instruction stringlengths 12 30k |
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I was looking my material on Fourier transforms and I came across the following statements. In my notations, $L > 0$, $I$ denotes the identity matrix and $\mu$ a self-adjoint matrix. The sum:
$$\sum_{p \in \frac{2\pi}{L}(\mathbb{Z} + \frac{1}{2})}\bigg{(}(-ipI + \mu)^{-1} + \frac{1}{ip}I\bigg{)} = \sum_{p \in \frac{2\... |
What does it mean to sum over the principal part? |
Given Hermitian matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
Given Hermitian matrices $H$ and $K$ related by $AH=K$, $H$ nonnegative, is it necessarily true that $A$ is normal? $H$ may be assumed to be positive if that helps.
Edit: Additional answer when $K$ and $H$ are both positive definite Hermitian.
We have $AH=HA^*$
$A$ is invertible and has no eigenvalue on the n... |
A pillar is leaning towards east and a and b are the angles of elevation of the top of the pillar
from two points due west of the pillar at distance a and b respectively. Find the angle between the pillar and the horizontal.
I started by fixing the directions,but don't west and east directions lie on a line? Then... |
A pillar is leaning towards east and $\alpha$ and $\beta$ are the angles of elevation of the top of the pillar
from two points due west of the pillar at distance $a$ and $b$ respectively. Find the angle between the pillar and the horizontal.
I started by fixing the directions,but don't west and east directions li... |
A pillar is leaning towards east and $\alpha$ and $\beta$ are the angles of elevation of the top of the pillar
from two points due west of the pillar at distances $a$ and $b$ respectively. Find the angle between the pillar and the horizontal.
I started by fixing the directions,but don't west and east directions l... |
I was watching this video https://www.youtube.com/watch?v=cHvYlrud8j8&ab_channel=AndrewMcCrady and in the proof, he states that when $0 \lt \delta = \min (1, \epsilon, \frac{\epsilon}{|L| + 1}) $,
$(L + \epsilon)(1 - \delta) - \delta \lt \frac{f(\alpha)}{g(\alpha)} \lt (L+\epsilon) + \delta$
implies that
$ L - ... |
I have a question about the definition of Cauchy sequence in $\sigma$-finite measure, e.g. Lebesgue measure. The reason why I have this question is because I was taught that
for a Cauchy sequence $(f_n)$ we can assume its limit $f$ for all $x\in A$, and zero otherwise where each $f_n$ is in $L^{\infty}$.
My quest... |
In the book (Anders Klarbring and Peter W. Christensen, "An introduction to Structural Optimization"), the *Pareto optimality* has been briefly mentioned which is a method used in the case when we have an optimization problem that involves several objective functions to be minimized simultaneously:
$\text{minimize}... |
I am looking for good math book on integrable systems, could you recommend any?
Thanks
|
I have a problem of $\cos(a)=t$ for $270\leq a\leq360$<br><br>$a)\cos(90+a)$<br>for obvious , my answer is the answer is <br>$\cos(90)\cos(a)-\sin(90)\sin(a)$ and left me with <br>$=-\sin(a)$<br><br> But what is the proof $\sin(a)=\sqrt{(1-\cos^2(a)}$<br>My solution:<br>
$\sin^2(a)+\cos^2(a)=1$<br>$\sin^2(a)=1-\cos^2(... |
There exists implemented algorithm which compute the [Hardy-Littlewood maximal function](https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function) at least for reasonably simple non-trivial cases? E.g. piecewise linear functions? |
I have a question about the definition of Cauchy sequence in $\sigma$-finite measure, e.g. Lebesgue measure. The reason why I have this question is because I was taught that
for a Cauchy sequence $(f_n)$ we can assume its limit $f$ for all $x\in A$, and zero otherwise where each $f_n$ is in $L^{\infty}$.
My quest... |
Helly's theorem is stated as:
**Helly's theorem**:
Let $C_1, \cdots, C_n \subseteq \mathbb{R}^d$ be convex sets such that any $d+1$ of them intersects. Then all $C_i$ have a common point.
Nerve theorem is stated as:
**Nerve theorem**:
If $U$ is a good cover ($U$ is a good cover if $U$ is a finite family of ... |
It is well-known that $(\text{Ord},+,0)$ and $(\text{Ord},\cdot,1)$ are monoids, but I haven't found references on these structures or other simpler ones (like $(\omega_1,+,0)$). For example, it would be useful for me to have references on submonoids, generators and similar things in these monoids.
Does anyone know ... |
I've been trying to compute the number of non-equivalent cusps of $\Gamma_0(3) \cap \Gamma(2)$. My approach so far has been the following: I believe that this is finite index subgroup of $\Gamma_0(3)$ so it should have the same cusps if I'm not mistaken.I wanted to work with these cusps and see when could they be equiv... |
Here is Theorem 7.20, in Chap. 7, in the book _Mathematical Analysis - A Modern Approach To Advanced Calculus_ by Tom M. Apostol, 2nd edition:
> Assume that $\alpha \nearrow$ on $[a, b]$. If $f \in R(\alpha)$ and $g \in R(\alpha)$ on $[a, b]$ and if $f(x) \leq g(x)$ for all $x$ in $[a, b]$, then we have
$$
\int_a... |
Amy and Beck are playing 'taking the stones game'. There are 18 stones on the table, and the two people take stones in turns. The first move of the starting player can take 1 to 4 stones. For the rest of the game, each player takes a number of stones from 1 up to twice the number of stones his/her opponent just took. ... |
Taking stones game beginning with 1 to 4 stones in a 2 player game. If we started with 18 stones, is the a winning strategy for the first player? |
I have an $n$ dimensional parallelizable manifold $M$. I know that *it is* the cartesian product of two parallelizable manifolds $M_1$ and $M_2$ but I *do not know* these two manifolds, not even their dimensions $n_1$ and $n_2$.
Note: the factorization into $M_1\times M_2$ could be non-unique. Please, assume that it... |
Let $A$ be the generator matrix of a continuous-time Markov chain. This means that $A$ has positive off-diagonal elements $A_{ij} > 0$, $i \ne j$, and row sums $\sum_j A_{ij}$ equal to $0$. For example, $A$ could be
$$
A = \left(
\begin{matrix}
-7 & 4 & 3 \\
1 & -2 & 1 \\
3 & 5 & -8
\end{matrix}
\right).
$$
I... |
Let $L = sl(2, F)$, and take the standard basis
$$x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \qquad y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \qquad h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
The multiplication table is $[xy] = h$, $[hx] = 2x$, $[hy] = -2y$.
So far so good, I understood w... |
Given the following state equation of a mechanical structure:
$K(x) u = F(x)$
where $K$ is the stiffness matrix of size $n \times n$, $x$ is the design variable, $u$ is the state variable (displacement vector) , and $F$ is the force vector.
If $K$ was invertible then $u = K^{-1} ~F$
But, if $K$ is not inve... |
I have an $n$ dimensional parallelizable manifold $M$. I know that *it is* the cartesian product of two parallelizable manifolds $M_1$ and $M_2$ but I *do not know* these two manifolds, not even their dimensions $n_1$ and $n_2$.
Note 1: the factorization into $M_1\times M_2$ could be non-unique. Please, assume that ... |
I'm new to using Geometric Brownian Motion, so I'm not sure if what I've done is correct.
Be the Geometric Brownian Motion $dS_t = \mu S_tdt + \sigma S_t dW_t$, $H$ a Heaviside, and $p_r, r_k$ constants, then
$$
\begin{align}
{\rm I\kern-.3em E} H\Big(\frac{p_r}{S_m}-r_k\Big)\frac{S_T}{S_m}=& {\rm I\kern-.3e... |
In *Quantum Computing: From Linear Algebra to Physical Realizations*, pg. 103, it states:
Let us consider the summation
$$
\frac{1}{2^n}\sum_{x=0}^{2^n-1}(-1)^{x \cdot y}
$$
with a fixed $y \in S_n$ Clearly it vanishes since $x \cdot y = 0$ for half of $x$ and $x\cdot y = 1$ for the other half unless $y=0$.
... |
Why does a summation vanish for a constant f(x) in the Deutsche-Josza algirthm? |
How to prove that $S = \sum_{i=1}^{2^n} \frac{1}{2^n + i} \gt \frac12$ for some natural number $n$? |
In my physics class we have learned to calculate a desired launch angle to allow a projectile to hit a target given the target’s distance away and the initial velocity.
Now in this case the initial velocity of the projectile occurs at the axis of rotation of the so called “cannon” that is launching the projectile. ... |
I would like to calculate $$\mathbb{E} \left(X_s X_t \right), s < t $$ where the $X_t$ is an exponential martingale or geometric brownian motion:
$$ X_t = X_0 \exp\left[\int_0^t \sigma(u) d W(u) - \frac{1}{2} \int_0^t \sigma^2(u)du\right]$$
Is my derivation that it equal to $X_0^2$ correct? I am not sure about my... |
I'm reading a pretty old paper. The authors deem "the most appropriate symbolism [...] that of Language II of Carnap, augmented with various notations drawn from Russell and Whitehead, incuding the *Principia* convention for dots." A few sentences later, they say
> [...] we shall intrudce a functor $S$, whose value ... |
Assume $P_{1}$ and $P_{2}$ are both positive definite. Let $A = P_{1}(P_{1} + P_{2})^{-1}$.
Can we conclude that for any vector $x$ of compatible size, the absolute value of each components of $Ax$ is less than that of $x$?
Thank you in advance |
Consider a function $f(x)$ with a period of 2L defined within the range $(L_1, L_2)$.
Will the Fourier series remain the same, regardless of the range in which I define the function $f(x)$?
For example, I can define $f(x)$ in the range $(0, 2L)$, $(\frac{L}{2}, \frac{3L}{2})$, or $(-L, L)$, but the Fourier series... |
Does Fourier series of a function with period (2L) remains the same regardless of the specific range within that period in which function is defined? |
I have four equations:
1) $n\geq 5k+2$
2) $n_1\geq 5k_1+1$
3) $n_2\geq 5k_2 +1$
4) $n=n_1+n_2$
What I want to get eventually is $k_1+k_2=k$, but it seems like I cannot say that as I get
$n\geq 5(k_1+k_2)+2$ and $n\geq 5k+2$, which does not necessarily mean $k_1+k_2=k$.
Is there another way to do this? Tha... |
You essentially build up the game from the bottom up.
We don't have exact winning positions because it also depends on how many stones the previous player took.
I've gotten you started looking at having 4, 5, 6, 7, 8 stones left. Can you complete it to 18 stones?
- Since the game is symmetric in moves, let $X... |
Proof that matrix A is invertible iff A is row equivalent to an upper diagonal matrix with no 0 on the diagonal? |
>Suppose functions $y$ and $f$ are non-negative and continuous on the interval $<a,b>$ (where $<$ can be $( $ or $[ $ and $>$ can be $] $ or $) $), and let $\lambda > 0$. If the inequality
$
y(x) \leq f(x) + \lambda \Big| \int_{x_0}^x y(s) \, ds \Big|, \quad x, x_0 \in (a,b)
$
holds, then
$
y(x) \leq f(x) + \lam... |
(Solution intentionally left incomplete) I've gotten you started looking at having 4-13 stones left. **Can you complete it to 18 stones?**
You essentially build up the game from the bottom up.
You might think that this is complicated because don't have exact / clean-cut winning positions because it also depend... |
(Solution intentionally left incomplete) I've gotten you started looking at having 4-13 stones left. **Can you complete it to 18 stones?**
You essentially build up the game from the bottom up.
You might think that this is complicated because don't have exact / clean-cut winning positions because it also depend... |
Let $M$ be a complex manifold and $I$ be the almost complex structure on $TM_{\mathbb{R}}$, and $E$ be a rank $k$ distribution stable under $I$. Let $\psi\colon U \to V$ be a submersion whose fibre are maximal integral manifold of $E$, or say $\ker \psi_{p,*}= E_p$. Then we have a almost complex structure on $V_{\psi(p... |
Amy and Beck are playing 'taking the stones game'. There are 18 stones on the table, and the two people take stones in turns. The first move of the starting player can take 1 to 4 stones. For the rest of the game, each player takes a number of stones from 1 up to twice the number of stones his/her opponent just took. ... |
You essentially build up the game from the bottom up.
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions, since gameplay also depends on how many stones the ... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following [$\erf^{-1}(z)$ series](https://www.wolframalpha.com/input?i=series+expansion+inverf%28x%29):
>$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via [Lagrange inversion](https://en.m.wikipedia.org/... |
We have $C = 2^\mathbb{N}$ and we identify the Cantor set with the image of C via:
$
t:C \rightarrow [0,1], \left(x_n\right)_n \mapsto \sum_{n=0}^{\infty}\frac{2x_n}{3^n}
$\
Now if $x,y\in t(C)$ and $\lvert x - y \rvert < 3^{-N}$ for some $N\in \mathbb{N}$ then the first $N$ digits of $x$ and $y$ coincide.
Now... |
It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$ denotes the linear interpolation $x_1 \rightarrow x_2$, then
$$S(\Delta x) = \left(1 , \Delta x, \frac{\Delta x^{\ot... |
I have a simple system and I am searching for the solution for f(t):
$$\frac{\partial f(t)}{\partial t} = c_1 \left( f(t) + g(t) + c_2 \right)$$.
It turns out that, in this system $g$ can be related to $f$ by a pure time delay:
$$g(t) = f(t-a)u(t-a)$$
where $u(t)$ is the Heaviside function.
Applying the ... |
You essentially build up the game from the bottom up.
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions, since gameplay also depends on how many stones the ... |
Let $n$ be a large positive integer and let $X$ be a random element of hypercube $\{0,1\}^n$ such that $\mathbb P(|X|= \ell) \propto \ell^{-\beta}$ for all $\ell \in \{0,1,\ldots,n\}$. Here $\beta \ge 0$ is a constant. Fix $x \in \{0,1\}$, $r \in (0,1)$, and let $B_n(x;rn)$ be the Hamming ball of radius $rn$ around $x... |
For context: there are Schwartz distributions called Pauli-Jordan (or Schwinger) functions which come up in quantum field theory. Given $m>0$, $x_0\in\mathbb{R}$ and $\vec{x}\in \mathbb{R}^3$, $D_m^{\pm}(x_0,\vec{x})\in \mathcal{S}(\mathbb{R}^4)'$ is defined by
\begin{align*}
D_m^\pm(x_0,\vec{x}):=f&\mapsto \frac{1}... |
You essentially build up the game from the bottom up.
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions, since gameplay also depends on how many stones the ... |
Let $n$ be a large positive integer and let $X$ be a random element of hypercube $\{0,1\}^n$ such that $\mathbb P(|X|= \ell) \propto \ell^{-\beta}$ for all $\ell \in \{0,1,\ldots,n\}$. Here $\beta \ge 0$ is a constant. Fix $x \in \{0,1\}^n$, $r \in (0,1)$, and let $B_n(x;rn)$ be the Hamming ball of radius $rn$ around ... |
I am examining the Proth numbers $n = h\cdot 2^k+1$, where $k$ is an integer and $h<2^k$ an odd integer.
Now I want to find the first Proth number with a million digits.
But I hve no idea how I could solve this.
So I know that we get the number of digits of $n$ by $\lfloor \log_{10}(n)\rfloor + 1$.
So I write
$$
... |
You essentially build up the game from the bottom up.
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions, since gameplay also depends on how many stones the ... |
You essentially build up the game from the bottom up.
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions, since gameplay also depends on how many stones the ... |
This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning.
Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). Consider $\bigcap_{n>0} I^n = \underset{\longrightarrow}{\textrm{lim}} \, I^n$ as a colimit. I would like to apply the $\t... |
You essentially build up the game from the bottom up.
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions, since gameplay also depends on how many stones the ... |
The solution is intentionally left incomplete. I've gotten you started almost till the very end. **Can you complete it to 18 stones?**
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / ... |
I am reading through ["State Estimation for Robotics" by Timothy Barfoot][1] and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions:
$$
C = (C_2C_1^T)^{\alpha}C_1, \alpha \in [0,1]
$$
$$
C , C_1, C_2 \in SO(3),
$$
are rotation matrices,
$$
C_... |
I am reading through ["State Estimation for Robotics" by Timothy Barfoot][1] and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions:
$$
C = (C_2C_1^T)^{\alpha}C_1, \alpha \in [0,1]
$$
$$
C , C_1, C_2 \in SO(3),
$$
That is, we have a rotation ma... |
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions on $n$ stones, since gameplay also depends on how many stones the previous player took. That is fine, we **simpl... |
I am reading through ["State Estimation for Robotics" by Timothy Barfoot][1] and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions:
$$
C = (C_2C_1^T)^{\alpha}C_1, \alpha \in [0,1]
$$
$$
C , C_1, C_2 \in SO(3),
$$
That is, we have a rotation ma... |
If you want a specific matrix $g$, you can take
$$
g=\left[ \begin{array}{cc}
1&0\\
\sqrt{2}&1\end{array}\right].
$$
Here is the reason. Let $U_\infty$ denote the subgroup of strictly upper triangular matrices in $G$. Then for $g\in G$,
$$
U_\infty g = g U, \quad U=g^{-1}U_\infty g.
$$
The unique fixed-point... |
Let $A$ be the generator matrix of a continuous-time Markov chain. This means that $A$ has positive off-diagonal elements $A_{ij} > 0$, $i \ne j$, and row sums $\sum_j A_{ij}$ equal to $0$. For example, $A$ could be
$$
A = \left(
\begin{matrix}
-7 & 4 & 3 \\
1 & -2 & 1 \\
3 & 5 & -8
\end{matrix}
\right).
$$
I... |
I am reading through ["State Estimation for Robotics" by Timothy Barfoot][1] and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions:
$$
C = (C_2C_1^T)^{\alpha}C_1, \alpha \in [0,1]
$$
$$
C , C_1, C_2 \in SO(3),
$$
That is, we have a rotation ma... |
I am reading through ["State Estimation for Robotics" by Timothy Barfoot][1] and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions:
$$
C = (C_2C_1^T)^{\alpha}C_1, \alpha \in [0,1]
$$
$$
C , C_1, C_2 \in SO(3),
$$
That is, we have a rotation ma... |
In a script I am currently reading it is used at some point that the convex funciton $f = \ln (p(\exp (x_1 ), \ldots , \exp( x_n)))$ can be minimized in polynomial time under linear inequality constraints. Here $p$ is a polynomial in $n$ variables having nonnegative coefficients.
I have read multiple times before th... |
I haven't done math in awhile so am a bit fuzzy on how to set up the problem. Let's say there's a test that students take and they can score anywhere between 0 and 10000. Let's say the distribution of test scores follows a Beta PDF, let's call this function PDF(X). And let's say there are 200 students.
There's also... |
How do I combine a PDF with another continuous function and then do a sum product? |
The Poisson equation $\nabla^2 u = \Delta u = f$.
When $f = 0$ we obtain the Laplacian equation which to me has an intuitive interpretation with the mean-value property. However, how can we form an intuitive understanding of the Poisson equation from the function $f$? How does $f$ impact $u$? |
I am trying to understand consequences of the Cobordism Hypothesis in dimension 1, following section 4.2 of Lurie's "On the Classification of Topological Field Theories". Especially, I want to understand the '$S^1$ action' on the dimension of a dualizable object.
Given a symmetric monoidal $(\infty, 1)$-category ... |
Given that you have Math Olympiad background, I assume that you're familiar with setting up winning positions. You might think that doesn't work here because we don't have exact / clean-cut winning positions on $n$ stones, since gameplay also depends on how many stones the previous player took. That is fine, we **simpl... |
Assume $P_{1}$ and $P_{2}$ are both positive definite. Let $A = P_{1}(P_{1} + P_{2})^{-1}$.
Can we conclude that for any vector $x$ of compatible size, the absolute value of each components of $Ax$ is less than that of $x$?
This problem may be solved if we can check the infinity norm of $A$ is less than 1, but I ca... |
I have a simple logic equivalence expression. I'm thinking there must be a name for this principal of OR inversion. Can anyone help put a name to this idea? Link to expression calc below.
$$(p\lor q)\iff \neg(\neg p\land\neg q)$$
https://www.emathhelp.net/en/calculators/discrete-mathematics/truth-table-calculat... |
Too long for a comment.
The brief search showed several related papers and even questions at Mathematics StackExchange. I list them below. Unfortunately, now I have no time to study them.
[Eck] M.W. Ecker, *[When Does a Sum of Positive Integers Equal Their Product?](https://www.researchgate.net/publication/27018... |
I am trying to understand consequences of the Cobordism Hypothesis in dimension 1, following section 4.2 of Lurie's "On the Classification of Topological Field Theories". Especially, I want to understand the '$S^1$ action' on the dimension of a dualizable object.
Given a symmetric monoidal $(\infty, 1)$-category ... |
According to the Wikipedia article on complex Lie algebras https://en.m.wikipedia.org/wiki/Complex_Lie_algebra , any complex Lie algebra $\mathfrak{g}$ that is isomorphic to its conjugate $\overline{\mathfrak{g}}$ admits a real form.
So as I understand it, this isomorphism exists if and only if there is an antilinea... |
Too long for a comment.
The brief search showed several related papers and even questions at Mathematics StackExchange. I list them below. Unfortunately, now I have no time to study them.
[Eck] M.W. Ecker, *[When Does a Sum of Positive Integers Equal Their Product?](https://www.researchgate.net/publication/27018... |
It is well-known that $(\text{Ord},+,0)$ and $(\text{Ord},\cdot,1)$ are monoids, but I haven't found references on these structures or other simpler ones (like $(\omega_1,+,0)$). For example, it would be useful for me to have references on submonoids, generators and similar things in these monoids.
Does anyone know ... |
I'm reading the proof of Proposition $2$ of the Appendix [here][1].
The proposition states
> Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $(P_n)_{n\ge0}$ be the $d$-color Pòlya urn random process having $S\cdot Id$ as replacement matrix and $(\alpha_... |
Why does Wolframalpha and Symbolab say the value of the definite integral from 0 to 8 for 1 divided by cube root of x with respect to x is 6 when the function is not continuous in the given interval?
[ftc][1]
[1]: https://i.stack.imgur.com/TsorS.png |
Glitch in computing definite integral? |
I haven't done math in awhile so am a bit fuzzy on how to set up the problem. Let's say there's a test that students take and they can score anywhere between 0 and 10000. Let's say the distribution of test scores follows a Beta PDF, let's call this function PDF(X). And let's say there are 200 students.
There's also... |
Why does Wolframalpha and Symbolab say the value of the definite integral from 0 to 8 for 1 divided by cube root of x with respect to x is 6 when the function is not continuous in the given interval?
[screenshot][1]
[1]: https://i.stack.imgur.com/TsorS.png |
I wonder if it's true that if $X_1,...,X_n,...$ are independent and nonnegative random variables and that $\sum_{i=1}^\infty E[X_i]=\infty$, then $\sum_{i=1}^\infty X_i=\infty$ a.s.?
I suspect so because its true if $X_i=1_{A_i}$, which is the 2nd Borel-Cantelli lemma. If $X_i=exp(\lambda_i)$, then its also true, wh... |
Is it true that nonnegative independent random variables with sum of expectations$=\infty$ also has sum$=\infty$ a.s.? |
Why doe Wolframalpha and Symbolab say
$$\int_0^8\frac1{\sqrt[3]x}dx=6$$
when the function is not continuous in the given interval?
[screenshot][1]
[1]: https://i.stack.imgur.com/TsorS.png |
Is fiber product of $M_s$ , $M_t$ over $M_{st}$ isomorphic to $M$? |
I am reading the RankNet to LambdaMART paper : https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/MSR-TR-2010-82.pdf , where the author makes a particular claim in equation (1). They start from a cost function and then explains it. Then, they say "This gives" <the actual gradient>.
I've been trying... |
Glitch in computing $\int_0^8\frac{dx}{\sqrt[3]x}$? |
According to the Wikipedia article on complex Lie algebras https://en.m.wikipedia.org/wiki/Complex_Lie_algebra , any complex Lie algebra $\mathfrak{g}$ that is isomorphic to its conjugate $\overline{\mathfrak{g}}$ admits a real form.
So as I understand it, this isomorphism exists if and only if there is an antilinea... |
Why do Wolframalpha and Symbolab say
$$\int_0^8\frac1{\sqrt[3]x}dx=6$$
when the function is not continuous in the given interval?
[screenshot][1]
[1]: https://i.stack.imgur.com/TsorS.png |
Why doe Wolframalpha and Symbolab say
$$\int_0^8\frac1{\sqrt[3]x}dx=6$$
when the function is not continuous in the given interval?
[screenshot][1]
[1]: https://i.stack.imgur.com/TsorS.png |
Why do Wolframalpha and Symbolab say
$$\int_0^8\frac1{\sqrt[3]x}dx=6$$
when the function is not continuous in the given interval?
[screenshot][1]
[1]: https://i.stack.imgur.com/TsorS.png |
Why do Wolframalpha and Symbolab say
$$\int_0^8\frac1{\sqrt[3]x}dx=6$$
when the function is not continuous in the given interval?
[screenshot][1]
[1]: https://i.stack.imgur.com/TsorS.png |
I have to find the minimal polynomial for $\cos(2\pi/5)$ and $\sin(2\pi/5)$ and I'd like to know what is wrong with my attempt:
$\cos(4\pi/5)=\cos(6\pi/5) = 4\cos^3(2\pi/5)-3\cos(2\pi/5)$, that gives $\cos^3(2\pi/5)= (1/4)\cos(4\pi/5)+(3/4)\cos(2\pi/5)$
$\cos^0(2\pi/5)=1$
$\cos(2\pi/5)=\cos(2\pi/5)$
$\co... |
Suppose a space $Y$ is given along two subspaces $Y_1,Y_2$ s.t. $Y=Y_1^{\circ}\cup Y_2^{\circ}$ and the intersection $Y_0=Y_1\cap Y_2$ is non-empty. If $(Y_1,Y_0)$ is $p$-connected and $(Y_2,Y_0)$ is $q$-connected for some $p,q\ge0$, then the inclusion induces isomorphisms $\pi_n(Y_1,Y_0,\ast)\rightarrow\pi_n(Y,Y_2,\as... |
We can calculate the minimal polynomial of $ 2\cos(\frac{2\pi}{7})=\zeta_7+\frac{1}{\zeta_7}$ over $\mathbb Q$ as $x^3+x^2-2x-1$ and simlary for $2\cos(\frac{2\pi}{5})=\zeta_5+\frac{1}{\zeta_5 }$.
Now my question is :
>Is there any way to calculate the minimal polynomial of $\zeta_7+\frac{1}{\zeta_7}+\zeta_5+\fr... |
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