inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate \( \omega(X,Y) \) for the differential form \( \omega = 2x \, dx \wedge dy + y^2 \, dx \wedge dz \) and the vector fields \( X = x^2y \frac{\partial}{\partial y} + x \frac{\partial}{\partial z} \) and \( Y = x \frac{\partial}{\partial y} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Using dimensional analysis, find the value of the exponent 'm' in the equation:
\[ \int \frac{x^m \, dx}{\sqrt{a^2 - x^2}} = b^n \sin^{-1} \left[\frac{x}{a}\right] \]
where 'a' and 'b' are constants with dimensions of length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of isomorphism classes of countable models of the empty theory over one binary relation. In other words, how many countable unlabeled graphs are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Solve the equation \( 2x - e^{-x} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3517}
Evaluate the limit: \[ \lim_{n \to \infty} \frac{n! F_n(1)}{\ln n} \] where $F_0(x) = \ln x$ and for $n \geq 0$ and $x > 0$, $F_{n+1}(x) = \int_0^x F_n(t) \, dt$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given two independent arrival processes $N_t = X_t + Y_t$, where $X_t$ and $Y_t$ each occur with probability rate $0.5$, find the expected value $E[N_t|X_t = n]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + \frac{t}{2}}
Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3430}
Calculate the line integral \( I = \int \limits_\Gamma x^2 \, dy - 2xy \, dx \), where \( \Gamma \) is the part of the curve \( x^4 - y^4 = 6x^2y \) from point \( A(-4\sqrt{2}, 4) \) to point \( B(0,0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{44}{3}}
Given $X \sim N(0,1)$ and $Y \sim N(X, 1)$, where $Y-X$ is independent of $X$, calculate the probability $P(Y<0 \mid X>0)$. This probability is relevant in the context of pricing a Slalom option in a Normal model without drift.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the number of positive integer solutions \((a, b, c, d)\) for the following system of equations:

\[\sqrt{2020}(\sqrt{a} + \sqrt{b} )= \sqrt{ (c+ 2020)(d+ 2020)}\]
\[\sqrt{2020}(\sqrt{b} + \sqrt{c} )= \sqrt{ (d+ 2020)(a+ 2020})\]
\[\sqrt{2020}(\sqrt{c} + \sqrt{d} )= \sqrt{ (a+ 2020)(b+ 2020)}\]
\[\sqrt{2020}(\sqrt{d} + \sqrt{a} )= \sqrt{ (b+ 2020)(c+ 2020)}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Determine the number of elements \( \alpha \in \mathbb{F}_{3^5} \) such that the field extension \( \mathbb{F}_{3}(\alpha) \) is equal to \( \mathbb{F}_{3^5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Evaluate the integral \( \int_{0}^{2\pi}\frac{\cos(3x)}{5-4\cos(x)}dx \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Evaluate the limit: $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\lambda_1+\lambda_2x)^n-x^n}{1-x}\,dx,$$ where $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln \lambda_2}
Compute the value of the function defined by the following recursive relations: \( f(x,0)=f(x-1,1) \), \( f(0,y) = (y+1) \mod 5 \), and \( f(x,y) = f(x-1, f(x,y-1)) \). Find \( f(333,3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the Generalized Directional Derivative (Clarke Derivative) of the function \( f(x) = -|x| \) at the point \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|v|}
Determine the range of \( \lambda > 0 \) for which the integral \( \int _0 ^\infty \frac{e^{\sin x}\sin 2x}{x^\lambda} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
In trapezium $ABCD$, the sides $AB$ and $CD$ are parallel. A semicircle with diameter $AB$ touches $CD$ and passes through the midpoints of diagonals $AC$ and $BD$. Determine the measure of angle $\angle BOE$, where $O$ is the center of the semicircle and $E$ is the intersection point of $AC$ with the semicircle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Let $H$ be the line at infinity in $\mathbb{CP}^{2}$, and let $P$ and $Q$ be distinct points on $H$. Let $X$ be the blow-up of $\mathbb{CP}^{2}$ at $P$ and $Q$; let $E_{1}$ and $E_{2}$ be the exceptional divisors over $P$ and $Q$, respectively, and let $L$ be the proper transform of $H$. What is the self-intersection number of $L$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the limit of the integral over the semicircle as \( R \to \infty \):

\[ \lim_{R\to\infty}\int_{\Gamma_R}\frac{e^{izt}}{\pi\cos(iz)}dz \]

where \( t > 0 \) and \( \Gamma_R \) is the upper semicircle of radius \( R \) in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Alison is eating 2401 grains of rice for lunch. She eats th
 rice in a very peculiar manner: every step, if she has only
 one grain of rice remaining, she eats it. Otherwise, she
 nds the smallest positive integer \( d > 1 \) for which she ca
 group the rice into equal groups of size \( d \) with none left
 over. She then groups the rice into groups of size \( d \) ,eats
 one grain from each group, and puts the rice back into a
 single pile. How many steps does it take her to finish all
 her rice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
17}
Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a
 time in random order. Given that the youngest person leaves the meeting sometime before the oldest
 person leaves the meeting, the probability that the third, fourth, and fifth people to leave the meeting do so in order of their ages
 (youngest to oldest) is \( \frac{m}{n} \) ,where \( \mathrm{m} \) and \( \mathrm{n} \) are relatively prime positive integers. Find \( m + n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25
}
Evaluate the limit: \[ \lim_{x\to \infty} \frac{e^{\frac{1}{x^2}}-1}{2\tan^{-1} (x^2)-\pi} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let
\[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\]
where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate
\[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Let $A$ be a $2\times2$ invertible matrix with real entries such that $\det(A) = k$. Given that $\det(A + k \operatorname{adj}(A)) = 0$, find the value of $\det(A - k \operatorname{adj}(A))$. \( \operatorname{adj}(A) \) denotes the adjugate of matrix $A$. \( k \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int_{0}^{2\pi} e^{ib \sin{\theta}} \, d\theta \) where \( b \) is a constant, using analytic methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi J_0(b)}
Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
180
}
Determine the number of trees on the vertex set \( \{v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8\} \) that have at least two vertices with a degree of at least 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102200}
Evaluate the infinite series \( \sum_{n = 1}^\infty \left( \frac{1}{n} - \frac{1}{n + x} \right) \) for \( x = 0.500 \). Round your answer to 3 decimal places and ensure the precision is at least 12 significant digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.614}
Determine the number of elements in the quotient ring \( \mathbb{Z}[\sqrt{-19}]/I \), where \( I=(18+\sqrt{-19}, 7) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given $X \sim \text{Binomial}(n, p)$ and $Y \sim \text{Bernoulli}\left(\frac{X}{n}\right)$, find the conditional expectation $E[X \mid Y]$. Express your answer in terms of $n$, $p$, and $Y$. \( \text{Note: } Y \text{ is a Bernoulli random variable whose parameter depends on } X. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Y + (n - 1)p}
A square contains \( K \) points (\( K > 2 \)). Determine the minimum number of triangles required to divide the square such that each triangle contains at most one point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2K + 2}
Given a call graph $G = (V, E)$ where each vertex $v \in V$ represents a phone number and each edge $\{v, w\} \in E$ indicates a call between $v$ and $w$, a set $S \subseteq V$ of authorized seeds with $|S| = 300$ is monitored. Calls can be investigated for vertices within a distance of 3 hops from any seed. If each phone number has interacted with exactly 100 other numbers, determine the tightest upper bound on the number of phone numbers that can be queried without a warrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{297030300}
Determine the number of distinct squares of elements in the alternating group $A_4$, considering that it has 12 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Define a function $f$ on the unit interval $0 \le x \le 1$ by the rule
\[f(x)
= \begin{cases}
		    1-3x &  \text{if } 0 \le x < 1/3 \, ; \\
			  3x-1 &  \text{if } 1/3 \le x < 2/3 \, ; \\
			  3-3x &  \text{if } 2/3 \le x \le 1 \, .
		  \end{cases}\]Determine $f^{(2018)}(1/730)$. Recall that $f^{(n)}$ denotes the $n$th iterate of $f$; for example, $f^{(3)}(1/730) = f(f(f(1/730)))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{730}}
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
$a_1=2021^{2021}$
$0 \le a_k < k$ for all integers $k \ge 2$
$a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0
 
}
Calculate the expected number of times the random draw function \( R(n) \), which selects integers between \( 0 \) and \( n-1 \) (inclusive), must be applied starting with \( 10^{100} \) until the integer \( 0 \) is obtained.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100 \ln 10 + \gamma}
Find the derivative of the matrix product $C = AB$ with respect to the matrix $B$, where $A \in \mathbb{R}^{n \times d}$ and $B \in \mathbb{R}^{d \times m}$. Express your answer in terms of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_m \otimes A}
Determine the value of $n$ for which two $n \times n$ matrices are similar if they have the same minimal polynomial and the same characteristic polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If a square matrix $B$ has $k$ distinct nonzero eigenvalues, what is the minimum possible nullity of $B$? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit \( \lim_{x\to\infty}\left(\frac{\max\{f(x),g(x)\}}{\min\{f^4(x),g^4(x)\}}\right) \) given that \( \lim_{x\to\infty}f(x)=\lim_{x\to\infty}g(x)=\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the minimum number of positive integers $a$ and $b$ such that every integer greater than a certain threshold can be expressed as a non-negative integer combination of $a$ and $b$. What is this minimum number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a sequence of vehicles, two-thirds of all trucks have a car behind them, and one-fifth of all cars have a truck behind them. What proportion of the vehicles are cars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{13}}
Find the minimum value of \(|z|\) for complex numbers \(z\) satisfying the equation \(|(z+i)^2-3| = 2|z-i|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \sqrt{2}}
In a survey, $82.5\%$ of respondents believed that mice were dangerous. Of these, $52.4\%$ incorrectly thought that mice commonly caused electrical fires. Given that these 27 respondents were mistaken, determine the total number of people surveyed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Compute the pullback of the 2-form $\alpha(x, y) = xy \, dx \, dy$ under the function $f:\mathbb{R}^2 \to \mathbb{R}^2$ given by $f(x, y) = (e^{2x}, xy)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x^2y e^{4x} dx dy}
计算下列实变函数定积分．

$\int _ { - \infty } ^ { \infty } \frac { x \sin m x } { 2 x ^ { 2 } + a ^ { 2 } } d x ( m > 0 , a > 0 )$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$e ^ { - m a / \sqrt { 2 } } \pi / 2 ,$}
Determine the number of ring homomorphisms from the ring of real numbers to itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
10 play kabadi, 35 play kho kho only, and some people play both games. In total, there are 45 players. How many people play both games?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
You cook a pot of beef stew for you and your roommate. Normally, it lasts 7 days between the two of you with each of you eating equal portions. Now, 3 guests are staying with you and your roommate, and everyone will be eating from the stew. You observe that your roommate and the guests eat 1.5 times more than you do. How many days will the beef stew now last if everyone eats their respective portions daily?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of \( k \) such that the solutions to the differential equation \( y'' + 2y = \cos(kt) \) are unbounded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\angle BAC = 60^{\circ}$, $D$ is a point on side $BC$ such that $AD = \sqrt{7}$, and $BD:DC = 2c:b$, then the minimum value of the area of $\triangle ABC$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
Milo rolls five fair dice, which have 4, 6, 8, 12, and 20 sides respectively, and each one is labeled $1$ to $n$ for the appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{248}
Determine the Euler characteristic of a Y-shaped pipe, which has three openings that converge together.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
On each of the five cards, a number is written. The cards are lying on the table with the numbers facing down. By paying a ruble, you can point to any three cards, and you will be told the sum of the numbers written on them. For what minimum price can you certainly determine the sum of all five numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Define the function $f(x)$ on $\mathbb{R}$ such that $f(0)=0$, $f(x)+f(1-x)=1$, $f\left(\frac{x}{5}\right)=\frac{1}{2}f(x)$, and for $0 \leq x_1 < x_2 \leq 1$, $f(x_1) \leq f(x_2)$. Find the value of $f\left(\frac{1}{2007}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{32}}
Use differentials to approximate the percentage decrease in mass of an object moving at a velocity of 0.85c as its velocity decreases to 0.83c, according to Einstein's Special Theory of Relativity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.1\%}
Find the value of the double integral $$\int_{-1}^{1}\int_{-1}^{1}|x+y|\, {\rm d}x\, {\rm d}y$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8/3}
Misha has thought of a five-digit number with all distinct digits. Igor wants to guess this number. In each turn, Igor can choose several positions of the number, and Misha will reveal the digits in those positions in random order. What is the minimum number of turns Igor needs to guarantee that he can determine the entire number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the smallest non-negative integer \( n \) such that the ring \( S_n = C^n([0,1], \mathbb{R}) \), consisting of all continuously \( n \)-times differentiable functions on the interval \([0,1]\), is not an integral domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The number of lines that are equidistant from point A (-3, 2) and B (1, 1) and have a distance of 2 is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Count the number  $N$  of all sets  $A:=\{x_1,x_2,x_3,x_4\}$  of non-negative integers satisfying  $$ x_1+x_2+x_3+x_4=36 $$  in at least four different ways.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9139}
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{def}$ 是偶数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let \( I \) be the center of the inscribed circle in triangle \( ABC \). It is given that \( CA + AI = BC \). Determine the value of the ratio \( \frac{\widehat{BAC}}{\widehat{CBA}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider the set \( A = \{1, 2, 3, 4, \cdots, 2^n\} \), where \( n \geq 2 \). If the sum of two elements in set \( A \) is a power of 2, then exactly one of them belongs to subset \( B \). Determine the number of such subsets \( B \) in set \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n+1}}
A cube with edge length 8 is balanced on one of its vertices on a horizontal table such that the diagonal from this vertex through the interior of the cube to the farthest vertex is vertical. When the sun is directly above the top vertex, the shadow of the cube on the table is a regular hexagon. The area of this shadow can be written in the form \( a \sqrt{b} \), where \( a \) and \( b \) are positive integers and \( b \) is not divisible by any perfect square larger than 1. Find the value of \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
All minions either have $1$ or $2$ eyes, and have one of $4$ possible hairstyles. They are all thin and short, fat and short, or thin and tall. Gru doesn’t want to have any $2$ minions that look exactly alike, so what is the maximum possible amoount of minions can he get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
24
}
Evaluate the limit \(\lim\limits_{n \to \infty} \frac{(n+3)(n+1)^{n+1}}{(n+2)^{n+2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Let \( f \) be a holomorphic function on the unit disk \( \mathbb{D} \) such that \( |f(z)| < 1 \) for all \( |z| < 1 \). Given that \( f(0) = \frac{1}{2} \), find a bound for \( |f'(0)| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Solve the recurrence relation given by \( T(1) = 1 \) and \( T(n) = 3T\left(\sqrt[3]{n}\right) + \log_{2}(\log_{2}n) \) using the master theorem, assuming \( n = 2^k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(\log n)}
Let BC be the latus rectum of the parabola \(y^2 = 4ax\) with vertex A. Determine the minimum length of the projection of BC on a tangent drawn in the portion BAC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}a}
Calculate the expected number of tails when tossing a fair coin until you achieve $n$ consecutive heads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 1}
Evaluate the sum: $$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1} \sin(nx)}
Determine the smallest integer $n$ such that the error in approximating $\sin(x)$ using its Taylor series expansion up to the $n$th term is less than $10^{-8}$ for all $x$ in the interval $(-0.5, 0.5)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the maximum number of elements of order 5 in a group of order 80 using Sylow's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
How many ways are there to win tic-tac-toe in $\mathbb{R}^{n}$? (That is, how many lines pass through three of the lattice points $(a_{1}, \ldots, a_{n})$ in $\mathbb{R}^{n}$ with each coordinate $a_{i}$ in $\{1,2,3\}$? Express your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\left(5^{n}-3^{n}\\right) / 2"]}
Find a linear transformation \( T: V \rightarrow W \) where \( V = C^\infty(-\infty, \infty) \), the vector space of real-valued continuous functions with continuous derivatives of all orders on \((-\infty, \infty)\), and \( W = F(-\infty, \infty) \), the vector space of real-valued functions defined on \((-\infty, \infty)\). The kernel of \( T \) should be \( P_3 \), the space of polynomials of degree \( \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{T(f) = f^{(4)}}
Find the height \( m \) of the horizontal line that divides the volume of the solid obtained by rotating the region bounded by \( y = x^2 \), \( y = 9 \), and the \( y \)-axis, around the vertical line \( x = 3 \), into two equal parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.18}
Evaluate the limit: $$\lim_{q\rightarrow 1} (1-q^{2}){\psi^2(q)}$$ using a q-continued fraction for the Ramanujan theta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Evaluate the limit using Taylor series: $$\lim_{h\to0}\frac{f(a-5h^2)-f(a+3h)}{h}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3f'(a)}