inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the covariance between $X$ and $X^3$ for a random variable $X$ that follows a normal distribution $N(\mu, \mu^2)$ using the moment generating function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\mu^4}
Let $V$ be the vector space of all polynomials over a field $F$. Consider a subspace $U$ of $V$ with $2 \leq \dim U \leq n$. Given distinct points $x_1, x_2 \in F$, determine the dimension of the subspace $W \cap U$, where $W$ is the subspace of polynomials that vanish at $x_1$ and $x_2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim U - 2}
How many permutations $f \in S_8$ commute with the permutation $g = (1\,2\,3)(4\,5\,6)(7\,8)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Evaluate the integral \( \int_{0}^{\pi/2}{\mathrm dx\over (1+\tan x)^6} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{60}}
Determine the chromatic number of the circulant graph $Ci_{2n}(1,n-1,n)$, where the vertex set is $V=\{a_0, \dots , a_{n-1}, b_0, \dots , b_{n-1}\}$ and the edge set is $E=\{(a_i,b_i), (a_i, a_{i+1\pmod{n}}), (b_i, b_{i+1 \pmod{n}}), (a_i, b_{i+1 \pmod{n}}), (b_i, a_{i+1 \pmod{n}} )\}$ for $0\le i <n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the intersection point of the tangent planes to the surface $S$ in $\Bbb R^3$ given by the graph of $f(x,y) = xy^2 − 2y^2 + e^x$ at the points $(x, y) = (0, 0)$, $(0, 1)$, and $(0, 2)$. Provide the coordinates of the intersection point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 0, -1)}
A pool table has 6 pockets. There are 10 red balls, one brown ball, and one black ball. How many ways are there to distribute the balls into the pockets if each pocket can contain no more than 9 balls and the black ball should not be in the same pocket as the brown ball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89610}
Evaluate the contour integral \( \int_C \frac{e^z}{z^2-4} \, dz \), where \( C \) is the circle defined by \( |z-1| = 1 \) with positive orientation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi i e^2}{4}}
Let \(X_1, X_2, X_3, \ldots\) be a sequence of i.i.d. uniform \((0,1)\) random variables. Calculate the value of \[ \lim_{n \to \infty} P\left(-\ln(1-X_1) - \ln(1-X_2) - \cdots - \ln(1-X_n) \geq n\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Classify the following partial differential equation in cylindrical coordinates as elliptic, parabolic, or hyperbolic:

\[ \frac{\partial^2 \phi}{\partial z^2} + \frac{\partial \phi}{\partial z} + \frac{\partial^2 \phi}{\partial r^2} + \left ( \frac{1}{r} \right )\frac{\partial \phi}{\partial \theta} = 0 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{parabolic}
Evaluate the limit: $$ \lim_{{M \to +\infty}} \frac{1}{M} \int_1^M M^{\frac{1}{x}} \, \mathrm{d}x $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( F(x,y,z) = (y+\sin(x),z^2+\cos(y),x^3) \) and \( \mathcal{C} \) be the curve parametrized by \( r(t)=(\sin(t),\cos(t),2\sin(2t)) \) with \( t \) ranging in \([0,2\pi)\). Use the curl theorem to evaluate \( \int_\mathcal{C} F \cdot ds \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Calculate the value of the following summation: \[ \sum_{k=0}^{13} \binom{26}{2k}\frac{(2k)!}{k!2^k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{532985208200576}
Determine the number of distinct equivalence classes for the relation \( \sim \) on \( \mathbb{Z}_n \) defined by \( a \sim b \) if and only if there exists a unit \( k \in U_n = \mathbb{Z}_n^{\times} \) such that \( a = kb \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tau(n)}
Consider the parabola given by the equation \( y^{2} = 2px \) where \( p > 0 \). The parabola has a focus \( F \) and a directrix \( l \). Points \( A \) and \( B \) lie on the parabola such that \( \angle AFB = \frac{\pi}{3} \). Let \( M \) be the midpoint of segment \( AB \), and let \( N \) be the projection of \( M \) onto the directrix \( l \). Determine the maximum value of the ratio \( \frac{|MN|}{|AB|} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find \( \lim_{n \rightarrow \infty} \mathbb{P}(X_1 + \ldots + X_n \leq 0) \), where the independent binary random variables \(X_k\) take values \(\pm 1\) with probabilities \((1 \pm k^{-1/2})/2\) for \(k = 1, 2, \ldots\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(-2)}
In triangle \( ABC \), \( \angle A = 45^\circ \) and \( \angle B = 60^\circ \). The feet of the altitudes from vertices \( A, B, \) and \( C \) are \( A_1, B_1, \) and \( C_1 \) respectively. The midpoints of sides \( AB, BC, \) and \( CA \) are \( C_0, A_0, \) and \( B_0 \) respectively. A circle passing through these midpoints has center \( F \). Consider the first angle trisectors of \( \angle A_0 F A_1, \angle B_0 F B_1, \angle C_0 F C_1 \), which are the rays \( F X, F Y, F Z \) such that \( \angle X F A_1 = 2 \angle A_0 F X \), \( \angle Y F B_1 = 2 \angle B_0 F Y \), and \( \angle Z F C_1 = 2 \angle C_0 F Z \). Find the angles between the rays \( F X, F Y, \) and \( F Z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Let $p$ be a prime number such that $p \equiv 3 \pmod{4}$. Determine the remainder when the sum $\sum_{i=1}^{p-1}\left(\frac{i^4+1}{p}\right)$ is divided by $p$, where $\left(\frac{i^4+1}{p}\right)$ denotes the Legendre symbol.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p-2}
Determine the largest number $k = k(n)$ such that every real $n \times n$ matrix is orthogonally equivalent to a matrix with at least $k$ zero entries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n - 1)}
Given a continuous and $2\pi$-periodic function $f: \mathbb{R} \to \mathbb{R}$, if for any positive integer $p$ in the range $[0, n-1]$, both integrals $\int_{0}^{2\pi} f(t) \cos(pt)\,dt$ and $\int_{0}^{2\pi} f(t) \sin(pt)\,dt$ are zero, determine the minimum number of roots $f$ has on the interval $[0, 2\pi]$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Four lines are drawn on a plane such that no two are parallel and no three are concurrent. Lines are then drawn joining the points of intersection of these four lines. How many new lines are formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral \( \int_C \frac{e^z}{z} \, dz \), where \( C \) consists of two circles centered at the origin with radii \( R = 2 \) and \( R = 1 \). Determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i}
Calculate the convolution of the function \( \frac{\sin(t)}{t} \) with the constant function 1. Express your answer as a function or an expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Consider forming a subset $X$ of $\mathbb{R}$ by the following process: for each $x \in \mathbb{R}$, toss a fair coin. If the result is heads, include $x$ in $X$. Assume that the coin tosses are independent. What is the probability that $X$ is a Lebesgue measurable subset of $\mathbb{R}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Problem 6. Point $A$ on the plane is located at the same distance from all points of intersection of two parabolas given in the Cartesian coordinate system on the plane by the equations $y=-3 x^{2}+2$ and $x=-4 y^{2}+2$. Find this distance.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\sqrt{697}}{24}"]}
Given that $x_1(t) = e^t$ is a solution to the differential equation $$tx''-(2t+1)x'+(t+1)x=0,$$ find a second linearly independent solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t^2 e^t}
Find a vector $X$ such that $U \cap W = \text{span}\{X\}$, where $U = \langle (1,1,3),(-5,-3,-15) \rangle$ and $W = \langle (4,4,9),(-44,-34,-120) \rangle$. Provide the vector $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(14, 9, 42)}
Evaluate the sum \( \sum_{n=1}^{101} \frac{\left(\frac{n}{101}\right)^3}{1 - 3\left(\frac{n}{101}\right) + 3\left(\frac{n}{101}\right)^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Determine the number of solutions for the matrix equation \(X^n = \begin{pmatrix}3&6\\ 2&4\end{pmatrix}\) when \(n\) is an odd positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the order of the normalizer of the subgroup of elementary abelian 2-groups in the affine general linear group $AGL(3,2)$ within the symmetric group $S_8$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1344}
Find the range of values for $c$ such that the ellipses given by \( \frac{u^2}{(c+\frac{1}{c})^2}+\frac{v^2}{(c-\frac{1}{c})^2}=1 \) cover the entire plane, except for a specific segment on the $u$-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1) \cup (1, \infty)}
## [ Right triangle with an angle of $30^{\circ}$ ]

In a right triangle $ABC$ with acute angle $A$ equal to $30^{\circ}$, the bisector $BD$ of the other acute angle is drawn.

Find the distance between the centers of the inscribed circles of triangles $ABD$ and $CBD$, if the shorter leg is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\sqrt{96-54\\sqrt{3}}}{3}"]}
Evaluate the limit \( \lim_{{n \to \infty}} \frac{n}{\sqrt[n]{n!}} \) without using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Evaluate the integral \( \int_V \sin(\sqrt{x^2+y^2}) \, dx \, dy \) where \( V = \{(x,y) : \pi^2 \le x^2+y^2 \le 4\pi^2\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6\pi^2}
Determine the value of \( r > 0 \) such that with probability one, for all sufficiently large \( n \), the inequality \( M_n \le r\sqrt{\log n} \) holds, where \( M_n = \max\{|B_t - B_{n-1}| : n - 1 \le t \le n\} \) and \( B_t \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Find the orthogonal trajectories of the family of curves given by the equation \(x^2 + 2y^2 = k^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = Cx^2}
Evaluate the limit \( \lim_{x \to \infty} x[1-F(x)] \), where \( F(x) \) is a cumulative distribution function and \( 1-F(x) \) is the tail distribution function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the product of all non-zero elements in a finite field \( \mathbb{F}_{q} \), where \( q \) is a power of a prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the Galois group of the field extension $F_{p^2}$ over $\mathbb{Z}_p$, where $F_{p^2}$ is the field with $p^2$ elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
Find the largest natural number \( n \ge k \) such that any selection of \( k \) numbers from a set of \( n \) natural numbers results in a composite sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k - 1}
Consider a topological space $(X, \tau)$ where $D$ is a closed subspace of $X$. Suppose $D$ is equipotent to $\mathbb{R}$ with a discrete induced topology, and $X$ is normal. Determine the number of continuous functions from a countable subset of $D$ to the interval $[0,1]$. Express your answer in terms of cardinality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Given a standard Brownian motion \( B_t \) starting at zero, and a non-negative stochastic process \( X_t \) that satisfies the stochastic differential equation \( dX_t = \frac{1}{X_t} dt + dB_t \), calculate the expected value \( E[\sigma] \), where \( \sigma = \inf \{ t \geq 0 : X_t = 1 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Determine the limit poset of the sequence \(\mathbf{0}\rightarrow\mathbf{1}\rightarrow\mathbf{2}\rightarrow\cdots\) in the category \(\mathbf{Posets}\) of posets and monotone functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{0}}
Find the non-zero constant \( c \) such that the integral \( \int_{-1}^\infty \frac{e^{x/c}}{\sqrt{|x|}(x+2)}dx \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Determine the values of $p$ for which the gradient of the function $A \mapsto \|A\|_p^p$ is locally Lipschitz.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \geq 2}
Determine the number of invertible $3 \times 3$ matrices over $\mathbb{Z}/3\mathbb{Z}$ that commute with the matrix $A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Consider a uniform probability distribution over the interval \([0, 1]\). Determine the expected number of trials needed to select numbers from this distribution such that their product is less than \(\frac{1}{e}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The domain of the function $f(x)=\\log_\frac{1}{2} (\\log_4(\\log_\frac{1}{4} (\\log_{16}(\log_{\frac{1}{16} }{x} ))))$ is an interval of length $\frac{m}{n} $, where $m$ and $n$ are relatively prime positive integers.  What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
271}
Determine the total number of Generalized Hermitian Gell-Mann Matrices in dimension $d$, given that they consist of the $h_k^d$ matrices, where $1 \leq k \leq d$, and the $f_{k,j}^d$ matrices, where $1 \leq k, j \leq d$. How many matrices are there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d^2}
Given a graph $G$ composed of two cycles $C_x$ and $C_y$ that share an edge, where $x, y \ge 3$, find a formula for the number of spanning trees of $G$. The graph $G$ has $x + y - 1$ edges. Provide the formula in terms of $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{xy - 1}
Determine the number of Boolean functions $f(x_1, x_2, \dots, x_n)$ from $\{0,1\}^n$ to $\{0,1\}$ that can be constructed using only the XOR and AND operations, without using OR.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^n}}
In the binomial expansion of \(\left(\frac{\sqrt{x+1}}{2 \sqrt[4]{x}}\right)^{n}\), the terms are arranged in descending powers of \(x\). If the coefficients of the first two terms form an arithmetic sequence, determine the number of terms in this expansion that have integer exponents of \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the domain of the self-adjoint extension of the Laplace-Beltrami operator $-\Delta$ when it is considered as an unbounded operator on $L^2(M)$, where $M$ is a complete Riemannian manifold.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H^2(M)}
Calculate the derivative of \( \text{Tr}(X^TP^TPX) \) with respect to the matrix \( P \), where \( X \) and \( P \) are real matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 P X X^T}
Calculate the flux of the vector field \( F = (x^3, y^3, z^3) \) through the surface of the sphere defined by \( x^2 + y^2 + z^2 = x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{5}}
Determine if the random variable $T$, which follows a Bernoulli distribution with parameter $p \in (0,1)$ and is defined as:

$$T= \left\{ \begin{aligned} &1 &\text{ with probability of }1-p,\\ &1-\frac{1}{p} &\text{ with probability of }p. \end{aligned} \right.$$ 

is a sub-Gaussian random variable. If it is, find the variance proxy $\sigma^2$ in terms of $p$. Provide your answer for $\sigma^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4p^2}}
Determine the distribution of the random variable $X = W(s) + W(s + t)$, where $W(t)$ is a Wiener process with $W(0) = 0$. Assume that for $t \le s$, $W(s) - W(t)$ follows a normal distribution with mean $0$ and variance $(s - t)$. Provide the distribution of $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathcal{N}(0, 4s + t)}
Consider a scenario with $n$ people, each having a unique ball. They place their balls into a box, and the balls are mixed. Each person then draws a ball from the box. If a person draws their own ball, they leave; otherwise, they participate in the next round. What is the expected number of rounds until everyone has retrieved their own ball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
The heights of certain plants are normally distributed with a mean of 110 cm and a variance of 1050 cm². Two plants are chosen at random. What is the probability that the height of one plant is at least 1.5 times the height of the other plant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.347}
Determine the radical of the ideal \((x^3-y^6, xy-y^3)\) in \(\mathbb{C}[x,y]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - y^2)}
Determine whether there exists a positive constant \( c \in \mathbb{R} \) such that for any nonzero vectors \( a, b \in \mathbb{R}^n \), the inequality \[ \frac{1}{2}\|a-b\|^2 - \frac{1}{\pi}\sqrt{\|a\|^2\|b\|^2 - \langle a,b\rangle^2} + \langle a,b\rangle\frac{\cos^{-1}(\rho)}{\pi} \geq c\|a-b\|^2 \] holds, where \( \rho = \frac{\langle a,b\rangle}{\|a\|\,\|b\|} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Identify the elements in the quotient ring $k[x,y]/(x{y^2})$ that are nilpotent but not units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(xy)}
Using Taylor's inequality, find an upper bound on the error of the approximation \( f(x) = x\ln(x) \approx T_3(x) \) on the interval \([1/2, 3/2]\), where \( T_3(x) \) is the third-degree Taylor polynomial about \( x=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{24}}
Evaluate the triple integral \( \int\limits_0^5 \int\limits_{\sqrt{25-x^2}}^{-\sqrt{25-x^2}} \int\limits_{\sqrt{25-x^2-z^2}}^{-\sqrt{25-x^2-z^2}} \frac{1}{x^2+y^2+z^2} \,\mathrm dy\,\mathrm dz\,\mathrm dx \) by converting to spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\pi}
What minimal number of numbers from the set \( \{ 1,2,\ldots ,{1982}\} \) should be deleted to provide the property:
 none of the remained numbers equals to the product of two other remained numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
43
}
Let \( f(z) \) be a function that is analytic in the unit disk \( U = \{z:|z|<1\} \) and continuous on the closed unit disk. Suppose that \( \frac{f(z)}{z^2} \) can be extended to be analytic in the open unit disk \( U \), including at the origin. Given that \( |f(z)| \leq 6 \) in the closed unit disk, determine the maximal possible value of \( f(0.4 + 0.5i) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{123}{50}}
The first three terms of a geometric sequence are the integers \( a,{720} \) , and \( b \) , where \( a < {720} < b \) . What is the sum of the
 digits of the least possible value of \( b \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
21}
Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Let $E_p,E_q, E_{p,q}$ be the exceptional divisors over $p,q$, and $L$, respectively. $E_p$ and $E_q$ are isomorphic to the blow-up of a point in $\mathbb{P}^2$, and $E_{p,q}\cong \mathbb{P}^1\times\mathbb{P}^1$. Let $\sigma_{p,q}\subset E_{p,q}$ be the class of the $\mathbb{P}^1$ which is not contracted by $\pi$. Given that $E_p\cdot \sigma_{p,q} = E_q\cdot\sigma_{p,q} = 1$, find the intersection number $E_{p,q}\cdot \sigma_{p,q}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the integral: $$I=\int_{-\infty}^{\infty}\arcsin\left(\frac{1}{\cosh x}\right)\,dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4G}
Simplify the expression: $$\lim_{x\to0}\left(\prod^{\frac{1}{x}-1}_{i=1}\frac{1}{\sec\frac{xi\pi}{2}}\right)^x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the Fréchet derivative of the functional $\|Au-f\|^2$, where $A$ is a bounded linear operator on an infinite-dimensional Hilbert space. Express your answer in terms of $A$, $u$, and $f$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2A^*(Au - f)}
How much larger is 3↑↑↑3 compared to itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given two reinforced concrete buildings A and B in a seismic region, an impending earthquake might be strong (S), moderate (M), or weak (W) with probabilities P(S) = 0.02, P(M) = 0.2, and P(W) = 0.78. The probabilities of failure for each building if these earthquakes occur are 0.20, 0.05, and 0.01, respectively. If building A fails, the probability that building B will also fail is 0.50, 0.15, and 0.02 for the three types of earthquakes. Calculate the probability that the earthquake was not strong given that building A has failed and building B has survived.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.890}
Evaluate the integral: \[ f(u) = \int_{-\infty}^{\infty} \frac{e^{-itu}}{\sqrt{1+t^2}} \, dt \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 K_0(|u|)}
Find the coefficient of \(x^{30}\) in the expansion of \(\left(\frac{1-x^{11}}{1-x}\right)^3\left(\frac{1-x^{21}}{1-x}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1111}