MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
values |
|---|---|---|
736,044 | A number divided by $q$ leaves remainder $r$. Find $r+q$. | Ambiguous |
2,339,487 | rove that whenever there exists an $[N,K,D]$-linear code over $\mathbb{F}_{q^{n-1}}$, there exists an $[nN,(n-1)K,2D]$-linear code over $\mathbb{F}_q$. | Unambiguous |
4,044,721 | Find the limit of $\lim_{x\to 0}\left(\frac{\cos x}{\cos 2x}\right)^\frac{1}{x^2}$ | Ambiguous |
2,666,204 | How do I optimize $(\pi R_1 R_2)h$ when I don't know what the surface area should be? | Ambiguous |
1,071,811 | Why is the Zariski topology on an algebraic variety over an algebraically closed field not Hausdorff? | Unambiguous |
2,260 | Prove by mathematical induction that the sum of the first $n$ positive integers is $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$. | Unambiguous |
128,823 | limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$ | Ambiguous |
2,006,377 | Average integral increasing | Ambiguous |
20,813 | What is the intuitionistic construction of the unit circle in constructive mathematics? | Unambiguous |
3,367,250 | For this equation, does it matter which variable we treat as independent? | Ambiguous |
4,236,641 | What is the best way to find the closest vector in a set to a given one? | Ambiguous |
721,784 | Absolute value of complex exponential | Ambiguous |
2,844,637 | I need to calculate $\cos(\pi/20)$ with an error less than $10^{-5}$ using a series. Is my answer correct? | Ambiguous |
835,454 | $f: SL_2(\mathbb{R}) \to GL_4(\mathbb{R})$ show that $Im(f)=SL_4(\mathbb{R})$ | Ambiguous |
568,493 | How do you find the number of common codes for [1,5], [4,8] and [3,7], [2,6]? | Ambiguous |
1,768,578 | How to find the relation between $a$, $c$ and $b$ to make my density function integral $1$? | Ambiguous |
3,815,586 | If a function is differentiable almost everywhere, what is its regularity? | Ambiguous |
2,931,443 | If $\sum c_n x^n = (\sum a_n x^n)(\sum b_n x^n)$, does it follow that $\sum c_n = (\sum a_n)(\sum b_n)$? | Ambiguous |
1,283,678 | Prove $||\lambda x_1 + (1-\lambda) x_2 - y|| \leq ||x_1 - y||$ | Ambiguous |
1,942,246 | I need to prove trigonometric equation | Ambiguous |
2,620,801 | Proof Group Homomorphism: $\varphi(e)= e'$ & $\varphi(\hat{a})= \widehat{\varphi(a)}$ | Ambiguous |
960,632 | Evaluating product of exponent and polynomial | Ambiguous |
697,675 | multiplying a matrix by a row vector | Ambiguous |
2,360,710 | What is the numerical value or a series expansion formula for the natural logarithm of $\pi$, $\ln(\pi)$? | Unambiguous |
587,560 | What is the purpose of the base case in a proof by mathematical induction? | Unambiguous |
2,038,540 | Permutations with men and women alternating | Ambiguous |
1,663,674 | Does my genome exist in the digits of pi? | Ambiguous |
2,315,737 | Is checking a norm on a function space just the triangle inequality and the other two rules? | Ambiguous |
3,969,765 | Evaluate the integral $\int_0^\infty \frac{e^{-tx^2}-e^{-x^2}}{x}dx$ for $t > 0$. | Unambiguous |
2,205,890 | In what mathematical context or under what non-standard ordering could the inequality $\frac{1}{2} < \frac{1}{4}$ be true? | Unambiguous |
1,072,741 | Is an open $n$-ball homeomorphic to $\mathbb{R}^{n}$? | Ambiguous |
62,473 | What is the definition of a quasifinite morphism of schemes? | Unambiguous |
824,000 | If a function $f: X \to X$ satisfies $f(f(x)) = x$ for all $x \in X$, does it follow that $f$ is bijective? | Unambiguous |
2,165,873 | For $f(x)=\begin{cases}2,&x>2\\3,&x<2\end{cases}$ and $g(x)=\begin{cases}-2,&x\ge2\\2,&x<2\end{cases}$, find $g(x)-f(x)$, addressing whether the result is defined at $x=2$. | Unambiguous |
264,371 | Recommend challenging but engaging mathematics books suitable for advanced undergraduate students that cover topics beyond the standard curriculum. | Unambiguous |
3,111,260 | Error evaluating $ \lim_{x\to 0}\frac{x-\tan x}{x^3} $ | Ambiguous |
1,176,875 | Find all integer solutions $(a, b, c)$ to the equation $a + b + c = abc$. | Unambiguous |
4,094,225 | Given point $K$ in a triangle, is it possible to construct a line through $K$ that bisects the area of the triangle? | Ambiguous |
13,582 | Show that a continuous function has a fixed point | Ambiguous |
3,735,022 | Does knowing the Gauss curvature is $1/R^2$ help prove the curve's curvature is $\geq 1/R$? | Ambiguous |
587,560 | What is the purpose of the first test in an inductive proof? | Ambiguous |
1,108,298 | Why is the canonical map $M \otimes_A N \to M \otimes_{A/I} N$ an isomorphism? | Ambiguous |
3,979,952 | Are $\sum_{i=1}^{n} \xi_i / i$ and $\max\xi_n$ equally distributed? | Ambiguous |
665,341 | Using only a compass fixed at a radius of $5$ inches and a straightedge, construct a circle of radius $4$ inches. | Unambiguous |
4,067,963 | Prove that the sum of the reciprocals of the integers in the set $M(P)$, where $M(P)$ consists of all positive integers whose prime factors are in the set $P$, converges and find its sum in terms of the prime zeta function. | Unambiguous |
1,048 | Compare the different definitions of trigonometric functions using right triangles, the unit circle, and power series. | Unambiguous |
675,332 | On the set $S = \mathbb{R}\setminus\{-1\}$ with operation $a * b = a+b+ab$, solve $2*x*3 = 7$ for $x$. | Unambiguous |
2,605,427 | Prove that for any connected graph $G$, there exists a partition of the vertex set into two sets $V_1$ and $V_2$ such that the induced subgraphs $G[V_1]$ and $G[V_2]$ are both connected. | Unambiguous |
3,729,113 | What is the definition and region of convergence for a Dirichlet series of a complex variable? | Unambiguous |
290,435 | Find five positive integers whose reciprocals sum to $1$ | Ambiguous |
261,680 | Find integer values of $a,b$ such that $k=a^3-b^2$ is a perfect square. | Ambiguous |
2,042,807 | Construct a homeomorphism between the unit circle $S^1$ and the boundary of a square in $R^2$. | Unambiguous |
288,432 | Let a power series have radius of convergence 1 and diverge at $x=1$. Does it diverge as $x\to 1$? | Ambiguous |
4,812,185 | How to prove what the Wikipedia page says about the entry wise norm limit being the maximum norm? | Ambiguous |
1,490,059 | What is the definition of the curl of a 3D vector field in vector calculus? | Unambiguous |
3,468,876 | Calculate the mean, variance, and marginal distribution of the random variable X = Y1 + Y2, where Y1 and Y2 are independent standard normal random variables. | Unambiguous |
331,345 | How do I find the solutions for the equation $2x^2 - 5x + 3 = 0$? | Unambiguous |
295,318 | Are all algebraic vector bundles on the affine line over a principal ideal domain (PID) trivial? | Unambiguous |
219,774 | Finding the periods of functions... | Ambiguous |
172,468 | Why does MatLab generate polynomials for cyclic codes for all $n,k$ I try dividing $x^n-1$? | Ambiguous |
4,117,494 | Volume of tetrahedron Solid | Ambiguous |
2,007,373 | What are the fundamental geometric intuitions used to understand concepts in undergraduate linear algebra and multivariable calculus? | Unambiguous |
2,897,989 | Prove that the dilation operator on $L^2(\mathbb{R})$ is a bounded linear operator. | Unambiguous |
4,567,071 | How can you show the results from substituting $x=a\cos{\theta}$ and $x=a\sin{\theta}$ are the same when finding $\int{\frac{16}{\sqrt{9-x^2}}}$? | Ambiguous |
730,358 | How do I graph a general sine function of the form y = a sin(bx - c) + d? | Unambiguous |
2,394,582 | Show that $f(x) = x^3 - x$ is an odd function. | Unambiguous |
3,839,814 | Probability that $\text{det}(A)$ is an even number. | Ambiguous |
2,160,552 | Do elementary row operations change the eigenvalues of a square matrix? | Unambiguous |
4,016,729 | Find all real values of theta such that $(\sin(\theta) - \cos(\theta))(2+\sin(\theta) \cos(\theta)) \leq 2$. | Unambiguous |
315,869 | Sequence of sets and neighborhoods | Ambiguous |
742,878 | Modeling with Exponential and Logarithmic Functions | Ambiguous |
1,496,959 | How to show $f_n$ converges uniformly to $f'$ if the derivative is bounded by $3|x-y|$? | Ambiguous |
1,745,083 | How do you find the shortest path between two fixed points in a 2D plane that avoids a given linear obstacle? | Unambiguous |
124,287 | Is the map $f(a+bi)=a$ a homomorphism of rings? | Ambiguous |
4,221,432 | Near-Pythagorean triplets: What are the general solutions to $a^2+b^2=c^2-1$? | Ambiguous |
2,540,002 | Let $$S=\sum_{i=0}^N \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(i+0.5)^2/(2\sigma^2)}$$. Any ideas? | Ambiguous |
765,044 | Challenge: Demonstrate a Contradiction in Leibniz' differential notation | Ambiguous |
1,896,195 | Under what conditions is the endomorphism ring of a module free as a module over the base ring? | Unambiguous |
1,882,798 | Given constants $X, Y, P$, determine whether there exist positive integers $a, b$ such that $(aX+bY) \bmod P = 0$. | Unambiguous |
2,540,359 | Does the fact that both examples and counterexamples exist mean a property is undecidable from the axioms? | Ambiguous |
3,516,041 | Third order Taylor polynomial of $f:(0, \infty) \times \mathbb{R} \to \mathbb{R}$ given by $f(x,y) = x^y$ | Ambiguous |
2,128,278 | How do I compute the indefinite integral of the hyperbolic sine function $\sinh(x)$? | Unambiguous |
4,087,621 | In which of the intervals is $\sqrt{12}$ | Ambiguous |
458,261 | How do I rewrite a triple integral with different limits for $x+y-2z=2$? | Ambiguous |
4,044,721 | Find the limit of $\lim_{x\to 0}\left(\frac{\cos x}{\cos 2x}\right)^\frac{1}{x^2}$ | Unambiguous |
1,310,890 | What is $\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$? | Ambiguous |
195,504 | Prove that for a linear functional, the kernel has codimension one. | Ambiguous |
1,438,319 | Prove that $\lim_{x\to 0} \frac{e^x-1}{x}=1$ using the definition of the exponential function or power series, without using L'Hôpital's rule or derivatives. | Unambiguous |
2,597,743 | How do I compute the indefinite integral of a function of the form $f(x) = x^n e^{-ax}$? | Unambiguous |
2,093,882 | Problem in solving a question related to finite difference. | Ambiguous |
2,555,463 | How do I find the coordinates of a point in a 2D plane that is equidistant from two given points and a given line? | Unambiguous |
2,339,487 | Does putting $n-1$ copies of a codeword after itself prove there is a linear code? | Ambiguous |
12,139 | How many binary relations on a set of size n are both symmetric and reflexive? | Unambiguous |
3,084,662 | How to convert a complex number $z = a + bi$ to polar form (calculate the magnitude r and the argument theta). | Unambiguous |
1,007,952 | For a random variable $X$ with pdf $f(x)=\frac12 e^{-|x|}$, what can we say about it? | Ambiguous |
608,477 | What is the general method for a proof by mathematical induction, and can you provide a step-by-step example? | Unambiguous |
4,569,632 | In $\triangle ABC$, $BD$ is a median, $\angle DAB=15$ and $\angle ABD=30$. Find $\angle ACB$. | Ambiguous |
1,002,480 | Is the Minkowski sum of a finite collection of compact line segments in Euclidean space compact? | Unambiguous |
2,782,880 | Proof check: AB not invertible | Ambiguous |
25,648 | Prove that if two integers $a$ and $b$ are congruent modulo $n$, then $\gcd(a, n) = \gcd(b, n)$. | Unambiguous |
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