New answers tagged gm.general-mathematics
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Each mathematician has only a few tricks
A trick that is used daily is Zorn's Lemma. Sure, every mathematician knows it, but it certainly helped prove non-trivial propositions and it is in daily use. I would consider it in a list of the Top ...
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Every mathematician has only a few tricks
The trivial cohomology box trick
When trying to solve a problem, prove that:
...
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Each mathematician has only a few tricks
A trick/technique that I like (and used) a lot is the formal geometry approach (after Gelfand-Kazhdan) for passing from a local to a global result.
Let $X$ be a $d$-dimensional manifold. There is a an ...
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Each mathematician has only a few tricks
In my personal field, applied optimal transport for PDEs, we often play the following game, so much so that some of my colleagues and I actually call it Brenier's trick (after Yann Brenier): When ...
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When is 4 qualitatively different than $n\leq 3$?
4 is the smallest composite number.
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4
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Each mathematician has only a few tricks
In all 200+ pages of my category theory notes, there were essentially three tricks I used in proofs:
Prove that two arrows are both the arrow induced by a universal property, so they're the same ...
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2
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When is 2 qualitatively different from 3?
Equivalence relations with at most two classes on a fixed set $E$ have a natural group-structure (identify
them with the quotient group formed by maps from $E$ into $\{\pm 1\}$ modulo constant maps (...
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2
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When is 2 qualitatively different from 3?
The Bethe lattice (infinite symmetric tree) grows exponentially for degree $\geq 3$ and at most linearly for $\leq 2$.
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When is 2 qualitatively different from 3?
$a^n b^n$ is context-free. $a^n b^n c^n$ isn't.
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When is 2 qualitatively different from 3?
A 2-input gate cannot be universal for reversible computing.
A 3-input gate can. For example, the Fredkin (CSWAP) gate or Toffoli (CCNOT) gate.
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1
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When is 2 qualitatively different from 3?
A cardinal $\kappa$ is regular iff $\forall a (\lvert a\rvert < \kappa \land (\forall b \in a) (\lvert b\rvert < \kappa) \to \lvert\bigcup a\rvert < \kappa)$.
That is, every union of fewer ...
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2
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When is 2 qualitatively different from 3?
$p = 2$ is often an exception in number theory to statements about primes $p$. One of the most basic and yet nontrivial examples of this phenomenon is the primitive root theorem:
For all primes $p$, ...
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When is 2 qualitatively different from 3?
Every ideal in a polynomial ring is isomorphic (after potentially adding variables) to a trinomial ideal (i.e., one generated by trinomials). But binomial ideals are very special. See the classic ...
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3
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When is 2 qualitatively different from 3?
$\mathbb{R}^d\setminus\{0\}$ is simply connected for $d\ge 3$, but not for $d=2$.
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3
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When is 2 qualitatively different from 3?
$0/1$ square matrix permanent modulo $2$ is in polynomial time but for modulo $3$ it is not.
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3
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When is 2 qualitatively different from 3?
Hilbert scheme of points on a smooth surface is irreducible. But for dimension greater than two, these are completely wild.
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4
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When is 2 qualitatively different from 3?
Two-variable logic is decidable. Three-variable logic is undecidable.
For a close study of this boundary, see An excursion to the border of decidability: between two- and three-variable logic by Fiuk ...
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When is 2 qualitatively different from 3?
Two Variables Are Not Enough:
Let n be the smallest integer such that every closed lambda term beta converts to one with at most n bound variables. We show that n = 3.
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3
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When is 4 qualitatively different than $n\leq 3$?
For each $d \le 3$, there are plenty of smooth projective plane curves of degree $d$ with infinitely many rational points, but for $d \ge 4$ there are none.
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2
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17 camels trick
Here's an example from elementary linear algebra.
Lemma: If $A$ is $n \times m$ and $B$ is $m \times n$, then the characteristic polynomials of $AB$ and $BA$ are related by:
$$ t^m \det(\lambda I_n - ...
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3
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When is 4 qualitatively different than $n\leq 3$?
When g ≤ 3, every principally polarized abelian variety of dimension g is the Jacobian of a curve of genus g, but most are not for g ≥ 4.
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8
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When is 4 qualitatively different than $n\leq 3$?
For $n\leq3$, all groups of order $n$ are isomorphic, but there are two nonisomorphic groups of order $4$.
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1
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17 camels trick
In a way one could claim that all of formalized mathematics works that way. As already observed by David Hilbert a century ago, when one formalizes a piece of mathematics, spurious ideal entities ...
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When is 2 qualitatively different from 3?
Let there be $k$ square matrices, $A_1, A_2, \dots, A_k : \mathbb{R}^{n \times n}$, let $f$ be the monoid morphism $\{1, 2, \dots, k\}^* \to \mathbb{R}^{n \times n}$ generated by $f(i) = A_i$ for $1 \...
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3
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When is 2 qualitatively different from 3?
There is no infinite squarefree word on a binary alphabet, but there is one on an $n$-ary alphabet for $n>2$.
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3
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When is 2 qualitatively different from 3?
A finite cell complex of cohomological dimension (including twisted cohomology) $n>2$, is homotopy equivalent to a finite cell complex of dimension $n$. It is unknown if this is true for $n=2$.
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5
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When is 4 qualitatively different than $n\leq 3$?
The Lie algebra $\operatorname{so}(4)$ splits into a direct product; $\operatorname{so}(3)$ doesn't.
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1
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17 camels trick
Derivation of "sine-tangent relations" from quadratic Gauss sums of some orders involves reversing not one but two manipulations: one involving a multiplier term and the other involving the ...
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4
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When is 2 qualitatively different from 3?
A two-player zero-sum game is easy to solve in polynomial time. It's essentially a convex optimization problem.
A three-player zero-sum game is PPAD-hard to solve, i.e. just as difficult as a general ...
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10
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When is 4 qualitatively different than $n\leq 3$?
The word problem is decidable in every $\leq3$-manifold group, but can be undecidable in $4$-manifold groups.
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2
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When is 4 qualitatively different than $n\leq 3$?
Distinct tilings of space by both the measure polytope and the cross polytope exist only in four dimensions (the measure and cross polytopes are identical in one or two dimensions).
Measure polytopes ...
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3
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When is 2 qualitatively different from 3?
There's an important example from algorithmic complexity theory.
The Boolean satisfiability problem (SAT) consists,
given $n$ Boolean variables $x_1, \dots, x_n$,
in asking whether we can assign truth ...
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4
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When is 2 qualitatively different from 3?
The Jordan–Schönflies theorem does not hold for 3 dimensions: there is a subset of $\mathbb R^3$ that is homeomorphic to the 3-ball but its exterior is not simply connected. However, such analogous ...
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