Glossary: G

Definitions in this project whose names start with G. The whole glossary has the other sections.

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G def

If $1=d_1<\cdots <d_{\tau(n)}=n$ are the divisors of $n$ (here Nat.nth (· ∣ n) i is $d_{i+1}$), then $$G(n) = \sum_{1\leq i<\tau(n)}\frac{d_i}{d_{i+1}}.$$ · Erdos673.G

G abbrev

A variant of Alpöge/Fable's counterexample: a polynomial self-map of k³ with Jacobian determinant 1 which is not injective. · JacobianConjecture.G

g def

g n is the largest k such that every set mapping f on the pairs of an n-element set with f(x, y) ∉ {x, y} admits an independent set of size at least k. · Erdos1025.g

g def

g d n is the least m such that every m points in ℝ^d determine at least n distinct distances, i.e. the least m with n ≤ minimalDistinctDistances (EuclideanSpace ℝ (Fin d)) m. · Erdos1089.g

g def

Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. · Erdos1095.g

g def

Let $g(n)$ be the maximal $k$ such that there exist integers $1 \le a_1, \dotsc, a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. · Erdos357.g

g def

Let $a$ be an infinite sequence of integers. Let $g(n)$ count the number of solutions to $$n=\sum_{u\leq i\leq v}a_i.$$ such that the sum has at least two terms. · Erdos358.g

g def

For any $k\geq 2$ let $g_k(n)$ denote the maximum value of $(a_1+\cdots+a_k)-n$ where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$. · Erdos400.g

g def

The largest Frobenius number of a $k$-element subset of $\{1,\ldots,n\}$ whose elements have greatest common divisor one. · Erdos433.g

g def

g N is the size of the largest A ⊆ {1, …, N} such that lcm(a, b) ≤ N for all a, b ∈ A. · Erdos441.g

g def

The largest size of a set $A\subseteq\{1,\ldots,N\}$ on which Euler's totient function is non-decreasing. · Erdos49.g

g def

g n is the largest t such that there exist integers 2 ≤ a 0 < a 1 < ⋯ < a (t - 1) < n whose greatest prime factors are strictly decreasing. · Erdos648.g

g def

g k n is the maximal number of edges possible on a graph with n vertices which does not contain a cycle with k chords incident to a vertex on the cycle. · Erdos767.g

g def

g k n = g_k(n) is the largest size of a subset A ⊆ {1, …, n} in which every m has fewer than k representations m = a₁ a₂ with a₁ < a₂ ∈ A. · Erdos796.g

g def

def g (k : ℕ) (n : ℕ) : ℕ · Erdos817.g

g def

Let $g(n)$ count the number of $m$ such that $\phi(m)=n$. · Erdos821.g

g def

g n is the maximum number of different sizes of cliques (maximal complete subgraphs) of a graph on n vertices. · Erdos927.g

g def

The maximum size of a Zhao-solution-free subset of $[N]$. · Green16.g

g def

$g(2n) = 0, g(2n+1) = c(2n+1)$ · OeisA100434.g

GaloisRealization structure

structure GaloisRealization.{u, v} (K : Type u) (G : Type v) [Field K] [Group G] · InverseGalois.GaloisRealization

galoisRepresentation def

The natural Galois action on geometric non-singular points with their discrete topology. · WeierstrassCurve.Affine.galoisRepresentation

gameLength def

The minimax value of the primitive-set saturation game from p, with Prolonger to move. · Erdos872.gameLength

GamePos structure

A game position consists of the already claimed set and the still unclaimed pool. · Erdos872.GamePos

gameValueAux def

Auxiliary finite minimax recursion with an explicit fuel bound. · Erdos872.gameValueAux

Gamma def

The set $\{a^k b^l : k,l \geq 0\}$. · Erdos246.Gamma

gamma def

The asymptotic constant $\gamma$ defined in [Aa19] p.579. · Green24.variants.gamma

gammaSubgroup def

The subgroup $\Gamma_{m \oplus n} \le \mathrm{S}_m \times \mathrm{S}_n$ consisting of all pairs $(\sigma_1, \sigma_2)$ of permutations with equal signature, i.e. $\mathrm{sgn}(\sigma_1) = \mathrm{sgn}(\sigma_2)$. · Arxiv.«2605.12342».gammaSubgroup

gaps def

The gaps $a_i - a_{i-1}$ between consecutive entries of coprimeList n. · Erdos235.gaps

gapSubsequence def

Helper definition for extracting a finite subsequence (pattern) as a list. · OeisA1223.gapSubsequence

GapSumSq def

The squared-gap sum ∑_{b_i < x} (b_{i+1} - b_i)², where b_i enumerates the positive integers not divisible by any element of A. · Erdos489.GapSumSq

gaussianMeasureInf def

The infinite-dimensional Gaussian measure γ∞ on ℝ^ℕ, defined as the countable product of standard Gaussian measures. · Green54.gaussianMeasureInf

gaussProd def

$Z_n = \prod_{k=1}^n (1 + ik)$, in the Gaussian integers. · Arxiv.«2607.05739».gaussProd

generalDet def

Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^{2^m} + (j+1)^{2^m}$ is prime, and $0$ otherwise. · OeisA71524.generalDet

generalRecurrence def

Stoll's general Graham–Pollak-type recurrence in base $g$, with real parameters $a$, $b$, and $\varepsilon$. Its even-index differences are used to read base-$g$ digits. · Erdos482.generalRecurrence

generatedSet def

The set of integers which eventually appear in the sequence, which is the union of all $A_n$. · Erdos424.generatedSet

genMatrix def

The general 2x2 matrix [(1,a); (1,b)]. · OeisA108306.genMatrix

genMyc def

The generalised Mycielskian $M_s(G)$. $M_2(G)$ is the classical Mycielskian, and $M_1(G)$ adds a vertex joined to everything. · Erdos750.genMyc

gerversSofa def

Gerver's sofa is the sofa according to the rotation path GerversSofa.p. · MovingSofa.gerversSofa

getAllDigitsMsf def

Concatenate a list of numbers into a list of decimal digits in most-significant-first order. · OeisA87571.getAllDigitsMsf

getAnswers def

Find answers by inspecting an InfoTree · Google.getAnswers

getCategoryStats def

def getCategoryStats : m (Category → Nat) · ProblemAttributes.getCategoryStats

getDesc def

def AMS.getDesc (a : AMS) : CoreM String · AMS.getDesc

getDocstringText def

def getDocstringText (stx : TSyntax ``Command.declModifiers) : String · LatexDocstringLinter.getDocstringText

getFormalProofTag def

Get the formal proof tag for a given declaration, if any. · ProblemAttributes.getFormalProofTag

getFormalProofTags def

def getFormalProofTags : m (Array FormalProofTag) · ProblemAttributes.getFormalProofTags

getFormalProofTagsFor def

Return every formal-proof tag attached to declName. · FormalProofLinter.getFormalProofTagsFor

getIdent def

Extracts the module identifier from an ImportRef. · ImportLinter.ImportRef.getIdent

getProofConditions def

Get the unproven hypotheses a given declaration's formal proof assumes. Empty when there is no formal proof or the proof is unconditional. · ProblemAttributes.getProofConditions

getStatementTags def

def getStatementTags : m (Std.HashMap Category (Array CategoryTag)) · ProblemAttributes.getStatementTags

getSubjectTags def

def getSubjectTags : m (Array SubjectTag) · ProblemAttributes.getSubjectTags

getTags def

def getTags : m (Array CategoryTag) · ProblemAttributes.getTags

ghzState abbrev

The standard $d$-dimensional GHZ state on $3$ parties. · OpenQuantumProblem35.ghzState

ghzState4 abbrev

The standard $d$-dimensional GHZ state on $4$ parties. · OpenQuantumProblem35.ghzState4

GivesWagstaffPrime def

A Wagstaff prime is a prime number of the form $(2^p+1)/3$. · Mersenne.Nat.GivesWagstaffPrime

gnorm def

The norm ‖y‖ = max (|y|, |y⁻¹|) on GL n ℝ, with |·| the sup norm on matrix entries. This is the norm used to define slow increase. · Matrix.GeneralLinearGroup.gnorm

go def

Given a finite Sidon set A and a lower bound m, go finds the smallest number m' ≥ m such that A ∪ {m'} is Sidon. If A is empty then this returns the value m. Note that the lower bound is required to avoid 0 being a … · Finset.greedySidon.go

gompertzConstant def

Gompertz constant $$\delta = -e * \int_1^∞ e^{-t}/t dt \approx 0.59634$$ · gompertzConstant

goodmanBound def

The Goodman bound $$\sum_{k=1}^{p} \frac{2k (n+p)!}{(p-k)!\,(p+k)!\,(n-p-1)!\,(n^2-k^2)} |b_k|,$$ the conjectured upper bound for $|b_n|$. · GoodmanConjecture.goodmanBound

goodSets def

The sets $B\subseteq X$ which intersect every set in $\mathcal{F}$, yet contain none of them. · Erdos1027.goodSets

grade def

The degree-n part K^M_n(F)/ℓ of K^M_*(F)/ℓ: the additive subgroup generated by the symbols of degree n. · MilnorK.grade

graph def

The union graph: distinct vertices are adjacent iff they lie in a common clique. · Erdos19.EFLConfig.graph

graph abbrev

The counterexample: a path on vertices $0,\ldots,12$, with triangles attached at vertices $1$ and $11$. · WrittenOnTheWallII.GraphConjecture65.Counterexample.graph

Graphon structure

A graphon is a symmetric, measurable function W : [0, 1] × [0, 1] → [0, 1]. The domain is the unit interval squared, and the image is contained in the unit interval. · LimitObjects.Graphon

graphonEdgeDensity def

The edge density of a graphon $W$ on $[0, 1]^2 \to [0, 1]$ equipped with the Lebesgue measure volume. · LimitObjects.graphonEdgeDensity

graphonHomDensity def

The homomorphism density $t(H, W)$ of a finite simple graph $H$ in a graphon $W$ on $[0, 1]$ equipped with the Lebesgue measure volume. · LimitObjects.graphonHomDensity

graphonHomIntegrand def

The integrand for the homomorphism density $t(H, W)$ of a simple graph $H$ in a graphon $W$ given a configuration x : V → [0, 1]. · LimitObjects.graphonHomIntegrand

graphRamsey def

The two-color Ramsey number graphRamsey G H is the minimum number of vertices n such that every 2-coloring of the edges of the complete graph on n vertices contains a copy of G in the first color or a copy of H in the second … · SimpleGraph.graphRamsey

graphSquare def

The square of a graph G, denoted G²: two distinct vertices are adjacent iff their distance in G is at most 2. · SimpleGraph.graphSquare

graphSquareRadius def

The radius of $G^2$ (the graph square): the minimum eccentricity over all vertices of graphSquare G. · WrittenOnTheWallII.GraphConjecture146.graphSquareRadius

greedySidon def

greedySidon is the sequence obtained by the initial set $\{1\}$ and iteratively obtaining the next smallest integer that preserves the Sidon property of the set. This gives the sequence 1, 2, 4, 8, 13, 21, 31, .... · Finset.greedySidon

greedySidonBelow def

The greedy Sidon set in {1, …, N}: starting from ∅, iterate through 1, …, N and include x if and only if A ∪ {x} remains Sidon. Alternatively, this is precisely the set of elements in the greedy Sidon sequence that are ≤ N. · Finset.greedySidonBelow

greedyUnitFractionRem def

Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x$: choose the minimal $n\in A$ not used so far such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$. · Erdos282.greedyUnitFractionRem

Green36Property def

A variant of the simultaneous double product property, as stated in [Gr24, Problem 36]. · Green36.Green36Property

GreenSawhneyBound def

The upper bound function from [GrSa25]. · Green22.GreenSawhneyBound

gridPoints def

The grid $\{1,\ldots,n\}^2$, viewed as a set of points of the plane. · Erdos798.gridPoints

gromovProduct def

The Gromov product of y and z with respect to x. · Metric.gromovProduct

groundSet def

The ground set $X=\bigcup_{A\in\mathcal{F}}A$ of a finite family of finite sets. · Erdos1027.groundSet

groupDiam def

The diameter of a finite group $G$, defined as the maximum diameter of the Cayley graphs $\Gamma(G, A)$ over all generating sets $A$ of $G$. · BabaiSeressConjectures.groupDiam

groupNorm def

The wordLength with respect to a generating family P defines a norm on G. · Group.Generators.groupNorm

GrowthCondition def

A sequence of integers a satisfies the growth condition if $\liminf a_n^{\frac{1}{2^n}} > 1$. · Erdos1051.GrowthCondition

GrowthFunction def

The GrowthFunction of a group G with respect to a set S counts the number of group elements that can be reached by words of length at most n in S. · GromovPolynomialGrowth.GrowthFunction

guaranteedMaxAPLength def

The largest length of an arithmetic progression guaranteed to be contained in $A+A$ for $A \subseteq \{1, \dots, N\}$ with density $\alpha$. · Green51.guaranteedMaxAPLength

guaranteedMaxCosetDim def

The largest dimension of a coset guaranteed to be contained in $2A$ for $A \subseteq \mathbb{F}_2^n$ with density $\alpha$. · Green51.guaranteedMaxCosetDim