MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
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365,782 | Prove that $-\int \frac{d^3p}{(2\pi)^3}\, p\,\frac{\partial f(p,t)}{\partial p} = 3F$, where $F = \frac{1}{(2\pi)^3}\int d^3p\, f(p,t)$, using integration by parts. | Unambiguous |
1,283,817 | Given data points $(x_1,y_1),\dots,(x_n,y_n)$ and the model $f(x)=ae^{2x}+be^x+c$, find the transformation of the points $(x_i,y_i)$ needed to fit $f(z)=az^2+bz+c$ via least squares under the substitution $z=e^x$. | Unambiguous |
2,539,486 | How do I solve for the min of $\int_{-1}^1| x^3-a-bx-cx^2|^2 dx$ and the max of $\int_{-1}^1 x^3 g(x) dx$ given the constraints on $g$? | Ambiguous |
2,961,334 | How to prove a solution to a functional equation is unique? | Ambiguous |
574,551 | Evaluate $\oint_C P\,dx + Q\,dy$ where $P(x,y)=\frac{y+x}{x^2+y^2}$ and $Q(x,y)=\frac{y-x}{x^2+y^2}$, and $C$ is the circle $x^2+y^2=a$ oriented clockwise, given that $P_y=Q_x$ but $(0,0)$ is a discontinuity. | Unambiguous |
1,941,590 | How do you use Cramer's rule to solve a system of linear equations using determinants? | Unambiguous |
2,158,739 | True or False: Every 3-dimensional linear subspace of the vector space of $2 \times 2$ real matrices $\mathbb{R}^{2 \times 2}$ contains at least one invertible matrix? | Unambiguous |
258,546 | How to find the parabolic curve equation from length $L$ and area $S$ based on my picture? | Ambiguous |
3,328,473 | How to find the numerical solutions for $2x - \tan(x) = 0$ in the interval $[0, 2\pi]$? | Unambiguous |
4,268,142 | A coin is tossed three times, defining two related random variables from the outcomes. Find their joint pmf and show they're dependent. | Ambiguous |
2,881,207 | How does the irreducibility of a polynomial over a finite field depend on the size of the field? | Unambiguous |
1,037,122 | Prove that for any real number $\lambda$, the integral $\int_0^\infty e^{-x} \sin(\lambda x) \, dx = \frac{\lambda}{1+\lambda^2}$. | Unambiguous |
2,166,855 | How to find the maximum value of the objective function $x + y$ subject to a set of linear constraints? | Unambiguous |
2,049,971 | Prove that the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cannot be transformed into the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ by any real coordinate transformation, without substituting $b \to ib$. | Unambiguous |
3,786,586 | How to solve an inequality involving the sum of two absolute value expressions, such as $|ax + b| + |cx + d| < k$? | Unambiguous |
41,706 | What are some practical applications of matrix multiplication in computer science and engineering? | Unambiguous |
3,111,111 | If $X$ is a topological space, does the property still hold for the product of two continuous functions? | Ambiguous |
2,022,771 | In mathematical learning, does the complexity of notation generally decrease as one advances to more abstract levels of study? | Unambiguous |
1,980,947 | Which value is greater, $2^{\sqrt{2}}$ or the mathematical constant $e$? | Unambiguous |
609,404 | Find all values of $c \in \mathbb{Z}_5$ such that the quotient ring $\mathbb{Z}_5[x]/\langle x^3+2x+c\rangle$ is a field. | Unambiguous |
2,539,486 | (a) Using the best approximation theorem, compute $\min_{a,b,c} \int_{-1}^1 |x^3 - a - bx - cx^2|^2\, dx$. (b) Find $\max \int_{-1}^1 x^3 g(x)\, dx$ subject to $\int_{-1}^1 g(x)\,dx = \int_{-1}^1 xg(x)\,dx = \int_{-1}^1 x^2 g(x)\,dx = 0$ and $\int_{-1}^1 |g(x)|^2\,dx = 1$. | Unambiguous |
3,087,356 | Why is the definition of prime numbers written to include 2 | Ambiguous |
3,399,461 | A polynomial defined by two squared terms is divided by two linear factors, giving known remainders. Find the coefficients. | Ambiguous |
4,107,862 | How should I think about the concept of location for a point? | Ambiguous |
744,139 | Prove by induction that $2\cdot 8^n + 3\cdot 15^n + 2$ is divisible by $7$ for all $n \in \mathbb{N}$. | Unambiguous |
4,749,159 | Equation exercise involving nth-roots in Serge Lang's Basic Mathematics. | Ambiguous |
795 | How to prove the theorem of Riemann asked by Kenny? | Ambiguous |
942,204 | Looking for the logic of a sequence from convolution of probability distributions | Ambiguous |
4,835,425 | Revisiting: Adding an optimizer to a nonlinear set of equations | Ambiguous |
2,237,243 | What are good "math habits" that have improved your mathematical practice? | Ambiguous |
1,406,770 | Iterated square roots over finite field. When do we hit a nonresidue? | Ambiguous |
3,472,866 | Are $\sigma^2$ and $\mu$ random variables? | Ambiguous |
3,806,534 | If $\lim_{x\to 0}(f(x) + \frac{1}{f(x)}) = 2$ and $\lim_{x\to 0} f(x)$ exists, prove that $\lim_{x\to 0} f(x) = 1$. | Unambiguous |
4,325,846 | How do I formally show that the Zariski tangent space of the intersection of two closed subschemes is the intersection of the tangent spaces? | Ambiguous |
2,530,549 | What are some mental math techniques for performing complex calculations without a calculator? | Unambiguous |
164,588 | How to find the general coefficient of the Cauchy product of two power series? | Unambiguous |
1,838,743 | In the figure, two circles intersect at $P$ and $Q$... | Ambiguous |
788,217 | A hammer and a nail cost $1.10, and the hammer costs one dollar more than the nail. How much does the nail cost? | Ambiguous |
216,344 | Does the equation $ax + by = ab$ have any solutions for $\gcd(a,b)=1$? | Ambiguous |
385,679 | A matrix satisfying $AB-BA=B$ | Ambiguous |
925,677 | I need the coordinates for an octagon. | Ambiguous |
2,159,465 | Are two polygons homeomorphic? | Ambiguous |
582,557 | Determine whether $\sum_{n=2}^\infty \frac{1}{n\ln(n^3)}$ converges or diverges. | Unambiguous |
4,336 | If eigenvalues are positive, is the matrix positive definite? | Ambiguous |
531,128 | Dimension of a subspace of finite-dimensional product space $V$ equals $\dim V - 1$ | Ambiguous |
2,388,784 | What is the intuitive explanation for the formula of variance in probability theory? | Unambiguous |
1,165,928 | What is the general formula for the k-th derivative of $G \circ F$? | Ambiguous |
4,768 | every permutation is either even or odd,but not both | Ambiguous |
1,997,424 | What is the rigorous definition of the differential of a function in multivariable calculus? | Unambiguous |
2,239,918 | Can someone help me with this: 2^(log(2x,x+2))+3^(log(2,x+3))=sqrt(-1-x) | Ambiguous |
354,134 | What is the logical negation of the statement "$A\subseteq B$" in set theory? | Unambiguous |
1,871,195 | Given a function $a(x)$, is $$\max\left(\frac{1}{2}a'(x)\right) > \max\left(\frac{1}{3!}a''(x)\right) > \cdots$$ always true? | Ambiguous |
422,707 | Finding eigen values of a binary matrix with diagonal elements are all 0s and non-diagonals are 1s | Ambiguous |
747,433 | What are the standard tests for determining the absolute convergence of an infinite series? | Unambiguous |
3,300,982 | I proved that if a symmetric real matrix commutes with all orthogonal matrices, it is a scalar multiple of the identity. Now how do I do this for orthogonal matrices? | Ambiguous |
1,674,072 | How do you find the greatest integer whose square root differs from another integer's square root by at least one? | Ambiguous |
188,703 | For $f(x,y,z) = e^{\sqrt{xy}+\sqrt{xz}+\sqrt{zy}}$, determine whether a Taylor series expansion exists at $(0,0,0)$, given that the partial derivatives there take the indeterminate form $0/0$. | Unambiguous |
2,072,845 | Why do different methods of factoring a quadratic trinomial, such as the AC method and trial and error, always yield the same linear factors? | Unambiguous |
730,358 | Sketch $\tan x$, solve $\cot x=0.15$, and evaluate $\csc x$ given $\cot x=0.15$. | Ambiguous |
853,164 | Provide an example of a binary relation on a set that is not reflexive, not symmetric, and not transitive. | Unambiguous |
3,934,173 | Can any permutation of elements be represented as a series of rotations in some n-dimensional space? | Ambiguous |
265,703 | Under what conditions does every element of a monoid occur exactly once in its Cayley table? | Unambiguous |
4,067,963 | What is the sum of $\sum_{n\in M(P)} \frac{1}{n}$ where $M(P)$ contains numbers with prime divisors in $P$? | Ambiguous |
2,540,359 | What is the formal definition of an undecidable problem in computability theory? | Unambiguous |
3,519,278 | $6$ persons enter the train during the journey with $6$ different tickets of the same class. How many different sets of ticket may they have had? | Ambiguous |
3,310,472 | Is the P versus NP problem classified as a topic within the field of Operations Research? | Unambiguous |
1,072,836 | Is every symmetric matrix diagonalizable, and what are the possible eigenvalues of an orthogonal matrix? | Ambiguous |
2,096,678 | Suppose $A^2 + B^2 = AB$ and $AB - BA$ is invertible. What can be said about $n$? | Ambiguous |
1,798,274 | Least value of $a$ for which $4ax^2 + \frac{1}{x} \geq 1$ | Ambiguous |
488,020 | Prove that the set of signed circuits of an oriented matroid is invariant under reorientation of its elements. | Unambiguous |
110,579 | In how many ways can a batsman score exactly 100 runs using only 6s and 4s, where the order of scoring matters? | Unambiguous |
1,249,731 | Let $X=(X_1,\dotsc,X_n)$ be a random vector with mean $0$. What is $\mathbb{E}[X_1\cdots X_n]$? | Ambiguous |
646,238 | How do I convince my students that the choice of variable of integration is irrelevant? | Ambiguous |
3,125,432 | Trouble with telescoping algorithm question | Ambiguous |
3,285,074 | Are there other functions that satisfy $|f(x)-f(y)|=|x-y|$? | Ambiguous |
664,657 | I'm trying to connect $\epsilon$ and $\delta$ for $|f(x) - f(c)| < \epsilon$. How do I do this? | Ambiguous |
2,360,710 | Formula for Natural logarithm of $\pi$ | Ambiguous |
1,731,822 | Given some unlabeled 2D data points, how do I construct a k-nearest-neighbour graph and its Laplacian? | Ambiguous |
2,146,495 | For $f(t)=\frac{2}{1+t^2}$, compute the convolution $(f*f)(t) = \int_0^t f(u)f(t-u)\,du$. | Unambiguous |
4,341,781 | The integral of a function along any circle is zero. What does this mean, and what follows from it? | Ambiguous |
227,317 | Explaining $\cos^\infty$ | Ambiguous |
1,067,717 | How do you get $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$? | Ambiguous |
108,308 | Apostol proof divides by zero? | Ambiguous |
2,062,239 | Numerical algorithms for approximating the polar decomposition of a matrix | Unambiguous |
687,510 | A shape is cut into 3 equal parts. Show one part contains 2 points at distance at least one. | Ambiguous |
330,556 | $a^{n}$: is it arithmetic or geometric, or neither? | Ambiguous |
3,791,855 | How does $x^2-8x+17=0$ have nonreal solutions? | Ambiguous |
988,784 | Why is it a common misconception that non-real numbers consist only of square roots of negative numbers? | Unambiguous |
2,394,502 | Under what conditions does a complex number $z=x+iy$ in the third quadrant imply that the ratio $\frac{\bar{z}}{z}$ also lies in the third quadrant? | Unambiguous |
3,098,928 | For a fixed real number $x > 0$, does the sequence $x^{1/n}$ converge as $n$ approaches infinity, where $n$ is a natural number? | Unambiguous |
2,823,907 | Prove that $\sum_{n=2}^\infty \frac{1}{n\ln(n^3)}$ diverges using the comparison test. | Unambiguous |
3,059,714 | How does one evaluate the multiplication $f(2)\cdot f(3)\cdot f(4)\cdots f(15)$ by formulating? | Ambiguous |
2,594,206 | Prove that there is no $2 \times 2$ real matrix $A$ such that $A^2 = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}$. | Unambiguous |
639,890 | Minimum value of $\cos(A-B)+\cos(B-C)+\cos(C-A)$? | Ambiguous |
225,922 | An example for a calculation where imaginary numbers are used but don't occur in the question or the solution. | Ambiguous |
983,405 | Prove the inequality $\|\nabla f\|_p \le C (\|\nabla \times f\|_p + \|\nabla \cdot f\|_p)$ for vector fields $f$ in $L^p$ spaces on $\mathbb{R}^3$. | Unambiguous |
4,693,000 | Prove that $\frac{1}{\zeta(3)} < 2\gamma - 1$, where $\zeta(s)$ is the Riemann zeta function and $\gamma$ is the Euler-Mascheroni constant. | Unambiguous |
724,075 | Where can I find the book 'Here's a Cool Problem: A Collection of Short Questions with Clever Solutions'? | Unambiguous |
1,302,479 | I'm having trouble with this function: $$f(x)=\begin{cases}1, & \text{if } x\in \Bbb Q\\ -1, & \text{if } x\in \Bbb R\backslash \Bbb Q. \end{cases}$$ | Ambiguous |
26,918 | Why is the mathematical constant $\pi$ defined as the ratio of a circle's circumference to its diameter rather than its ratio to the radius? | Unambiguous |
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