inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the number of points with integer coordinates that lie inside the region bounded by the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) and the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Nick is mixing two solutions. He has $100$ mL of a solution that is $30\%$ $X$ and $400$ mL of a solution that is $10\%$ $X$. If he combines the two, what percent $X$ is the final solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(14\%\)}
Determine the number of distinct ways to select four 3-element subsets from a 6-element set such that each pair of subsets shares exactly one common element, considering distinct arrangements only (ignoring rotations and reflections).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A graph has 20 vertices and does not contain a $K_7$. If each edge is colored either green or yellow such that there are no monochromatic triangles, determine the maximum number of edges this graph can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Let $G$ be a group of order 30 with a center $Z(G)$ of order 5. Determine the isomorphism class of the quotient group $G/Z(G)$. To which commonly known group is $G/Z(G)$ isomorphic?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Find a specific solution to the equation $x = \log(x) + c$ without using numerical methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-W\left(-e^{-c}\right)}
The function f is defined on the set of integers and satisfies
$\bullet$ $f(n) = n - 2$, if $n \ge 2005$
$\bullet$ $f(n) = f(f(n+7))$, if $n <  2005$.
Find $f(3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2006}
Evaluate the limit \( \lim_{n \to \infty} \frac{1}{n}\int_{0}^{n\pmod {2\pi}} \arcsin(\sin(t))dt \), where \( n\pmod {2\pi} \) represents the remainder when \( n \) is divided by \( 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the conditional covariance of $B_s$ and $B_t$ given $B_1$, where $B$ is a one-dimensional Brownian motion and $s, t \in [0,1]$. Provide your answer as a function of $s$ and $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(s, t) - st}
Let $p_1, \dots, p_r$ be $r$ distinct odd primes. Define the group \( G = \prod_{i=1}^r \mathrm{SL}_3(\mathbb{F}_{p_i}) \). Determine the maximum possible rank of a subgroup of \( G \), denoted as \( \max \{d(H) : H \leq G\} \), where \( d(K) \) is the smallest cardinality of a generating set of a group \( K \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2r}
Find the roots of the polynomial $Y^3 - Y + 1$ in the ring $\mathbb{F}_3[Y]/(Y^3 - Y^2 + 1)$. Provide your answer as a set of elements in this ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{Y + 2Y^2, 1 + Y + 2Y^2, 2 + Y + 2Y^2\}}
Let $M$ be a Von Neumann algebra and $\phi$ be a completely bounded (c.b.) map on $M$. Consider the Hilbert space $H$ and the map $Id\otimes \phi$ on $B(H)\overline{\otimes} M$, where $B(H)$ is the algebra of bounded linear operators on $H$ and $\overline{\otimes}$ denotes the minimal tensor product of operator spaces. Determine if $Id\otimes \phi$ is a completely bounded map, and if so, find its completely bounded norm $\|Id\otimes \phi\|_{cb}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|\phi\|_{cb}}
Determine the smallest value of \( c \) such that the series \( \sum_{n=1}^{\infty}\frac{a_n}{S_n^c} \) converges, where \( a_n > 0 \), \( S_n = \sum_{k=1}^{n}a_k \), and \( \lim_{n \rightarrow \infty} S_n = +\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ appears in $W$ if there are indices $k\le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called special if $Ua, Ub, aU, bU$ all appear in $W$. (The empty word is considered special) You are given that there are no special words of length greater than 2015.Let $N$ be the minimum possible number of special words. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
535
}
Compute the surface integral \( \iint_S{\mathbf{F} \cdot \mathbf{n}} \, dS \) for the hemisphere \( S \) defined by \( x^2 + y^2 + z^2 = 1 \) with \( z \ge 0 \), where \( \mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} \) and \( \mathbf{n} \) is the unit outward normal to \( S \). Use the parameterization \( \mathbf{r}(u, v) = \sin u \cos v \mathbf{i} + \sin u \sin v \mathbf{j} + \cos u \mathbf{k} \) to evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{3}}
Determine the values of \( \beta \) for which the series \( \sum_{n=1}^{\infty} \left( \sqrt{1+\frac{(-1)^n}{n^\beta}} - 1 \right) \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta > \dfrac{1}{2}}
Evaluate the integral: $$\int^{(2,2,2)}_{(1,1,1)}5x^4y^{15}z^{15}dx+15x^5y^{14}z^{15}dy+15x^5y^{15}z^{14}dz$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{35} - 1}
Calculate the limit of the function \( f(x) = \sum_{n=0}^{+\infty} x^{n^2} \) as \( x \to -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Calculate the average distance between two randomly chosen points from the Cantor set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
A young baseball fan wants to collect a complete set of 262 baseball cards. The cards are distributed randomly, one per package of chewing gum. The fan buys two packages of chewing gum each day. On average, how many days will it take the fan to collect the complete set of cards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{805}
Find the sum of all possible values of $b$ in the equation $p(x)=x^3+ax+b$ if $r$ and $s$ are the roots of $p(x)$ and $r+4$, $s-3$ are the roots of $q(x)=x^3+ax+b+240$. Express your answer in the form of $|\frac{k}{5}|$ where $|.|$ stands for the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
What is the best known upper bound on the index of an abelian subgroup in a discrete subgroup of the Euclidean group of isometries, \( \isom(\mathbb{R}^n) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n n!}
Determine the number of points at which a line in \(\mathbb{CP}^n\) (a copy of \(\mathbb{CP}^1\)) intersects with \(\mathbb{CP}^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the integral \( \int_{\lambda} \frac{dz}{(z^2-1)^2} \), where \( \lambda \) is a path in \( \mathbb{R}^2 - \{1,-1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A uniform container of height $6m$ is full of water and has three identical drainage pipes $A$, $B$, and $C$. Each pipe can independently empty the container (the part above it) in $T$ minutes. Pipe $A$ is fixed at the base, while pipe $B$ is fixed at a height of $5m$. Pipe $C$ is fixed somewhere between them. If all three pipes together can empty the container in $\frac{2T}{3}$ minutes, determine the height at which pipe $C$ is fixed. Assume no variations due to pressure differences at different heights.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider the set $S(R)$ of all representations $\rho : \mathbb{Z}/2\mathbb{Z} \to GL(2, R)$ such that $\rho(0+2\mathbb{Z}) \neq \rho(1+2\mathbb{Z})$, where $R$ is a ring and $GL(2, R)$ is the group of 2x2 invertible matrices with entries from $R$. Let $R = \mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime. Determine whether the following property $(P(\rho, R))$ holds for every, some, or no $\rho \in S(R)$: There exist $v, w \in R^2$ such that for all $g \in \mathbb{Z}/2\mathbb{Z}$, $\rho(g)(R\cdot v) = R\cdot v$, $\rho(g)(R\cdot w) = R\cdot w$, and $R\cdot v + R\cdot w = R^2$. Provide your answer as "every", "some", or "no".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{some}
Find a function $g(x)$ such that for all positive integers $k$, the following conditions hold: $g^{\underline{2k}}(x) = x$ and $g^{\underline{2k + 1}}(x) = g(x)$. Here, $g^{\underline m}(x)$ denotes the composition of $g(x)$ with itself $m$ times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-x}
Calculate the order of the Weyl group $E_6$ using elementary group theory.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51840}
Given a $202 \times 202$ matrix $A$ with $||A||_2 = 100$ and $||A||_F = 101$, find the sharpest lower bound on the 2-norm condition number $\kappa(A) = ||A||_2 \, ||A^{-1}||_2$. Use the theorem that states $||A||_2 = \sigma_1$ (the largest singular value), $||A^{-1}||_2 = \sigma_r^{-1}$ (where $\sigma_r$ is the smallest singular value), and $||A||_F = \sqrt{\sigma_1^2 + \ldots + \sigma_r^2}$, with $r = \text{rank}(A)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A fair die has two faces each of brown, black, and white colors. Let Y be the random variable representing the number of tosses needed for all three colors to appear at least once, given that two of the three colors appear in the first k-1 tosses and all colors appear in the first k tosses. Calculate the expected value of Y, E[Y].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{2}}
Find the largest integer \( n \) such that \( 64n\log_2 n < 8n^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Compute the sum \( \sum_{n=1}^{2023} (-1)^{\Omega(n)} \left\lfloor \frac{2023}{n} \right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Given a sequence of independent and identically distributed (i.i.d.) uniform random variables $X_1, X_2, \dots$ on the interval $(0,1)$, let $N$ be the minimum value of $n \geq 2$ for which $X_n < X_{n-1}$. Calculate the expected value of $N$, denoted as $E(N)$. \( \text{Express your answer in terms of } e. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Evaluate the integral: $$\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\theta}
Given that $f: D \Rightarrow \mathbb{C}$ is analytic in a domain $D$ and $a$ is a singularity of $f$, with the condition that there exist $r > 0$ and positive numbers $M$ and $N$ such that:
\[
N|z-a|^{-\frac{5}{2}} \leq |f(z)| \leq M|z-a|^{-\frac{7}{2}},
\]
for $0 < |z-a| < r$, determine the order of the pole at $a$. The order is known to be higher than 2 and less than or equal to 4. Determine whether the order is 3 or 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a small town, there are $n\times n$ houses indexed by $(i, j)$ for$1\leq i,j\leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c\leq\frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[\/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters?
$\text{A house indexed by }(i,j)\text{is a neighbor of a house indexed by}$
$(k,l)\mathrm{if}|i-k|+|j-l|=1.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$n^2 - n$

}
There are 15 passengers, each with an assigned seat on a flight. Each passenger can either take their correct seat or a different one. Calculate the probability that at least one passenger sits in their assigned seat.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{1}{e}}
Determine the number of integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 25\) subject to the constraints:
\(-2 \le x_1 \le 5\), \(1 \le x_2 \le 4\), \(-2 \le x_3 \le 6\), \(3 \le x_4 \le 7\), \(2 \le x_5 \le 8\), and \(3 \le x_6 \le 7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3025}
Find the maximum value of the function:
\[ f(y) = \max_{\mu>0} \left[ \exp\left(\frac{-n\mu^{2}}{\sigma^{2}}\right) \exp\left(\frac{2\mu}{\sigma^{2}}\sum_{k=1}^{n}y_{k}\right) \right] \]
where \( y_{k} \) is a random variable with a normal distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In an acute triangle \(ABC\), the altitudes \(AD\) and \(BE\) intersect at point \(H\). The circumcircle of triangle \(ABH\) intersects sides \(AC\) and \(BC\) at points \(F\) and \(G\) respectively. If \(DE = 5\) cm, find the length of \(FG\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the value of \(\alpha > 0\) for which the function \(f(z) = |z|^\alpha z\) belongs to the class \(C^m(\mathbb{C}, \mathbb{C})\), where \(m\) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > m - 1}
Evaluate the limit: \[ \lim_{n\to \infty}\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{\frac{1}{2}}\cdots(2)^{\frac{1}{n}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{\pi^2}{12}}}
Determine the number of positive integers \( n < 180 \) such that \((n^3 + 3n + 1)^{180} \equiv 1 \pmod{180}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Evaluate \( \lim_{\epsilon \to 0} \left(\sup_{z \in \partial B(x,\epsilon)} |G(x,z)|\right) \) for \( n \geq 3 \), where \( G(x,z) = \phi (z-x) - w^{x}(z) \) and the fundamental solution for \( n \geq 3 \) is given by \( \phi(z-x) = \frac{1}{(n-2)w_{n}} \cdot \frac{1}{|z-x|^{n-2}} = \frac{1}{(n-2)w_{n}} \cdot \frac{1}{\epsilon^{n-2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $\\triangle ABC$ be a triangle such that the angle bisector of $\\triangle BAC,$ the median from $B$ to side $AC,$ and the perpendicular bisector of $AB$ intersect at a single point $X.$ If $AX = 5$ and $AC = 12,$ compute $a+b$ where $BC^2=\\tfrac{a}{b}$ and $a,b$ are coprime positive integers.\n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
715}
Given a 3-fold covering $X \to Y$ of a manifold $Y$ ramified over $m$ disjoint submanifolds, and knowing that the Euler characteristic of $Y$ is $\chi(Y) = 1$, determine the Euler characteristic of $X$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are an equal number of red, yellow, green, and blue cards. One card is placed in a box, and the rest are randomly selected. It is observed that red cards are most frequently selected, followed by yellow, green, and blue. What is the most probable color of the card in the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{blue}
Evaluate the integral: \[ \int_0^1{\dfrac{\left(2x^3-3x^2\right)f'(x)}{f(x)}}\,dx \] given that the function $f(x)$ is differentiable on $[0, 1]$ and satisfies the conditions $f(1) = 1$ and $f(x)f(1-x) = e^{x^2 - x}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{10}}
A family buys two insurance policies with independent losses uniformly distributed on the interval from $0$ to $10$. One policy has a deductible of $1$, and the other has a deductible of $2$. The family experiences exactly one loss under each policy. Calculate the probability that the total benefit paid to the family does not exceed $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{59}{200}}
Find the values of $k$ for which the function $f(x) = x^k|x|$ is thrice differentiable at $x = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \geq 3}
Find the value of $n$ such that \( \int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the directional derivative of the function \( f(x,y,z) = \sqrt{x^2 + y^2 + z^2} \) at the point \((0,0,0)\) in the direction of the vector \( v = \left(\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the splitting field of the polynomial \(x^4 + x + 1\) over the finite field \(\mathbb{F}_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{2^4}}
Transform the quadratic form \( F(x) = x_1^2 - 3x_3^2 - 2x_1x_2 + 2x_1x_3 - 6x_2x_3 \) into its normal form over the real number field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y_1^2 - y_2^2}
Find all possible values of \(|z^z|\) using the polar form of the complex number \(z\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Determine the points of discontinuity of the function \( f(x) = \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor 2rx \rfloor}{n^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the interval of values for $p$ such that the integral \( \int_1^{\infty} \frac{\log{x}}{x^p} \,dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Consider a Markov chain model for rolling a fair six-sided die. What is the probability that the sequence "1 followed by 2" occurs before the sequence "two consecutive sixes"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{13}}
Find the simplified form of the sum \(\sum_{k=r}^{n} {k\choose r} p_{n,k}\), where \(p_{n,k}\) represents the number of permutations of \([n]\) with exactly \(k\) fixed points, and is given by \(p_{n,k}={n\choose k}!(n-k)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n!}{r!}}
Find the value of \( n \) that makes the equation \( \binom{n}{n/2} \approx 2^n\sqrt{\frac{2}{0.02 \pi}} \) true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the sum of integer solutions to the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\) within the interval \([-25, 25]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{290}
Evaluate the limit \( \lim_{N \to \infty} \sum_{n=1}^{N} \frac{1}{(n+1)} \sum_{i=1}^{n} \frac{1}{i(n+1-i)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\zeta(3)}
An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive $\sqrt{3}$’s in a row of twelve $\sqrt{3}$’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to $12\sqrt{3}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dfrac{15}{512}$}
Determine the fundamental group of the complex projective space \( \mathbb{P}^n(\mathbb{C}) \) for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider eight light bulbs placed on the eight lattice points in space that are $\sqrt{3}$ units away from the origin. Each light bulb can be either on or off. If two light bulbs that are at most 2 units apart are both on, they explode. Determine the number of configurations where no explosions occur.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Let $n$ be a positive integer. Suppose that the diophantine equation$\\z^n=8x^{2009}+23y^{2009}\\$ uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Given the characteristic polynomial of a linear operator \( S \) as \( p(z) = (z-1)^2(z-2)^2 \), determine the minimal polynomial of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(z - 1)^2(z - 2)^2}
Determine the number of distinct values of \(z^{\frac{m}{n}}\) when \(m\) and \(n\) are integers such that \(n \neq 0\) and \(m\) and \(n\) are relatively prime. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
A square number in the decimal system has the form: $\overline{a b 1 a b}$. Which is this square number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["76176"]}
Given the function \( f = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)\cdots (x-k) \), let \( S \) be the sum of all real roots of \( f \) and \( T \) be the sum of all real roots of \( f' \). Find the value of \( S-T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Let $W$ be a finite-dimensional vector space over $\mathbb{R}$. Let $S:W\to W$ be a linear transformation with minimal polynomial $x^6(x+2)^2$. Determine the minimal polynomial of $S^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 (x + 8)^2}
Determine the maximum number of unique seating plans possible for 15 people in a row of 15 chairs, where two plans are considered identical if they share the same adjacent quadruples. For instance, if the people are seated left to right as 1, 2, 3, 4, ..., 15, then the quadruples 1234, 2345, ..., 12131415 cannot appear in any other plan. What is this maximum number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2730}
Simplify the expression: $$\frac{\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Find a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all real numbers \( x \) and \( y \), the equation \( f(f(x) + y) = 2x + f(f(y) - x) \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = x + c}
Determine the number of homomorphisms from the symmetric group $S_4$ to the cyclic group $\mathbb{Z}_6$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the value of \( \alpha \) that minimizes the condition number \( k(A) = \Vert A \Vert \Vert A^{-1} \Vert \) for the matrix \( A = \begin{bmatrix} 0.1\alpha & 0.1\alpha \\ 1 & 1.5 \end{bmatrix} \), assuming the \( \infty \)-norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5}