MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
values |
|---|---|---|
721,784 | What is the absolute value of the complex exponential function $e^{iz}$ where $z$ is a real number? | Unambiguous |
2,581,976 | How do I find the limit of a function as x approaches a specific value? | Unambiguous |
1,645,747 | Compute the derivative of $x$ with respect to $\frac{1}{x}$. | Unambiguous |
997,152 | Why does the General Leibniz Rule for the nth derivative of a product have the same binomial coefficient structure as the Binomial Theorem for raising a sum to the nth power? | Unambiguous |
1,475,726 | How to find three sets of three collinear points in a given 2D point set? | Unambiguous |
981,781 | $$f(x)=x^3-x^2-14x+24$$ | Ambiguous |
1,040,128 | How do I evaluate the improper integral $\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$ using trigonometric substitution? | Unambiguous |
2,278,214 | Calculate the Shannon entropy of the random variable representing the number of heads obtained when flipping a fair coin ten times. | Unambiguous |
2,751,401 | If $a\mid b$ and $a>0$ then $(a,b)=a$ | Ambiguous |
4,057,665 | Are these topologies the same? | Ambiguous |
1,466,559 | How to find the angle between two vectors using the dot product formula derived from the law of cosines? | Unambiguous |
1,602,977 | What's an example of an infinitesimal? | Ambiguous |
1,899,905 | How can physical quantities that are modeled by irrational numbers be realized or measured in the physical world? | Unambiguous |
2,604,608 | Prove that the constant function $f(x) = 2a + 3b$ is continuous for all x in its domain, where a and b are real constants. | Unambiguous |
1,761,660 | Find the value of $a$ such that the range of the function $f(x) = ax(1-x)$ contains the interval $[0,1]$. | Unambiguous |
1,695,878 | Let $A$ and $B$ be square matrices of order n so that $AB = BA$. Prove $\mathrm{rank}(A+B) \leq \mathrm{rank}(A) + \rm{rank}(B) - \mathrm{rank}(AB)$. | Ambiguous |
4,269,673 | Is there a notation to indicate that a variable in this equation is constant? | Ambiguous |
127,376 | In the context of the extended real number system, why is the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right defined as $\infty$, while the limit from the left is $-\infty$? | Unambiguous |
3,905,420 | Would a certain cyclic or dihedral group be an example? How do we find its subgroups? | Ambiguous |
2,138,866 | What is the value of the definite integral $\int_{0}^{\pi/4} \frac{\tan^2 x}{1+x^2} \text{d}x$ from the 2015 MIT Integration Bee? | Unambiguous |
1,688,323 | How to find the coefficients of a quadratic polynomial given three coordinate points using the method of undetermined coefficients or differentiation? | Unambiguous |
13,582 | Prove that every continuous function from a closed interval $[a, b]$ to itself has a fixed point. | Unambiguous |
254,561 | Provide a proof that $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$ for all real numbers $x$. | Unambiguous |
1,222,768 | Why are vector valued functions 'well-defined' when multivalued functions aren't? | Ambiguous |
738,236 | How do I calculate the area of a triangle with vertices defined by the position vectors $a$, $b$, and $c$ in 3D space using the cross product? | Unambiguous |
2,597,743 | Can a certain type of integral be solved using the gamma function method? | Ambiguous |
2,152,735 | What is the definition of an open set in the context of a general topological space? | Unambiguous |
2,820,510 | How to find the rotation quaternion from the angles between coordinate axes $x_1, y_1, z_1$ and $x_2, y_2, z_2$? | Ambiguous |
2,365,371 | Prove $ \frac{1}{x^{2n}}+x^{2n}$ is an integer for all $n$. | Ambiguous |
401,477 | Let $A$ be a commutative free algebra of rank $2$ over a ring. Is $A$ isomorphic to a quotient of a polynomial ring by an irreducible element? | Ambiguous |
2,717,459 | What is the formula for the expected value of a weighted sum of positive random variables? | Unambiguous |
2,639,179 | Am I wrong about the Monty Hall Problem? | Ambiguous |
1,176,215 | Prove by induction that $3^{3n+1} + 2^{n+1}$ is divisible by 5 for all non-negative integers $n$. | Unambiguous |
1,271,168 | What is the difference between the square of a logarithm, $(\log x)^2$, and the logarithm of a square, $\log(x^2)$? | Unambiguous |
1,376,320 | Prove that for integers $a$ and $b$, if $a+b$ and $ab$ have the same parity, then both $a$ and $b$ must be even. | Unambiguous |
620,187 | What do these symbols mean? | Ambiguous |
2,401,656 | Compute the value of ${34}^{429} \mod 431$. | Unambiguous |
1,290,374 | Can you provide an intuitive explanation and a proof for both parts of the Fundamental Theorem of Calculus? | Unambiguous |
19,236 | Efficient numerical algorithms to compute the Euler gamma function for real and complex arguments | Unambiguous |
1,194,644 | Is every integral extension of commutative rings a finite extension? | Unambiguous |
2,619,473 | Does the one-sided derivative of a real power series always exist at the boundary of its interval of convergence if the series converges there? | Unambiguous |
846,706 | What strategies can I use to find a conformal map from a lens shape to the first quadrant? | Ambiguous |
547,928 | What is the minimum value of $a+b+\frac{1}{ab}$ if $a^2 + b^2 = 1$? | Ambiguous |
2,273,478 | Translate "There is exactly one book that no students read" into first-order predicate logic using predicates $\text{book}(y)$, $\text{student}(x)$, and $\text{reads}(x,y)$. | Unambiguous |
1,435,003 | How do I calculate the total number of empty bins in the balls and bins problem given n balls and m bins? | Unambiguous |
973,713 | What are the common notation styles and conventions for the second-derivative symbol in calculus? | Unambiguous |
2,489,152 | Why is the bracket always zero? | Ambiguous |
3,005,021 | Proof Verification: Differentiability implies continuity. | Ambiguous |
2,096,678 | Provide a set of practice exercises on basic matrix arithmetic, including addition, subtraction, and multiplication. | Unambiguous |
2,618,534 | Compute the indefinite integral of $\int \frac {1+\sin (x)}{1+\cos (x)} dx$ with respect to $x$. | Unambiguous |
159,505 | Find the value of $x$ that solves the linear congruence $3x + 7 \equiv 6 \pmod{13}$. | Unambiguous |
12,139 | Number of relations that are both symmetric and reflexive | Ambiguous |
114,107 | Determine the rotation necessary to transform one point on a sphere to another. | Ambiguous |
1,899,905 | How irrational quantities physically exist in nature? | Ambiguous |
1,894,173 | Sum of reciprocal of primes | Ambiguous |
3,628 | Find a general formula for the n-th derivative of the function $f(x)=\frac {x} {x^2-1}$. | Unambiguous |
1,314,138 | How do I prove my integral limit is $0$ using the substitution $xt=u$? | Ambiguous |
332,632 | What is the probability that a person which has a positive test result in AIDS, is indeed infected with AIDS? | Ambiguous |
1,383,373 | Provide the solution and explanation for Question 55 from the GRE Mathematics Subject Test form 1268. | Unambiguous |
2,452,316 | Evaluating a contour integral using Cauchy's generalised integral formula | Ambiguous |
2,705,618 | Evaluate $\lim_{n\to\infty} \frac{e^{1/n} + e^{2/n} + \cdots + e^{n/n}}{n}$. | Unambiguous |
3,021,170 | Provide an example of a sequence of Lebesgue integrable functions on $R$ that converges pointwise to a limit function, but where the integral of the limit is not equal to the limit of the integrals. | Unambiguous |
4,569,632 | In $\triangle ABC$, $BD$ is a median, $\angle DAB=15$ and $\angle ABD=30$. Find $\angle ACB$. | Unambiguous |
3,303,672 | Line intersecting circle from origin | Ambiguous |
92,765 | Find all complex roots of the polynomial equation $z^4 + 2z^3 + 6z - 9 = 0$ | Unambiguous |
2,138,847 | If $f$ is a holomorphic function on the unit disk $\mathbb{D}$ such that $|f(z)| \le 1$ for all $z \in \mathbb{D}$ and $f(1/2) + f(-1/2) = 0$, prove that $|f(0)| \le 1/4$ and specify if a Möbius transformation is required for the proof. | Unambiguous |
685,435 | Trying to get a bound on the tail of the series for $\zeta(2)$ | Ambiguous |
820,424 | What are the standard methods and laws for simplifying Boolean functions in digital logic? | Unambiguous |
3,117,241 | Complex function to Taylor and Laurent series | Ambiguous |
1,516,096 | When does $A^TA=0$ imply $A=0$? | Ambiguous |
1,950,367 | Does equality of the complexifications imply equality of the algebras? | Ambiguous |
551,350 | Find all complex solutions to the equation $x^4+24x^3+18x^2-27=0$. | Unambiguous |
1,735,792 | Prove that for a homogeneous polynomial $P$ of degree $k$ in $n$ complex variables, $$P(z_1,\dots,z_n)=\frac{1}{k!}\sum_{\epsilon_1,\dots,\epsilon_k\in\{0,1\}}(-1)^{k-\sum_{j=1}^k \epsilon_j}P\left(a+\left(\sum_{j=1}^k\epsilon_j\right)z\right)$$ for any $a\in\mathbb{C}^n$. | Unambiguous |
1,145,215 | Consider a matrix $A$ that is full rank. Is $A(A^\top A)^{-1}A^\top = I$? | Ambiguous |
2,987,994 | Provide a list of examples of definite integrals solvable using the Feynman trick (Leibniz integral rule). | Unambiguous |
462,723 | What are some examples of mathematical games suitable for undergraduate students that can be used for teaching discrete mathematics or game theory? | Unambiguous |
184,593 | Is it necessary for the angle $\theta$ to be measured in radians for the limit $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ to hold? | Unambiguous |
2,186,187 | How do you calculate the first Fourier coefficient $a_0$ for a periodic function in its Fourier series representation? | Unambiguous |
574,465 | How do you find the complex roots of the polynomial equation $x^4 - 16 = 0$ to show they are $2$, $-2$, $2i$, and $-2i$? | Unambiguous |
2,289,082 | Compute the definite integral $\int_1^3\frac{\ln x}{x^2+3}\ dx$ | Unambiguous |
269,798 | Is it true that $E(X^a) = \int x^a g(x)\,dx$? | Ambiguous |
4,683,394 | What is the category-theoretic concept that unifies affine hull, convex hull, closure, and interior as intersections of all sets with a given property containing a set? | Unambiguous |
4,398,943 | What does 'abstract' mean in this context? | Ambiguous |
4,367,689 | Does there exist an ordered field extension of the rational numbers $\mathbb{Q}$ in which $\mathbb{Q}$ is not a dense subset? | Unambiguous |
1,867,732 | Gradient and Hessian of function on matrix domain | Ambiguous |
791,508 | Let $A, B, X$ be subgroups of $G$ with $A \subset X \subset AB$. Prove that $A(B\cap X) = X$. | Unambiguous |
1,950,039 | What is the identity property of multiplication by one in arithmetic? | Unambiguous |
4,073,406 | Infinite subset of natural numbers | Ambiguous |
823,613 | What is the time complexity in $\Theta$-notation in terms of $n$ for the Merge Sort algorithm? | Unambiguous |
1,896,195 | Let $M$ be a module over a ring. If $\operatorname{Hom}(M,M)$ is free, what can we conclude about $M$? | Ambiguous |
3,300,982 | Prove that if a real orthogonal matrix A commutes with all real orthogonal matrices of the same dimension, then A must be a scalar multiple of the identity matrix. | Unambiguous |
3,178,788 | What are the methods to determine the type of stationary point of a single-variable function without calculating the second derivative? | Unambiguous |
1,587,079 | How do I prove the map $(i,j) \mapsto (i+j, ij_0 - i_0 j)$ is surjective? | Ambiguous |
3,764,582 | Is there a convenient implementation in R to compute my incomplete gamma function integral? | Ambiguous |
2,918,597 | Find angle of which a cannon of fixed length has to fire | Ambiguous |
1,651,728 | Prove that the union and intersection of two recursively enumerable sets are also recursively enumerable. | Unambiguous |
3,487,811 | Evaluate the limit $\lim_{x \to 0} {1-x\cot x\over x^2}$ | Unambiguous |
1,302,479 | Prove that the limit of the Dirichlet function does not exist at any point in its domain. | Unambiguous |
3,493,088 | $\mathbb{C} \cup \{ \infty \}$ and $\mathbb{R} \cup \{ -\infty, +\infty \}$ | Ambiguous |
2,520,262 | Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges. | Ambiguous |
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