Autodata: An agentic data scientist to create high quality synthetic data
Paper • 2606.25996 • Published • 21
uid stringlengths 11 11 | content stringlengths 112 51.3k | question stringlengths 1 13.2k | reasoning stringlengths 1 49.8k | answer stringlengths 1 2.11k ⌀ | question_model stringclasses 13
values | reasoning_model stringclasses 2
values | filter_qwen bool 2
classes | filter_teutonic_king bool 2
classes | category stringclasses 122
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item_000001 | Let $ \textbf{Set} $ denote the category of sets and functions. Consider two functors $ F, G : \textbf{Set}^{op} \to \textbf{Set} $ defined as follows:
- $ F(X) = \text{Hom}_{\textbf{Set}}(X, \textbf{2}) $, where $ \textbf{2} = \text{{0, 1}} $ is the discrete two-element set.
- $ G(X) = \text{Sub}(X) $, the set of all... | Let $ \textbf{Set} $ denote the category of sets and functions. Consider two functors $ F, G : \textbf{Set}^{op} \to \textbf{Set} $ defined as follows:
- $ F(X) = \text{Hom}_{\textbf{Set}}(X, \textbf{2}) $, where $ \textbf{2} = \text{{0, 1}} $ is the discrete two-element set.
- $ G(X) = \text{Sub}(X) $, the set of all... | Step 1: Define the component maps. For each set $X$, define
\[
\theta_X : F(X)=\operatorname{Hom}_{\mathbf{Set}}(X,\mathbf 2)\longrightarrow G(X)=\operatorname{Sub}(X)
\]
by
\[
\theta_X(\chi)=\chi^{-1}(\{1\})=\{x\in X\mid \chi(x)=1\}.
\]
Thus a function $\chi:X\to\{0,1\}$ is sent to the subset of $X$ on which it takes ... | \theta_X(\chi)=\chi^{-1}(\{1\})\text{ defines a natural isomorphism }F\cong G. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Category Theory | false |
item_000004 | Suppose we are estimating the definite integral $ I = \frac{1}{2} \times \text{erf} ig( \frac{1}{2} ig) $ using a Monte Carlo estimator based on $ N $ independent and uniformly distributed random samples from the interval $ [0, 1] $. Let $ ar{X}_N $ be the sample mean of the function $ f(x) = \frac{1}{\frac{1}{2} \t... | Suppose we are estimating the definite integral $ I = \frac{1}{2} \times \text{erf} ig( \frac{1}{2} ig) $ using a Monte Carlo estimator based on $ N $ independent and uniformly distributed random samples from the interval $ [0, 1] $. Let $ ar{X}_N $ be the sample mean of the function $ f(x) = \frac{1}{\frac{1}{2} \t... | Step 1: The integral is I = (1/2) erf(1/2). The Monte Carlo estimator uses samples U_i ~ Uniform[0,1] and computes the sample mean of g(U_i) = erf(U_i/2), since I = E[erf(U/2)] for U~Uniform[0,1].
Step 2: The problem defines f(x) = [erf(x/2)] / [(1/2) erf(1/2)] = g(x)/I. Therefore E[f(U)] = E[g(U)]/I = I/I = 1.
Step 3:... | \mathbb{E}[|\bar{X}_N - 1|] \approx \sqrt{\frac{2\,\operatorname{Var}(f(x))}{\pi N}} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Applied Mathematics | false |
item_000005 | A small town has 1,000 residents and is experiencing an outbreak of a contagious disease. The town is divided into three compartments: Susceptible (S), Infected (I), and Recovered (R). At time t = 0, 990 individuals are susceptible, 10 are infected, and 0 are recovered. The dynamics are modeled as a discrete-time SIR m... | A small town has 1,000 residents and is experiencing an outbreak of a contagious disease. The town is divided into three compartments: Susceptible (S), Infected (I), and Recovered (R). At time t = 0, 990 individuals are susceptible, 10 are infected, and 0 are recovered. The dynamics are modeled as a discrete-time SIR m... | Step 1: Define the discrete-time SIR model from the stated assumptions.
Let S_t, I_t, R_t be the compartment sizes at day t. Each day:
- New infections = S_t * [1 - (1 - 0.002)^{I_t}]
- New recoveries = I_t * 0.2
- S_{t+1} = S_t - (new infections)
- I_{t+1} = I_t + (new infections) - (new recoveries)
- R_{t+1} = R_t + ... | 743 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Applied Mathematics | false |
item_000006 | Let $ a, b, c $ be positive integers such that $ a^2 + b^2 = c^2 $ and $ ab = 2(c - a - b) $. Prove that $ c $ is divisible by 5.
Assume such positive integers exist.
From a^2+b^2=c^2 and a,b>0, we have c^2=a^2+b^2<a^2+b^2+2ab=(a+b)^2.
Since c>0, taking square roots gives c<a+b.
Thus c-a-b<0, so 2(c-a-b)<0.
But ab>0, s... | Let $ a, b, c $ be positive integers such that $ a^2 + b^2 = c^2 $ and $ ab = 2(c - a - b) $. Prove that $ c $ is divisible by 5. | Assume such positive integers exist.
From a^2+b^2=c^2 and a,b>0, we have c^2=a^2+b^2<a^2+b^2+2ab=(a+b)^2.
Since c>0, taking square roots gives c<a+b.
Thus c-a-b<0, so 2(c-a-b)<0.
But ab>0, so ab=2(c-a-b) equates a positive integer to a negative integer, impossible.
Therefore no positive integers a,b,c satisfy both give... | \text{No such positive integers exist; the claim is vacuously true.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Number Theory | false |
item_000007 | Let $ \textbf{Set} $ be the category of sets and functions. Define two functors $ F, G: \textbf{Set} \to \textbf{Set} $ as follows:
- For any set $ X $, let $ F(X) = X \times X $ and for any function $ f: X \to Y $, let $ F(f): X \times X \to Y \times Y $ be the function $ F(f)(x, x') = (f(x), f(x')) $.
- For any set ... | Let $ \textbf{Set} $ be the category of sets and functions. Define two functors $ F, G: \textbf{Set} \to \textbf{Set} $ as follows:
- For any set $ X $, let $ F(X) = X \times X $ and for any function $ f: X \to Y $, let $ F(f): X \times X \to Y \times Y $ be the function $ F(f)(x, x') = (f(x), f(x')) $.
- For any set ... | Step 1: Identify the functors. We have $F,G:\mathbf{Set}\to\mathbf{Set}$ where $F(X)=X\times X$ with $F(f)(x,x')=(f(x),f(x'))$, and $G(X)=X$ with $G(f)=f$ (the identity functor).
Step 2: Define the natural transformation. For each set $X$, define $\theta_X:F(X)=X\times X\to G(X)=X$ by $\theta_X(x,x')=x$ (the first pro... | \theta:F\to G\text{ is defined by }\theta_X(x,x')=x\text{ (first projection); it is natural since }\theta_Y(f(x),f(x'))=f(x)=f(\theta_X(x,x'))\text{ for all }f:X\to Y. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Category Theory | false |
item_000008 | Let $ f(x) = \frac{\tan(x)}{x} $ for $ 0 < x \frac{\tan(2)}{2} $, the value of the integral $ I = \frac{\tan(x)}{x} \tan(x) $ with $ n = 20 $ subintervals (i.e., $ 10 $ parabolic segments). Use the error bound for the composite Simpson's rule, which is given by $ \frac{(b - a) h^4}{180} \tan(x) $, the maximum of the fo... | Let $ f(x) = \frac{\tan(x)}{x} $ for $ 0 < x \frac{\tan(2)}{2} $, the value of the integral $ I = \frac{\tan(x)}{x} \tan(x) $ with $ n = 20 $ subintervals (i.e., $ 10 $ parabolic segments). Use the error bound for the composite Simpson's rule, which is given by $ \frac{(b - a) h^4}{180} \tan(x) $, the maximum of the fo... | The problem statement contains severe OCR/encoding corruption, but the intended problem is recognizable: estimate the maximum error in the composite Simpson's rule approximation of
\[ I = \int_0^2 \frac{\tan x}{x}\,dx \]
using $n=20$ subintervals ($10$ parabolic segments), with the error bound
\[ |E| \le \frac{(b-a)h^4... | \text{No finite error bound exists; the integral diverges due to the pole of } \tan x \text{ at } x=\pi/2 \in (0,2). | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Numerical Analysis | false |
item_000009 | Let $(X, \textbf{X},
u)$ be a finite measure space. Consider a sequence of measurable functions $(f_n)$ defined on $X$ such that for each $n \to \textbf{X}$ and all $x \to X$, $|f_n(x)| \to \frac{1}{1 + n^2 x^2}$. Suppose further that $f_n(x) \to f(x)$ pointwise for all $x \to X$. Show that the limit function $f$ is i... | Let $(X, \textbf{X},
u)$ be a finite measure space. Consider a sequence of measurable functions $(f_n)$ defined on $X$ such that for each $n \to \textbf{X}$ and all $x \to X$, $|f_n(x)| \to \frac{1}{1 + n^2 x^2}$. Suppose further that $f_n(x) \to f(x)$ pointwise for all $x \to X$. Show that the limit function $f$ is i... | Step 1: Identify the domination. For all $n \in \mathbb{N}$ and $x \in X$, we have $|f_n(x)| \leq \frac{1}{1+n^2x^2} \leq 1$. Thus the constant function $g(x) = 1$ dominates every $f_n$.
Step 2: Verify $g \in L^1(\nu)$. Since $(X, \mathbf{X}, \nu)$ is a finite measure space, $\nu(X) < \infty$, so $\int_X g\,d\nu = \nu(... | 0 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | false | Measure Theory | false |
item_000011 | Consider the set $ X = [0, 1] \times [0, 1] $ equipped with the metric $ d((x_1, y_1), (x_2, y_2)) = |x_1 - x_2| + |y_1 - y_2| $. Let $ A
eq X $ be a non-empty open set in the topology induced by $ d $. Define the metric space $ (A, d_A) $, where $ d_A $ is the restriction of $ d $ to $ A $. Prove that $ (A, d_A) $ is... | Consider the set $ X = [0, 1] \times [0, 1] $ equipped with the metric $ d((x_1, y_1), (x_2, y_2)) = |x_1 - x_2| + |y_1 - y_2| $. Let $ A
eq X $ be a non-empty open set in the topology induced by $ d $. Define the metric space $ (A, d_A) $, where $ d_A $ is the restriction of $ d $ to $ A $. Prove that $ (A, d_A) $ is... | Step 1: State the relevant definitions. A metric space $(A, d_A)$ is complete if every Cauchy sequence in $A$ converges to a limit that belongs to $A$. A sequence $(p_n)$ is Cauchy if for every $\epsilon > 0$ there exists $N$ such that $d_A(p_n, p_m) < \epsilon$ for all $n, m \ge N$. A set $A$ is open in $X = [0,1] \ti... | (A, d_A) \text{ is not complete} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Topology | false |
item_000012 | Let $ f(x, y, z) = x^4 + y^4 + z^4 - 4xyz $. Determine the global minimum of $ f $ over the convex set defined by $ x + y + z = 3 $ and $ x, y, z
eq 0 $. Justify your answer by verifying the convexity of the function over the domain and confirming the nature of the critical point.
Step 1: The constraint is $x+y+z=3$ w... | Let $ f(x, y, z) = x^4 + y^4 + z^4 - 4xyz $. Determine the global minimum of $ f $ over the convex set defined by $ x + y + z = 3 $ and $ x, y, z
eq 0 $. Justify your answer by verifying the convexity of the function over the domain and confirming the nature of the critical point. | Step 1: The constraint is $x+y+z=3$ with $x,y,z\neq 0$. The feasible set is the plane $x+y+z=3$ excluding the three lines where one coordinate is zero. This set is not convex (removing lines from a plane disconnects it), but the problem asks to minimize over this set; we analyze the full plane and then exclude boundary... | -1 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | false | false | Optimization | false |
item_000013 | Let G be a connected simple graph with 12 vertices and 18 edges. Suppose that for each vertex v in G, the subgraph induced by the neighbors of v is connected. Furthermore, the chromatic number of G is 4. What is the minimum number of edges that must be removed to disconnect G, under the constraint that the chromatic nu... | Let G be a connected simple graph with 12 vertices and 18 edges. Suppose that for each vertex v in G, the subgraph induced by the neighbors of v is connected. Furthermore, the chromatic number of G is 4. What is the minimum number of edges that must be removed to disconnect G, under the constraint that the chromatic nu... | Let me reason through this step-by-step.
1. **Analyze the graph structure.** G is a connected simple graph with n=12 vertices, m=18 edges, chromatic number χ(G)=4, and for every vertex v, the subgraph induced by N(v) is connected.
2. **Use the neighbor-connectivity condition.** For every vertex v, G[N(v)] is connecte... | 3 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | false | true | Graph Theory | false |
item_000014 | Given the set of points $(0, 2), (1, 5), (2, 14), (3, 37)$, construct the Lagrange interpolating polynomial $P(x)$ of degree 3 that passes through these points. Evaluate $P(2.5)$ and round your final result to the nearest hundredth. Justify the accuracy of your result using a relevant error bound from numerical analysi... | Given the set of points $(0, 2), (1, 5), (2, 14), (3, 37)$, construct the Lagrange interpolating polynomial $P(x)$ of degree 3 that passes through these points. Evaluate $P(2.5)$ and round your final result to the nearest hundredth. Justify the accuracy of your result using a relevant error bound from numerical analysi... | Step 1: Nodes are x0=0, x1=1, x2=2, x3=3 with y0=2, y1=5, y2=14, y3=37.
Step 2: Lagrange basis polynomials are L0(x)=-(x-1)(x-2)(x-3)/6, L1(x)=x(x-2)(x-3)/2, L2(x)=-x(x-1)(x-3)/2, L3(x)=x(x-1)(x-2)/6.
Step 3: Thus P(x)=2L0(x)+5L1(x)+14L2(x)+37L3(x)=(4/3)x^3-x^2+(8/3)x+2.
Step 4: Evaluate the basis weights at x=2.5: L0(... | 23.25 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Numerical Analysis | false |
item_000015 | Let $(X, \frak{M},
u)$ be a finite measure space, and define for each $n
otin \frak{M}$, the function $f_n: X \to b{R}$ by $f_n(x) = \frac{1}{n} \times b{1}_{A_n}(x)$, where $A_n
otin \frak{M}$ are pairwise disjoint subsets of $X$. Define $g_n(x) = \frac{1}{n^2} \times b{1}_{A_n}(x)$ and $G(x) = \frac{1}{n^2} \ti... | Let $(X, \frak{M},
u)$ be a finite measure space, and define for each $n
otin \frak{M}$, the function $f_n: X \to b{R}$ by $f_n(x) = \frac{1}{n} \times b{1}_{A_n}(x)$, where $A_n
otin \frak{M}$ are pairwise disjoint subsets of $X$. Define $g_n(x) = \frac{1}{n^2} \times b{1}_{A_n}(x)$ and $G(x) = \frac{1}{n^2} \ti... | Step 1: Clarify the setup. We have a finite measure space $(X, \mathfrak{M}, \nu)$ with $\nu(X) < \infty$. The sets $A_n$ are stated to be non-measurable ($A_n \notin \mathfrak{M}$) with $\nu^*(A_n) = 1/n$, and pairwise disjoint.
Step 2: Check measurability of the functions. Since each $A_n \notin \mathfrak{M}$, the i... | \text{The problem is ill-posed: } A_n \notin \mathfrak{M} \text{ makes } f_n, h_m \text{ non-measurable, so } \nu(f_n h_m) \text{ is undefined and DCT does not apply.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Measure Theory | false |
item_000016 | Let $ A $, $ B $, and $ C $ be three distinct points on the coordinate plane with coordinates $ A = (a, 0) $, $ B = (0, b) $, and $ C = (0, 0) $, where $ a $ and $ b $ are positive real numbers. Let $ D $ be a point on segment $ AB $ such that $ CD $ is perpendicular to $ AB $. Let $ E $ be the midpoint of $ CD $, and ... | Let $ A $, $ B $, and $ C $ be three distinct points on the coordinate plane with coordinates $ A = (a, 0) $, $ B = (0, b) $, and $ C = (0, 0) $, where $ a $ and $ b $ are positive real numbers. Let $ D $ be a point on segment $ AB $ such that $ CD $ is perpendicular to $ AB $. Let $ E $ be the midpoint of $ CD $, and ... | 1. The line through A=(a,0) and B=(0,b) has equation x/a + y/b = 1.
2. Since CD is perpendicular to AB and C=(0,0), D is the foot of the perpendicular from C to AB.
3. The circle with diameter CD has center E, the midpoint of CD, and radius ED.
4. Because E lies on CD, the radius ED is collinear with CD. Since CD⊥AB, w... | No such point F exists; the problem is ill-posed. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | false | true | Geometry | false |
item_000017 | Let $(X, \textbf{M},
u)$ be a finite measure space. Define for each $n
eq m$, a function $f_{n,m}: X \to [0,1]$ such that $f_{n,m}(x) = 1$ if $x
otin A_n \triangle A_m$, and $f_{n,m}(x) = 0$ otherwise, where $A_n$ and $A_m$ are measurable sets with $
u(A_n \triangle A_m) < \frac{1}{n+m}$. Consider the sequence $g_n(... | Let $(X, \textbf{M},
u)$ be a finite measure space. Define for each $n
eq m$, a function $f_{n,m}: X \to [0,1]$ such that $f_{n,m}(x) = 1$ if $x
otin A_n \triangle A_m$, and $f_{n,m}(x) = 0$ otherwise, where $A_n$ and $A_m$ are measurable sets with $
u(A_n \triangle A_m) < \frac{1}{n+m}$. Consider the sequence $g_n(... | Step 1. For every x\in X, since \mathbf{1}_{A_n}(x)\in\{0,1\}, we have g_n(x)=\frac1n\mathbf{1}_{A_n}(x)\in\{0,1/n\}. Hence 0\le g_n(x)\le 1/n.
Step 2. Given \varepsilon>0, choose N>1/\varepsilon. Then for all n\ge N and all x\in X, |g_n(x)-0|\le 1/n<\varepsilon. Thus g_n\to0 pointwise, hence almost everywhere.
Step 3.... | \text{The statement is false: }g_n\text{ converges to }0\text{ pointwise (hence a.e.) and in }L^1. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Measure Theory | false |
item_000018 | Let $ T: \text{C}[0,1] \to \text{C}[0,1] $ be the bounded linear operator defined by $ (Tf)(x) = \frac{1}{1+x^2} \times \frac{1}{\text{Vol}(B_1(x))} \times \text{avg}_{y \text{ in } B_1(x)} f(y) $, where $ B_1(x) \text{ is the interval } [x - 0.1, x + 0.1] \text{ intersected with } [0,1] $, and $ \text{Vol}(B_1(x)) $ i... | Let $ T: \text{C}[0,1] \to \text{C}[0,1] $ be the bounded linear operator defined by $ (Tf)(x) = \frac{1}{1+x^2} \times \frac{1}{\text{Vol}(B_1(x))} \times \text{avg}_{y \text{ in } B_1(x)} f(y) $, where $ B_1(x) \text{ is the interval } [x - 0.1, x + 0.1] \text{ intersected with } [0,1] $, and $ \text{Vol}(B_1(x)) $ i... | Step 1: For any f with ||f||_sup = 1, we have |f(y)| <= 1 for all y in [0,1]. Thus |(Tf)(x)| <= (1/(1+x^2)) * (1/Vol(B_1(x))) * avg_{y in B_1(x)} |f(y)| <= 1/(1+x^2) <= 1. Hence ||T|| <= sup_{x in [0,1]} 1/(1+x^2) = 1.
Step 2: To show ||T|| >= 1, take f_0 = 1 (the constant function). Then ||f_0||_sup = 1.
Step 3: For f... | 1 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | false | false | Functional Analysis | false |
item_000019 | Consider the category **Set** of sets and functions. Let **C** be a small category with three objects **A**, **B**, and **C**, and non-identity morphisms **f: A → B**, **g: B → C**, and **h: A → C**, such that **g ∘ f = h**. Define two functors **F, G: C → Set** as follows:
- **F(A) = {a1, a2}, F(B) = {b1, b2, b3}, F(... | Consider the category **Set** of sets and functions. Let **C** be a small category with three objects **A**, **B**, and **C**, and non-identity morphisms **f: A → B**, **g: B → C**, and **h: A → C**, such that **g ∘ f = h**. Define two functors **F, G: C → Set** as follows:
- **F(A) = {a1, a2}, F(B) = {b1, b2, b3}, F(... | Step 1: Identify the naturality conditions. For each morphism in C, we need commutativity:
- For f: A→B: η_B ∘ F(f) = G(f) ∘ η_A
- For g: B→C: η_C ∘ F(g) = G(g) ∘ η_B
- For h: A→C: η_C ∘ F(h) = G(h) ∘ η_A
Step 2: Propose the component maps by matching corresponding elements:
- η_A(a1) = x1, η_A(a2) = x2
- η_B(b1) = y1... | \eta_A(a_1)=x_1,\ \eta_A(a_2)=x_2;\quad \eta_B(b_1)=y_1,\ \eta_B(b_2)=y_2,\ \eta_B(b_3)=y_3;\quad \eta_C(c_1)=z_1,\ \eta_C(c_2)=z_2 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Category Theory | false |
item_000020 | Let $ A, B, C $ be arbitrary sets. Prove from the axioms of set theory that $ A ackslash (B ackslash C) = (A ackslash B) ackslash C $ if and only if $ A ackslash C = A $. Use only first-order logic and definitions of set operations in your proof.
We prove the equivalence using first-order logic and the definition ... | Let $ A, B, C $ be arbitrary sets. Prove from the axioms of set theory that $ A ackslash (B ackslash C) = (A ackslash B) ackslash C $ if and only if $ A ackslash C = A $. Use only first-order logic and definitions of set operations in your proof. | We prove the equivalence using first-order logic and the definition of set difference: $X\setminus Y = \{x : x\in X \land x\notin Y\}$.
Step 1 (Translate to membership logic):
$x\in A\setminus(B\setminus C) \iff x\in A \land x\notin(B\setminus C) \iff x\in A \land \neg(x\in B \land x\notin C) \iff x\in A \land (x\noti... | A\setminus(B\setminus C)=(A\setminus B)\setminus C \iff A\setminus C=A | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Logic and Set Theory | false |
item_000021 | Let $ A $ be a $ 3 \times 3 $ matrix defined as follows:
$$ A = egin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ -a & -b & -c
cend{bmatrix}, $$
where $ a, b, c $ are real constants. Suppose the characteristic polynomial of $ A $ is given by $ \text{det}(A -
u I) = -
u^3 - c
u^2 - b
u - a $. Determine all eigenvalues $
u $ ... | Let $ A $ be a $ 3 \times 3 $ matrix defined as follows:
$$ A = egin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ -a & -b & -c
cend{bmatrix}, $$
where $ a, b, c $ are real constants. Suppose the characteristic polynomial of $ A $ is given by $ \text{det}(A -
u I) = -
u^3 - c
u^2 - b
u - a $. Determine all eigenvalues $
u $ ... | Step 1: The characteristic polynomial is given as det(A - νI) = -ν³ - cν² - bν - a. Setting this equal to zero gives -ν³ - cν² - bν - a = 0, which simplifies to ν³ + cν² + bν + a = 0.
Step 2: Thus the eigenvalues of A are precisely the roots of the monic cubic polynomial p(ν) = ν³ + cν² + bν + a, where a, b, c are arbi... | \text{The eigenvalues } \nu \text{ are the three roots of } \nu^3 + c\nu^2 + b\nu + a = 0. \text{ Explicitly, } \nu_k = -\frac{c}{3} + \omega^k \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} + \omega^{-k} \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}, \ k=0,1,2, \text{ where } p = b - \... | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Linear Algebra | false |
item_000022 | Given the points (1, 3), (2, 7), (4, 19), and (5, 29), construct the Lagrange interpolating polynomial $ P(x) $ of degree 3 that passes through these points. Use this polynomial to estimate the value of $ P(3) $. Additionally, provide an upper bound on the error of this estimate under the assumption that the function $... | Given the points (1, 3), (2, 7), (4, 19), and (5, 29), construct the Lagrange interpolating polynomial $ P(x) $ of degree 3 that passes through these points. Use this polynomial to estimate the value of $ P(3) $. Additionally, provide an upper bound on the error of this estimate under the assumption that the function $... | Step 1: The four data points are (x0,y0)=(1,3), (x1,y1)=(2,7), (x2,y2)=(4,19), (x3,y3)=(5,29). The Lagrange basis polynomials for x=3 are:
L0(3)=((3-2)(3-4)(3-5))/((1-2)(1-4)(1-5))=((1)(-1)(-2))/((-1)(-3)(-4))=2/(-12)=-1/6
L1(3)=((3-1)(3-4)(3-5))/((2-1)(2-4)(2-5))=((2)(-1)(-2))/((1)(-2)(-3))=4/6=2/3
L2(3)=((3-1)(3-2)(3... | P(3)=12,\ \text{error bound}=2 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Numerical Analysis | false |
item_000023 | Let $ A $ be a $ 2 \times 2 $ real matrix satisfying the equation $ A^3 - 3A^2 + 5A = I $, where $ I $ is the identity matrix. Suppose further that the trace of $ A $ is 3. Determine the determinant of $ A $.
Let d = det A. Since tr A = 3, the characteristic polynomial is x^2 - 3x + d.
By Cayley-Hamilton, A^2 = 3A - dI... | Let $ A $ be a $ 2 \times 2 $ real matrix satisfying the equation $ A^3 - 3A^2 + 5A = I $, where $ I $ is the identity matrix. Suppose further that the trace of $ A $ is 3. Determine the determinant of $ A $. | Let d = det A. Since tr A = 3, the characteristic polynomial is x^2 - 3x + d.
By Cayley-Hamilton, A^2 = 3A - dI.
Then A^3 = A(3A - dI) = 3A^2 - dA = (9-d)A - 3dI.
Substituting into A^3 - 3A^2 + 5A = I gives (5-d)A = I.
If d = 5, this says 0 = I, impossible; otherwise A = (1/(5-d))I is scalar.
The trace condition gives ... | No such matrix exists | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Linear Algebra | false |
item_000024 | A toy factory produces two types of wooden toys: cars and trains. Each car requires 2 units of wood and 1 hour of labor, while each train requires 1 unit of wood and 2 hours of labor. The factory has 10 units of wood and 10 hours of labor available each day. Each car yields a profit of $3, and each train yields a profi... | A toy factory produces two types of wooden toys: cars and trains. Each car requires 2 units of wood and 1 hour of labor, while each train requires 1 unit of wood and 2 hours of labor. The factory has 10 units of wood and 10 hours of labor available each day. Each car yields a profit of $3, and each train yields a profi... | Step 1: Define variables. Let x = number of cars, y = number of trains.
Step 2: Formulate the linear program. Maximize profit P = 3x + 4y subject to: 2x + y ≤ 10 (wood), x + 2y ≤ 10 (labor), x ≥ 0, y ≥ 0.
Step 3: Identify corner points of the feasible region: (0,0), (5,0), (0,5), and the intersection of 2x + y = 10 and... | \text{Maximum profit } = \frac{70}{3} \text{ dollars, producing } \frac{10}{3} \text{ cars and } \frac{10}{3} \text{ trains.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Optimization | false |
item_000025 | A fair six-sided die is rolled repeatedly until the sum of all the rolls is at least 100. Let X be the number of rolls required. Given that the first roll is a 6, compute the conditional probability that X is less than or equal to 20.
Let S_n be the sum after n rolls. Given the first roll is 6, we need P(X ≤ 20 | first... | A fair six-sided die is rolled repeatedly until the sum of all the rolls is at least 100. Let X be the number of rolls required. Given that the first roll is a 6, compute the conditional probability that X is less than or equal to 20. | Let S_n be the sum after n rolls. Given the first roll is 6, we need P(X ≤ 20 | first roll = 6).
X ≤ 20 means S_20 ≥ 100. Given the first roll is 6, this is equivalent to the sum of the remaining 19 rolls being at least 94.
Let Y_1, ..., Y_19 be the remaining 19 rolls, each uniform on {1,...,6}. We need P(Y_1 + ... +... | \dfrac{164449490501}{609359740010496} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Probability | false |
item_000026 | Let $ X = [0,1] \times [0,1] $ be the unit square in $ b{R}^2 $, equipped with the topology induced by the Euclidean metric. Consider the open cover $ \frak{U} = igcup_{n=2}^\fty \frak{U}_n $, where each $ \frak{U}_n = igrace{(x,y) igm| x
eq \frac{1}{n}, \text{ and } d((x,y), (0,0)) < \frac{1}{n} igrace} $. Pro... | Let $ X = [0,1] \times [0,1] $ be the unit square in $ b{R}^2 $, equipped with the topology induced by the Euclidean metric. Consider the open cover $ \frak{U} = igcup_{n=2}^\fty \frak{U}_n $, where each $ \frak{U}_n = igrace{(x,y) igm| x
eq \frac{1}{n}, \text{ and } d((x,y), (0,0)) < \frac{1}{n} igrace} $. Pro... | Let \(U_n=\{(x,y)\in X: x\neq 1/n \text{ and } \|(x,y)\|<1/n\}\), interpreting the proposed cover as the collection \(\{U_n:n\ge2\}\).
Definition. An open cover of \(X\) is a collection of open subsets of \(X\) whose union is \(X\). A finite subcover is a finite subcollection whose union is still \(X\). A space is com... | \text{The given family is not an open cover of }X\text{ (hence has no finite subcover), but }X\text{ is compact.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Topology | false |
item_000027 | Let $X = [0,1]$ with the standard topology inherited from $b{R}$, and let $Y$ be the set $b{Q} \times [0,1]$ with the subspace topology inherited from $b{R}^2$ with the standard topology. Prove or disprove: $X$ and $Y$ are homeomorphic.
Definition: A homeomorphism is a bijection f: X -> Y such that both f and f^{-1}... | Let $X = [0,1]$ with the standard topology inherited from $b{R}$, and let $Y$ be the set $b{Q} \times [0,1]$ with the subspace topology inherited from $b{R}^2$ with the standard topology. Prove or disprove: $X$ and $Y$ are homeomorphic. | Definition: A homeomorphism is a bijection f: X -> Y such that both f and f^{-1} are continuous. Equivalently, f is a bijection and U is open in X iff f(U) is open in Y.
Definition: A topological space is path-connected if for any two points a,b there is a continuous map g:[0,1]->space with g(0)=a, g(1)=b.
Definition: ... | X and Y are not homeomorphic. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Topology | false |
item_000028 | Let triangle $ ABC $ have vertices $ A(0, 0) $, $ B(1, 0) $, and $ C(a, b) $, where $ a > 0 $, $ b > 0 $. Let $ D $ be the midpoint of $ AB $, and let $ E $ be the midpoint of $ AC $. A point $ P $ lies on segment $ DE $ such that $ \tan(\theta) = 2 $, where $ \theta $ is the angle between vector $
ightarrow{AP} $ and... | Let triangle $ ABC $ have vertices $ A(0, 0) $, $ B(1, 0) $, and $ C(a, b) $, where $ a > 0 $, $ b > 0 $. Let $ D $ be the midpoint of $ AB $, and let $ E $ be the midpoint of $ AC $. A point $ P $ lies on segment $ DE $ such that $ \tan(\theta) = 2 $, where $ \theta $ is the angle between vector $
ightarrow{AP} $ and... | 1. The midpoints are D=(1/2,0) and E=(a/2,b/2).
2. Parametrize any point P on segment DE by P(t)=((1-t)/2+ta/2, tb/2), where 0<=t<=1.
3. The area of triangle ABC is [ABC]=1/2|det(B,C)|=b/2.
4. The area of triangle APC is [APC]=1/2|det(P,C)|=1/2|b x_P - a y_P|=b(1-t)/4.
5. Hence [APC]/[ABC]=(1-t)/2, which equals 1/2 onl... | \text{The statement is false.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Geometry | false |
item_000029 | Let $ C[0,1] $ be the space of continuous real-valued functions on the interval $[0,1]$ equipped with the supremum norm. Define a bounded linear functional $ \tau $ on $ C[0,1] $ by $ \tau(f) = \frac{1}{2}fig(\frac{1}{2}ig) + \frac{1}{2}fig(\frac{3}{4}ig) $. Use the Riesz representation theorem to find a function $... | Let $ C[0,1] $ be the space of continuous real-valued functions on the interval $[0,1]$ equipped with the supremum norm. Define a bounded linear functional $ \tau $ on $ C[0,1] $ by $ \tau(f) = \frac{1}{2}fig(\frac{1}{2}ig) + \frac{1}{2}fig(\frac{3}{4}ig) $. Use the Riesz representation theorem to find a function $... | The problem is ill-posed. The Riesz representation theorem applies to a Hilbert space, but C[0,1] with the supremum norm is not a Hilbert space. Moreover, the given inner product <f,g> = (1/2)(f(0)g(0)+f(1)g(1)) is degenerate on C[0,1]: any continuous function vanishing at 0 and 1 has zero norm, so it is not a genuine ... | \text{No such } g \text{ exists; the problem is ill-posed.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Functional Analysis | false |
item_000030 | A board game features a sequence of states labeled $ S_0, S_1, S_2, S_3, $ and $ S_4 $. The game starts at $ S_0 $, and from any state $ S_i $, the game transitions to $ S_{i+1} $ with probability $ \frac{1}{2} $, or it transitions back to $ S_0 $ with probability $ \frac{1}{2} $, for all $ 0
eq i < 4 $. When the game... | A board game features a sequence of states labeled $ S_0, S_1, S_2, S_3, $ and $ S_4 $. The game starts at $ S_0 $, and from any state $ S_i $, the game transitions to $ S_{i+1} $ with probability $ \frac{1}{2} $, or it transitions back to $ S_0 $ with probability $ \frac{1}{2} $, for all $ 0
eq i < 4 $. When the game... | Let T be the move on which S4 is first reached. A forward transition is a success with probability 1/2; a reset is a failure. Reaching S4 requires four consecutive successes, since any failure returns progress to S0.
Since B={T=6} and A={T<=6}, and B is a subset of A, we have Pr(B|A)=Pr(T=6)/Pr(T<=6).
For T=4: the firs... | \frac{1}{4} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Probability | false |
item_000031 | Let $ A $, $ B $, and $ C $ be sets. Prove that $ A \times (B \triangle C) = (A \times B) \triangle (A \times C) $, where $ \triangle $ denotes the symmetric difference.
1. Let (x,y) be arbitrary. By definition of Cartesian product, (x,y)∈A×(B△C) iff x∈A and y∈B△C.
2. By definition of symmetric difference, y∈B△C iff ex... | Let $ A $, $ B $, and $ C $ be sets. Prove that $ A \times (B \triangle C) = (A \times B) \triangle (A \times C) $, where $ \triangle $ denotes the symmetric difference. | 1. Let (x,y) be arbitrary. By definition of Cartesian product, (x,y)∈A×(B△C) iff x∈A and y∈B△C.
2. By definition of symmetric difference, y∈B△C iff exactly one of y∈B or y∈C holds, i.e. (y∈B)⊕(y∈C).
3. Therefore (x,y)∈A×(B△C) iff x∈A ∧ [(y∈B)⊕(y∈C)].
4. Using the logical distributive law p∧(q⊕r) ≡ (p∧q)⊕(p∧r), this is ... | A \times (B \triangle C) = (A \times B) \triangle (A \times C) | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | false | false | Logic and Set Theory | false |
item_000032 | Let $ (X, d) $ be the metric space defined by $ X = (0, 1] $ and $ d(x, y) = |\tan(\frac{\theta}{2})| $, where $ \theta = \frac{\tan^{-1}(x) - \tan^{-1}(y)}{1 + \tan(\frac{\tan^{-1}(x)}{2}) \tan(\frac{\tan^{-1}(y)}{2})} $. Prove or disprove: $ (X, d) $ is a complete metric space.
Let me carefully parse the problem. The... | Let $ (X, d) $ be the metric space defined by $ X = (0, 1] $ and $ d(x, y) = |\tan(\frac{\theta}{2})| $, where $ \theta = \frac{\tan^{-1}(x) - \tan^{-1}(y)}{1 + \tan(\frac{\tan^{-1}(x)}{2}) \tan(\frac{\tan^{-1}(y)}{2})} $. Prove or disprove: $ (X, d) $ is a complete metric space. | Let me carefully parse the problem. The metric is given by $d(x,y)=|\tan(\frac{\theta}{2})|$ where
$$\theta=\frac{\tan^{-1}(x)-\tan^{-1}(y)}{1+\tan(\frac{\tan^{-1}(x)}{2})\tan(\frac{\tan^{-1}(y)}{2})}.$$
Step 1: Identify the structure. Let $\phi:(0,1]\to S^1\setminus\{-1\}$ be the inverse stereographic-type map $\phi(... | The space $(X,d)$ is not complete. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Topology | false |
item_000033 | Consider the space $ C[0,1] $ of all continuous real-valued functions on the interval $ [0,1] $, equipped with the norm $ \|f\| = ∞ ∞ f(0) ∞ ∞ $. Prove that this normed space is not a Banach space.
Step 1: Interpret the given functional as \(\|f\|=|f(0)|\).
Step 2: A Banach space must be a complete normed space; in par... | Consider the space $ C[0,1] $ of all continuous real-valued functions on the interval $ [0,1] $, equipped with the norm $ \|f\| = ∞ ∞ f(0) ∞ ∞ $. Prove that this normed space is not a Banach space. | Step 1: Interpret the given functional as \(\|f\|=|f(0)|\).
Step 2: A Banach space must be a complete normed space; in particular, a norm must satisfy \(\|f\|=0 \iff f=0\).
Step 3: Consider \(f(x)=x\), which is continuous on \([0,1]\) and nonzero since \(f(1)=1\).
Step 4: Compute \(\|f\|=|f(0)|=|0|=0\).
Step 5: Thus a ... | Not a Banach space, since ||f||=|f(0)| is not a norm. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Functional Analysis | false |
item_000034 | Solve the first-order linear differential equation:
$$ t \frac{dy}{dt} + 2y = \frac{\tan t}{t} $$
with the initial condition $ y(1) = 0 $. Provide the solution in a closed-form expression for $ y(t) $.
Step 1: Identify the equation as a first-order linear ODE: $t\,\frac{dy}{dt} + 2y = \frac{\tan t}{t}$. We use the i... | Solve the first-order linear differential equation:
$$ t \frac{dy}{dt} + 2y = \frac{\tan t}{t} $$
with the initial condition $ y(1) = 0 $. Provide the solution in a closed-form expression for $ y(t) $. | Step 1: Identify the equation as a first-order linear ODE: $t\,\frac{dy}{dt} + 2y = \frac{\tan t}{t}$. We use the integrating factor method.
Step 2: Divide through by $t$ (assuming $t>0$) to put it in standard form: $\frac{dy}{dt} + \frac{2}{t}y = \frac{\tan t}{t^2}$.
Step 3: Compute the integrating factor: $\mu(t) = e... | y(t) = \frac{1}{t^2}\ln\!\left(\frac{\cos 1}{\cos t}\right) | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Ordinary and Partial Differential Equations | false |
item_000035 | Let $ G $ be a connected simple graph with 10 vertices and 18 edges. Suppose that each edge is assigned a color such that no two edges with the same color share a common vertex. Determine the minimum number of colors required to ensure that for any subset $ S $ of edges of size at most 4, the graph $ G' = (V, E acksla... | Let $ G $ be a connected simple graph with 10 vertices and 18 edges. Suppose that each edge is assigned a color such that no two edges with the same color share a common vertex. Determine the minimum number of colors required to ensure that for any subset $ S $ of edges of size at most 4, the graph $ G' = (V, E acksla... | The sum of degrees of $G$ is $2|E|=36$, so the average degree is $36/10=3.6$.
Hence some vertex $v$ has degree $d(v)\le 3$.
Let $S$ be the set of all edges incident to $v$; then $|S|=d(v)\le 3\le 4$.
After deleting $S$, the vertex $v$ has no incident edges, so $v$ is isolated in $G'=(V,E\setminus S)$.
Since $G'$ has $1... | \text{No finite number of colors exists} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Graph Theory | false |
item_000036 | Determine whether the series $ \frac{1}{2} + \frac{1}{3} inom{1}{1} + \frac{1}{4} inom{2}{1} + \frac{1}{5} inom{3}{1} + \frac{1}{6} inom{4}{1} + \frac{1}{7} inom{5}{1} + \frac{1}{8} inom{6}{1} + \frac{1}{9} inom{7}{1} + \frac{1}{10} inom{8}{1} + \frac{1}{11} inom{9}{1} + \frac{1}{12} inom{10}{1} + \frac{1}{13... | Determine whether the series $ \frac{1}{2} + \frac{1}{3} inom{1}{1} + \frac{1}{4} inom{2}{1} + \frac{1}{5} inom{3}{1} + \frac{1}{6} inom{4}{1} + \frac{1}{7} inom{5}{1} + \frac{1}{8} inom{6}{1} + \frac{1}{9} inom{7}{1} + \frac{1}{10} inom{8}{1} + \frac{1}{11} inom{9}{1} + \frac{1}{12} inom{10}{1} + \frac{1}{13... | The series is: $\frac{1}{2} + \sum_{n=1}^{\infty} \frac{1}{n+2}\binom{n}{1}$.
Since $\binom{n}{1} = n$, the general term (for $n \geq 1$) is $a_n = \frac{n}{n+2}$.
We check the limit of the terms: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n}{n+2} = 1 \neq 0$.
By the nth-term (divergence) test, since the t... | \text{The series diverges.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Calculus | false |
item_000037 | Let $ (X, d) $ be a metric space, and define a function $ f: X \to
eal $ by $ f(x) = d(x, A) $, where $ A
eq \text{\textemptyset} $ is a subset of $ X $. Consider the subset $ B = races{x \text{ in } X : f(x) \text{ is irrational}} $. Determine whether $ B $ is open, closed, or neither in the metric topology of $ X ... | Let $ (X, d) $ be a metric space, and define a function $ f: X \to
eal $ by $ f(x) = d(x, A) $, where $ A
eq \text{\textemptyset} $ is a subset of $ X $. Consider the subset $ B = races{x \text{ in } X : f(x) \text{ is irrational}} $. Determine whether $ B $ is open, closed, or neither in the metric topology of $ X ... | 1. Definitions: In a metric space, U is open if for every x in U there is ε>0 such that the open ball {y in X : d(x,y)<ε} is contained in U. A set C is closed if X\C is open, equivalently if every limit in X of a sequence from C belongs to C.
2. The map f(x)=d(x,A)=inf_{a in A} d(x,a) is 1-Lipschitz because |d(x,A)-d(y... | No universal classification; B can be neither open nor closed (e.g. X=R, A={0}), and in special cases it can be clopen. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Topology | false |
item_000039 | Consider a dynamical system governed by the stochastic differential equation dX_t = -X_t dt + σ dW_t, where W_t is a standard Wiener process and σ > 0 is a constant. Suppose we are interested in estimating the expected value of the first exit time τ from the interval [-1, 1], starting from X_0 = 0, using a Monte Carlo ... | Consider a dynamical system governed by the stochastic differential equation dX_t = -X_t dt + σ dW_t, where W_t is a standard Wiener process and σ > 0 is a constant. Suppose we are interested in estimating the expected value of the first exit time τ from the interval [-1, 1], starting from X_0 = 0, using a Monte Carlo ... | Let τ_1,…,τ_N be independent copies of the first exit time τ, with mean μ=E[τ] and Var(τ)=4.
The Monte Carlo estimator is τ_N=(1/N)∑_{i=1}^N τ_i, so E[τ_N]=μ and Var(τ_N)=4/N.
By the Monte Carlo central limit theorem, (τ_N−μ)/(2/√N) is approximately standard normal.
For 95% confidence, require P(|τ_N−μ|<0.1)≥0.95, so 0... | 1537 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | false | true | Applied Mathematics | false |
item_000040 | Let $(\tilde{X}, \tilde{
u})$ be a finite measure space. Define a sequence $(f_n)_{n=1}^{\tilde{
ho}}$ in $L^1(\tilde{X}, \tilde{
u})$ such that for each $n
otin \tilde{
ho}$, $f_n$ is a simple function with disjoint supports, where $\tilde{
u}(\text{supp}(f_n)) = 1/\tilde{
ho}$ and $\tilde{
u}(\text{supp}(f_n)) < \ti... | Let $(\tilde{X}, \tilde{
u})$ be a finite measure space. Define a sequence $(f_n)_{n=1}^{\tilde{
ho}}$ in $L^1(\tilde{X}, \tilde{
u})$ such that for each $n
otin \tilde{
ho}$, $f_n$ is a simple function with disjoint supports, where $\tilde{
u}(\text{supp}(f_n)) = 1/\tilde{
ho}$ and $\tilde{
u}(\text{supp}(f_n)) < \ti... | Step 1 (Construction). Let $(X,\mu)=([0,1],\lambda)$ be Lebesgue measure. For each $n\ge 1$, write $n=2^k+j$ with $k\ge 0$ and $0\le j<2^k$. Define
$$f_n=\frac{2^k}{n}\,\mathbf{1}_{[j/2^k,\,(j+1)/2^k]}.$$
Each $f_n$ is simple. Within each generation $k$, the supports $[j/2^k,(j+1)/2^k]$ are pairwise disjoint. Moreover ... | f_n\to 0\text{ in }L^1([0,1],\lambda),\text{ but }f_n\text{ does not converge a.e.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Measure Theory | false |
item_000041 | Consider Laplace's equation $
abla^2 u = 0 $ on the rectangular region $ 0 < x < 1 $, $ 0 < y < 2 $, with the following Dirichlet boundary conditions:
- $ u(0, y) = \frac{\tan^{-1}(y)}{1 + y^2} $ for $ 0
e y
e 2 $,
- $ u(1, y) = 0 $ for $ 0 < y < 2 $,
- $ u(x, 0) = 0 $ for $ 0 < x < 1 $,
- $ u(x, 2) = 0 $ for $ 0 <... | Consider Laplace's equation $
abla^2 u = 0 $ on the rectangular region $ 0 < x < 1 $, $ 0 < y < 2 $, with the following Dirichlet boundary conditions:
- $ u(0, y) = \frac{\tan^{-1}(y)}{1 + y^2} $ for $ 0
e y
e 2 $,
- $ u(1, y) = 0 $ for $ 0 < y < 2 $,
- $ u(x, 0) = 0 $ for $ 0 < x < 1 $,
- $ u(x, 2) = 0 $ for $ 0 <... | We solve Laplace's equation $\nabla^2 u = u_{xx}+u_{yy}=0$ on $0<x<1$, $0<y<2$ with homogeneous Dirichlet data on $y=0$, $y=2$, and $x=1$, and nonhomogeneous data on $x=0$.
Step 1: Separation of variables. Since $u=0$ at $y=0$ and $y=2$, we choose the $y$-eigenfunctions satisfying $Y(0)=Y(2)=0$. Let $u(x,y)=X(x)Y(y)$.... | u(x,y)=\sum_{n=1}^{\infty}\frac{\sinh\!\bigl(\frac{n\pi}{2}(1-x)\bigr)}{\sinh\!\bigl(\frac{n\pi}{2}\bigr)}\left[\int_0^2 \frac{\tan^{-1}(s)}{1+s^2}\sin\!\Bigl(\frac{n\pi s}{2}\Bigr)\,ds\right]\sin\!\Bigl(\frac{n\pi y}{2}\Bigr) | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Ordinary and Partial Differential Equations | false |
item_000042 | Consider a feedback control system where the output $ y(t) $ is governed by the differential equation $ \frac{dy}{dt} = -k(y - r(t)) $, where $ k > 0 $ is a constant and $ r(t) $ is a reference input. The reference input is a periodic function defined as $ r(t) = A \text{ for } 0 \text{ to } T/2 $, and $ r(t) = -A \tex... | Consider a feedback control system where the output $ y(t) $ is governed by the differential equation $ \frac{dy}{dt} = -k(y - r(t)) $, where $ k > 0 $ is a constant and $ r(t) $ is a reference input. The reference input is a periodic function defined as $ r(t) = A \text{ for } 0 \text{ to } T/2 $, and $ r(t) = -A \tex... | The system equation is y'(t) + k y(t) = k r(t), where r(t) is a square wave of period T with r(t)=A on [0,T/2) and r(t)=-A on [T/2,T). Its mean over one period is \bar r = (A T/2 - A T/2)/T = 0.
Since k>0, the homogeneous response decays like e^{-kt}, so the initial condition y(0)=0 only produces a transient. Thus as ... | 0 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | false | false | Applied Mathematics | false |
item_000043 | Let $ S $ be the set of all subsets $ A \times A $ of $
atnum \times
atnum $ such that $ A
eq
atnum $ and $ A $ is infinite. Let $ T $ be the set of all subsets $ B \times B $ of $
atnum \times
atnum $ where $ B $ is finite. Determine whether the cardinality of $ S $ is strictly greater than, equal to, or strictl... | Let $ S $ be the set of all subsets $ A \times A $ of $
atnum \times
atnum $ such that $ A
eq
atnum $ and $ A $ is infinite. Let $ T $ be the set of all subsets $ B \times B $ of $
atnum \times
atnum $ where $ B $ is finite. Determine whether the cardinality of $ S $ is strictly greater than, equal to, or strictl... | Step 1: Interpret the sets. We have $S = \{A \times A : A \subseteq \mathbb{N},\ A \neq \mathbb{N},\ A \text{ infinite}\}$ and $T = \{B \times B : B \subseteq \mathbb{N},\ B \text{ finite}\}$.
Step 2: Compute $|T|$. The collection of all finite subsets of $\mathbb{N}$ is countable, i.e., has cardinality $\aleph_0$. Th... | |S| > |T| | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Logic and Set Theory | false |
item_000044 | Define two categories as follows:
- Let **C** be the category whose objects are finite sets and whose morphisms are functions between them.
- Let **D** be the category whose objects are commutative monoids and whose morphisms are monoid homomorphisms.
For each object $ X $ in **C**, define $ F(X) $ as the free commut... | Define two categories as follows:
- Let **C** be the category whose objects are finite sets and whose morphisms are functions between them.
- Let **D** be the category whose objects are commutative monoids and whose morphisms are monoid homomorphisms.
For each object $ X $ in **C**, define $ F(X) $ as the free commut... | Let **C** be the category of finite sets and functions, and **D** the category of commutative monoids and monoid homomorphisms.
For a finite set $X$, let $F(X)$ denote the free commutative monoid on $X$. Concretely, $F(X)$ consists of finite formal sums $\sum_{x\in X} n_x x$ with $n_x\in\mathbb{N}$, with addition defi... | F:\mathbf{C}\to\mathbf{D}\text{ is a functor.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Category Theory | false |
item_000045 | Consider the first-order ordinary differential equation:
$$ \frac{dy}{dx} = \frac{2x}{y}, \text{ with the initial condition } y(1) = 2. $$
Solve the differential equation and determine the explicit solution $ y(x) $ in closed form.
1. Method: Separation of variables.
2. Separate the variables: y dy = 2x dx.
3. Integra... | Consider the first-order ordinary differential equation:
$$ \frac{dy}{dx} = \frac{2x}{y}, \text{ with the initial condition } y(1) = 2. $$
Solve the differential equation and determine the explicit solution $ y(x) $ in closed form. | 1. Method: Separation of variables.
2. Separate the variables: y dy = 2x dx.
3. Integrate both sides: ∫ y dy = ∫ 2x dx, which gives y^2 / 2 = x^2 + C.
4. Multiply by 2 to simplify: y^2 = 2x^2 + 2C. Let K = 2C, so y^2 = 2x^2 + K.
5. Substitute the initial condition y(1) = 2: 2^2 = 2(1)^2 + K, so 4 = 2 + K, which gives K... | y = \sqrt{2x^2 + 2} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Ordinary and Partial Differential Equations | false |
item_000046 | Let $ A
eq \text{\textup{Borel}} $ be a subset of $ [0, 1] $, and consider the function $ f: [0, 1] \to \text{\textup{R}} $ defined by $ f(x) = egin{cases} 1, & x
otin A \ 0, & x
otin A \text{ but } x \text{ is rational} \text{.} \text{ (This is a contradiction, but consider the function to be 0 on all of } A^c \te... | Let $ A
eq \text{\textup{Borel}} $ be a subset of $ [0, 1] $, and consider the function $ f: [0, 1] \to \text{\textup{R}} $ defined by $ f(x) = egin{cases} 1, & x
otin A \ 0, & x
otin A \text{ but } x \text{ is rational} \text{.} \text{ (This is a contradiction, but consider the function to be 0 on all of } A^c \te... | Step 1: Parse the definitions. The function is $f = \mathbf{1}_A$ (indicator of $A$) and $f_n = \mathbf{1}_{A_n}$, where each $A_n$ is Lebesgue measurable and $m^*(A_n \triangle A) \to 0$.
Step 2: Note that $\{x : |f_n(x) - f(x)| > 0\} = A_n \triangle A$, so $f_n \to f$ in outer measure: $m^*(\{|f_n - f| > \epsilon\})... | f \text{ is Lebesgue measurable.} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Measure Theory | false |
item_000047 | Two groups of students participated in a learning experiment: Group A used a traditional textbook, while Group B used an interactive digital platform. After one month, each student took a standardized test with a maximum score of 100. Group A had 25 students, with a sample mean of 72 and sample standard deviation of 8.... | Two groups of students participated in a learning experiment: Group A used a traditional textbook, while Group B used an interactive digital platform. After one month, each student took a standardized test with a maximum score of 100. Group A had 25 students, with a sample mean of 72 and sample standard deviation of 8.... | 1. Let μ_A and μ_B be population mean scores for Group A and Group B. Test H0: μ_B − μ_A = 0 (or μ_B ≤ μ_A) against H1: μ_B > μ_A at α = 0.05.
2. With equal variances, use the pooled two-sample t-test with df = n_A + n_B − 2 = 25 + 30 − 2 = 53.
3. Pooled variance: s_p^2 = [(24)(8^2) + (29)(9^2)]/53 = (1536 + 2349)/53 =... | Reject H0: t = 2.156 > critical t_{0.95,53} = 1.674, so the digital platform significantly improves performance. | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Statistics | false |
item_000048 | A rare disease affects 1 in 2000 people in a population. A test for the disease has the following properties: if a person has the disease, the test is positive 99% of the time; if a person does not have the disease, the test is negative 98% of the time. A person is selected at random and tested twice for the disease. T... | A rare disease affects 1 in 2000 people in a population. A test for the disease has the following properties: if a person has the disease, the test is positive 99% of the time; if a person does not have the disease, the test is negative 98% of the time. A person is selected at random and tested twice for the disease. T... | Let D be the event that the person has the disease, so P(D)=1/2000 and P(D^c)=1999/2000.
Assuming the two test results are conditionally independent given disease status:
P(+,-|D)=0.99*0.01=99/10000.
P(+,-|D^c)=0.02*0.98=196/10000.
By Bayes' theorem,
P(D|+,-)=[P(D)P(+,-|D)]/[P(D)P(+,-|D)+P(D^c)P(+,-|D^c)].
Substituting... | \frac{99}{391903} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Probability | false |
item_000049 | Let $ ( [0,1], \text{Borel}([0,1]), m ) $ be the unit interval equipped with the Borel $ \text{σ} $-algebra and Lebesgue measure $ m $. Define a sequence of functions $ \textstyle f_n(x) = \frac{\text{sign}(x - 1/n)}{\text{ln}(1 + 1/n)} $ for each $ x \text{ in } [0,1] $ and $ n \text{ in } \text{N} \text{ with } n \te... | Let $ ( [0,1], \text{Borel}([0,1]), m ) $ be the unit interval equipped with the Borel $ \text{σ} $-algebra and Lebesgue measure $ m $. Define a sequence of functions $ \textstyle f_n(x) = \frac{\text{sign}(x - 1/n)}{\text{ln}(1 + 1/n)} $ for each $ x \text{ in } [0,1] $ and $ n \text{ in } \text{N} \text{ with } n \te... | For each n, f_n is Borel measurable since sign(x-1/n) is Borel measurable and the denominator is a positive constant.
Partition [0,1] as [0,1/n) and [1/n,1]. On [0,1/n), sign(x-1/n)=-1; on [1/n,1], sign(x-1/n)=1. The point {1/n} has Lebesgue measure 0, so it does not affect the integral. Hence
\int_{[0,1]} f_n\,dm
= \... | +\infty\ \text{and no }g\in L^1([0,1])\text{ exists; DCT does not apply} | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Measure Theory | false |
item_000050 | Given a fixed point $ A(1, 0) $ and a fixed circle $ C $ centered at the origin with radius $ 2 $, find the locus of all points $ P $ such that the distance from $ P $ to $ A $ is equal to the distance from $ P $ to the closest point on the circle $ C $. Express your answer as a set of points or an equation in the coor... | Given a fixed point $ A(1, 0) $ and a fixed circle $ C $ centered at the origin with radius $ 2 $, find the locus of all points $ P $ such that the distance from $ P $ to $ A $ is equal to the distance from $ P $ to the closest point on the circle $ C $. Express your answer as a set of points or an equation in the coor... | Let P = (x, y). The distance from P to A(1,0) is sqrt((x-1)^2 + y^2).
The closest point on circle C (center origin, radius 2) to P lies along the line from the origin through P. If r = sqrt(x^2+y^2), the distance from P to the closest point on C is |r - 2|.
Setting the distances equal: sqrt((x-1)^2 + y^2) = |sqrt(x^2... | (x - \tfrac{1}{2})^2 + \tfrac{4}{3}y^2 = 1 | Qwen/Qwen3-32B-TEE | z-ai/glm-5.2 | true | true | Geometry | false |
End of preview. Expand in Data Studio
Dendrite Synth Multi-Domain
A verified, difficulty-filtered, style-amplified synthetic corpus of question / reasoning / answer triples spanning the 14 MMLU-Pro categories - mathematics, computer science, natural sciences, chemistry, physics, engineering, health, law, business, economics, psychology, philosophy, history and others (expanded to 122 fine-grained categories and 695 subcategories). Problems are written by a pool of generator models, solved with explicit reasoning by GLM-5.2, verified by an LLM judge, and gated so that only problems a weak model fails but a strong model solves are kept. The dataset contains approximately 500M tokens.
Dataset summary
| Subset | File | Rows | Unique UIDs | Categories | Subcategories |
|---|---|---|---|---|---|
| Dendrite Synth Multi-Domain | dendrite_synth_multi_domain.jsonl | 140,847 | 140,847 | 122 | N/A |
| Dendrite Synth Multi-Domain QA | dendrite_synth_multi_domain_qa.jsonl | 138,316 | 138,316 | 122 | 694 |
| Dendrite Synth Multi-Domain CONVERSATION | dendrite_synth_multi_domain_conversation.jsonl | 140,350 | 140,350 | 122 | 695 |
| Dendrite Synth Multi-Domain REWRITE | dendrite_synth_multi_domain_rewrite.jsonl | 140,047 | 140,047 | 122 | 694 |
| Dendrite Synth Multi-Domain TEXTBOOK | dendrite_synth_multi_domain_textbook.jsonl | 138,802 | 138,802 | 122 | 694 |
The Dendrite Synth Multi-Domain had a ~4x style amplification of the main subset produced by GPT-OSS-120B, with one row per structural form: original Q/A, conversation, rewrite and textbook passage.
Data fields (main subset)
| Field | Description |
|---|---|
| uid | Globally unique record id (normalized-problem-text hash based) |
| content | Full rendered record text |
| question | Problem statement |
| reasoning | Step-by-step solution produced by GLM-5.2 |
| answer | Final answer |
| question_model | Model that authored the problem |
| reasoning_model | Model that produced the reasoning / answer (GLM-5.2) |
| filter_qwen | Weak-model difficulty signal from Qwen/Qwen3-4B |
| filter_teutonic_king | Weak-model difficulty signal from dendriteholdings/teutonic-5ev3rrdend-cl004 |
| category | Category label |
| contam | Benchmark-contamination flag (stub) |
Filters
filter_qwen and filter_teutonic_king are informational, not row filters - every problem is kept regardless of filter outcome, so downstream users can pick their own difficulty threshold.
| Field | Value | Entries | Share |
|---|---|---|---|
| Qwen filter | True |
124,186 | 88.17% |
| Qwen filter | False |
16,661 | 11.83% |
| Teutonic King filter | True |
114,007 | 80.94% |
| Teutonic King filter | False |
26,840 | 19.06% |
Structural categories
| Structural category | Entries | Share of structure subset |
|---|---|---|
conversation |
140,350 | 25.17% |
rewrite |
140,047 | 25.12% |
textbook |
138,802 | 24.90% |
qa |
138,316 | 24.81% |
The four structural-category counts sum to 557,515.
Dataset Composition
Subject coverage
The 122 fine-grained categories are linked to the 14 MMLU-Pro categories below. The pair count is the number of distinct subcategories observed with each fine-grained category in The Dendrite Synth Multi-Domain.
| MMLU-Pro category | Fine-grained categories (subcategory pairs) |
|---|---|
| Mathematics | Abstract Algebra (6); Algebra (7); Applied Mathematics (5); Calculus (6); Category Theory (4); Complex Analysis (6); Discrete Math (9); Functional Analysis (4); Geometry (9); Graph Theory (6); Linear Algebra (6); Logic and Set Theory (5); Measure Theory (6); Number Theory (6); Numerical Analysis (6); Optimization (5); Ordinary and Partial Differential Equations (7); Probability (8); Statistics (6); Topology (6) |
| Computer science | Algorithms and Complexity (6); Automata and Formal Languages (5); Computer Architecture (5); Data Structures (5); Databases (5); Discrete Math and Boolean Logic (5); Information Theory and Cryptography (5); Networking (5); Operating Systems (5) |
| Natural sciences | Astronomy (5); Biochemistry and Metabolism (6); Cell Biology (6); Ecology (6); Environmental Science (5); Evolution (6); Genetics (6); Geography (5); Microbiology (6); Molecular Biology (6) |
| Chemistry | Acid-Base and pH (5); Atomic Structure and Periodicity (5); Chemical Equilibrium (5); Chemical Kinetics (5); Electrochemistry (5); Gas Laws (5); Stoichiometry (5); Thermochemistry (5) |
| Physics | Classical Mechanics (6); Electromagnetism (5); Modern and Nuclear Physics (5); Quantum Mechanics (5); Special Relativity (5); Thermodynamics and Statistical Mechanics (5); Waves and Optics (5) |
| Engineering | Chemical & Process Engineering (6); Circuit Analysis (6); Control Systems (6); Fluid Mechanics (11); Materials Science (6); Signals & Systems (6); Statics & Mechanics of Materials (6); Thermodynamics & Heat Transfer (6) |
| Health | Clinical Diagnostics (6); Epidemiology & Biostatistics (6); Human Anatomy (6); Medical Microbiology (6); Nutrition (6); Nutrition and Everyday Health Facts (5); Pathology (6); Pharmacology & Dosage (6); Physiology (9) |
| Law | Administrative Law (6); Civil Procedure (6); Constitutional Law (6); Contracts (6); Criminal Law (6); Evidence (6); Property (6); Torts (6) |
| Business | Accounting and Business Facts (5); Corporate Finance (6); Financial Accounting (6); Investments and Valuation (6); Managerial and Cost Accounting (6); Marketing Metrics (6); Microeconomics for Business (6); Operations and Supply Chain (6); Statistics for Business (7) |
| Economics | Consumer Theory and Demand (6); Elasticity and Welfare (6); Game Theory (6); International Trade (6); Macroeconomic Models (6); Market Structures (6); Money and Banking (6); Producer Theory and Costs (6) |
| Psychology | Biopsychology & Neuroscience (6); Cognitive Psychology (6); Developmental Psychology (6); Learning & Conditioning (6); Personality & Abnormal Psychology (6); Research Methods & Statistics (6); Social Psychology (6) |
| Philosophy | Ancient Philosophy (6); Epistemology (6); Ethics and Moral Theory (6); History of Modern Philosophy (6); Logic (6); Metaphysics (6); Philosophy of Mind (6); Political Philosophy (6) |
| History | Ancient Civilizations (6); Classical Antiquity (6); Early Modern and Renaissance (6); History of Asia and Africa (6); Medieval History (6); Modern European History (6); US History (6); World Wars (6) |
| Other | General Science Facts (5); Measurement and Unit Conversion (6); Miscellaneous General Knowledge (5) |
These 14 rows cover all 122 fine-grained categories and all 715 category-subcategory pairs.
Subcategory taxonomy
The Dendrite Synth Multi-Domain contains 715 category–subcategory pairs, with 4 to 11 subcategories per category (mean 5.86). Subcategory counts sum to the full 557,515-entry inventory. Examples of the largest subcategories are:
| Category | Subcategory | Entries | Within category |
|---|---|---|---|
| Statistics | bayesian_stats |
3,603 | 19.3% |
| Statistics | descriptive |
3,339 | 17.9% |
| Statistics | anova |
3,296 | 17.7% |
| Number Theory | cryptographic |
3,249 | 18.9% |
| Linear Algebra | transformations |
3,209 | 19.0% |
| Probability | stochastic_processes |
3,179 | 19.0% |
Final dataset design
The released dataset uses the following pipeline: generate -> difficulty-score -> amplify -> partial verification -> contamination-flag -> tokenize. This design is based on the paper - Autodata: An agentic data scientist to create high quality synthetic data (https://arxiv.org/abs/2606.25996)
- Generate: sample a specification from a per-category catalog; a generator model writes the problem, GLM-5.2 produces the reference reasoning and answer, and an LLM judge verifies it.
- Difficulty-score: evaluate problems with Qwen/Qwen3-4B and dendriteholdings/teutonic-5ev3rrdend-cl004. Their outcomes are stored as informational signals; they do not determine whether a row is included in the released main subset.
- Amplify: use GPT-OSS-120B to restyle each problem into four structural forms: qa, conversation, rewrite, and textbook.
- Partially verify: re-check amplified variants with checkable answers when they pass through the full verification path. Fast-path variants are retained without this additional check.
- Tokenize: deduplicate the separately released training corpus and pack it into uint32 shards with a manifest using the silx-ai/Quasar-10B tokenizer.
Generation and amplification run as a provider-split fleet: GLM-5.2 endpoints handle solution and judging work, while GPT-OSS-120B handles amplification. Dedicated GLM-5.2 endpoints supplement shared capacity because judging is the principal throughput constraint. The release uses a single strong judge rather than a diversified verifier ensemble.
Source pools were deduplicated by normalized-problem-text hash (not item_id, which is not globally unique across independent runs). A benchmark contamination screening wasn't done past a hook and any remnants of it are stubs.
Models used
Question Generator models Used
| Question-model label | Entries | Share |
|---|---|---|
deepseek/deepseek-v4-flash |
38,616 | 27.42% |
google/gemma-4-31b-it |
30,215 | 21.45% |
deepseek/deepseek-v4-pro |
27,939 | 19.84% |
xiaomi/mimo-v2.5-pro |
15,379 | 10.92% |
nvidia/nemotron-3-ultra-550b-a55b |
12,126 | 8.61% |
Qwen/Qwen3-32B-TEE |
6,418 | 4.56% |
qwen/qwen3.6-35b-a3b |
4,027 | 2.86% |
openai/gpt-oss-120b |
3,918 | 2.78% |
google/gemma-4-31B-turbo-TEE |
1,455 | 1.03% |
minimax/minimax-m2.5 |
614 | 0.44% |
qwen/qwen3-32b |
67 | 0.05% |
z-ai/glm-5.2 |
48 | 0.03% |
Qwen/Qwen3-14B |
25 | 0.02% |
The five largest question-model groups account for approximately 88.23% of the cleaned main subset.
Pipeline models Used
| Role | Model |
|---|---|
| Solver / verifier / judge | glm-5.2 (shared + dedicated FP8 vLLM endpoints) |
| Amplifier | gpt-oss-120B |
| Weak filters | Qwen/Qwen3-4B, dendriteholdings/teutonic-5ev3rrdend-cl004 |
| Tokenizer | silx-ai/Quasar-10B |
Reasoning is off by default (enable_thinking: false) for the judge - roughly 1.9x cheaper and faster, at a small cost in re-solve rigor.
Intended uses
- Mid-training / continued pretraining on reasoning-dense STEM and general-knowledge text.
- SFT on question -> reasoning -> answer traces.
- Difficulty-aware curriculum construction using the weak-model filter columns.
Citation
@misc(dendrite_synth_multi_domain,
title = The Dendrite Synth Multi-Domain,
author = Dendrite,
year = 2026,
note = Synthetic question/reasoning/answer corpus across MMLU-Pro categories
)
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