Glossary: B

Definitions in this project whose names start with B. 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

B def

$B_n = \operatorname{Im} Z_n$. · Arxiv.«2607.05739».B

B def

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

B def

B r n is the $r$-full part of $n$: the product of prime powers $p^a \| n$ with $a \geq r$. · Erdos367.B

B def

def B : ℝ · MovingSofa.GerversSofa.B

B def

Let $P$ be the largest prime power dividing k. Then $k / P > P$. · OeisA80170.B

B abbrev

The second-order constant $\zeta(2/3)/\zeta(2)$, where $\zeta(2/3)$ is given by analytic continuation. · PowerfulNumbersDensity.B

b def

$b(2n) = -c(2n+1)$, $b(2n+1) = c(2n)$ · OeisA100434.b

b def

A005156 (offset 0): the number of $(2n+1) \times (2n+1)$ alternating sign matrices symmetric about the vertical axis (VSASMs). · OeisA109074.b

b def

Sequence A004125: sum of remainders $n \bmod k$ for $1 \le k \le n$. · OeisA129365.b

b def

Auxiliary sequence $b(n) = a(2n-1)$. · OeisA3161.b

b def

Auxiliary sequence $b(n) = a(2n-1)$. · OeisA3162.b

b def

Denominators of a sequence of fractions converging to $e$. This sequence is defined by the same recurrence relation as a but with initial values $b(1)=1, b(2)=2$. $$b(1) = 1, b(2) = 2$$ For $n > 2$: $$b(n) = \begin{cases} … · OeisA340737.b

b def

b n is the product of the first n terms of the Euclid-Mullin sequence. · OeisA945.b

b def

b G is the number of vertices of a largest induced bipartite subgraph of G. Returned as a real number. · SimpleGraph.b

B2 def

A set A ⊆ ℕ is said to be a B₂[g] set if for all n, the equation a + a' = n, a ≤ a', a, a' ∈ A has at most g solutions. This is defined in [ESS94]. · Erdos158.B2

balancedColoringThreshold def

The least size above which a coloring gives both colors a positive share of every induced complete uniform hypergraph. Positivity at α = 0 retains the Ramsey convention. · Hypergraph.balancedColoringThreshold

base abbrev

The base (10) used for digit reversal. · LychrelNumbers.base

baseEntropy def

The entropy $E(\xi, b)$ of the base $b$ expansion of $\xi$. · Real.baseEntropy

baseSeq def

The sequence $b_n$ such that $A227582(n) = b_{n-1}$ for $n \ge 1$. This is the 0-indexed solution to the linear recurrence in $\mathbb{Z}$. · OeisA227582.baseSeq

baseΦ def

The generating family of functions for Φ(G). The functions fᵢ : G × G → ℂ are bounded by 1 in sup norm. · Green57.baseΦ

baseΦ' def

The generating family of functions for Φ′(G), where the third kernel depends only on x₁ + x₂. · Green57.baseΦ'

BatemanHornPartialProduct def

The partial Euler product of the Bateman-Horn constant of a set of polynomials S, taken over the primes $p < N$: $$\prod_{p < N} (1 - \frac{1}{p})^{-|S|} (1 - \frac{\omega_p(S)}{p})$$ where $\omega_p(S)$ is the number of residue classes … · BatemanHornConjecture.BatemanHornPartialProduct

BB def

BB(n) is the n-th Busy Beaver number. This is the maximum shifts function, not the "number of ones function" · BusyBeaver.BB

bb84Family def

The BB84 family of four qubit states. · OpenQuantumProblem23.bb84Family

bddProdUpper abbrev

If A be a set of natural numbers and let x be real, then A.bddProdUpper x is the finite upper-triangular set of pairs of elements of A that are ≤ x. Specifically, it is the set {(n, m) | n ∈ A, n ≤ x, m ∈ A, m ≤ x, n < m} · Erdos442.Set.bddProdUpper

bealConjecture def

def bealConjecture : Prop · BealConjecture.bealConjecture

bellState abbrev

The standard $d$-dimensional Bell state. · OpenQuantumProblem35.bellState

bernsteinTail def

Cumulative sum $J_{n,k}(x) = \sum_{j=k}^n p_{n,j}(x)$. · VoronovskajaTypeFormula.bernsteinTail

best def

Computes the best outcome for the current player from a list of possible outcomes. Win > Draw > Loss. · CatchUp.Outcome.best

bestLower def

The best-known lower bound [ESS89]. · Green25.bestLower

bestLower def

Best-known lower bound [Ra38]. · Green46.bestLower

bestUpper def

The best-known upper bound [ESS89]. · Green25.bestUpper

bestUpper def

Best-known upper bound [Iw78]. · Green46.bestUpper

beta def

The even-step coefficient of the Rabinowitz–Gilbert/Stoll binary recurrence. · Erdos482.beta

betaConst def

The constant $\beta = \log(5)/(2\log(\phi)) - 1$. · OeisA105565.betaConst

betweenCount def

The number of Farey fractions of order n strictly between x and y. · Erdos1005.betweenCount

beurlingInteger def

A Beurling integer is a number of the form ∏ i, (a i) ^ (k i) for a given sequence a and a finitely-supported sequence of naturals k. · beurlingInteger

BeurlingIntegers def

The set of Beurling integers are numbers of the form ∏ i, (a i) ^ (k i), where k has finite support. · BeurlingIntegers

bezierBernstein def

Bézier–type Bernstein operator: $$ (B_{n,\alpha} f)(x) = \sum_{k=0}^{n} f\!\left(\frac{k}{n}\right) \left( J_{n,k}(x)^{\alpha} - J_{n,k+1}(x)^{\alpha} \right) $$ · VoronovskajaTypeFormula.bezierBernstein

bezierBias def

The proposed first-order bias constant for the Bézier--Bernstein operator: \[ \mu_\alpha = \int_0^\infty \bigl((1-\Phi(t))^\alpha + \Phi(t)^\alpha - 1\bigr)\,dt. \] Here $\Phi$ is Mathlib's cumulative distribution function of the standard … · VoronovskajaTypeFormula.bezierBias

bfs_dist_aux def

def bfs_dist_aux (G : SimpleGraph α) [DecidableRel G.Adj] (target : α) : ℕ → ℕ → Finset α → ℕ · SimpleGraph.bfs_dist_aux

bfsIterate def

Iterate G.bfsStep at most n times, stopping as soon as no new vertex shows up. · SimpleGraph.bfsIterate

bfsStep def

One round of breadth-first search: G.bfsStep s consists of the vertices of s together with their neighbours. · SimpleGraph.bfsStep

binaryDigit def

The binary digits of a normalized real $t \in [1, 2)$, indexed from one and including the leading digit. The value at index zero is padding. · Erdos482.binaryDigit

binaryPattern def

The binary representation of a natural number $n$, most significant bit first. For $n = 0$, this is $[0]$. · OeisA76141.binaryPattern

binaryWeight def

The binary weight of a natural number: the number of ones in its binary expansion. · TuDengConjecture.binaryWeight

bitDecode def

The decoding function of the canonical BitstringEncoding of α. · BitstringEncoding.bitDecode

bitEncode def

The encoding function of the canonical BitstringEncoding of α. · BitstringEncoding.bitEncode

BitstringEncoding class

A canonical encoding of a type as bitstrings (List Bool). · BitstringEncoding

bloch def

The Bloch vector associated to the first column of a qubit basis matrix. · OpenQuantumProblem13.Qubit.bloch

blochConstant def

The Bloch constant $B$ is the largest radius such that every holomorphic function on the unit disk with $f'(0) = 1$ has a schlicht (univalent) disk of that radius in its image. · Bloch.blochConstant

blochRadius def

The Bloch radius $B_f$ of a function $f$ is the supremum of radii of schlicht disks in the image of the unit disk under $f$: the restriction of f maps an open subdomain injectively and onto the disk. Requiring an open subdomain and … · Bloch.blochRadius

BlochVec abbrev

The real Bloch-vector space for qubits. · OpenQuantumProblem13.Qubit.BlochVec

blockComplexity def

The block complexity $p(n, a)$ of a sequence $a$, that is, the number of distinct blocks $a_k a_{k+1} \cdots a_{k+n-1}$ of $n$ consecutive terms of $a$. It is ⊤ when $n \ge 1$ and $a$ takes infinitely many values. · SymbolicDynamics.blockComplexity

blockEntropy def

The entropy of a sequence $a$, $$E(a) = \lim_{n \to \infty} \frac{\log p(n, a)}{n}.$$ The limit exists because $n \mapsto \log p(n, a)$ is subadditive, so it agrees with the limsup used here. The value is ⊤ exactly when $a$ takes … · SymbolicDynamics.blockEntropy

blockSensitivity def

Global block sensitivity of f, maximum block sensitivity of f over all inputs. · StrongSensitivityConjecture.blockSensitivity

blockSensitivityAt def

Local block sensitivity bs(f,x), maximum size of a collection of sensitive, disjoint blocks for f at x. · StrongSensitivityConjecture.blockSensitivityAt

blockSizeProfile def

The nondecreasing block-size sequence, retaining repeated sizes. · Finset.blockSizeProfile

blockSizeProfiles def

All block-size sequences of pairwise balanced designs on n labeled vertices. · Hypergraph.blockSizeProfiles

blowupC5 def

The blow-up of the 5-cycle $C_5$: replace each vertex of $C_5$ with an independent set of $n$ vertices, and connect two vertices iff their corresponding vertices in $C_5$ are adjacent. The vertex set is $\mathbb{Z}/5\mathbb{Z} \times {0, … · Erdos23.blowupC5

bookNumber def

The size of the largest book in G: the greatest number of triangles sharing a single edge. Finset.sup gives 0 on a graph with no edges, which is the right answer there. · Erdos80.bookNumber

BorsukConjecture def

Borsuk's conjecture in dimension n: every bounded subset of $\mathbb{R}^n$ with at least two points can be partitioned into $n + 1$ sets of strictly smaller diameter. · Borsuk.BorsukConjecture

BoseChowlaConjecture def

The conjecture that the size of the set $A\subseteq \{1,\ldots,N\}$ is asymptotically $N^{1/r}$. · Erdos241.BoseChowlaConjecture

boundaryCount def

Given a family $F$, for a subset $S$ of $X$, we write $h(S)$ as the number of subsets $T$ in $X$ such that (1) $T$ differs from $S$ in exactly one element (2) Exactly one set among $S$ and $T$ belongs to $F$. · Mathoverflow10799.boundaryCount

BoundedDiscCover structure

An $r$-bounded disc cover of a subset of a metric space $M$ is an indexed family of closed discs whose radii sum to at most $r$. · Erdos509.BoundedDiscCover

boundedDiscCover_empty def

def boundedDiscCover_empty [Nonempty M] (r : ℝ) (hr : 0 < r) : (BoundedDiscCover (∅ : Set M) r (PUnit : Type v)) · Erdos509.boundedDiscCover_empty

BPP def

The class BPP is the set of decision problems $L$ such that there exists a polynomial $p$ over ℕ and a poly-time Turing machine computing a relation $R$ where for all $x$, a uniformly random string $r$ of length $p (|x|)$ satisfies $R (x, … · ComplexityTheory.BPP

bumpAt def

Bump coordinate b of vertex v: add 1 to the b-th component. · ClaudesCycles.bumpAt

BunyakovskyCondition def

Bunyakovsky condition A condition on polynomials in the Schinzel and Bunyakovsky conjectures. Holds for nonconstant irreducible polynomial with positive leading coefficient. · BunyakovskyCondition

Bvas structure

A branching vector addition system of dimension d. · BranchingVAS.Bvas