MathSE ID int64 795 4.87M | Math Question stringlengths 23 305 | Tag stringclasses 2
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|---|---|---|
137,352 | What are the domain and range of the map f in Devaney's definition of chaos on a metric space? | Unambiguous |
2,561,078 | How does the triangle inequality apply to the proof for f-g using epsilon and delta? | Ambiguous |
735,461 | For a completely positive map $f: M_n \to M_m$ with $f(a) = I_m$ for some $a \in M_n$, prove or disprove that there exists a positive element $b \in M_n$ such that $f(b) = I_m$. | Unambiguous |
2,461,048 | What is the cyclic subgroup generated by -1? | Ambiguous |
578,647 | What is an intuitive explanation of the Fourier transform and how is it used to analyze signals in the frequency domain? | Unambiguous |
2,430,945 | Decide if the statement " $n^2-1$ is multiple of $4$ if and only if $n-1$ is multiple of $4$" is true or false. | Unambiguous |
1,403,539 | Prove that $\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ for $x >0$. | Unambiguous |
510,991 | Prove that every continuous function defined on a compact metric space is uniformly continuous. | Unambiguous |
2,904,744 | Prove that $\lim_{x\to 1} f(x) = 3$ where $f(x) = \frac{x^3-1}{x-1}$ for $x \neq 1$. | Unambiguous |
982,186 | Decompose the expression $\frac{1}{4n^2-1}$ into partial fractions. | Unambiguous |
4,087,621 | Between which two consecutive integers does the value of $\sqrt{12}$ lie? | Unambiguous |
697,675 | How do you multiply a matrix by a row vector from the left? | Unambiguous |
1,025,321 | Find the derivative $\frac{dy}{dx}$ using implicit differentiation for the equation $\ln(1+xy) = xy$. | Unambiguous |
3,313,255 | What is the definition and standard form of Sum of Products (SOP) in boolean algebra? | Unambiguous |
578,647 | In the Fourier transform, what does the cross term mean? | Ambiguous |
3,843,282 | finding quotient group relationship | Ambiguous |
3,372,191 | Asymptotic behavior of $a_{n+1}=a_n + f(a_n)$ | Ambiguous |
2,694,508 | Find the range of values of $k$ such that both roots of $x^2 - 2kx + k^2 - 1 = 0$ lie in the interval $(-2, 4)$. | Unambiguous |
3,735,022 | What is the formula for the curvature of a Frenet curve constrained to lie on a sphere? | Unambiguous |
4,870,757 | If I know the dimension, depth, and codimension of $R$ and $I$, can I determine the depth of $I$? | Ambiguous |
166,794 | Proof that the set of irrational numbers is uncountable | Unambiguous |
4,127,695 | Evaluate the sum $\sum_{r=0}^n 2^{n-r} \binom{n+r}{r}$ for non-negative integers $n$ and $r$. | Unambiguous |
2,949,838 | Can every basis of a finite dimensional vector space be orthogonal? | Ambiguous |
7,391 | How do I evaluate the definite integral $\int_0^{\pi} \sin(2x)\sin(x) dx$? | Unambiguous |
1,315,088 | Evaluate the limit $\lim_{x \to 0} \left( \frac{1}{\sin(x)\arctan(x)} - \frac{1}{\tan(x)\arcsin(x)} \right)$ | Unambiguous |
36,134 | Modern Equivalent of Carr's Synopsis? | Ambiguous |
2,526,693 | Evaluate the limit: $\lim_{x \to 0^+} x^{\sqrt{x}}$ | Unambiguous |
432,999 | Prove that the limit of the sequence $\frac{n}{2^n}$ as $n$ approaches infinity is 0. | Unambiguous |
1,572,032 | Why is the CDF of $Y_n$ equal to $(\frac{t- \theta}{\theta})$ and what are the restrictions on $t$? | Ambiguous |
2,186,358 | What is the correct term for $I^n f(x)$ where $I$ is the integral operator? | Ambiguous |
606,172 | If the sequence $(x_n)$ of real numbers converges and $\lim_{n \to \infty} (2x_{n+1} - x_n) = x$, is it true that $\lim_{n \to \infty} x_n = x$? | Unambiguous |
4,468,673 | If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the maximum value of the product $xy$? | Unambiguous |
166,794 | I found a proof arguing that the irrationals are countable based on the density of the rationals. I know this is wrong, but where is the mistake in the logic? | Ambiguous |
1,191,610 | Prove that the composition of two group homomorphisms is also a group homomorphism. | Unambiguous |
4,437,668 | What are some interesting examples of elementary proofs by mathematical induction for students? | Unambiguous |
1,354,331 | Does there exist a function that is discontinuous at a point but has a continuous derivative on its domain? | Unambiguous |
970,133 | Prove that the infinite series $\sum_{n=1}^\infty \frac{n^2(n-1)}{2^n}$ converges to 20. | Unambiguous |
419,859 | Prove that the graph $y=x^{p/q}$ has a vertical tangent. | Ambiguous |
105,575 | What is the difference between the soundness and completeness theorems in first-order logic? | Unambiguous |
456,595 | Prove: $\frac{1}{1^2} +\frac{1}{2^2} + \cdots + \frac{1}{n^2} + \cdots = \sum_{n=1}^\infty \frac{1}{n^2} < 2$ | Ambiguous |
3,075,859 | What is the next step after showing $(c^k)^t = (id)$ for cycle orders? | Ambiguous |
2,538,727 | How to use Cantor's diagonal to prove my space is not sequentially compact? | Ambiguous |
217,966 | What is the relationship between monomorphisms and pullbacks in category theory? | Unambiguous |
3,117,992 | Could different physical systems follow different underlying mathematics? | Ambiguous |
181,367 | Provide a counterexample to show that a pseudocompact topological space is not necessarily a compact space. | Unambiguous |
854,380 | How to distinguish between walking on a sphere and walking on a torus? | Ambiguous |
3,087,356 | Why is the number 2 included in the definition of prime numbers instead of being excluded as the only even prime? | Unambiguous |
4,468,673 | If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the greatest possible value of $xy$? | Ambiguous |
4,060,048 | How do I solve integrals of the form integral of (polynomial * exponential) dx using integration by parts? | Unambiguous |
1,024,830 | What is the intuitive meaning of a compact set in the context of a metric space or topology? | Unambiguous |
3,172,708 | How to efficiently update the inverse of a lower triangular matrix after changing 4 individual elements? | Unambiguous |
1,133,113 | How do we define arc length? | Ambiguous |
2,490,690 | In how many ways can a payment of $107$ shekels be made using coins/bills of denominations $1$, $10$, and $50$ shekels, assuming an unlimited supply of each? | Unambiguous |
1,711,154 | Prove that f is differentiable at x=0 | Ambiguous |
551,148 | Why is $2x^3 + x$, where $x \in \mathbb{N}$, always divisible by 3? | Ambiguous |
2,304,892 | Probability of rolling any set of size k on n dice with m sides | Ambiguous |
1,576,678 | Can my sine formula with $\pi/6$ be put into Excel somehow to plot the graph? | Ambiguous |
2,982,578 | Prove that every finitely generated module over a ring $R$ is a homomorphic image of a free module. | Unambiguous |
1,619,762 | What are the equivalent statements of the Generalized Riemann Hypothesis (GRH) expressed in terms of the summatory functions of the Möbius and Liouville functions? | Unambiguous |
1,882,798 | Is there a mathematical rule to check if positive numbers $a$ and $b$ satisfy my modulo $P$ equation? | Ambiguous |
289,779 | I'm looking at the sequence $a_n = 3^n - 2^{n+100}$. How do I handle this? | Ambiguous |
1,320,951 | Prove that the intersection of any two cyclic subgroups of a group is also a cyclic subgroup. | Unambiguous |
2,449,459 | What's wrong with this fake proof that $2+2=5$? | Ambiguous |
1,017,503 | Prove by induction that $\sum_{k=0}^n k\cdot k! = (n+1)! - 1$ for all $n \in \mathbb{N}$. | Unambiguous |
3,490,887 | Question on existence of square root function! | Ambiguous |
3,791,238 | Why is a curve homeomorphic to a knot? | Ambiguous |
2,284,995 | Expected count of "double records" | Ambiguous |
3,468,876 | Suppose $Y_1$ and $Y_2$ are independent and $X=Y_1+Y_2$. What is the distribution of $X$? | Ambiguous |
243,106 | How to apply $\epsilon-\delta$ definition to a 2-valued function? | Ambiguous |
2,396,246 | One tank holds more than another by some amount. When the smaller one is about 2/3 full, it holds roughly as much as half the larger one. What is the capacity? | Ambiguous |
303,933 | Evaluate the limit $\lim_{n \to \infty} \frac{(\log(n))^a}{n^b}$ for positive real constants $a$ and $b$. | Unambiguous |
975,836 | How to solve linear recurrence relations using the method of repeated substitution? | Unambiguous |
164,244 | Prove that if a subgroup of index p is the smallest prime dividing the order of a finite group, then it is a normal subgroup. | Unambiguous |
2,344,993 | What are some unexpected real-world applications or occurrences of the natural logarithm in mathematics and science? | Unambiguous |
781,320 | Why does this curve cross the y-axis if $\cos(\sin x)$ is never zero? | Ambiguous |
2,834,788 | Is the EulerâMascheroni constant $\gamma$ known to be irrational? | Unambiguous |
3,247,615 | How do I evaluate the polynomial P(x) at x = -2? | Unambiguous |
1,329,195 | How to calculate a percentage increase when the result is above 100%? | Unambiguous |
2,449,459 | Where is the logical fallacy in the common fake proof that $2+2=5$? | Unambiguous |
3,017,476 | Interpreting an agreement function | Ambiguous |
1,169,872 | Is there a positional numeral system base $b$ such that every rational number has a terminating representation? | Unambiguous |
2,923,438 | How does the RSA cryptosystem work, and what are the steps for key generation, encryption, and decryption? | Unambiguous |
3,114,691 | How to find $\lim\limits_{n\to\infty} A^n$? | Ambiguous |
1,559,640 | How to find all natural numbers $n$ for quadratic permutations with a perfect square sum? | Ambiguous |
3,699,937 | The cyclic subgroups of $p^2$ order non-cyclic group are normal | Ambiguous |
551,148 | Prove that for every natural number $x$, the expression $2x^3 + x$ is always divisible by 3. | Unambiguous |
3,537,073 | Help with proof by induction, please! I may not be understanding how to work with factorials. | Ambiguous |
3,537,073 | Can you provide a step-by-step example of how to use proof by induction to prove a summation formula involving factorials? | Unambiguous |
1,647,675 | A Quadrilateral and A Triangle in a Trapizium | Ambiguous |
2,999,909 | In an arbitrary category, is the hom-set $Hom(X,Y)$ always equipped with an Abelian group structure? | Unambiguous |
1,466,540 | Find the $\lim_{x\to \infty} (\ln(x)-x)$ | Unambiguous |
4,247,524 | Finding the probability that at least 8 houses in a row are of same colour. | Ambiguous |
486,600 | Given that ai (where i is the imaginary unit) is a root of the polynomial $x^3-bx^2+a^2x-a^2b=0$, prove this fact and find the other two roots in terms of $a$ and $b$. | Unambiguous |
308,644 | Do I need to include the definition of union in this proof? | Ambiguous |
2,396,246 | How do I calculate the volume of water a cylindrical tank can hold given its radius and height? | Unambiguous |
3,560,940 | If the weights of three different items are 10g, 25g, and 5g, how do I divide 1kg among them? | Ambiguous |
2,561,078 | Prove that if functions $f$ and $g$ are continuous at a point $x$, then their difference $f-g$ is also continuous at $x$. | Unambiguous |
1,568,161 | Evaluating $\lim\limits_{x \to \pi} \frac{(\pi -x)\sin x}{1 + \cos x}$ | Ambiguous |
1,074,072 | Is there a notion dual to that of a path in a topological space? | Ambiguous |
2,664,926 | Evaluate the complex power $(-\sqrt{3} + i)^9$ using De Moivre's theorem. | Unambiguous |
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