Glossary: D

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

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D def

def D (n : ℕ) : Set (ℤ × ℤ × ℤ) · CongruentNumber.D

D def

D a b is the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$. · Erdos305.D

d def

Gilbreath's nth difference, $d^n$ Let $d^0(n) = p_n$ and $d^k(n) = |d^{k-1}(n+1) - d^{k-1}(n)|$ · Gilbreath.d

d def

$d(n)$ starts with $(2, 4, -10, -24)$ and satisfies the same recurrence as a · OeisA100434.d

d def

The difference sequence $d(n) = a(n) - a(n-1)$ for $n \ge 2$. · OeisA166944.d

D_A def

The set of numbers $D(A)$ which occur infinitely often as $a_1 - a_2$ with $a_1, a_2 \in A$. · Erdos332.D_A

decAux def

Auxiliary decision procedure for Reachable, taking a fuel argument f bounding m. · Mathoverflow75792.Reachable.decAux

DecidableAllFactorialSubtractionsComposite def

def DecidableAllFactorialSubtractionsComposite (n : ℕ) : Prop · Erdos1059.DecidableAllFactorialSubtractionsComposite

DecidableDistinctSubsetSums def

The predicate that all (finite) subsets of A have distinct sums, decidable version · Erdos350.DecidableDistinctSubsetSums

decidableFactorialsLessThanN def

def decidableFactorialsLessThanN (n : ℕ) : Finset ℕ · Erdos1059.decidableFactorialsLessThanN

DecidableIsFactorial abbrev

abbrev DecidableIsFactorial (d : ℕ) : Prop · Erdos1059.DecidableIsFactorial

decimalAppend def

Append the decimal digits of b to those of a. · OeisA37274.decimalAppend

decimalDigitCount def

The number of decimal digits of a natural number, counting zero as one digit. · OeisA37274.decimalDigitCount

DecisionComplexityClass abbrev

The type of complexity classes. We define these as sets of decision problems. · ComplexityTheory.DecisionComplexityClass

DecisionProblem abbrev

The type of decision problems. · ComplexityTheory.DecisionProblem

Decreasing def

A family F of sets is Decreasing if it is closed under taking subsets. · Chvatal.Decreasing

DedekindZetaRH def

The Riemann hypothesis for the Dedekind zeta function $\zeta_K$ of a number field $K$: every zero of (the analytic continuation of) $\zeta_K$ in the critical strip $0 < \operatorname{Re} s < 1$ has real part $\frac{1}{2}$. · ArtinPrimitiveRootsConjecture.DedekindZetaRH

deficiency def

If defined, the deficiency is the count of $0 \le i < k$ such that $n - i$ is $k$-smooth, that is, divisible only by primes $\le k$. · Erdos1093.deficiency

DeGiorgi_conclusion def

The conclusion to De Giorgi's conjecture: if $u : ℝ^n → ℝ$ is a bounded classical solution to $Δ u + u - u^3 = 0$ satisfying $∂₁u > 0$ everywhere, then the level sets of $u$ are hyperplanes. · DeGiorgi.DeGiorgi_conclusion

degreeFreq def

The number of vertices of G having degree d. · SimpleGraph.degreeFreq

degreeL2Norm def

The length of a graph: the square root of the sum of the squares of degrees. · SimpleGraph.degreeL2Norm

degreeMultiset def

The multiset of degrees of a graph. · SimpleGraph.degreeMultiset

degreeSequence def

The degree sequence of a graph, sorted in nondecreasing order. · SimpleGraph.degreeSequence

degreeSequenceMultiplicity def

The maximum number of occurrences of any term of the degree sequence of G. · SimpleGraph.degreeSequenceMultiplicity

DegreesProduct def

The product of degrees of polynomials in a finite set. · BatemanHornConjecture.DegreesProduct

delimit def

Make a bitstring self-delimiting: each payload bit b becomes the two bits [true, b], and the block is terminated by false. · BitstringEncoding.delimit

deltaGap def

The $n$th gap of loneliness $\delta_n$: the infimum of deltaTuple over all $n$-tuples of distinct nonzero integer velocities. · LonelyRunnerConjecture.deltaGap

deltaTuple def

For an $n$-tuple of distinct integer velocities $v_1,\dots,v_n$, deltaTuple v is the maximal value of $\min_i \|t v_i\|_{\mathbb{R}/\mathbb{Z}}$ over time. · LonelyRunnerConjecture.deltaTuple

density def

The density d(X) of a topological space X: the least cardinality of a dense subset, plus ℵ₀. · TopologicalSpace.density

density_zero def

A set of natural numbers has density 0. · Erdos16.density_zero

descDegreeSequence def

The descending degree sequence of $G$, used as the starting point of the Havel-Hakimi reduction. · WrittenOnTheWallII.GraphConjecture291.descDegreeSequence

diagonalEmbedding def

The diagonal embedding of the rational points G(ℚ) = GL n ℚ into G(𝔸_f) × G(ℝ) = GL n 𝔸ᶠ[ℤ, ℚ] × GL n ℝ, through the inclusions of ℚ into the finite adeles and into ℝ. · Matrix.GeneralLinearGroup.diagonalEmbedding

diagonalGraphRamsey def

The diagonal graph Ramsey number R(G, G). · SimpleGraph.diagonalGraphRamsey

diagonalHom def

The multiplicative group homomorphism (Rˣ)ⁿ →* GL(n, R) given by mapping a vector x to Matrix.diagonal x. · Matrix.GeneralLinearGroup.diagonalHom

diagonalRamsey def

The diagonal classical Ramsey number R(k) = R(K_k, K_k). · SimpleGraph.diagonalRamsey

diagonalState def

The diagonal $n$-party state: the uniform superposition over constant computational-basis strings. · OpenQuantumProblem35.diagonalState

diagonalSubgroup def

The group of invertible diagonal matrices. · Matrix.GeneralLinearGroup.diagonalSubgroup

diagonalSubgroup def

The group of invertible diagonal matrices with determinant 1. · Matrix.SpecialLinearGroup.diagonalSubgroup

digitSeq def

The n-th digit (0-indexed) after the radix point in the base-b expansion of x. · NormalNumber.digitSeq

digraphEdgeFinset def

The directed edge finset of a finite digraph. · LimitObjects.digraphEdgeFinset

DilationEquivSimilar def

Two finite sets of points in $\mathbb{R}^2$ are similar if one can be mapped to the other by a DilationEquiv. · Erdos91.DilationEquivSimilar

directionLift def

A deterministic projective lift of a direction in $\mathbb{R}^3$ to a pair of complex numbers. · AtiyahSutcliffe.directionLift

disjoint_without_infinite_claim def

A natural but FALSE relaxation of erdos_602.variants.disjoint: drop the hypothesis that each A i is infinite. The original disjoint variant requires (∀ i, (A i).Infinite). Without it, the claim is false. · Erdos602.disjoint_without_infinite_claim

distanceMatrix def

The distance matrix $D_{ij} = d(x_i, x_j)$ of a finite metric space. · Arxiv.«2607.05349».distanceMatrix

distanceMultiplicity def

The multiplicity of the distance d determined by points, that is, the number of unordered pairs of distinct points at distance d apart. · distanceMultiplicity

distanceSet def

The set of distances determined by a finite set of points in a metric space. · distanceSet

DistancesSeparated def

If two distances in A differ, they differ by at least 1. · Erdos100.DistancesSeparated

distanceValues def

The finite set of distinct distances determined by a configuration. · Erdos959.distanceValues

distavg def

Average distance from all vertices to a given set. · SimpleGraph.distavg

distAvgSet def

The average distance between distinct vertices of a set S: $\operatorname{dist}_{\operatorname{avg}}(S)$ is the mean of $\operatorname{dist}_G(u, v)$ over all ordered pairs $(u, v)$ of distinct vertices of $S$. Returns 0 when $S$ … · SimpleGraph.distAvgSet

distEven def

distEven G v counts the vertices at even distance from v in G. This is DeLaVina's dist_even(v) invariant appearing in several WOWII conjectures. Distance zero is even, so v itself is always counted. · SimpleGraph.distEven

distinctDistances def

Given a finite set of points in a metric space, we define the number of distinct distances between pairs of points. · distinctDistances

distinctDistancesFrom def

Given a finite set of points in a metric space, we define the number of distinct distances between a given point and all other points. · distinctDistancesFrom

DistinctSubsetSums def

The predicate that all (finite) subsets of A have distinct sums. · Erdos350.DistinctSubsetSums

distMaxSet def

The maximum distance between two vertices of a set S: $\operatorname{dist}_{\max}(S) = \max\{\operatorname{dist}_G(u,v) \mid u, v \in S\}$. Returns 0 when S is empty or a singleton. · SimpleGraph.distMaxSet

distMin def

The minimum distance between distinct vertices of $S$: $\min \{\operatorname{dist}_G(u, v) \mid u, v \in S, u \ne v\}$. Returns 0 when $S$ contains fewer than two vertices. · SimpleGraph.distMin

distToNearestInt def

The distance from a real number to the nearest integer. · distToNearestInt

distToSet def

Distance from a vertex to a finite set. · SimpleGraph.distToSet

divergence def

def divergence (v : ℝ^n → ℝ^n) (x : ℝ^n) : ℝ · NavierStokes.divergence

divisorProduct def

The product term in the denominator of the general conjecture: $$\prod_{i = 1..r, i \text{ coprime to } 30} (30n - i)$$ We define this in ℤ to handle the $n=0$ case where $30n-i$ in the product might be negative. · OeisA211417.divisorProduct

DivisorSumSet def

DivisorSumSet t is the set of natural numbers n such that t can be represented as a sum of distinct divisors of n. · Erdos859.DivisorSumSet

DlabLt def

Dlab's order on a group $G$ of order automorphisms of $\mathbb{R}$: $f < g$ if $f(x) < g(x)$ at some point $x$ and every point $y$ with $g(y) < f(y)$ lies to the right of $x$. For elements of a Dlab group, this says that $f$ is below $g$ … · Kourovka.«21.149».DlabLt

Dmax def

$D(b)=\max_{1\leq a<b}D(a,b)$. · Erdos305.Dmax

dominationNumber def

The domination number of a graph G is the minimum size of a dominating set. It is 0 if there are no vertices. · SimpleGraph.dominationNumber

dp def

dp G is the number of diametrical pairs of G: the number of unordered pairs {u, v} of vertices at distance diam(G). · WrittenOnTheWallII.GraphConjecture36.dp