User Eevee Trainer

28 votes

9 answers

1k views

Various ways to calculate $\int \sin(x) \cos(x) \, \mathrm{d}x$

asked Sep 28, 2021 at 2:52

19 votes

5 answers

4k views

A curiosity: how do we prove $\mathbb{R}$ is closed under addition and multiplication?

real-analysis group-theory field-theory real-numbers abelian-groups

asked Dec 12, 2018 at 2:36

13 votes

3 answers

3k views

Numbers such that they equal the product of their own digits

elementary-number-theory recreational-mathematics decimal-expansion

asked Dec 27, 2018 at 5:36

13 votes

9 answers

406 views

Determinants of matrices defined by the minimum/maximum indices of their entries

linear-algebra matrices determinant alternative-proof

asked Dec 21, 2020 at 2:37

9 votes

1 answer

239 views

How does one solve for $x$ in the equality $a^x = bx + c$?

inequality logarithms exponential-function special-functions lambert-w

asked Aug 14, 2020 at 9:38

8 votes

1 answer

428 views

Proving the pullback of monics is monic.

proof-verification category-theory

asked Oct 15, 2018 at 22:36

7 votes

1 answer

124 views

Solution verification: Showing that $\|A\|_\infty = \max_{1 \le i \le n} \sum_{k=1}^n |a_{i,k}|$ for $A \in \Bbb R^{n\times n}$

linear-algebra matrices solution-verification normed-spaces

asked Jan 16, 2021 at 0:33

7 votes

1 answer

137 views

Showing a biconditional statement about function lim sups in $\Bbb R^n$, and codifying the intuition into a proof

real-analysis sequences-and-series proof-writing metric-spaces limsup-and-liminf

asked Jan 27, 2021 at 6:19

6 votes

2 answers

289 views

Visualizing the norm of a bounded linear functional

functional-analysis metric-spaces normed-spaces visualization

asked Jun 14, 2021 at 22:58

6 votes

5 answers

263 views

Showing $f(t) = t^4 + 2t^2 + 9$ is reducible over $\Bbb Q$

abstract-algebra proof-verification polynomials factoring irreducible-polynomials

asked Apr 4, 2019 at 8:58

6 votes

1 answer

121 views

Definitions of the Stratonovich integral and why the "average" definition is arguably correct

probability stochastic-processes stochastic-calculus brownian-motion stochastic-integrals

asked Apr 6 at 22:24

5 votes

2 answers

244 views

What is the difference between "$=$" and "$\equiv$"?

notation

asked Jan 10, 2019 at 3:36

5 votes

1 answer

81 views

Proof verification: An arrow which is monic under a faithful functor is itself monic

proof-verification category-theory functors monomorphisms

asked Jan 23, 2019 at 2:36

5 votes

2 answers

2k views

Why is the negation of the statement $\exists x P(x)$ given by $\forall x (\neg P(x))$ and not $\not \exists x P(x)$?

logic notation predicate-logic quantifiers

asked Jan 1, 2019 at 0:07

5 votes

2 answers

432 views

Riemann-Stieltjes integral of a continuous function w.r.t. a step function

real-analysis integration solution-verification riemann-integration stieltjes-integral

asked Feb 10, 2021 at 4:28

5 votes

1 answer

142 views

Showing a summation identity for $1$, possibly tied to Legendre polynomials

integration sequences-and-series fourier-series legendre-polynomials legendre-functions

asked Oct 3, 2020 at 2:22

4 votes

1 answer

60 views

Given a particular matrix $A \in \operatorname{GL}_n(F)$, what is an easy way to determine $A^n$ $\forall n \in \Bbb Z$?

linear-algebra matrices exponentiation

asked Nov 8, 2020 at 5:33

4 votes

2 answers

147 views

Completing a proof on integrability: $f$ is Riemann integrable under the mesh definition iff $\sup L_\Gamma = \inf U_\Gamma$

real-analysis integration supremum-and-infimum riemann-integration riemann-sum

asked Feb 16, 2021 at 3:57

4 votes

4 answers

112 views

Why do the vertices of $f(x) = ax^2 + bx + c$, when fixing $a$ and $c$ but varying $b$, lie on $g(x) = -ax^2 + c$?

algebra-precalculus quadratics graphing-functions

asked Aug 26, 2021 at 20:14

4 votes

1 answer

201 views

Why do we notate the greatest common divisor of $a$ and $b$ as $(a,b)$?

elementary-number-theory notation math-history gcd-and-lcm

asked Dec 29, 2018 at 23:35

4 votes

3 answers

102 views

Constructing an isomorphism of group products

group-theory finite-groups group-isomorphism

asked Jan 31, 2019 at 3:11

4 votes

1 answer

249 views

A quick question about complex integrals and Cauchy's integral formula

complex-analysis complex-integration

asked Oct 12, 2018 at 7:48

4 votes

1 answer

88 views

How to solve a specific complex integral: $\int_M \frac{(6z+1)^5 \cos(3z+1)}{(3z+1)^2}dz$

complex-analysis complex-integration

asked Oct 26, 2018 at 4:20

3 votes

3 answers

216 views

$Set$, $Set^{op}$, categorical properties, and proving why these categories aren't isomorphic.

category-theory

asked Oct 24, 2018 at 3:25

3 votes

1 answer

291 views

Proof sketch/verification - proving a pullback in a diagram is an equalizer.

proof-verification category-theory

asked Oct 18, 2018 at 0:35

3 votes

1 answer

277 views

Why do we have the present order of operations, and how do hyperoperations fit in?

math-history hyperoperation

asked Feb 28, 2019 at 0:57

3 votes

1 answer

664 views

Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?

sequences-and-series riemann-zeta divergent-series natural-numbers zeta-functions

asked Dec 18, 2018 at 5:33

3 votes

1 answer

58 views

Generalizing the apothem characterization of area for regular polygons, to convex regular polyhedra (and more?).

geometry area volume polytopes

asked Mar 3, 2021 at 3:52

3 votes

1 answer

103 views

Showing $\sum_{k=0}^{n+1} \binom n k \frac{(-1)^k}{(n+k)(n+k+1)} = \sum_{k=0}^{n+1} \binom {n+1} k \frac{ (-1)^k}{n+k}$

sequences-and-series definite-integrals summation binomial-coefficients beta-function

asked Jun 22, 2020 at 3:35

3 votes

1 answer

41 views

Continuity of a bilinear form on $H^1(0,2)$

integration inequality continuity sobolev-spaces bilinear-form

asked Apr 4 at 7:37

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82 Questions

Various ways to calculate $\int \sin(x) \cos(x) \, \mathrm{d}x$

A curiosity: how do we prove $\mathbb{R}$ is closed under addition and multiplication?

Numbers such that they equal the product of their own digits

Determinants of matrices defined by the minimum/maximum indices of their entries

How does one solve for $x$ in the equality $a^x = bx + c$?

Proving the pullback of monics is monic.

Solution verification: Showing that $\|A\|_\infty = \max_{1 \le i \le n} \sum_{k=1}^n |a_{i,k}|$ for $A \in \Bbb R^{n\times n}$

Showing a biconditional statement about function lim sups in $\Bbb R^n$, and codifying the intuition into a proof

Visualizing the norm of a bounded linear functional

Showing $f(t) = t^4 + 2t^2 + 9$ is reducible over $\Bbb Q$

Definitions of the Stratonovich integral and why the "average" definition is arguably correct

What is the difference between "$=$" and "$\equiv$"?

Proof verification: An arrow which is monic under a faithful functor is itself monic

Why is the negation of the statement $\exists x P(x)$ given by $\forall x (\neg P(x))$ and not $\not \exists x P(x)$?

Riemann-Stieltjes integral of a continuous function w.r.t. a step function

Showing a summation identity for $1$, possibly tied to Legendre polynomials

Given a particular matrix $A \in \operatorname{GL}_n(F)$, what is an easy way to determine $A^n$ $\forall n \in \Bbb Z$?

Completing a proof on integrability: $f$ is Riemann integrable under the mesh definition iff $\sup L_\Gamma = \inf U_\Gamma$

Why do the vertices of $f(x) = ax^2 + bx + c$, when fixing $a$ and $c$ but varying $b$, lie on $g(x) = -ax^2 + c$?

Why do we notate the greatest common divisor of $a$ and $b$ as $(a,b)$?

Constructing an isomorphism of group products

A quick question about complex integrals and Cauchy's integral formula

How to solve a specific complex integral: $\int_M \frac{(6z+1)^5 \cos(3z+1)}{(3z+1)^2}dz$

$Set$, $Set^{op}$, categorical properties, and proving why these categories aren't isomorphic.

Proof sketch/verification - proving a pullback in a diagram is an equalizer.

Why do we have the present order of operations, and how do hyperoperations fit in?

Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?

Generalizing the apothem characterization of area for regular polygons, to convex regular polyhedra (and more?).

Showing $\sum_{k=0}^{n+1} \binom n k \frac{(-1)^k}{(n+k)(n+k+1)} = \sum_{k=0}^{n+1} \binom {n+1} k \frac{ (-1)^k}{n+k}$

Continuity of a bilinear form on $H^1(0,2)$