Glossary: H

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

A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · X · Y · Z · Other

H def

H n is the minimum, over all colourings f of pairs from {1, …, n} with values in {-1, 1}, of the largest imbalance of a subset X ⊆ {1, …, n}. · Erdos1028.H

H def

Let $H(n)$ be the minimum integer $m$ such that there is a function $f: \mathcal{P}(X) \to X$ where $X$ is a finite set of size $n$, such that for every subset $Y \subseteq X$ with $|Y| \ge m$, the set $\{f(A) : A \subseteq Y\}$ covers $X$. · Erdos624.H

H def

H p q n is H(n) = H(n; p, q): the largest m such that every (p, q)-locally dense graph on n vertices must contain a complete graph on m vertices Kₘ. · Erdos667.H

H def

$H_k(n)$ is the maximal $r$ such that if $A\subset\mathbb{N}$ has $\lvert A\rvert=n$ and $\| 1_A\ast 1_A\|_\infty \leq k$ then $A$ contains a Sidon set of size at least $r$. · Erdos772.H

H def

The graph $H_k$ on the vertices $x, y_1, \ldots, y_k, z_{ij}$ ($i < j$): $x$ is adjacent to every $y_i$, and each pair $y_i, y_j$ is adjacent to the vertex $z_{ij}$. · Erdos926.H

H def

The $n$-th harmonic number $H_n = \sum_{k=1}^n \frac{1}{k}$ as a rational number. · OeisA64169.H

h def

$h_c(n)$: the minimum of maxCollinear P over all admissible P. It is ⊤ when no admissible configuration exists. · Erdos102.h

h def

h n is the number of congruence classes of minimisers for n points, as an element of ℕ∞: the supremum of the cardinalities of finite families of pairwise incongruent minimisers, which is ⊤ when there are infinitely many classes. · Erdos103.h

h def

For $\alpha>1$ and the divisors $1=d_1<\cdots<d_{\tau(n)}=n$ of $n$, $$h_\alpha(n) = \sum_i \left( \frac{d_{i+1}}{d_i}-1\right)^\alpha.$$ · Erdos1099.h

h def

Let $h(n)$ be the smallest $k$ such that $\{1,\ldots,n\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. · Erdos160.erdos_160.h

h abbrev

Let $h(N)$ be the maximum size of a Sidon set in $\{1, \dots, N\}$. · Erdos30.h

h def

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

h def

The auxiliary function $h(n) = n + τ(n)$ (where $τ(n) counts the number of divisors of $n$) · Erdos414.h

h def

Let $h n$ be the minimal number such that $2 ^ n - 1, \dots, h(n) ^ n - 1$ are collectively coprime. · Erdos770.h

h def

h n is $h(n)$: the smallest clique number of a graph on $n$ vertices in which every set of $7$ vertices contains a triangle. The complete graph satisfies the condition, so the set is never empty and sInf is a true minimum. · Erdos813.h

h def

If $n! = \prod_{i}p_i^{k_i}$ is the factorization into distinct primes, then we define $h(n)$ to be the number of distinct exponents $k_i$. · Erdos912.h

h def

Let $h(n)$ count the number of powerful integers in $[n^2, (n + 1)^2)$. · Erdos942.erdos_942.h

h def

$h(n)$ is the minimum number of distinct distances determined by any $n$-point set in $\mathbb{R}^2$ in general position (no three collinear, no four cocyclic). · Erdos98.h

H12 def

Hadamard constructs a 12 x 12 matrix ... · Hadamard.H12

h2 def

For a graph G on Fin n, h2 G is the smallest number of edges that need to be added to G so that the resulting supergraph has diameter at most 2 (every two distinct vertices are adjacent or have a common neighbour) and is still … · Erdos618.h2

H5 def

$H_5$ is $C_5$ with two vertex-disjoint chords (5 vertices, 7 edges). Also known as $K_4^*$ (the graph obtained from $K_4$ by subdividing one edge). · Erdos567.H5

HallConjectureExp def

def HallConjectureExp (e : ℝ) : Prop · Hall.HallConjectureExp

HallIneq def

def HallIneq (C : ℝ) (e : ℝ) : Prop · Hall.HallIneq

hallway def

The hallway is the union of its horizontal and vertical sides. · MovingSofa.hallway

haltingNumber def

def haltingNumber : ENat · Turing.BusyBeaver.Machine.haltingNumber

HaltsAfter def

The predicate that a machine starting at configuration s stops after at most n steps, i.e. reaches a configuration from which there are no defined transitions. · Turing.BusyBeaver.Machine.HaltsAfter

hammingBall def

The Hamming ball of radius $r$ in $\mathbb{F}_2^n$. · Green40.hammingBall

hammingBallFinset def

def hammingBallFinset (n r : ℕ) : Finset (𝔽₂ n) · Green40.hammingBallFinset

hankelMatrix def

The $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry $a(i+j)$. · OeisA53175.hankelMatrix

harmonicBlock def

The block harmonic sum $\sum_{n\leq k\leq m}\frac{1}{k}$. For n = 1 this is Mathlib's harmonic, see harmonicBlock_one. · harmonicBlock

harmonicDen def

The denominator (in lowest terms) of the partial harmonic sum $\sum_{a \leq n \leq b}\frac{1}{n}$. · Erdos290.harmonicDen

harmonicIndex def

The harmonic index of G, defined as ∑_{uv ∈ E(G)} 2 / (deg u + deg v), where deg is the degree of a vertex. This is a standard topological index in chemical graph theory, introduced by Fajtlowicz. · SimpleGraph.harmonicIndex

harrisHypergraph def

The 3-uniform hypergraph on Fin 9 given by Harris: the 27 triples containing exactly one vertex from each of the three parts {0, 1, 2}, {3, 4, 5}, {6, 7, 8}, together with the triple {0, 1, 2}. A triple is a transversal of … · Erdos794.harrisHypergraph

HasAnalyticContinuation def

The $L$-series of E has the analytic continuation L: L is analytic on $\mathbb{C}$ and agrees with E.LSeries on $\operatorname{Re} s > 3/2$, where the series converges absolutely. · HasseWeil.HasAnalyticContinuation

HasAvoidingChromaticCardinal def

Some hypergraph of the same uniformity and chromatic cardinal κ omits F. · UniformHypergraph.HasAvoidingChromaticCardinal

HasBalancedSubsets def

A 2-coloring of $K_n$ (represented by graph $G$ on $\mathrm{Fin}\; n$) is balanced on subsets of size at least $m$ with parameter $\alpha$ if every $X \subseteq [n]$ with $|X| \geq m$ contains more than $\alpha \binom{|X|}{2}$ edges of … · Erdos563.HasBalancedSubsets

HasBorsukCover def

HasBorsukCover k s means that the set s can be covered by k sets, each of strictly smaller extended diameter than s. · Borsuk.HasBorsukCover

HasBoundedDensity def

A set $\Lambda \subseteq \mathbb{R}$ has bounded density if the number of points of $\Lambda$ in any unit open interval is uniformly bounded: $\sup_{x \in \mathbb{R}} \#(\Lambda \cap (x, x + 1)) < \infty$. · WeakTiling.HasBoundedDensity

HasBoundedGaps def

A set $S \subseteq \mathbb{Z}$ has bounded gaps if it is syndetic, meaning there is a uniform bound $M$ such that every interval of length $M$ contains an element of $S$. · Erdos332.HasBoundedGaps

HasCasasAlveroProp def

A polynomial P satisfies the Casas-Alvero property if it shares a factor with each of its Hasse derivatives up to order d-1, where d is the degree of P. · CasasAlvero.HasCasasAlveroProp

HasCasasAlveroPropᵣ def

A stronger version of the Casas-Alvero property, which requires that the polynomial P shares a root with each of its Hasse derivatives up to order deg P - 1. The subscript r indicates "root" in the definition. · CasasAlvero.HasCasasAlveroPropᵣ

HasCentreOfSymmetry def

$K$ has a centre of symmetry: some $c$ with $K$ invariant under reflection in $c$. · Arxiv.«2607.03582».HasCentreOfSymmetry

HasClassNumber def

The class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-k})$ equals $n$. · OeisA38552.HasClassNumber

HasCM def

An elliptic curve $E$ over $\mathbb{Q}$ has complex multiplication over $\overline{\mathbb{Q}}$ if its $j$-invariant belongs to the set of thirteen rational CM $j$-invariants. · SatoTateConjecture.HasCM

HasCollinearTriple def

True if set $A$ contains 3 distinct collinear points over $R$. · Erdos193.HasCollinearTriple

HasCompactdegreeSequence def

The degree sequence of G is compact if it satisfies IsCompactSequenceOn for all valid indices k such that k + 2 < Fintype.card α. · SimpleGraph.HasCompactdegreeSequence

HasCompleteImage def

The predicate that the rational polynomial P has a complete image. · Erdos351.HasCompleteImage

HasCompleteMOLS def

A complete set of mutually orthogonal latin squares (MOLS) of order n consists of n - 1 latin squares of order n, pairwise orthogonal to each other. · LatinSquare.HasCompleteMOLS

HasCompleteMUBs def

There exists a complete set of $d + 1$ mutually unbiased bases in $\mathbb{C}^d$. · OpenQuantumProblem13.HasCompleteMUBs

HasConstantOverlapSq def

A family has constant pairwise squared overlap $c$ if every two distinct members have squared overlap $c$. · OpenQuantumProblem23.HasConstantOverlapSq

HasConvexNGon def

The set P contains a convex n-gon. See also IsConvexPolygon. · EuclideanGeometry.HasConvexNGon

hasCorrectCopyright def

Check whether a file, given as a String, is prefixed with the correct copyright header. · CopyrightLinter.hasCorrectCopyright

HasCosetHole def

A set $A$ has a coset hole of size $L$ if there exists a subspace $W$ and a vector $v$ such that the affine space $v + W$ has size at least $L$ and is disjoint from $A$. · Green32.HasCosetHole

HasCountableRamseyEscape def

The countable Ramsey escape property for the pair $(G_1, G_2)$: every $G_1$-free graph H on a Type-valued vertex type has a countable edge-colouring in which every colour class is $G_2$-free. · SimpleGraph.HasCountableRamseyEscape

HasCountableSouslinNumber class

A space has countable Souslin number when its Souslin number is at most ℵ₀. · WeaklyFirstCountable.HasCountableSouslinNumber

HasCycleWithIncidentChords def

G contains a cycle with k chords incident to a vertex on the cycle. · Erdos767.HasCycleWithIncidentChords

HasDensity def

A set S in an order β where all intervals bounded above are finite is said to have density α : ℝ (relative to a set A) if the proportion of x ∈ S such that x < n in A tends to α as n → ∞. · Set.HasDensity

HasDensityJump def

A density increase on growing induced vertex sets for every growing host sequence. The boolean chooses the strict or weak lower bound at the initial density. · Hypergraph.HasDensityJump

HasDimension def

G has dimension n: the least m for which G embeds in ℝ^m with every edge a unit line segment. · SimpleGraph.HasDimension

HasDisjointUnitDecomps def

There are k pairwise disjoint subsets of {1, ..., N} with reciprocal sum 1. · Erdos296.HasDisjointUnitDecomps

HasDistinctConjClassSizes def

A finite group G satisfies HasDistinctConjClassSizes if the map assigning to each conjugacy class the cardinality of its carrier is injective, i.e. distinct conjugacy classes have distinct sizes. · ConjugacyClassSizes.HasDistinctConjClassSizes

HasDistinctModuli def

No two classes of S share a modulus. · Erdos1188.HasDistinctModuli

HasDistinctSums def

def HasDistinctSums {ι α : Type*} [Preorder ι] [AddCommMonoid α] (a : ι → α) : Prop · Erdos357.HasDistinctSums

hasDocstring def

Whether the declaration modifiers contain a docstring. · CategoryDocstringLinter.hasDocstring

HasEqualDegreePathThree def

G contains two vertices of the same degree which are joined by a path of length 3. · Erdos816.HasEqualDegreePathThree

HasEqualIntervalSums def

A finite sequence A 0, …, A (s - 1) has two distinct (nonempty) index intervals [u, v) and [x, y) with the same sum. · Erdos1213.HasEqualIntervalSums

HasFabryGaps def

A sequence of natural numbers n₀ < n₁ < ... is said to have Fabry gaps if nₖ / k → ∞. This is the terminology adopted in [Wa01] and some other sources. · HasFabryGaps

HasFejerGaps def

A sequence of natural numbers n₀ < n₁ < ... is said to have Fejér gaps if ∑' 1 / nₖ < ∞. This is the terminology adopted in [Wa01] and some other sources. · HasFejerGaps

HasFew3APs def

HasFew3APs A means that $A \subset \mathbb{N}$ is a set for which there exists some $\epsilon > 0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\epsilon n$ which contains no three-term arithmetic … · Erdos847.HasFew3APs

HasFiniteConcentration def

M has finite concentration, written $\mathrm{con}(M) \neq \infty$ in the source: some gauging for M exists. · Arxiv.«2607.05349».HasFiniteConcentration

HasFinitePrimeCoveringSet def

A covering set for a positive odd integer $k$ is a finite set of primes $P$ such that every number of the form $k \cdot 2^n + 1$ is divisible by at least one prime in $P$. · Erdos1113.HasFinitePrimeCoveringSet

HasFiniteRamseyProperty def

The finite Ramsey property for the pair $(G_1, G_2)$: for every $n \geq 1$, there exists a $G_1$-free graph H on some vertex type in Type (universe 0) such that every n-edge-colouring of H contains a monochromatic copy of G_2. · SimpleGraph.HasFiniteRamseyProperty

HasFiniteTransversal def

A family of finite sets has a transversal with at most k vertices. · Set.HasFiniteTransversal

HasFullTorsionAction def

Every additive automorphism of $E[p]$ is induced by a Galois automorphism. For elliptic $E$ and prime $p$, this means that the mod-$p$ representation is surjective. · SerreUniformity.HasFullTorsionAction

HasGallaiTriangle def

A Gallai triangle (or ordinary triangle) in a collection A of lines: three lines from A which intersect in three points, and each of these intersection points only intersects two lines from A. · Erdos209.HasGallaiTriangle

HasGap def

A set $A$ has a gap of length $L$ if there exists $x$ such that $x, x+1, \dots, x+L-1$ are all not in $A$. · Green32.HasGap

HasGδSingletons class

A space where all singletons are Gδ sets. · HasGδSingletons

HasHamiltonianArcDecomposition def

The arcs of the cube digraph on (ZMod m)³ can be decomposed into three directed Hamiltonian cycles: there exist three permutations, each forming a directed Hamiltonian cycle, such that every arc (v, bumpAt b v) belongs to exactly one … · ClaudesCycles.HasHamiltonianArcDecomposition

HasHypergraphChromaticNumber def

The weak chromatic number is exactly k. · Finset.HasHypergraphChromaticNumber

HasHypergraphMatching def

A collection of k pairwise vertex-disjoint edges. · Finset.HasHypergraphMatching

HasHyperplaneLevelSets def

The level sets of $u : ℝ^n → ℝ$ are hyperplanes. This is expressed by stating that there exists some affine subspace with rank $n - 1$ which coincides with the level set. · DeGiorgi.HasHyperplaneLevelSets

HasIntDensity def

A set of integers has two-sided natural density α. · Set.HasIntDensity

HasIntermediateGrowth def

The sequence $m$ eventually grows at least as fast as $\exp(n^{\alpha})$, i.e., super-exponential growth when $\alpha > 1$, and stretched-exponential when $0 < \alpha < 1$. · Bugeaud06.HasIntermediateGrowth

HasLargeGapDilate def

The generalized problem: for a prime $p$ and a set $A \subset \mathbb{Z}/p\mathbb{Z}$ of size $\lfloor \omega(p) \rfloor$, is there a dilate of $A$ containing a gap of length $\lfloor 100p/\omega(p) \rfloor$? · Green32.HasLargeGapDilate

HasLebesgueCoveringDimensionEq def

A topological space X has Lebesgue covering dimension exactly n if it has covering dimension at most n but not at most n - 1. · HasLebesgueCoveringDimensionEq

HasLebesgueCoveringDimensionLE def

A topological space X has Lebesgue covering dimension at most n if every finite open cover of X admits a finite open refinement in which no point of X is contained in more than n + 1 elements. · HasLebesgueCoveringDimensionLE

HasLipschitzBoundary def

Ω has Lipschitz boundary if every boundary point p has a neighbourhood U in which, after some rigid motion σ (used to choose a "vertical" axis), Ω coincides with the strict subgraph of a Lipschitz function g of the first … · PompeiuProblem.HasLipschitzBoundary

HasLocalIndependence def

A graph G on n vertices in which every induced subgraph on m vertices has an independent set of size at least t. · Erdos804.HasLocalIndependence

HasLocalUniformization def

A valuation subring 𝒪 of F admits local uniformization over k if some affine model of F over k inside 𝒪 is regular at the centre of 𝒪: there is a finitely generated k-subalgebra A of F with A ⊆ 𝒪 and fraction field … · ValuationSubring.HasLocalUniformization

HasLogDensity def

A set A of natural numbers has logarithmic density d if the sequence $(1 / \log n) \cdot \sum_{k \in A, k \le n} (1/k)$ converges to d. Logarithmic density is a weaker notion than natural density: if a set has natural density d, … · Set.HasLogDensity

HasManyNonIsomorphicInducedSubgraphs def

G contains at least k many induced subgraphs which are not pairwise isomorphic, that is, there is a family of at least k many sets of vertices whose induced subgraphs are pairwise non-isomorphic. · Erdos1036.HasManyNonIsomorphicInducedSubgraphs

HasMaximallyMixedFirstReduction def

A state has maximally mixed reduction on the first $m$ parties. · OpenQuantumProblem35.HasMaximallyMixedFirstReduction

HasMediumDivisor def

m has a divisor strictly between n and 2n. · Erdos450.HasMediumDivisor

HasMeromorphicContinuation def

The $L$-series of E has the meromorphic continuation L: L is meromorphic on $\mathbb{C}$ and agrees with E.LSeries on $\operatorname{Re} s > 3/2$, where the series converges absolutely. · HasseWeil.HasMeromorphicContinuation

HasMicroscopicWeighting def

X admits a microscopic weighting when $\vec{w}(t)$ converges as $t \to 0^+$. · Arxiv.«2607.05349».HasMicroscopicWeighting

HasMinDist1 def

A set has minimum distance $1$ if all pairwise distances are at least $1$, and the minimum is achieved. · Erdos99.HasMinDist1

hasMinEquidistantProperty def

This property holds for a set of points A if every point x in A has at least k other points from A that are equidistant from x. · Erdos92.hasMinEquidistantProperty

HasMonochromaticSumProduct def

The monochromatic sum-product property: a colouring $c$ of $\{1, \ldots, N\}$ has a pair $(x, y)$ with $x, y \geq 3$ such that $x + y$ and $xy$ are both in $\{1, \ldots, N\}$ and receive the same colour. · Green22.HasMonochromaticSumProduct

HasMonotoneAP def

A function f : β → α has a monotone k-term arithmetic progression if there exists a choice of indices b 1 < b 2 < ... < b k such that the subsequence f (b i) forms an increasing or decreasing arithmetic progression of length k. · HasMonotoneAP

HasMUBs def

There exist $k$ mutually unbiased bases in $\mathbb{C}^d$. · OpenQuantumProblem13.HasMUBs

HasMulVCDimAtMost def

def HasMulVCDimAtMost : Prop · HasMulVCDimAtMost

HasMulVCNDimAtMost def

def HasMulVCNDimAtMost : Prop · HasMulVCNDimAtMost

HasNEquidistantPointsAt def

A set of points $A$ has n equidistant points at $p$ if there exist at least $n$ other points in $A$ that are equidistant from $p$. · Erdos97.HasNEquidistantPointsAt

HasNEquidistantPointsOn def

A set of points $A$ has n equidistant points on a set of points $B$ if for every point in $B$, there exist at least $n$ other points in $A$ that are equidistant from it. · Erdos97.HasNEquidistantPointsOn

HasNEquidistantProperty def

A set of points $A$ has n equidistant property if for every point in $A$, there exist at least $n$ other points in $A$ that are equidistant from it. · Erdos97.HasNEquidistantProperty

HasNoetherProperty def

We say that a rational extension L of K in the indeterminates ι has the Noether Property if, for every identification of L with the rational function field K(X_i : i ∈ ι), the fixed field L^H of every group H of … · NoetherProblem.HasNoetherProperty

hasNondecreasingPattern3 def

A list has a non-decreasing subsequence of length 3 (the pattern $x \le y \le z$). · OeisA52709.hasNondecreasingPattern3

HasNoUniqueRepresentation def

A set A has no unique representation in its sumset A + A if for every pair of elements a₁, a₂ ∈ A, there exist another pair of elements b₁, b₂ ∈ A such that a₁ + a₂ = b₁ + b₂ and {a₁, a₂} ≠ {b₁, b₂}. · HasNoUniqueRepresentation

HasNUnitDistancePointsAt def

A set of points $A$ has n unit distance points at $p$ if there exist at least $n$ other points in $A$ that are at unit distance from $p$. · Erdos97.HasNUnitDistancePointsAt

HasNUnitDistancePointsOn def

A set of points $A$ has n unit distance points on a set of points $B$ if for every point in $B$, there exist at least $n$ other points in $A$ that are at unit distance from it. · Erdos97.HasNUnitDistancePointsOn

HasNUnitDistanceProperty def

A set of points $A$ has n unit distance property if for every point in $A$, there exist at least $n$ other points in $A$ that are at unit distance from it. · Erdos97.HasNUnitDistanceProperty

HasOddCoeffs def

Polynomial.HasOddCoeffs f means that all coefficients of f : Polynomial ℤ are odd. · LehmerMahlerMeasureProblem.Polynomial.HasOddCoeffs

HasPolynomialGrowth def

A group has polynomial growth if there exists a finite generating set whose growth function is bounded above by a polynomial. · GromovPolynomialGrowth.HasPolynomialGrowth

HasPompeiuProperty def

A set Ω has the Pompeiu property if the only continuous f : ℝ^{N+1} → ℝ whose integral over every rigid-motion image σ(Ω) vanishes is the zero function. · PompeiuProblem.HasPompeiuProperty

HasPosDensity def

A set S in an order β where all intervals bounded above are finite is said to have positive density (relative to a set A) if there exists a positive α : ℝ such that S has density α (relative to a set A). · Set.HasPosDensity

HasPositiveDeriv def

The first partial derivative of $u : ℝ^n → ℝ$ is strictly positive. · DeGiorgi.HasPositiveDeriv

HasPrimeFactorsIn def

A positive integer $n$ has all its prime factors in the set $P$. By convention, $1$ satisfies this for any $P$ as it has no prime divisors. · Erdos269.HasPrimeFactorsIn

HasPropertyB def

A family (A_i)_{i ∈ I} of subsets of α has Property B if there exists a 2-colouring f : α → Fin 2 such that no A_i is monochromatic. · Erdos602.HasPropertyB

HasPropertyB def

A family F of finite sets has property B (equivalently, chromatic number 2) if there is a 2-colouring of the ground set under which no member of F is monochromatic. · Finset.HasPropertyB

HasPropertyP def

Property P : A set $A ⊆ ℕ $ has property P, if for all $n ≥ 1$ the set $ \{a ∈ A | n + a\text{ is squarefree}\}$ is finite. · Erdos1102.HasPropertyP

HasPropertyQ def

Property Q : A set $A ⊆ ℕ $ has property Q, if the set $\{n ∈ ℕ | ∀ a ∈ A, n > a\text{ implies }n + a\text{ is squarefree}\}$ is infinite. · Erdos1102.HasPropertyQ

HasRamseyProperty def

HasRamseyProperty N s t asserts: for every m, every coloring c : Finset (Fin m) → Bool, and every V : Finset (Fin m) with V.card ≥ N, the subset V contains either a subset of size s whose 2-element subsets are all … · Combinatorics.Diagonal.HasRamseyProperty

HasRedCopies def

Any unit-distance-avoiding colouring contains a congruent red copy of every set of n points. · Erdos214.HasRedCopies

HasRepBound def

A has at most k - 1 representations of every m (fewer than k). · Erdos796.HasRepBound

HasRepeatedDisjointUnion def

Two distinct disjoint pairs of edges with the same union. · Finset.HasRepeatedDisjointUnion

HasResolution def

A scheme X over a base S with structure morphism sX : X ⟶ S admits a resolution of singularities if there is an integral scheme Y, smooth over S, together with a proper morphism f : Y ⟶ X that is birational over X, i.e. … · AlgebraicGeometry.Scheme.HasResolution

HasSeparatedDistances def

A finite set $A$ of points has separated distances if all its pairwise distances differ by at least $1$: for any two different unordered pairs $\{a, b\} \neq \{c, e\}$ of distinct points of $A$, we have $\lvert d(a, b) - d(c, e)\rvert … · Erdos670.HasSeparatedDistances

HasSICPOVM def

There exists a SIC-POVM in dimension $d$. · OpenQuantumProblem23.HasSICPOVM

hasSorry def

Checks whether a syntax node contains sorry, admit, or sorryAx. · StubLinter.hasSorry

HasSubsetWithDistinctDistances def

The finite set S contains an n-point subset with all pairwise distances distinct. · Erdos1088.HasSubsetWithDistinctDistances

HasSuperPolynomialGrowth def

A group has superpolynomial growth if there exists a finite generating set whose growth function eventually dominates every polynomial in the growth-function preorder, up to linearly rescaling the radius. · GromovPolynomialGrowth.HasSuperPolynomialGrowth

HasUniqueExtension def

The property that the Diophantine 3-tuple $\{a, b, c\}$ extends to a 4-tuple $\{a, b, c, d\}$ with $d > \max(a, b, c)$ only via regularExtension. · DiophantineTuple.HasUniqueExtension

HasUniqueExtensionOfForall abbrev

The statement that every integral Diophantine triple of three distinct elements has a unique extension by a larger element. · DiophantineTuple.HasUniqueExtensionOfForall

HasVCDimAtMost def

A set family 𝒜 has VC dimension at most d if there are no families x of elements indexed by [d + 1] and A of sets of 𝒜 indexed by 2^[d + 1] such that x i ∈ A s ↔ i ∈ s for all i ∈ [d + 1], s ⊆ [d + 1]. · HasVCDimAtMost

HasZeroSubsetSum def

A finite set has a non-empty subset whose sum is zero. · Erdos540.HasZeroSubsetSum

havelHakimiIterate def

The Havel-Hakimi sequence of iterates: s i is the result of applying havelHakimiStep $i$ times to the descending degree sequence of $G$. · WrittenOnTheWallII.GraphConjecture291.havelHakimiIterate

havelHakimiStep def

Helper function: Performs one step of the Havel-Hakimi reduction on a degree sequence. Assumes the input list s is sorted descending. Removes the first element d, decrements the next d elements by 1, and re-sorts the list descending. · SimpleGraph.havelHakimiStep

havelHakimiZeroStep def

The first step $i$ (counting from $0$) at which a zero appears in the Havel-Hakimi reduction of the descending degree sequence of $G$, or in which the sequence has been emptied. We use sInf over the set of such $i$ so the definition is … · WrittenOnTheWallII.GraphConjecture291.havelHakimiZeroStep

headerToImportRefs def

Collects every import of a parsed header, with its modifiers and syntax. · ImportLinter.headerToImportRefs

heightLE def

The set of elliptic curves over ℚ with naïve height less than or equal to a given height. · EllipticCurveRank.RatEllipticCurve.heightLE

hesseFamily def

The Hesse qutrit SIC family. · OpenQuantumProblem23.hesseFamily

hesseS def

The common scale used in the Hesse qutrit SIC. · OpenQuantumProblem23.hesseS

hexagonal def

$p_6(z) = 2z^2 - z$ is the $z$-th hexagonal number. · OeisA160324.hexagonal

Hilbert17thProblemHomogenousPoly def

For the polynomial version, Hilbert showed that every nonnegative homogeneous polynomial in $n$ variables of degree $2d$ can be written as a sum of squares of polynomials if and only if - $n = 1$ - $n = 2$ - $d = 1$ - $(n, d) = (3, 2)$. · Hilbert17.Hilbert17thProblemHomogenousPoly

hilbertFunction def

The Hilbert function of a homogeneous ideal I of $k[x_1, \dots, x_n]$, sending d to $\dim_k I_d$. This is finite, since $I_d$ is a subspace of the finite dimensional space $S_d$. Two homogeneous ideals have the same Hilbert function in … · MvPolynomial.hilbertFunction

Hom structure

The type of digraph homomorphisms D →g D'. A function f : V → W is a digraph homomorphism if D.Adj u v → D'.Adj (f u) (f v). Note that f is a general mapping and is not required to be injective. · Digraph.Hom

homCount def

The number of homomorphisms from a digraph D to a digraph D'. · Digraph.homCount

homCount def

The number of graph homomorphisms H → G, as a natural number. For finite vertex types this is just the cardinality of the Hom type. · SimpleGraph.homCount

homDensity def

The homomorphism density $t_D(D')$ of a digraph $D$ in a digraph $D'$. It is defined as #Hom(D, D') / |W|^{|V|}. · Digraph.homDensity

homDensity def

The homomorphism density t(H, G) of a simple graph H in a simple graph G with finite vertex sets is #Hom(H, G) / |V(G)|^{|V(H)|}. · SimpleGraph.homDensity

HomogeneousSpace class

A topological space $X$ is called homogeneous if for all $x, y \in X$ there is homeomorphism $f : X \to X$ with $f(x) = y$. · Homogeneous.HomogeneousSpace

HomogeneousSpace class

A space is homogeneous if its homeomorphism group acts transitively. · HomogeneousSpace

HooleyHypothesis def

Hooley's hypothesis for $a$: the Riemann hypothesis holds for the Dedekind zeta function of the Kummer field $\mathbb{Q}(\zeta_k, a^{1/k})$, the splitting field of $X^k - a$ over $\mathbb{Q}$, for every squarefree $k$. This is the … · ArtinPrimitiveRootsConjecture.HooleyHypothesis

horizontalHallway def

The horizontal side of the hallway is $(-\infty, 1] \times [0, 1]$. · MovingSofa.horizontalHallway

HouseGraph abbrev

House Graph: Square 0-1-2-3-0 with roof 4 connected to 2,3. · WrittenOnTheWallII.Test.HouseGraph

Hypercube def

A graph on the power set of Fin n, where two sets are adjacent if they differ by a single element. · SnakeInBox.Hypercube

hypercube def

hypercube n is the n-dimensional hypercube graph Qₙ: the vertices are the n-bit vectors, and two vertices are adjacent when they differ in exactly one coordinate. · SimpleGraph.hypercube

HypergraphColorable def

The edge family admits a weak proper coloring with k colors. · Finset.HypergraphColorable

hypergraphDegree def

The number of hyperedges containing a vertex. · Finset.hypergraphDegree

hypergraphIndependenceNumber def

The largest cardinality of an independent vertex set in a finite ground type. · Finset.hypergraphIndependenceNumber

hypergraphInduce def

The edges induced on a vertex set. · Finset.hypergraphInduce

hypergraphRamsey def

The r-uniform hypergraph Ramsey number R_r(n). The smallest natural number m such that every 2-coloring of the r-subsets of a set of size m contains a monochromatic subset of size n. · Combinatorics.hypergraphRamsey