Matching Tag: spherical-harmonics
Let N = { n ∈ N : n ≢ 0 , 4 , 7 ( m o d 8 ) } \mathcal N=\{n\in\mathbb N:n\not\equiv 0,4,7\pmod{8}\} N = { n ∈ N : n ≡ 0 , 4 , 7 ( mod 8 )} E ( n ) = { x ∈ Z 3 : ∣ x ∣ 2 = n } \mathcal E(n)=\{\mathbf{x}\in\mathbb Z^3:|\mathbf{x}|^2=n\} E ( n ) = { x ∈ Z 3 : ∣ x ∣ 2 = n }
Let d ≥ 1 d\geq 1 d ≥ 1 λ = d − 1 2 \lambda=\frac{d-1}{2} λ = 2 d − 1 ( θ − t ) + = max { θ − t , 0 } ({\theta}-t)_+=\max\{{\theta}-t,0\} ( θ − t ) + = max { θ − t , 0 } δ ≥ λ + 1 \delta\geq\lambda+1 δ ≥ λ + 1 θ ∈ ( 0 , π ) {\theta}\in(0,\pi) θ ∈ ( 0 , π )
Symmetry conjecture. The limit
Simple angular dependence conjecture. The fundamental modes of B ( R ) \mathbb{B}(R) B ( R ) u 1 ( r , θ ^ ) u_1(r,\hat{\theta}) u 1 ( r , θ ^ )
Let H ℓ \cal H_\ell H ℓ ℓ \ell ℓ μ ( u ) \mu(u) μ ( u ) u u u
Integral positivity conjecture. The inequality F n λ , δ ( t ) > 0 F_n^{\lambda,\delta}(t)>0 F n λ , δ ( t ) > 0 n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0 t ∈ ( 0 , π ] t\in(0,\pi] t ∈ ( 0 , π ] δ ≥ λ + 1 \delta\geq\lambda+1 δ ≥ λ + 1
