inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $\rho_1$ and $\rho_2$ be two representations of a group $G$, with $V$ and $W$ as the corresponding $FG$-modules. The tensor product $\rho_1 \otimes \rho_2$ is defined by $\rho_1 \otimes \rho_2: G \to GL(V \otimes_F W)$, where $(\rho_1 \otimes \rho_2)(g)(v \otimes w) = (\rho_1(g)(v), \rho_2(g)(w))$. Suppose the $FG$-module $V \otimes_{FG} W$ affords the representation $\rho_3$. Determine if $\rho_3 \sim \rho_1 \otimes \rho_2$. Provide your answer as 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Evaluate the limit using L'Hopital's rule to determine if it diverges: $$\lim_{x \to 2} \frac{\int_1^{(x-2)^5} \cos(t^4)\, dt}{(x-2)^{18}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Given triangle $ABC$, reflect vertex $A$ over the perpendicular bisector of side $BC$, then reflect the resulting point over the perpendicular bisector of side $CA$ to obtain point $A'$. Similarly, reflect point $B$ over the perpendicular bisector of side $CA$, and then over the perpendicular bisector of side $AB$ to obtain point $B'$. Finally, reflect point $C$ over the perpendicular bisector of side $AB$, and then over the perpendicular bisector of side $BC$ to obtain point $C'$. If triangle $A'B'C'$ is congruent to triangle $ABC$, does this imply that triangle $ABC$ is equilateral? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Determine whether the limit \( \lim_{(x,y)\to(0,0)}\frac{x^3y^3}{x^8+y^4} \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether a countable set can have uncountably many infinite subsets such that the intersection of any two distinct subsets is finite. Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?

Proposed by Nathan Xiong
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$16$}
Let $\triangle ABC$ be a triangle with side lengths $a$, $b$, and $c$. Circle $\omega_A$ is the $A$-excircle of $\triangle ABC$, defined as the circle tangent to $BC$ and to the extensions of $AB$ and $AC$ past $B$ and $C$ respectively. Let $\mathcal{T}_A$ denote the triangle whose vertices are these three tangency points; denote $\mathcal{T}_B$ and $\mathcal{T}_C$ similarly. Suppose the areas of $\mathcal{T}_A$, $\mathcal{T}_B$, and $\mathcal{T}_C$ are $4$, $5$, and $6$ respectively. Find the ratio $a:b:c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$a:b:c = 22:25:27$}
Problem 4. For a positive integer $n$, two players A and B play the following game: Given a pile of $s$ stones, the players take turn alternatively with A going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $\mathrm{A}$ and $\mathrm{B}$ play perfectly, for how many values of $s$ the player A cannot win?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-1}
Determine the number of ordered pairs \((x, y)\) of real numbers such that \(-100\pi \le x \le 100\pi\), \(-100\pi \le y \le 100\pi\), \(x + y = 20.19\), and \(\tan x + \tan y = 20.19\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{388}
The Caterpillar owns five different matched pairs of socks. He keeps the ten socks jumbled in random order inside a silk sack. Dressing in the dark, he selects socks, choosing randomly without replacement. If the two socks he puts on his first pair of feet are a mismatched pair and the two socks he puts on his second pair of feet are a mismatched pair, then what is the probability that the pair he selects for his third set of feet is a mismatched pair?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dfrac{112}{125}$}
Let $U$ be an open disc centered at $0$ with radius $R > 0$ in the complex plane, and let $f: U \setminus \{0\} \rightarrow \mathbb{C}$ be a holomorphic function. Suppose that for some $a \in \mathbb{C}$, the radial limits of $f$ at $0$ exist and are equal to $a$ for any $\theta \in [0, 2\pi)$:
\[
\lim_{r \rightarrow 0} f(r e^{i \theta}) = a,
\]
where $r$ is a positive real number. Determine whether the limit of $f$ at $0$ exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}
Calculate the expected number of draws from a standard 52-card deck (with replacement) until you have exactly two Kings and two Jacks. Note that Kings and Jacks are not replaced once drawn, and the deck is shuffled after each draw. If you draw a third King or Jack, it is put back into the deck.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{845}{21}}
How many positive integers divide the number $10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Suppose a positive sequence $\{a_n\}_{n=1}^\infty$ has a convergent subsequence with $\lim\limits_{j\to\infty}a_{n_j}=\mu$. If for all $n>0$, $|a_{n+1}-a_n|<\frac{1}{n}$, determine whether $\lim\limits_{n\to\infty}a_{n}=\mu$ is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mu}
Given that Figure 1 is the graph of an even function \( f(x) \) and Figure 2 is the graph of an odd function \( g(x) \), determine the sum of the number of real roots of the equations \( f(f(x))=0 \), \( f(g(x))=0 \), \( g(g(x))=0 \), and \( g(f(x))=0 \). Let the number of real roots of these equations be \( a \), \( b \), \( c \), and \( d \) respectively. Find \( a + b + c + d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Evaluate the integral: \[ \int_0^2 \left((1+x^3)^{1/2} + (x^2+2x)^{1/3}\right) \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a countably infinite set, what is the minimum number of infinite subsets that must be added to a cofinite topology to transform it into a discrete topology?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
For the function \( p_k(x) = (x + \sqrt{x^2 - 4})^k - (x - \sqrt{x^2 - 4})^k \), where \( x > 2 \) and \( k, j \) are natural numbers with \( k > j \), determine if \( p_k(x) > p_j(x) \) holds true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p_k(x) > p_j(x)}
Find the value of $\beta^*$ that minimizes the Kullback-Leibler divergence when the observed random variables $X_1$ and $X_2$ are independent and normally distributed with means $\mu_1$ and $\mu_2$, and variances $\sigma_1^2$ and $\theta\sigma_1^2$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\theta}
Evaluate the principal value of the integral: \[ \text{p.v.} \int_{-\infty}^{\infty} \frac{x \sin 4x}{x^2 - 1} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \cos 4}
Given a ring $S$ with identity and a specified number of elements, determine the number of ring homomorphisms from $\mathbb{Q}[x]$ to $S$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\qquad$. $(a+b$ and $b+a$ are considered the same equation)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1536}
Find the remainder when the integer \( {\left(\sqrt{3} + 5\right)}^{103} - {\left(\sqrt{3} - 5\right)}^{103} \) is divided by 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the exact value of the norm \( \|T\| \) for the operator \( T: \ell^2 \to \ell^2 \) defined by:

\[\begin{cases} (Tx)_1 = 0, \\ (Tx)_n = -x_n + \alpha x_{n+1}, \quad n\ge 2 \end{cases} \]

where \( \alpha \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + |\alpha|}
Let $ABCD$ be a convex quadrilateral with positive integer side lengths, where $\angle{A} = \angle{B} = 120^{\circ}$, $|AD - BC| = 42$, and $CD = 98$. Determine the maximum possible value of $AB$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69}
Consider the probability space \(([0,1]^2, \mathcal{B}[0,1], Leb_2)\). Let \(X: [0,1]^2 \rightarrow \mathbb{R}\) and \(Y: [0,1]^2 \rightarrow \mathbb{R}\) be random variables defined by \(X(\omega_1, \omega_2) = \omega_1 + \omega_2\) and \(Y(\omega_1, \omega_2) = \omega_2 - \omega_1\). Find the value of the joint cumulative distribution function \(F_{X,Y}(0.58,0.35)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.154975}
Given that
 \( \frac{1}{2!{17}!} + \frac{1}{3!{16}!} + \frac{1}{4!{15}!} + \frac{1}{5!{14}!} + \frac{1}{6!{13}!} + \frac{1}{7!{12}!} + \frac{1}{8!{11}!} + \frac{1}{9!{10}!} = \frac{N}{1!{18}!} \)
 find the greatest integer that is less than \( \frac{N}{100} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137
}
Let $(X_1, X_2, \ldots)$ be an independent sequence of random 2-dimensional vectors, where for each $n$, $X_n$ is uniformly distributed on the square with vertices $[\pm n, \pm n]$. Calculate the probability \( P(\{w: |X_n(w)| \rightarrow \infty \ \text{as} \ n \rightarrow \infty\}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the largest integer $n$ such that $n! < 100^{n-1}$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{263}
\n\t Consider a three-person game involving the following three types of fair six-sided dice.\n\\begin{itemize}\n\t\\item Dice of type $A$ have faces labelled $2$, $2$, $4$, $4$, $9$, $9$.\n\t\\item Dice of type $B$ have faces labelled $1$, $1$, $6$, $6$, $8$, $8$.\n\t\\item Dice of type $C$ have faces labelled $3$, $3$, $5$, $5$, $7$, $7$.\n\\end{itemize}\nAll three players simultaneously choose a die (more than one person can choose the same type of die, and the players don\u0027t know one another\u0027s choices) and roll it.  Then the score of a player $P$ is the number of players whose roll is less than $P$\u0027s roll (and hence is either $0$, $1$, or $2$).\nAssuming all three players play optimally, what is the expected score of a particular player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{8}{9}$}
Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the number of zeros of the polynomial \( z^4 + 3z^2 + z + 1 \) that lie in the right half-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
$ P$ is the intersection point of diagonals of cyclic $ ABCD$. The circumcenters of $ \triangle APB$ and $ \triangle CPD$ lie on circumcircle of $ ABCD$. If $ AC + BD = 18$, then area of $ ABCD$ is ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \frac {81\sqrt 3}{4}$
}
Given a distributional equation $u=h$ with $u \in L^2$ and $h\in W^{-1,2}$, determine if $h\in L^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h \in L^2}
Let \( \omega \) be a complex number which satisfies the equation \( i \omega = \frac{u^2 -1}{u^2 +1} \) for some suitable complex number \( u \). Find the value of \( \tan^{-1}{\omega} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-i \ln u}
Evaluate the limit: $$\lim_{x \to 0} \left( 1 + \sin \left( \frac{3}{x} \right) \right)^x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a function $f(x)$ such that the $n$-th derivative of $f$ evaluated at $0$ is $n^{n-1}$, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-W(-x)}
In how many ways can 9 black and 9 white rooks be placed on a 6 × 6 chessboard such that no white rook can capture a black rook? A rook can capture another piece if it is in the same row or column as the other piece, with no other pieces between them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Find the limit of the sequence defined by the recurrence relation \(x_{n+1} = \alpha x_n + (1+\alpha) x_{n-1}\) for \(\alpha \in (0,2)\) and all \(n \geq 1\), in terms of \(\alpha\), \(x_0\), and \(x_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the equation \( \theta+\tan^{-1}\left(\frac{6}{4-x}\right)+\tan^{-1}\left(\frac{3}{x}\right)=\pi \), where \( x \) is a function of time \( t \), find the value of \( \frac{d\theta}{dt} \) when \( x=2 \) and \( \frac{dx}{dt}=4 \) cm/sec.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{65}}
In an exotic country, the National Bank issues coins that can take any value in the interval [0, 1]. Find the smallest constant $ c > 0 $ such that the following holds, no matter the situation in that country:

Any citizen of the exotic country that has a finite number of coins, with a total value of no more than 1000, can split those coins into 100 boxes, such that the total value inside each box is at most $ c $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$11 - \frac{1}{91}$
}
Identify a topological space whose fundamental group is isomorphic to \(\mathbb{Z}_3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L(3,1)}
Determine the number of subfields $K$ of the complex splitting field $L$ of the polynomial $f(x) = x^5 - 4x + 2$ over $\mathbb{Q}$, such that the degree of the extension $(K : \mathbb{Q}) = 2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the value of \( x \) such that the equation \( e^{e^{10^{10^{2.8}}}} = 10^{10^{10^{10^{x}}}} \) holds true, given that you can only change the topmost number on the right side while keeping all other numbers as 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.8}
Calculate the value of \( \displaystyle \sum ^{2020}_{n=1} n^{2019} \pmod{2019} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine all positive integers $n$ such that $n$ divides $a^{25} - a$ for every positive integer $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2730}
Calculate the expected value of the largest power of 2 that divides a randomly chosen non-zero positive even integer $E$, expressed as $E = A \cdot 2^n$, where $A$ is an odd number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x=0}
Find the elements of $U(L)$ that satisfy the grouplike element property for a Lie algebra $L$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $F$ be a finite field of order 32. Determine the number of solutions $(x, y) \in F \times F$ for the equation $x^2 + y^2 + xy = 0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine all real numbers \(x\) for which the series \(\sum_{n=1}^\infty \left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\frac{\sin nx}{n}\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Suppose $\alpha$ is a root of the polynomial $x^3 + x^2 + 1$ in the finite field $\mathbb{F}_2$, and let $K = \mathbb{F}_2(\alpha)$. If $f$ is an irreducible polynomial in $K[x]$ of degree 4, and $\beta$ is a root of $f$, consider the splitting field $L$ of $f$ over $K$. Given that $K$ has 8 elements, determine the number of elements in the field $L$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4096}
Given a function \(f: \mathbb{R} \to \mathbb{R}\) that satisfies the equation \[f(2xy) + f(f(x + y)) + f(x + y) = xf(y) + yf(x)\] for all real numbers \(x\) and \(y\), determine the sum of all possible absolute values \(|f(48)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{186}
The sports plane flew along a diamond-shaped route in windy weather. He flew through the first three sides of the rhombus in $a $, $b$ and $c$ hours, respectively. How long did it take him to cover the fourth side of the diamond? (The speed of an aircraft is a vector equal to the sum of two vectors: the aircraft own speed and the wind speed. Wind speed is a constant vector. The aircraft own speed is a vector of constant length).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$t = \dfrac{ac}{b}$}
Determine the interval of convergence for the series \(1+\frac{x}{2}+\frac{2! x^2}{3^2}+\frac{3!x^3}{4^3}+\ldots\) using an appropriate convergence test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-e, e)}
Determine the maximum number of axes of symmetry that a three-dimensional figure, consisting of three lines, can have if no two of the lines are parallel and no two coincide.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Solve the equation \(3^{\log_a x} + 3x^{\log_a 3} = 2\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{-\log_3 2}}
Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10999
}
Determine the number of terms required to approximate the sum of the series \(\sum_{n=0}^\infty \frac{(-1)^n(5)^{2n+1}}{(2n+1)(2n+1)!}\) to within an error of \(10^{-6}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the value of \(\left(\frac{1}{4}\right)!\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9064}
Is the conditional variance Var(X|\(\mathcal{P}(\Omega)\)) equal to 0 for any random variable X?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the vector space $[0,1]^\mathbb{N}$ of real sequences with values in the interval $[0,1]$ together with the product topology. Determine if the subspace consisting of all monotone sequences with values in $[0,1]$ is connected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Connected}
Determine the value of $f(0,0)$ for polynomials $f(x,y)$ and $g(x,y)$ in two variables, given that $g(0,0)=6$ and \((x^2+2xy+3y^2)f(x,y)=(4x^2+5xy+6y^2)g(x,y)\) holds for all real numbers $x$ and $y$ satisfying \(x^2y+6y=xy^3+5x^2+2x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Determine whether the $n$-th singular homology group of an oriented, compact, and connected $n$-dimensional smooth manifold with boundary, denoted as $H_n(M)$, is trivial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of ordered pairs of integers \((m, n)\) such that \(mn \geq 0\) and \(m^3 + n^3 + 99mn = 33^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements diﬀer by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Suppose $h$, $i$, $o$ are real numbers that satisfy the products $hi = 12$, $ooh = 18$, and $hohoho = 27$. Find the value of the product $ohio$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432
}
Consider a continuous and non-decreasing distribution function $F$ with an associated density function $f$. Let $S_1$ be the set where $F(s+\varepsilon)> F(s-\varepsilon)$ for all $\varepsilon>0$, and $S_2$ be the closure of the set where $f(s)>0$. Determine if these two sets are equal, i.e., is $S_1=S_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_1 = S_2}
Given a pair (N, S) in the Tonalpohualli calendar, where N is the number and S is the symbol, determine the original day number from 1 to 260 that corresponds to this pair. For example, find the day number that produces the pair (3, 12).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172}
Determine whether the series \( \sum\limits_{n=1}^{\infty}\frac{n^{n-1}}{n!e^n} \) converges to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the product measure \((\mu \times \nu)\) defined on \(\mathbb{R} \times \widetilde{\mathbb{R}}\), where \(\mu\) is the Lebesgue measure on \(\mathbb{R}\) and \(\nu\) is the counting measure on \(\widetilde{\mathbb{R}}\), the real line with the discrete topology. Let \(E = \{(x, x) \mid x \in [0, 1]\}\). Determine whether \((\mu \times \nu)(E) = 0\) when \(\widetilde{\mathbb{R}}\) is endowed with the discrete topology and the associated discrete \(\sigma\)-algebra.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the dimension of the symplectic group \(\operatorname{Sp}(2n, \mathbb{R})\), which is the group of invertible matrices \(A\) such that \(\omega = A^T \omega A\), where \(\omega = \begin{pmatrix} 0 & \mathit{Id}_n \\ -\mathit{Id}_n & 0 \end{pmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(2n + 1)}
Given a set of points $X$ in $\mathbb{R}^2$, define $g(X)$ as the geometric median of $X$, where $g(X) = \arg \min_{y \in \mathbb{R}^2} \sum_{x_i \in X} ||x_i - y||_2$. For two neighboring sets $X$ and $X'$ that differ by only one element ($X \sim X'$), determine the supremum of the Euclidean distance between their geometric medians, $\sup_{X':X \sim X'} ||g(X) - g(X')||_2$. Find this supremum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}