MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
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|---|---|---|
3,289,735 | Is the arc length always irrational between two rational points? | Ambiguous |
3,564,289 | What is the expected number of times a simple symmetric random walk on a 1D integer lattice returns to the origin? | Unambiguous |
2,381,284 | For $f_{a,c}(x)=\int_0^\infty\left[t^a(1+t)^{c-a-1}e^{-xt}-e^{-t}+e^{-xt}\right]\frac{dt}{t}$ with $a>0,x>0,c\in\mathbb{R}$, determine the sign of $f_{a,c}(x)$, given that $f_{a,c}'(x)<0$. | Unambiguous |
1,592,652 | Example of a relation that is symmetric and transitive, but not reflexive | Ambiguous |
984,271 | Provide an explicit isomorphism showing that $\mathbb{K} \otimes M_n(\mathbb{C})$ is isomorphic to $\mathbb{K}$, where $\mathbb{K}$ denotes the compact operators on a separable Hilbert space. | Unambiguous |
3,762,872 | Zeno's Achilles & Tortoise - Where exactly is the proof wrong? | Ambiguous |
1,489,146 | How is the imaginary unit $i$ formally defined in the context of complex numbers, and why is $i$ not equal to $-i$? | Unambiguous |
1,987,599 | It's known that a certain function behaves like $1/u$ near $u=0$. Are there sharper inequalities describing this behavior? | Ambiguous |
859,968 | Let $Y = \{(x, x/n) \in \mathbb{R}\times\mathbb{R} : x \in [0,1], n \in \mathbb{N}\}$ and $X = \bigcup_{n\in\mathbb{N}} [0,1]\times\{n\}$ with equivalence relation $(0,n) \sim (0,m)$ for all $n,m \in \mathbb{N}$. Prove or disprove that the quotient space $X/{\sim}$ is homeomorphic to $Y$. | Unambiguous |
1,204,425 | Solve the nonlinear ODE $y' + y^2 = 0$ using a power series solution $y = \sum_{n=0}^\infty a_n x^n$, given that substitution leads to $\sum_{n=1}^\infty a_n n x^{n-1} + \sum_{n=0}^\infty\left(\sum_{k=0}^n a_k a_{n-k}\right)x^n = 0$. | Unambiguous |
1,059,293 | Can someone explain how the proof for f(x) works? | Ambiguous |
956,087 | Provide an example problem and a step-by-step solution involving implicit differentiation of a curve defined by an equation in x and y. | Unambiguous |
3,023,583 | Does the infinite series $\sum_{n=1}^\infty \frac{n^2(-1)^n}{1+n^2}$ converge or diverge? | Unambiguous |
759,112 | Given that $Y - X^2$ is an irreducible polynomial, prove that the variety $V(Y-X^2)$ is irreducible. | Unambiguous |
959,137 | Finding missing coordinates in a non-linear graph? | Ambiguous |
2,715,973 | $$ \frac{n}{n+2}\sum_{k=1}^n\frac{k}{k+3}$$ | Ambiguous |
801,923 | Prove that for all integers $x$ and $y$, if $x + y + xy = 0$, then $(x, y) = (0, 0)$ or $(x, y) = (-2, -2)$. | Unambiguous |
1,503,414 | Why is the natural logarithm of 1 equal to 0? | Unambiguous |
3,035,983 | How can I find integers which satisfy $\frac{150+n}{15+n}=m$? | Ambiguous |
476,186 | Is there a category whose internal logic is paraconsistent? | Ambiguous |
108,191 | Why won't sine functions work as a counter example for my infinite integral proof? | Ambiguous |
401,477 | What is the definition of a $\mathbb{Z}/N\mathbb{Z}$-algebra? | Unambiguous |
1,135,959 | What is the standard way to represent a complex number as a $2 \times 2$ real matrix? | Unambiguous |
802,940 | How to calculate the numerical value of the sum from $k=1$ to $50$ of $k^{1/3}$? | Unambiguous |
1,035,200 | Let $p:\mathbb{N}\to\mathbb{N}$. What is $\lim p(n)$? | Ambiguous |
685,681 | Prove that a finite-dimensional vector space has a finite basis. | Unambiguous |
1,893,124 | If 16 players are divided into eight pairs, what is the probability that only one of S1 or S2 advances? | Ambiguous |
158,469 | Prove that if a linear endomorphism $\alpha$ on a finite-dimensional complex inner product space satisfies $\alpha^* = -\alpha$, then its eigenvalues are purely imaginary. | Unambiguous |
1,564,864 | What does the symbol # mean? | Ambiguous |
2,564,222 | Find the equations of the tangent lines drawn from the point $(11, 3)$ to the circle defined by $x^2 + y^2 = 65$. | Unambiguous |
1,295,010 | What are examples of true statements about natural numbers that can be proven using non-inductive methods, or that are independent of Peano arithmetic and thus cannot be proven by induction? | Unambiguous |
3,743,564 | Examples of classic coin weighing puzzles and their optimal solutions using a balance scale | Unambiguous |
783,173 | Evaluate the definite integral $\int_0^1 \frac{dx}{\sqrt{x(1-x)}}$ | Unambiguous |
4,814 | What are the names of directed graphs where every vertex has an in-degree of 2 and an out-degree of 2? | Unambiguous |
2,448,533 | How to evaluate a limit with arctan(x) | Ambiguous |
2,365,371 | If $x + frac{1}{x}$ is an integer, prove that $x^{2n} + frac{1}{x^{2n}}$ is an integer for all positive integers $n$. | Unambiguous |
1,232,385 | Let $S=\{\sqrt{D}: D\equiv 2,3\pmod4\}$ and $T=\left\{\frac{1-\sqrt{D}}{2}: D\equiv 1\pmod 4\right\}$. For $w\in S\cup T$ and the norm map $N:\mathbb{Z}[w]\to\mathbb{Z}$, prove that $\mathbb{Z}[w]/(a) \cong \mathbb{Z}_{N(a)}$ for $a\in\mathbb{Z}[w]$. | Unambiguous |
1,528,935 | How to prove a set is linearly independent without using a basis or FTA? | Ambiguous |
2,274,060 | If $x$ and $y$ are positive numbers, what is the minimum possible value of $(x+y)\left(\frac1x + \frac1y\right)?$ | Ambiguous |
1,788,328 | Calculate the numerical value of the product $\sin(10^\circ) \sin(50^\circ) \sin(70^\circ)$. | Unambiguous |
311,665 | Proof a graph is bipartite if and only if it contains no odd cycles | Ambiguous |
176,625 | For which positive integers $n$ is it true that there do not exist positive integers $x$ and $y$ such that $x^2 - y^2 = n$? | Unambiguous |
1,883,279 | What are examples of continuous functions that cannot be uniformly approximated by polynomials on a given interval? | Unambiguous |
405,972 | A basis for a subspace of $\mathbb R^3$ | Ambiguous |
4,734,106 | Is there an equation similar to square root, but faster for a computer to compute? | Ambiguous |
3,226,462 | On an infinitely sided die, is it possible to roll and get the same number twice? | Ambiguous |
529,215 | Is it true that if a sequence {a_n} converges, then the set of its values has exactly one limit point? | Ambiguous |
2,297,557 | Given three points $A, B, M$ in the plane, determine whether there is a unique point $R$ satisfying $d(R,A)\cdot d(R,B) = d(R,M)^2$, or find the curve formed by all such points $R$. | Unambiguous |
1,796,173 | I tried to expand the bracket $[X,Y]=XY-YX=Y$, how do I proceed to find $[Y,Z]$? | Ambiguous |
3,154,407 | Applying a function on any multiset of real numbers | Ambiguous |
2,591,017 | Unity roots that sum to 1 | Ambiguous |
1,382,185 | How many divisors of the combination of numbers? | Ambiguous |
151,810 | What is the probability of getting exactly 3 heads in 10 independent flips of a fair coin? | Unambiguous |
807,589 | For a pure tensor $v_1\otimes\cdots\otimes v_n \in V^{\otimes n}$, what is the standard terminology for the component vector $v_j$? | Unambiguous |
927,248 | Maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$ | Ambiguous |
1,286,651 | How do I compute the limit of a function of two variables as it approaches a point $(x, y)$? | Unambiguous |
2,666,326 | For a square matrix $A$ over the complex numbers, is it true that if $A^4 = I$, then $A^2 = I$ or $A^2 = -I$? | Unambiguous |
2,322,481 | Does the limit as $k$ approaches infinity of the ratio $\frac{\sum_{n=1}^{k} 2^{2 \cdot 3^n}}{2^{2 \cdot 3^k}}$ equal $1$? | Unambiguous |
4,093,371 | Show that a graph that is connected but not complete has vertices u,v and w such that uv and vw are edges but not uw. | Ambiguous |
1,565,903 | What are some simple-to-prove, counter-intuitive theorems in set theory that depend on the Axiom of Choice? | Unambiguous |
2,893,411 | Evaluate the sum $\sum\limits_{k=1}^n {\binom{n}{k} f(k)}$. | Ambiguous |
2,555,463 | Find the point equidistant from two points and a line | Ambiguous |
728,599 | How to evaluate the triple integral for region $E$ bounded by the spheres? | Ambiguous |
5,877 | Efficiently finding two squares which sum to a prime | Ambiguous |
3,111,260 | How do I evaluate $\lim_{x\to 0}\frac{x-\tan x}{x^3} $? | Unambiguous |
2,307,642 | How to find an orthonormal $T$ eigenbasis without matrices? | Ambiguous |
1,774,294 | Explain why $y^2 = 2x^2 + C$ is not equivalent to $y = \sqrt{2x^2} + C$. | Unambiguous |
3,322,357 | How is the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$ proved for square matrices $A$ and $B$ of the same size? | Unambiguous |
449,507 | Compute the indefinite integral of $\frac{\sqrt{x^2-4}}{x^4}$ with respect to $x$. | Unambiguous |
1,037,426 | Find $\lim\limits_{x \to \infty}{e^{-x}\sqrt{x}}$ | Ambiguous |
1,822,028 | In a normed vector space, is the closed convex hull of a weak*-compact subset of the dual space also weak*-compact? | Unambiguous |
2,959,054 | Can I use stars and bars for this? | Ambiguous |
398,593 | Calculate the length of curve $f(x)=\arcsin(e^x)$, check solution, please. | Ambiguous |
199,447 | Provide a proof that the derivative of the function $f(x) = e^x$ is $f'(x) = e^x$ using the limit definition of the derivative and without using the chain rule. | Unambiguous |
43,118 | How can I explain to a layperson why the 'staircase' proof that $\pi = 4$ is mathematically incorrect? | Unambiguous |
1,562,459 | Prove that $( 1 + n^{-2}) ^n \to 1$. | Ambiguous |
2,116,037 | Does $\sum_{n=1}^{\infty} \frac{\sin^2(nx)}{n}$ converge uniformly? | Ambiguous |
3,258,160 | Calculate $\iint_\Omega \left(\sqrt{\frac xa}+\sqrt{\frac yb}\right)\,dxdy$ | Ambiguous |
4,564,068 | Is there a closed-form formula for the sum of the first $n$ squares, $\sum_{k=1}^{n} k^2$? | Unambiguous |
1,086,318 | I'm looking for an explanation regarding an exponential function and a Taylor series approximation. Could you elaborate on how these relate in the context of a measurement? | Ambiguous |
68,897 | Let $A$ be positive definite and $B$ be full rank. Show that $BAB'$ is positive definite. | Ambiguous |
611,256 | How do you convert a decimal number to 8-bit two's complement binary representation? | Unambiguous |
1,410,163 | Show that $\lim_{(x,y)\to(0,0)} \frac{xy^2}{x^2+y^4}$ does not exist by evaluating the limit along the paths $y=x$ and $y=-x$. | Unambiguous |
1,982,170 | Probability of an event in relationship with a certain geometrical representation | Ambiguous |
763,272 | Which is greater: the value of $\sqrt{1001} - \sqrt{1000}$ or the fraction $\frac{1}{10}$? | Unambiguous |
1,372,825 | Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $ | Ambiguous |
1,117,045 | What are the standard methods and techniques for evaluating definite integrals given specific upper and lower limits of integration? | Unambiguous |
1,762,215 | Diamond and Shurman page 73 says $j'(\tau)$ doesn't vanish, how do I prove this? | Ambiguous |
405,972 | How do I find a basis for a subspace of $\mathbb{R}^3$ defined by a system of linear equations? | Unambiguous |
2,490,690 | In how many ways can you make 107 using 1, 10, and 50? | Ambiguous |
2,247,820 | For some general variable $c$, why is the limit always positive infinity except when $c=1$? What happens if $c=0$ or $c=-1$? | Ambiguous |
1,602,977 | What is an example of an infinitesimal in the context of non-standard analysis? | Unambiguous |
611,256 | What are the least and greatest decimal integers that can be stored in an 8-bit register? | Ambiguous |
1,502,938 | How to prove the sum equals $n/(n+1)$ using induction? | Ambiguous |
811,451 | Evaluate $\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}}$. | Unambiguous |
4,566,790 | Can someone help me understand eigenvalues? | Ambiguous |
1,008,311 | Prove that for a bounded linear operator $L$ on a Hilbert space $H$, the null space of $L^*$ is equal to the null space of $L L^*$. | Unambiguous |
262,371 | How do you calculate the area of a trapezoid given the lengths of its diagonals and the angle between them? | Unambiguous |
3,248,549 | Let $f$ be a continuous map then which of the following is correct | Ambiguous |
119,307 | Give an example of a Cauchy sequence that does not converge in the space of rational numbers $Q$. | Unambiguous |
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