $2n-2$ in two variables.... |
So I had an assignment due yesterday, regarding the probability of the attackers second highest dice roll, being strictly bigger than the defenders second highest dice roll, in the game Risk.
In the game, the attacker has three dices and the defender has two. The highest dice of the attacker is compared with the hi... |
I have been trying to show the following fact:
$$e -(1+\frac{1}{1!}+ ... + \frac{1}{n!})<\frac{1}{nn!}$$
My steps:
it seems reasonable to me to try to do this problem by induction, but when I tried it I came up with the following:
inductive hypothesis: satisfies for n.
for the case n+1 we will have to
$... |
Let $A$ be a $m\times n(m<n)$ matrix, $\mu>0$. $A$ is a full rank matrix. I am interested in showing that $$\lim_{\mu\to 0} (A^TA+\mu I)^{-1}A^T= A^T(AA^T)^{-1},$$
where $I$ is the identity matrix. Could you provide insights or a step-by-step proof for this limit expression?
|
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}_m$, $\ $ $\Delta^{\frac{m-1}{2}}f$ is a monogenic function, namely
$$
\Delta^{\frac{m-1}{2}}f\in\ker{\overline{\part... |
I haven't really understood parts of the proof given for $\textrm{Problem 6.2}$ of Steele's Cauchy-Schwarz Master Class.
>Suppose that $f : [a, b] → \mathbb{R} $ is strictly convex and show that if
>
>\begin{equation} f\left(\sum^{n}_{j=1}p_jx_j\right)=\sum_{j=1}^{n}p_jf(x_j)\end{equation}
where the positive r... |
I'm trying to determine the eigenvalues for the following matrix:
$$
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & -i & 0 & 0 \\
0 & 0 & i & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix}
$$
Where the first row and column share a single non zero... |
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 tails appears.
- The first time heads appears, the game ends and the play... |
Why Doesn't the St Petersburg Paradox Happen All the Time? |
I am trying to learn more about the Inverse Gaussian Distribution and its applications (e.g. First Passage Time).
First, define a **Brownian Motion with Drift**: If $X(t) = \mu t + W(t)$ represents a Brownian motion with drift $\mu$, where $W(t)$ is a standard Brownian motion (i.e. $W(t) \sim N(0,t)$). Here, the in... |
I usually meet with my friend Thomas to discuss engineering mathematics on Saturdays. We agreed to be taking turns when buying lunch (I buy this Saturday he buys the next saturday). On two Saturdays when he was supposed to buy he did not have the money so I bought instead. He was now owing me two meals. But something I... |
I need to show that $Ax\leq b$ for $A=\begin{pmatrix} 1 & -1\\ 1 & 1\end{pmatrix}$ and $b=(0, 0)^T$ is not total dual integral. So I have calculated that $Ax\leq b$ holds only when $x_1\leq 0$ and $x_1\leq x_2\leq -x_1$. I think we have an optimal solution for $\max c^Tx$ when $c=(1,1)^T$, and the optimal value is 0. S... |
\sum_{j=0}^\infty(-1)^j\frac{1 -e^{(aj^2 + b)M}}{1-e^{aj^2 +b}}? |
$\sum_{j=0}^\infty(-1)^j\frac{1 -e^{(aj^2 + b)M}}{1-e^{aj^2 +b}}$? |
It's well-known that if a continuous function taking values in a Hausdorff space is uniquely determined by its specification on a dense subset of the domain. Now, I contemplate on the necessity of Hausdorff-ness in this result:
It's clear that the result no longer holds if the codomain is just $T_0$, as is demonstra... |
Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus?
These can be in the form of resources (preferably accessible online) as well as directly writing out these results as answers.
If possible, it would be helpful to link to proofs of the results. These can be external p... |
Suppose that you initially have a container which has the shape of a paraboloid with equation
$ z = a x^2 + b y^2 $ where $ 0 \le z \le h $
Now you tilt this paraboloid by rotating it about any point (let's say the origin) about a horizontal axis whose unit vector is $v = (\cos \varphi, \sin \varphi, 0)$. Let t... |
The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. [this][1] MathSE post):
\begin{align}
&\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y)
\quad \Rightarrow \quad
\phi(x) = \frac{-1}{4 \pi |x-y|} \qquad
\\
&\mathbb{R}^2: \qquad \nabla^2 \phi(x) = \delta^2(x-y)
\quad ... |
In our bocce division, we have 14 teams (including us). Everyone plays each other 1x during the 13 week season (13 games for all). At the end of the 13 weeks, the top 6 teams advance to the playoffs. What is the minimum wins a team must have to land in the top 6? At this time no one is undefeated so the very best re... |
Bocce league: How many wins to advance to playoffs? |
I am trying to learn more about the Inverse Gaussian Distribution and its applications (e.g. First Passage Time).
First, define a **Brownian Motion with Drift**: If $X(t) = \mu t + W(t)$ represents a Brownian motion with drift $\mu$, where $W(t)$ is a standard Brownian motion (i.e. $W(t) \sim N(0,t)$). Here, the in... |
In our bocce division, we have $14$ teams (including us). Everyone plays each other $1$ time during the $13$ week season ($13$ games for all). At the end of the $13$ weeks, the top $6$ teams advance to the playoffs. What is the minimum wins a team must have to land in the top $6$? At this time no one is undefeated so... |
Suppose that you initially have a container which has the shape of a paraboloid with equation
$ z = a x^2 + b y^2 $ where $ 0 \le z \le h $
Now you tilt this paraboloid by rotating it about any point (let's say the origin) about a horizontal axis whose unit vector is $v = (\cos \varphi, \sin \varphi, 0)$. Let t... |
### Question:
Let $X$ a r.v. verifying $P(X=x)=0, \forall x \in \mathbb{R}$ and $F_X$ its repartition function. Prove that $F_X(X)$ follows an uniform distribution law $]0;1[$.
### Answer:
1- We write $0 \leq F_X(x)=P(X < x ) \leq 1$ which is a continuous and a not - decreasing function.
*$F_X(x)=P(X ... |
Generative artificial intelligence (a.k.a. GPT, LLM, generative AI, genAI) tools may not be used to generate content for $SiteName. The content you provide must either be your own original work, or your summary of the [properly referenced](/help/referencing) work of others. If your content is determined to have been w... |
What is this site’s policy on content generated by generative artificial intelligence tools? |
Generative artificial intelligence (a.k.a. GPT, LLM, generative AI, genAI) tools may not be used to generate content for $SiteName. The content you provide must either be your own original work, or your summary of the [properly referenced](/help/referencing) work of others. If your content is determined to have been w... |
**Determine which of two functions are Riemann integrable on $[0, 1]$:**
$$ f(x) =
\begin{cases}
\ln(x), &\text{$x \in (0,1]$}\\
42, &\text{x = 0}
\end{cases}
$$
$$ g(x) =
\begin{cases}
\left(1 + \frac{1}{n} \right)^n, &\text{$x \in \left( \frac{1}{2^n}, \frac{1}{2^{n-1}} \right] $}\\
e, &\tex... |
I was curious about how to define a linear function in a vector space whose basis is not countable. In this case, to have a concrete example, I though about the sequence space in $\mathbb{R}$, that is,
$$\mathbb{R}^\mathbb{N}=\prod^{\infty}\mathbb{R}$$
Imagine that we wanted to define a linear function $A_{i,j}$ ... |
You can also use the multilinearity of $f$. Can you see that
$$f(x_1,\ldots,x_n) = \sum_{j_1,\ldots,j_n=1}^n \left(\prod_{k=1}^n x_{k, j_k}\right)f(e_{j_1}, \ldots, e_{j_n})?$$
Now what happens if two of $j_k$ are the same? What happens if they are all distinct? Can you finish the proof? |
So right now i'm reading this paper: Distance-based detection of out-of-distribution silent failures for Covid-19 lung lesion segmentation, available here: https://arxiv.org/abs/2208.03217
In brief, they use a latent representation of the input to determine whether or not it is out of distribution using the mahalano... |
When to use chi square law for confidence intervals with mahalanobis distance? |
This is my starting equation for complex number, $ \hat{k} $
$$ \hat{k}^2 = \mu \epsilon \omega - \mu \sigma \omega^2 i $$
I assumed $ \hat{k} = \alpha + \beta i $ where $ \alpha , \beta \in \mathbb{R} $
I then derived the following:
$$ \alpha^4 - \mu \epsilon \omega \alpha^2 - \frac{1}{4} \mu^2 \sigma^2 \... |
The presentation link is broken, but I am still able to independently verify quite a bit.
Sanity checks:
- Dimensional analysis checks out
- Extreme/particular cases check out (when $p_1=1$ or all $p_n$ are equal give expected correct results)
Experimental data from my own hand-written code shows that for $... |
So right now i'm reading this paper: Distance-based detection of out-of-distribution silent failures for Covid-19 lung lesion segmentation, available here: https://arxiv.org/abs/2208.03217
In brief, they use a latent representation of the input to determine whether or not it is out of distribution using the mahalano... |
Let $H$ be a commutative and cocommutative Hopf algebra over an algebraically closed field $k$. I've read that if $H$ is _grouplike_ in the sense that it has no nonzero primitive elements, then $H$ is isomorphic to the group ring on its grouplike elements,but no reference or proof was given.
What is a proof of this ... |
Okay, I'm not very comfortable with either complex geometry or minimal surfaces, so bear with me. I've encountered the following theorem, but I can't find a proof:
Suppose $V\subset \mathbb{C}^n$ is a properly embedded complex submanifold, say of complex-dimension $k$. Here, $\mathbb{C}^n$ is equipped with the usual... |
Why the local mass-ratio in a complex submanifold of the Euclidean space is nonzero? |
Is $\int_0^{\pi}\log \sin \theta d\theta$ not well-defined?
On the third edition of Ahlfors' Complex Analysis, page 160 it states: As a final example we compute the special integral
\begin{equation*}
\int_0^{\pi}\log \sin \theta d\theta.
\end{equation*}
...
The same proof applies near the vertex $\pi$, and w... |
Is $\int_0^{\pi}\log \sin \theta d\theta$ not well-defined? |
I just learnt character theory and I am reading the paper https://academic.oup.com/jlms/article-abstract/66/3/623/811347 which is a quite beautiful paper. Section 4.3, on p. 631 contains an inequality which confuses me for a while: \
Let $P=\{\lambda \vdash n, \lambda \neq (1^n), (n)\}$ be the set of partition of the ... |
Let $F$ be a multiplicative generalized cohomology theory ($\tilde F$ will denote the reduced theory as usual).
Let $\sigma^0 \in F^0(pt) \simeq \tilde{F}^0(S^0)$, and let for any $n\in \mathbb{Z}$, $\sigma^n \in \tilde{F}^n(S^n)$ be the element associated to $\sigma^0$ under the suspension isomorphism.
In the... |
To reiterate multiple comments, the statement "Every bounded function on $[a, b]$ is Riemann Integrable on $[a, b]$" is false.
In response to the clarification, "I wanted to ask if the interval $[a, b]$ in the above question is the domain of $f$ or just any closed interval different from the domain of $f$?":
If w... |
So I have the ODE given by $-u''(x)=f(x;\lambda)$ for $x\in(0,1)$ with the boundary condition $u(0)=u(1)$ and the function is defined as $f(x;\lambda) = -\lambda e^{-\lambda x}$ for $\lambda>0$. This has a unique solution in $C^2_0[0,1]=\{g\in C^2[0,1] \mid g(0)=g(1)=0\}$. How can I show that $u(x)\leq 0$ for all $x\in... |
I do not claim to know how WolframAlpha works behind the scenes. This is merely an attempt to walk through its reasoning for the integral over the negative part of $[-1,1]$.
WA chooses branches of the [square root](https://www.wolframalpha.com/input?i=simplify+sqrt%28x%29+if+-1%3Cx%3C0) and [logarithm](https://www.w... |
Problem: Let $X$ be a metric space and $F$ a closed subset of $X$. Assume $f\in L^p(X)$ such that $f|_{F}=0$. Then show that there for every $\epsilon>0$, there exists a continuous function $g\in L^p(X)$ such that $g|_{F}=0$ and such that $||f-g||_{L^p}<\epsilon$
We may use standard facts, otherwise if false, I must... |
I would like to prove the following statement:
>Let $g:\mathbb{R}^n\to\mathbb{R}^n$ be contraction mapping. If function $f:\mathbb{R}\to\mathbb{R}^n$ satisfies:
$$f(t)+f'(t)=g(f(t))$$
then the limit $\lim_{t\to\infty}f(t)$ exists.
Below I write a sketch of my solution to this problem. My question is: is there som... |
Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a quadratic form:
$$\frac{1}{\sqrt 2}\frac{x_1 x_2^2 + x_1 x_3^2 + x_2 x_1^2 +x_2 x_3^2 + x_3 x_1^2 + x_3 x_2^2 - 6x_1 x... |
Can a oblique antiprism be constructed?
Intuitively, it would seem oblique antiprism exist: Take any right antiprism and translate one of the parallel faces within it's plane.
I'm baffled, though, because Google shows countless models and pictures of (right) antiprisms, but [not a single oblique antiprism](https:... |
Can a oblique antiprism be constructed? |
Minor, pedantic nitpick: a point is zero dimensional. The objects you are describing are points in an $n$-dimensional space. The points themselves are not $n$-dimensional.
With respect to notation, it sounds like you are writing notes or completing homework assignments. In that kind of context the notation can b... |
### Question:
Given a r.v. $Y \sim U]0;1[$ and $ \mu $ a probability measure on $B( \mathbb{R})$. Build a new r.v. $X$ of law $ \mu $ by composing $Y$ with a well chose function.
### Answer:
1- We define $0<t = F_X(x)=P(X<x)= \mu (] -\infty; x]) < 1 $
2- Our objective is to get $ F_X(F^{-1}_X(t)) = t... |
Show that $2+ \sqrt{-5}$ is not prime in $\mathbb{Z}\sqrt{-5}$.
Here is my approach:
If $2+ \sqrt{-5}$ is a prime in $\mathbb{Z}\sqrt{-5}$, then for all $r,s \in \mathbb{Z}\sqrt{-5}$ we would have:
$2+ \sqrt{-5} \mid rs \Rightarrow 2+ \sqrt{-5} \mid r$ or $2+ \sqrt{-5} \mid s$.
We can consider the function $N... |
Can you please tell me if my approach is correct or if there are other solutions :
**The statement of the problem** : Let $f: \mathbb R \rightarrow \mathbb R $ with the 2 properties :
1) $f(x+y) + f(x-y) = 2f(x)f(y) , \forall x , y \in \mathbb R$
2) There exists a "the smallest" strictly positive number **a*... |
Problem: Let $X$ be a metric space and let the borel measure be inner and outer regular and $F$ a closed subset of $X$. Assume $f\in L^p(X)$ such that $f|_{F}=0$. Then show that there for every $\epsilon>0$, there exists a continuous function $g\in L^p(X)$ such that $g|_{F}=0$ and such that $||f-g||_{L^p}<\epsilon$
... |
No.
You know that if $a$ divides $b$ in $\mathbb{Z}[\sqrt{-5}]$, then $N(a)$ divides $N(b)$ in $\mathbb{Z}$.
However:
1. You don't have an $N(a)$ and an $N(b)$. You have $N(a)$, $N(2+\sqrt{-5})$, and an integer $18$, and note that $N(a)$ divides $18$. So? You have not even shown an element $b\in\mathbb{Z}[\s... |
Does the set A have a maximum element?
${\displaystyle A =\{q\in \mathbb {Q} |q<a\}} $
Thoughts: I don't think it does. We can suppose that it has a maximum element, q. Then q is a real number and a is also a real number (and they are not equal), so infinite rational numbers exist between them, let's say one of t... |
Does the set ${\displaystyle A =\{q\in \mathbb {Q} |q<a\}} $ have a maximum element? |
So I have the ODE given by $-u''(x)=f(x;\lambda)$ for $x\in(0,1)$ with the boundary condition $u(0)=u(1)$ and the function is defined as $f(x;\lambda) = -\lambda e^{-\lambda x}$ for $\lambda>0$. This has a unique solution in $C^2_0[0,1]=\{g\in C^2[0,1] \mid g(0)=g(1)=0\}$. How can I show (using Green's function) that $... |
The original question was: Differentiate the given function with respect to x:
$$
f(x) = \dfrac{ax^2+bx+c}{\sqrt{x}}
$$
I simplified till
$$
\dfrac{\sqrt{x}\left(2ax+b\right)-\frac{ax^2+bx+c}{2\sqrt{x}}}{x}
$$
then the next step simplified it into
$$
\dfrac{2ax+b}{\sqrt{x}}-\dfrac{ax^2+bx+c}{2x^\frac{3}{2... |
I was wondering if I could get some input on my strategy for inverting a set of multivariate polynomial equations for the independent variables. My functions are as follows:
$$g_1(x,y) = a_0 + a_{1}x+a_{2}y + a_{3}xy + a_{4}y^2 \\
g_2(x,y) = b_0 + b_{1}x+b_{2}y + b_{3}xy + b_{4}y^2 $$
Now, what I want to get are ... |
Consider a number in its decimal representation that begins with an **odd number of consecutive digits of 4**, followed by a **single digit of 1**. An example of such a number would be 41, 4441, or any similar pattern extending with 4s. My question is: Can such a number ever be a **perfect square**?
To clarify, the ... |
Can a Number Comprised of Repeated 4s Followed by a Single 1 be a Square Number? |
[![enter image description here][1]][1]
this is from Munkres Topology and I'm confused about the extension of $j$ from $A$ to $A\cup C$ by the preceding lemma which is the below:
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/6EcpC.png
[2]: https://i.stack.imgur.com/b4GN3... |
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