Glossary: U

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

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U def

The Lucas sequence of the first kind $U_0(P, Q) = 0$, $U_1(P, Q)=1$, $U_{n+2}(P, Q)=PU_{n+1}(P, Q)-QU_n(P, Q)$ · LucasSequence.U

u def

The sequence $u_0 = 1$, $u_{n+1} = u_n(u_n + 1)$ of [ElPl21, Corollary 3]: $1, 2, 6, 42, 1806, \ldots$, a shifted copy of Sylvester's sequence. This is the sequence Erdos315.u of Erdős Problem 315. · Erdos148.u

u def

The sequence $u_1=1$, $u_{n+1}=u_n(u_n+1)$, indexed here so that u i is $u_{i+1}$: u 0 = 1, u 1 = 2, u 2 = 6, u 3 = 42, … (so u i + 1 is Sylvester's sequence $2, 3, 7, 43, \ldots$). · Erdos315.u

u def

u n is the normalized nth prime, defined as pₙ / (n+1) where pₙ is the nth prime (with 0.nth Nat.Prime = 2). · Erdos968.u

u def

Let m be a nonnegative integer and set $u(n)$ = the numerator of $$\sum{k=1}^{n} \frac{1}{k^{2m+1}} \binom{n}{k}^2 \binom{n+k}{k}^2$$ (seems like a typo in the OEIS entry: the sum starts with $k=0$ there. In order to avoid a division by … · OeisA357513.u

u0 def

The first entry of the first column of a qubit unitary basis matrix. · OpenQuantumProblem13.Qubit.u0

u1 def

The second entry of the first column of a qubit unitary basis matrix. · OpenQuantumProblem13.Qubit.u1

UMat abbrev

A unitary matrix representing an orthonormal basis of $\mathbb{C}^d$ via its columns. · OpenQuantumProblem13.UMat

uncovered def

The integers 1 ≤ m ≤ n which are not divisible by any element of A. · Erdos542.uncovered

uncoveredCount def

The number of integers below n k not covered by the first k congruence classes. · Erdos280.uncoveredCount

undelimit def

Parse one self-delimiting block from the front of the input, returning the payload and the remaining input. · BitstringEncoding.undelimit

undelimitBlocks def

Parse a sequence of self-delimiting blocks off the front of the input. · BitstringEncoding.undelimitBlocks

undelimitBlocksAux def

Parse a sequence of self-delimiting blocks, using fuel to bound the number of blocks. · BitstringEncoding.undelimitBlocksAux

uniformCoeff def

The common amplitude of the Bell and GHZ witnesses. · OpenQuantumProblem35.uniformCoeff

UniformHypergraph structure

A uniform Mathlib hypergraph whose vertex set is the entire ambient type. · UniformHypergraph

UniformlySparse def

Every window (x, x+y) has at most ε y integers with a divisor in (n, 2n). · Erdos450.UniformlySparse

UnimodularOn def

f is unimodular on (a, ∞) if it increases until some point M > a and decreases thereafter. · UnimodularOn

UnionFree def

A family F of finsets is union-free if there are no solutions to A ∪ B = C with A, B, C distinct members of F. · Finset.UnionFree

UniqueMinimizer def

The predicate on $n$ asserting all $A, B\subset \mathbb{R}^2$, with $\lvert A\rvert=n = \lvert B\rvert$, which minimise the number of distinct points for all sets with $n$ elements are similar. · Erdos91.UniqueMinimizer

uniqueSubgraphCount def

The number of distinct unique subgraphs of H: the number of isomorphism classes of graphs that occur exactly once as a subgraph of H. Each such class contains exactly one subgraph G ≤ H (uniqueness forbids a second isomorphic copy), … · SimpleGraph.uniqueSubgraphCount

UniqueUlamSum def

UniqueUlamSum a n m means that $m$ has a unique representation as $a(i) + a(j)$ with $i < j < n$. · Erdos342.UniqueUlamSum

unitaryDivisors def

The set of unitary divisors of $n$: divisors $d$ such that $\gcd(d, n/d) = 1$. · OeisA63880.unitaryDivisors

UnitCircle def

The unit circle as a subset of the Euclidean plane. · SquarePacking.UnitCircle

unitCircleCount def

The number of distinct unit circles containing at least three points of P. · Erdos104.unitCircleCount

unitDisk def

def unitDisk : Set ℂ · BrennanConjecture.unitDisk

UnitDistanceAvoiding def

Colouring the points of S blue and the points of Sᶜ red gives a red/blue colouring of $\mathbb{R}^2$; it is unit-distance-avoiding if no two blue points are distance $1$ apart. · Erdos214.UnitDistanceAvoiding

unitDistanceCounts def

The set of all possible numbers of unit distances for a configuration of $n$ points. · Erdos90.unitDistanceCounts

UnitDistanceEmbeddable def

G can be embedded in ℝ^n with every edge a unit line segment if there is an injective map from the vertices of G to ℝ^n sending any two adjacent vertices to points at distance 1. · SimpleGraph.UnitDistanceEmbeddable

UnitDistancePlaneGraph def

A unit distance graph in ℝ²: A graph where the vertices V are a collection of points in ℝ² and there is an edge between two points if and only if the distance between them is 1. · SimpleGraph.UnitDistancePlaneGraph

unitDistNum def

The number of pairs of points of a finite set s in a metric space that are distance 1 apart. · unitDistNum

unitEquilateralTriangleCount def

Number of unit equilateral triangles spanned by a finite point set. · Erdos755.unitEquilateralTriangleCount

unitFractionExpressible def

The set of k for which a / b can be expressed as a sum of k distinct unit fractions. · Erdos304.unitFractionExpressible

UnitInterval abbrev

The unit interval [0, 1]. · LimitObjects.UnitInterval

unitSphere abbrev

The unit n-sphere, defined as Metric.sphere 0 1 in EuclideanSpace ℝ (Fin (n + 1)). · Mathoverflow1973.unitSphere

UnitSquare def

The unit square as a subset of the Euclidean plane. · SquarePacking.UnitSquare

unitSquare def

The four vertices of a unit square. · Erdos214.unitSquare

unitSquare def

def unitSquare : Finset ℝ² · Erdos91.unitSquare

unitSquare def

The unit square. · Mathoverflow34145.unitSquare

unitSquare def

The unit square. · MovingSofa.unitSquare

UnitSquareCorners def

def UnitSquareCorners : Fin 4 → ℝ² · RationalDistanceProblem.UnitSquareCorners

univalentBlochConstant def

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

universalEnveloping abbrev

The universal enveloping algebra U(𝔤𝔩 n ℂ) of the complexified Lie algebra of GL n ℝ. The complexification of 𝔤𝔩 n ℝ is 𝔤𝔩 n ℂ = Matrix n n ℂ with its commutator bracket. · Matrix.GeneralLinearGroup.universalEnveloping

universalSpectrum def

def universalSpectrum (τ : ℝ) : ℝ · BrennanConjecture.universalSpectrum

universalSpectrumBounded def

def universalSpectrumBounded (τ : ℝ) : ℝ · BrennanConjecture.universalSpectrumBounded

UnrepWithout def

Integers not representable as a sum of exactly h elements of A while avoiding n. · Erdos330.UnrepWithout

upper_bound abbrev

A known upper bound obtained by constructing Wichmann's Rulers [Wi63]. · Erdos170.upper_bound

upperBarrier def

The "Barrier" function: n^(-7/6) used for the best upper bound [CPZ24]. · Erdos507.upperBarrier

upperBest def

Best-known upper bound [Be23, Theorem 5]. · Green27.upperBest

upperDensity def

Given a set S in an order β, where all intervals bounded above are finite, we define the upper density of S (relative to a set A) to be the limsup of the partial densities of S (relative to A) for b → ∞. · Set.upperDensity

usigma def

The sum of unitary divisors of $n$, denoted $\text{usigma}(n)$. · OeisA63880.usigma