orthogonalityProperty

P76852
predicate

Indicates that one entity possesses the property of being orthogonal (i.e., at right angles or having zero inner product) with respect to another entity or a given reference.

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All labels observed (14)

Label Occurrences
orthogonalityProperty canonical 5
orthogonalWithRespectTo 3
orthogonalityRelation 3

Description generation (PDg)

The one-sentence description above was generated by prompting gpt-5.1 with the predicate name and this instruction.

Instruction
Given a predicate that represents a relationship or action between entities, generate a one-sentence description explaining its meaning.  
# Instructions
Focus on describing the relationship, not the entities themselves. 
# Response Format
Begin the description with \' Indicates...\'
Input
Predicate: orthogonalityProperty
Generated description
Indicates that one entity possesses the property of being orthogonal (i.e., at right angles or having zero inner product) with respect to another entity or a given reference.

Sample triples (25)

Subject Object
Legendre symbol sum_{a mod p} (a/p)=0 ⓘ
Jacobi polynomials weight (1-x)^α (1+x)^β on [-1,1] via predicate surface "areOrthogonalWithRespectTo" ⓘ
Jacobi polynomials ∫_{-1}^1 (1-x)^α (1+x)^β P_m^{(α,β)}(x) P_n^{(α,β)}(x) dx = 0 for m ≠ n via predicate surface "hasOrthogonalityRelation" ⓘ
Ramanujan’s sum sum_{n mod q} c_q(n) = 0 for q > 1 via predicate surface "orthogonalityRelation" ⓘ
Ramanujan’s sum sum_{q ≥ 1} c_q(n) c_q(m) / φ(q) converges to a function of gcd(m,n) via predicate surface "orthogonalityRelation" ⓘ
Dirichlet characters sum over n mod N of χ(n) = 0 for non-principal χ ⓘ
Dirichlet characters sum over characters χ of χ(a)overline{χ(b)} = φ(N) if a ≡ b mod N ⓘ
Hermite functions Gaussian weight via predicate surface "orthogonalWithRespectTo" ⓘ
Hermite functions Lebesgue measure with Gaussian weight via predicate surface "orthogonalWithRespectTo" ⓘ
Hermitian forms (work on quadratic forms)
surface form: Hermitian form
x ⟂ y iff h(x,y)=0 via predicate surface "orthogonalityCondition" ⓘ
Bernoulli polynomials not orthogonal on standard intervals with usual weights ⓘ
Liouville function exhibits cancellation in many average sums over n ⓘ
Legendre polynomials interval [-1,1] via predicate surface "areOrthogonalOn" ⓘ
Legendre polynomials weight function w(x)=1 via predicate surface "areOrthogonalWithRespectTo" ⓘ
Legendre polynomials ∫_{-1}^1 P_m(x)P_n(x) dx = 2/(2n+1) δ_{mn} via predicate surface "satisfyOrthogonalityRelation" ⓘ
Sturm–Liouville problem ∫_a^b w(x) y_m(x) y_n(x) dx = 0 for m ≠ n via predicate surface "hasOrthogonalityRelation" ⓘ
Chebyshev polynomials of the first kind w(x) = (1 - x^2)^(-1/2) via predicate surface "orthogonalWithRespectToWeight" ⓘ
Chebyshev polynomials of the first kind [-1,1] via predicate surface "orthogonalOnInterval" ⓘ
Chebyshev polynomials of the first kind \int_{-1}^1 T_m(x) T_n(x) (1-x^2)^{-1/2} dx = 0 for m ≠ n via predicate surface "orthogonalityRelation" ⓘ
Gegenbauer polynomials weight function (1-x^2)^{\lambda-1/2} via predicate surface "orthogonalWithRespectTo" ⓘ
Gegenbauer polynomials [-1,1] via predicate surface "orthogonalOn" ⓘ
Gegenbauer polynomials \int_{-1}^1 (1-x^2)^{\lambda-1/2} C_m^{(\lambda)}(x) C_n^{(\lambda)}(x) dx = 0 for m \neq n via predicate surface "orthogonalityIntegral" ⓘ
Gegenbauer polynomials \lambda > -1/2 via predicate surface "orthogonalityCondition" ⓘ
Jack polynomials Jack inner product depending on α via predicate surface "orthogonalityWithRespectTo" ⓘ
Macdonald polynomials orthogonal with respect to a certain (q,t)-deformed scalar product via predicate surface "hasOrthogonality" ⓘ