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show that the set of 2x2 matix of the form
$$\begin{pmatrix}
a & b \\
c & 0 \\
\end{pmatrix}
$$
is a vector space with the standard operations
can i incorporate the idea that this certain matrix can be connected (a subspace) to the $M_{22}$ matrix? or not since it has not been stated in the problem and pro... |
I’m learning about Borel sets and I’m struggling to understand the following sentence.
> The intersection of every sequence of open subsets of $\mathbb{R}$ is a Borel set. However, the set of all such intersections is not the set of Borel sets (because it is not closed under countable unions).
Why isn’t the set o... |
Why is intersection of open sets not closed under countable unions? |
show that the set of 2x2 matix of the form
$$\begin{matrix}
a & b \\
c & 0 \\
\end{matrix}
$$
is a vector space with the standard operations
can i incorporate the idea that this certain matrix can be connected (a subspace) to the M_22 with real entries matrix, which has been proven in our class to be a vec... |
To the best of our knowledge,in two-dimensional [Complex space][1], there are N vectors in arbitrary directions. Using "Method One," by fixing one vector among the N vectors and independently rotating the remaining N-1 vectors, it is possible to maximize the magnitude of the composite vector obtained by rotating and su... |
Assume $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (let's denote this as $\mathcal{C}$) is a tensor-category in the sense of Deligne/J.S. Milne (https://www.jmilne.org/math/xnotes/tc2018.pdf).
Assume furthmore that for each object $X$ in $\mathcal{C}$, we have an adjunction $- \otimes X \dashv \underline{\text{Hom}... |
I saw [this question][1] and I was able to prove that $f(x)= e^x \left(\frac{x+1}{x^2+1}\right)^x \frac{(x+1)}{e^{2\arctan(x)}} $ but a new question came to my mind : Does the sequence $f_n(x):=\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$ converge uniformly to $f(x... |
Let $B$ be a ball of radius $R$ centered at $0$. I am considering the integral
$$\int_B \int_B \frac{1}{|x-y|^k}dxdy,$$ where $k$ is some real number (which can make the integral infinity). To calculate it explicitly, I want to do the obvious change of variable $r=|x-y|$. Here's my attempt:
If we only consider the... |
How to do change of variable to polar coordinates in higher dimensions double integral? |
Well of course $0^\sharp$ cannot be added by forcing. What I mean is the following. Suppose $0^\sharp$ exists. Let $\mathbb{P}=\mathrm{Add}(\omega,1)$ be the Cohen poset, and $L^\mathbb{P}$ be the class of $\mathbb{P}$-names in $L$. Let $W=\{\sigma[0^\sharp]:\sigma\in L^\mathbb{P}\}$ where $\sigma[0^\sharp]$ is the int... |
Whic constant makes the serie converges to 1? |
**1.** It is true that Glasser's Master Theorem (GMT) is not directly applicable in this case. However, the reason is not because the integrand is an even function. In fact, **GMT applies to any integrable or non-negative functions on $\mathbb{R}$**, as we can see from the following far-reaching generalization (which a... |
There is not necessarily any "middle" of a four-torus, nor is there necessarily any "hole" in any place you might call the "middle".
When we try to visualize space in cases where the space is not Euclidean, there's a strong inclination to try to do this by embedding the space in a Euclidean space of higher dimension... |
$$\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$$
Here, $\lfloor x \rfloor$ represents the floor or celling of $x$.
I tried using a graphing calculator (desmos) to plot the function $f(x) = \frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}$
from the graph (see image [here][1]), it is cl... |
How to compute the following limit? $\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$ |
**1.** It is true that Glasser's Master Theorem (GMT) is not directly applicable in this case. However, the reason is not because the integrand is an even function. In fact, **GMT applies to any integrable or non-negative functions on $\mathbb{R}$**, as we can see from the following far-reaching generalization (which a... |
Which constant makes the series converge to 1? |
Let $\mathcal{H},\mathcal{K}$ be Hilbert spaces ($\cong \mathbb{C}^2$ if necessary) and let $h_1,h_2$ denote semi-positive $\ge 0$ operators on $\mathcal{H}\otimes \mathcal{K}$. If $h_1,h_2$ are entangled, i.e., not a trivial product operator, then is this also true for $h_1 +h_2$? (not including the trivial case where... |
**1.** It is true that Glasser's Master Theorem (GMT) is not directly applicable in this case. However, the reason is not because the integrand is an even function. In fact, **GMT applies to any integrable or non-negative functions on $\mathbb{R}$**, as we can see from the following far-reaching generalization (which a... |
We know that to prove vectors $v_1,v_2,v_3$ linearly dependent we must find scalers $x_1,x_2,x_3$ not all equal to $0$ such that $x_1v_1+x_2v_2+x_3v_3=0$. But the doubt I have is on the other alternative condition,which says if one of them can be expressed as a linear combination of other $2$,then they are linearly dep... |
**1.** It is true that Glasser's Master Theorem (GMT) is not directly applicable in this case. However, the reason is not because the integrand is an even function. In fact, **GMT applies to any integrable or non-negative functions on $\mathbb{R}$**, as we can see from the following far-reaching generalization (which a... |
> Which of the following represents zeros of $r=2-4\sin 2 \theta $? (Multiple choice)
I don't understand this question. I figured out that $\theta$ is $30$, and I determined that the answer is $A$ (i.e. $\pi/6$ and $5\pi/6$). However, the answer sheet tells me that $D$ (i.e. $\pi/12$, $5\pi/12$, $13\pi/12$, $17\pi/1... |
I was reading the book *Introduction to the topological groups* by Taqdir Husain, he gives a proof for this theorem, but I had a problem understanding some part. The proof goes like this
It remains to show that the inversion mapping $x\to x^{-1}$ is continuous. So let $U$ be a open neighborhood of $e$. We wish to sh... |
I am implementing a finite difference scheme for the following PDE:
$$
u_t=u_{xx}+f(u)
$$
with the nonflux boundary condition and a given initial condition. The scheme is backward in time and central in space(which should be unconditionally stable) as
$$
\frac{u^{k+1}_i-u^k_i}{\Delta t}=\frac{u^{k+1}_{i-1}-2u^{k+... |
I don't understand this AP Precalculus question.
> Which of the following represents zeros of $r=2-4\sin 2 \theta $?
>
> A. $\;\dfrac\pi6, \dfrac{5\pi}{6}$
>
> B. $\;0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2}$
>
> C. $\;\dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}$
>
> D. $\;\dfrac{\pi}{1... |
To the best of our knowledge,in two-dimensional [Complex space][1], there are N vectors in arbitrary directions. Using "Method One," by fixing one vector among the N vectors and independently rotating the remaining N-1 vectors, it is possible to maximize the magnitude of the composite vector obtained by rotating and su... |
> Find the constant $c$ such that this series converges to $1$?
$$\sum^{\infty}_{n=1}\dfrac{c}{n!}$$
I tried with $1$, but I don't know if it converges exactly to $1$ cause I noticed that it might converges to $e$.
And I also found $c=\dfrac{1}{e-1}$, but I doubt it converges to $1$.
Thanks in advance. |
Which value of $c$ makes $\sum^{\infty}_{n=1}\dfrac{c}{n!}$ converge to 1? |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1} -P_n)<70000000$ where $P_n$ is the $n$the prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)=lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =(n+1)\log(n+1)-n\log n>\log n$ for sufficiently large $n$. Then how can it be 7... |
Consider a set $X$ of size $n$, and a size-$n$ family of sets $\mathcal{S}$. The sets in $\mathcal{S}$ have average size $r$, and their intersections are of size at most $k$. I'm trying to show that the *intersection graph* of $\mathcal{S}$ satisfies the inequality
$$|E|≥\frac{n}{k}\binom{r}{2}.$$
The intersection ... |
Given an $n$-element family $\mathcal{S}$ of average size $r$, is $\sum |S_i \cap S_j|\geq n\binom{r}{2}$? |
I have a nonlinear operator $\mathbf{A}(\theta): X \rightarrow Y$, where $X$, $Y$ are Banach space with $||\cdot ||_{X}$ and $||\cdot ||_{Y}$. $\theta \in \mathbb{R}^q$ is the parameter of $\mathbf{A}$. Fixed $\theta$, we can define the operator norm of $\mathbf{A}(\theta)$ as $||\mathbf{A}(\theta)||_{op} = max\{sup_{x... |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1} -P_n)<70000000$ where $P_n$ is the $n$ prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)=lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =(n+1)\log(n+1)-n\log n>\log n$ for sufficiently large $n$. Then how can it be 7000... |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1}-P_n)<70000000$, where $P_n$ is the $n$ prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)=lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =(n+1)\log(n+1)-n\log n>\log n$ for sufficiently large $n$. Then how can it be 7000... |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1}-P_n)<70000000$, where $P_n$ is the $n$ prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)=lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =(n+1)\log(n+1)-n\log n>\log n$ for sufficiently large $n$. Then how can it be less... |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1}-P_n)<70000000$, where $P_n$ is the $n$the prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)=lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =(n+1)\log(n+1)-n\log n>\log n$ for sufficiently large $n$. Then how can it be l... |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1}-P_n)<70000000$, where $P_n$ is the $n$th prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)=lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =(n+1)\log(n+1)-n\log n>\log n$ for sufficiently large $n$. Then how can it be le... |
Let $H$ one Hermitian matrix acts on $\mathbb{C}^d\otimes \mathbb{C}^d\otimes\mathbb{C}^d$. If I know this Hermitian matrix is invariant under the permutation of subsystems, what I can say about the structure of this Hermitian matrix $H$? For example, all linear combinations of tensor product of Hermitian matrices will... |
I have a list of numbers in radix $10$. My objective is to concatenate these numbers together to form one single number, and then convert it to radix $20$.
Unfortunately, the list is pretty extensive. The concatenated number is over $1$ million digits long, so converting it to radix $20$ afterwards is very inefficie... |
Why is this function almost Lipshchitz? |
The First method is feasible. The second method however, According to me, has some issues with it.
Issue 1:
Say you have a set $(S)$ of vectors in the Argand plane,
$$S=\left\{A_{k}e^{i\theta_{k}},\ \forall\ k∊\left\{1,2,3...n\right\}\right\}$$
And denoting
$$C_{all}=\sum_{k=1}^{n}A_{k}e^{i\theta_{k}}$$
Your ... |
We know the following proposition is true. The proof together with the specification of the eigenvectors of $\operatorname{ad}x$ is [here][1].
Let $x\in \operatorname{gl}(n,F)$ be diagonalizable with $n$ eigenvalues $a_1,\ldots,a_n$ in $F$. The eigenvalues of $\text{ad }x$, where $\operatorname{ad}x(y):=[x,y]=xy-yx$... |
What are the eigenvalues of $\operatorname{ad}x$ for non-diagonalizable $x$? |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1}-P_n)<70000000$, where $P_n$ is the $n$th prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)>lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =(n+1)\log(n+1)-n\log n>\log n$ for sufficiently large $n$. Then how can it be le... |
I saw a paper by Zhang Yitang that $lim_{n\to\infty}\inf(P_{n+1}-P_n)<70000000$, where $P_n$ is the $n$th prime.
But by prime number theorem, $lim_{n\to\infty}\inf(P_{n+1} -P_n)>lim_{n\to\infty}\inf(P_{n+1}) -lim_{n\to\infty}\sup(P_n) =lim_{n\to\infty}((n+1)\log(n+1)-n\log n)>lim_{n\to\infty}\log n$ for sufficientl... |
The First method is feasible. The second method however, According to me, has some issues with it.
Issue 1:
Say you have a set $(S)$ of vectors in the Argand plane,
$$S=\left\{A_{k}e^{i\theta_{k}},\ \forall\ k∊\left\{1,2,3...n\right\}\right\}$$
And denoting
$$C_{all}=\sum_{k=1}^{n}A_{k}e^{i\theta_{k}}$$
Your ... |
> Which of the following represents zeros of $r=2-4\sin 2 \theta $? (Multiple choice)
I don't understand this question. I figured out that $\theta$ is $30$, and I determined that the answer is
$A$ (i.e. $\pi/6$ and $5\pi/6$).
However, the answer sheet tells me that
$D$ (i.e. $\pi/12$, $5\pi/12$, $13\pi/12$,... |
What are the eigenvalues and eigenvectors of $\operatorname{ad}x$ for non-diagonalizable $x$? |
Consider a set $X$ of size $n$, and a size-$n$ family of sets $\mathcal{S}$. The sets in $\mathcal{S}$ have average size $r$, and their intersections are of size at most $k$. I'm trying to show that the *intersection graph* of $\mathcal{S}$ satisfies the inequality
$$|E|≥\frac{n}{k}\binom{r}{2}.$$
The intersection ... |
I saw a paper by Zhang Yitang that $\liminf_{n\to \infty}(P_{n+1}-P_n)<70000000$, where $P_n$ is the $n$th prime.
But by prime number theorem, $\liminf_{n\to \infty}(P_{n+1} -P_n)>\liminf_{n\to \infty}(P_{n+1}) -\limsup_{n\to \infty}(P_n) =\lim_{n\to\infty}((n+1)\log(n+1)-n\log n)>\lim_{n\to\infty}\log n$ for suffi... |
Consider a set $X$ of size $n$, and a size-$n$ family of sets $\mathcal{S}$. The sets in $\mathcal{S}$ have average size $r$, and their intersections are of size at most $k$. I'm trying to show that the *intersection graph* of $\mathcal{S}$ satisfies the inequality
$$|E|≥\frac{n}{k}\binom{r}{2}.$$
The intersection ... |
I’m reading the proof for the following lemma regarding the additivity of the Lebesgue outer measure:
> Suppose $A$ and $G$ are disjoint subsets of $\mathbb{R}$ and $G$ is
> open. Then $$|A∪G| = |A|+|G|.$$
>
> Proof: We can assume that $|G| < ∞$ because otherwise both $|A ∪ G|$
> and $|A| + |G| $ equal $∞$.
> ... |
In [this](https://math.stackexchange.com/questions/4378258/the-determinant-of-a-cartan-matrix-is-positive/4380678#4380678) answer, it is claimed that the determinant of a Cartan matrix is same as its "index" of the root lattice in the weight lattice.
Firstly, what is the equation which describes the definition of t... |
What is the exact definition of the "index" which matches the determinant of the Cartan matrix and how to prove the equivalence? |
I am researching on **decentralised storage network**. Where each **nodes** in the network will be connected to `d` other nodes. When any node shares a file it is propagated to all other nodes. I'm assuming we have a connected network. So, the shared data will reach all the nodes. But now there are some **malicious nod... |
Your intuition is correct. You only need to prove $\sum |S_i \cap S_j| \geq {r \choose 2}$. I'll give you a *hint* as you requested.
First of all, the correct summation you want is $\sum_{i<j}|S_i \cap S_j|$ to ensure no double counting of the edges in the intersection graph.
For any element $x \in X$, define $... |
Your intuition is correct. You only need to prove $\sum |S_i \cap S_j| \geq {r \choose 2}$. I'll give you a *hint* as you requested.
First of all, the correct summation you want is $\sum_{i<j}|S_i \cap S_j|$ to ensure no double counting of the edges in the intersection graph.
For any element $x \in X$, define $... |
Say that $\{X_n\}$ are none-negative random variables converging almost surely to zero, and $Y_n$ are a sequence of i.i.d random variable, non-negative, with finite $r$-moment for any $r>0$, and whose distribution does not depend on $n$.
Then $X_nY_n\rightarrow 0$ in probability (using Markov inequality), but is it tr... |
https://math.stackexchange.com/questions/2541155/bounded-variation-of-frac1f-when-inff0-f-bounded-variation
Let $f: \mathbb{R}\to \mathbb{R}$ be bounded variation on $[a,b]$ with $f(x)\neq 0$ for all $x$. Show that $1/f$ is bounded variation if and only if $\inf_x |f(x)|>0$.
---
For one direction, assume that ... |
How to prove that in another direction $\inf_x |f(x)|>0$? |
How active is the current research scene on the application of algebraic and differential topology to robotics? I have spent a decent portion of time researching and reading introductory papers, but have struggled to determine if this field is very active or not. I've come to understand that algebraic topology as a fie... |
I'm approaching localization of rings using multiplicatively closed sets, and the obvious case of when we take the complementary of a prime ideal of a ring, that is always multiplicatively closed.
>The question at hand asks to prove that the ring $k[x,y]/(xy^2,x^2y)$ localized at $(x)$ is (or isn't) isomorphic to th... |
Let $U$ be a bounded, connected, open subset of $\mathbb{R}^n$. Assume $1 \le p \le \infty$ .Let $f_n \to f$ be a convergent sequence in $L^p(U)$.
My question is, then,
**Q.1.** $ \lim_{n\rightarrow\infty} \left\Vert f_n \right\Vert_p =\left\Vert f\right\Vert_p<\infty $?
I found assoicated post : ht... |
In the textbook "Kac-Moody Groups, their Flag Varieties and Representation Theory" by Shrawan Kumar, he defines the minimal Kac-Moody groups, $\mathcal{G}_{min}$, as the group generated by a Torus, $T$, and one parameter groups, $\mathcal{U}_{\alpha}$, corresponding to real root spaces of the Lie algebra.
Later, he d... |
Consider four positive numbers a, b, c, d and addition modulo n.
I would like to determine the minimum number of constraints needed to express that:
1) $a + b + c = d$
2) $a + b + c$ does not wrap around the modulo n
I think I need at least 5 constraints, for instance:
1) $a \le a + b$
2) $b \le a + b$
3) ... |
Let $U$ be a bounded, connected, open subset of $\mathbb{R}^n$. Assume $1 \le p \le \infty$ .Let $f_n \to f$ be a convergent sequence in $L^p(U)$.
My question is, then,
**Q.1.** $ \lim_{n\rightarrow\infty} \left\Vert f_n \right\Vert_p =\left\Vert f\right\Vert_p<\infty $?
I found assoicated post : ht... |
I am a beginner in Koopman operators，and I want to apply it to the robotics. But I think the Koopman operator approximated by EDMD is more like a fitting tool. Because we always choose some dictionary functions $\Phi=\left \{ \phi_1(x),\phi_2(x),\dots,\phi_n(x) \right \} $, and the approximation of koopman operator i... |
I was wondering if the Kraft-McMillan inequality continues to hold for codes with infinite alphabets. The reason I'm confused about this is because I was thinking about a code from $\{0,1\}^*$ to $\{0,1\}^*$ (all strings consisting of 0's and 1's of all possible lengths), which simply maps each string to itself. This i... |
Consider four positive numbers a, b, c, d and addition modulo N.
I would like to determine the minimum number of constraints needed to express that:
1) $a + b + c = d$
2) $a + b + c$ does not wrap around the modulo N
I think I need at least 5 constraints, for instance:
1) $a \le a + b$
2) $b \le a + b$
3) ... |
Let $f:X\to Y$ be a projective morphism of noetherian schemes and let $\mathcal{F}$ be a coherent sheaf on $X$ which is flat over $Y$. If you read Hartshorne chapter 3 section 12 or [Vakil](https://math.stanford.edu/~vakil/216blog/FOAGfeb2124public.pdf) chapter 25, you can see the development of a wonderful theory whic... |
Given, $f(x,y),g(x,y)$ functions with $f(x,y)\gt g(x,y)\gt 1$, where $(x, y)\in [0,1]\times[0,1]$.
Now, irrespective of continuity of $f(x,y),g(x,y)$, can we say that there is an open ball $B$ about $(1,1)$ such that $\left(g^{-\frac{x+y}{x+xy}}-f^{-\frac{x+y}{y+xy}}\right)-(f-g)\lt0$ for $(x,y)\in B\cap ([0,1]\time... |
Define $$f(n):=\frac{\sum_{j=1}^n j^2\cdot j!}{99}$$
> Is $f(n)$ prime for only finite many positive integers $n\ge 10$ ?
Approach : If we find a prime number $q>11$ with $q\mid f(q-1)$ , then for every $n\ge q$ , we have $q\mid f(n)$ , therefore only finite many prime numbers of the form $f(n)$. I found however ... |
Can we show that $\frac{\sum_{j=1}^n j^2\cdot j!}{99}$ generates only finite many primes? |
I am beginner in group theory and I have learned cyclic groups yet. When I look at the books for definition, it is said that
> Let $G$ be a cyclic group and let a be a generator of $G$ so that $$G= \langle a \rangle = {\{ a^n \mid n \in Z \}}$$
I know that the definition could have been like $G={\{ an \mid n \... |
In many sources one finds a computation of the complex topological K-theory of the spheres $S^n$, but the real theory $KO(S^n)$ is usually not computed. Maybe it can be read off some more advanced stuff, but I am unable to do so. I would like to see a computation of, say $KO(S^2)$.
Can somebody give me a reference? |
What is the real K-theory of the spheres? |
I am researching on **decentralized storage network**, where each **node** in the network will be connected to $d$ other nodes.<br> When any node shares a file, it is propagated to all other nodes (I'm assuming we have a connected network). So, the shared data will reach all the nodes.<br> But suppose now that there ar... |
Is this method for constructing nilpotent matrices effective? |
show that the set of 2x2 matix of the form
$$\begin{pmatrix}
a & b \\
c & 0 \\
\end{pmatrix}
$$
is a vector space with the standard operations
can i incorporate the idea that this certain matrix can be connected (a subspace) to the $M_{2\times 2}$ with real entries matrix, which has been proven in our clas... |
Consider four positive numbers a, b, c, d and addition modulo N.
I would like to determine the minimum number of constraints needed to express that:
1) $a + b + c = d$
2) $a + b + c$ does not wrap around the modulo N
I think I need at least 5 constraints, for instance:
1) $a \le a + b$
2) $b \le a + b$
3) ... |
Wikipedia mentions [two-point Padé approximant](https://en.wikipedia.org/wiki/Padé_approximant#Two-points_Padé_approximant). I don't have access to the reference provided (Yoshiki Ueoka, *Introduction to multipoints summation...*).
I checked also chapter 8 (*The N-Point Padé Approximants*) in the George A. Baker's *... |
How to calculate two-point Padé approximant? |
I started playing with the functional equation $x^n f(1/x) = x$, with $n \in \mathbb N$. I found two solutions: $g(x)=a x^{n/2}$ and $h(x)=b(1+x+...+x^n)=b S_n(x)$.
I also found that a linear combination of $g,h$ is also a solution.
Any other solutions? More general ideas? |
Here's an exercise from Thomas Calculus, I need to do an implicit differentiation:
$$
x^3=\frac{2x-y}{x+3y}
$$
If I [enter this](https://www.wolframalpha.com/input?i=derivative+of+%28x%5E3%29%3D%282x-y%29%2F%28x%2B3y%29) in Wolfram Alpha, I get:
$$
y'(x) =\frac{3 x^4 + 18 x^3 y + 27 x^2 y^2 - 7 y}{7 x}
$$
... |
I was going through my textbook and one of the topics was identifying one-one and many-one functions the first 2 methods described were quite easy, doing so anytically or drawing the graph of the function however the third method i did not quite understand, the method is given as follows,
If for a continuous functio... |
Suppose that $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is $L$-Lipschitz continuous (for some $L>0$) and monotone, i.e.,
\begin{align}
(\forall x,y \in \mathbb{R}^{n}) \quad \| F(x) - F(y) \| \leq L \| x-y \|,
\end{align}
and
\begin{align}
(\forall x,y \in \mathbb{R}^{n}) \quad \langle F(x) - F(y), x-y \rangle \geq 0,
... |
I started playing with the functional equation $x^n f(1/x) = f(x)$, with $n \in \mathbb N$. I found two solutions: $g(x)=a x^{n/2}$ and $h(x)=b(1+x+...+x^n)=b S_n(x)$.
I also found that a linear combination of $g,h$ is also a solution.
Any other solutions? More general ideas? |
This is actually in reference to the question posed here
https://stackoverflow.com/posts/66285948/edit
but is more appropriate as a question to be posed on a non-coding site.
I provide a partial answer, but not the full answer, relaying the full question here as an unresolved question. Perhaps others can take the ... |
I've read in multiple books that a function $F$ is a cumulative distribution function for some random variable $X$ if and only if it satisfies three conditions:
1. $F$ is non-decreasing.
2. $F$ is normalized.
3. $F$ is right-continuous.
but I've never seen an analogous set of conditions for a probability mass fun... |
Is $f$ a probability mass function for a random variable $X$ if and only if it complies with these to properties? |
> Shuffle an ordinary deck of 52 playing cards containing four aces.
> Then turn up the cards from the top until the first ace appears. On
> the average how many cards are required to be turned before producing
> the first ace?
The question is answered [here][1] using symmetry. I have seen other such [questions][... |
I'm doing a course in first-order logic in mathematics and I've stumpled upon a question while preparing for the exam: <br>
Assume you have two formulas (well-formed) $\varphi, \psi$ and you have proved that $\psi \vdash \varphi$, i.e. there is a deduction from $\psi$ to $\varphi$. Now you get asked whether or not it ... |
Begin a deduction from a tautology, also make a tautology? |
I need some help with an equation system modulo some prime $p$. The two equations are of the form
$$
\sum_{i=1}^n x_i = 1 \bmod p
$$
and
$$
\sum_{i=1}^n x_i^2 = 1 \bmod p
$$
The $x_i$ are elements of $\mathbb{Z}_p$.
I want to *either* show that the only solutions are $x_i = 0$ for all $i \neq k$ and $x_k = 1... |
Usually, when we have a complete noncompact manifold, we can look at its universal cover so that many geometric property can be lifted to a simply-connected manifold. If we take a large compact subset of this complete noncompact manifold such that the complement has only one component with compact boundary (probably ... |
Let $a,b\in\mathbb{N}$.\
**Prove that if $1+ab\equiv 0[24]$ then $a+b\equiv 0[24]$**\
My attempts: \
$1+ab\equiv 0[24]\implies ab\equiv -1[24]$\
so : $ab\equiv 23[24]$\
Then there $k\in\mathbb {N}$ such as $ab=24k$ \
But how can I move to $a+b$?.
___
Thanks |
Define $P_\phi := e^{-iP\phi}$, where $P$ is a [Pauli matrix][1] with some overall phase factor and $\phi\in[0,2\pi)$. It is claimed (see Page 1 of [this paper](https://arxiv.org/pdf/1808.02892.pdf)) that if $P'P = -PP'$ i.e. we have two anticommuting Pauli matrices $P, P'$, then
$$P_{\frac{\pi}{4}}P'_{\phi} = (iPP'... |
The Lecturer demonstrate that $S:=\{(x_n)\in l^2:\sum^{\infty}_{n=1}|x_n|^2\leq 1\}\subset l^2$ is closed by showing every convergent sequence $(a_n)$ in $S$ has the limit $a^*\in S$.
Now I have a question :
Is this space itself $l^2:=\{(x)_n\in\mathbb{R}^{\infty}:\sum^{\infty}_{i=1}|x_n|^2<\infty\}$ closed and... |
Is this space $l^2:=\{(x)_n\in\mathbb{R}^{\infty}:\sum^{\infty}_{i=1}|x_n|^2<\infty\}$ closed and bounded? |
Let $n$ be a positive integer with $n\equiv 4\mod 6$ and define $p:=\frac{n^2+n+1}{3}$ (which is in this case a positive integer as well).
> Conjecture : $$p^2\mid n^n+(n+1)^{n+1}$$ for every $n$ of the given form.
The conjecture is true upto $n=10^8$
Trial : We have to show $(n^2+n+1)^2 \mid 9(n^n+(n+1)^{n+1}... |
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