The decays ψ2(3823)→γχc0,1,2, π+π−J/ψ, π0π0J/ψ, ηJ/ψ, and π0J/ψ are searched for using the reaction e+e−→π+π−ψ2(3823) in a 19 fb−1 data sample collected at center-of-mass energies between 4.1 4.7 GeV with BESIII detector. process ψ2(3823)→γχc1 is observed 9 energy range 4.3–4.7 GeV, which confirms previous observation but higher significance of 11.8σ, evidence ψ2(3823)→γχc2 found 3.2σ first tim...

Let φ be a compactly supported symmetric real-valued refinable function in L2(R) with a finitely supported symmetric real-valued mask on Z. Under the assumption that the shifts of φ are stable, in this paper we prove that one can always construct three wavelet functions ψ, ψ and ψ such that (i) All the wavelet functions ψ, ψ and ψ are compactly supported, real-valued and finite linear combinati...

Lemma 2. For any r∗ ∈ (r, r̃) and ε > 0, there exist η̄ > 0 and ρ̄ < r/r∗ such that for any (η, ρ) < (η̄, ρ̄), conditions (3)-(2) below admit a solution (x0, x̂, θ0, θ00) that satisfies θ0 ≤ θ00, |x0 − x∗| < ε, ̄̄θ0 − θ∗ ̄̄ < ε, ̄̄θ00 − θ∗∗ ̄̄ < ε, and x̂ < −1/ε. 1−Ψ(x−θ σ ) = r − (r − ρr∗)[Ψ(x 0−θ0 σ )−Ψ(x 0−θ00 σ )] (1) 1−Ψ( x̂−θ0 σ ) = r + [r∗ρ+ r∗(1− ρ) exp( r ∗−r η )− r][Ψ( x̂−θ 0 σ )−Ψ( x̂−θ 00 σ )] (2) θ0...

1 Mirror operator ∇ψ * In this section, we discuss properties of distance generating functions and their subdif-ferentials. Let ψ be a proper, closed, convex function, and suppose that X is the effective domain of ψ (i.e. X = {x ∈ R n : ψ(x) < ∞}). The subdifferential of ψ at x ∈ X is ∂ψ(x) = {z ∈ E * : ψ(y) − ψ(x) − z, y − x ≥ 0 ∀y ∈ X }.

For the problem ψ′′ κ2 = ψ − ψ +Aψ, A′′ = ψA, ψ′(0) = ψ(∞) = 0, A′(0) = A′(∞) = h, it is proved for type II superconductors (κ > 1/ √ 2) that (1) No solutions can exist for h ≤ 1/√2 other than the normal state ψ ≡ 0, A = hx+ C; (2) Positive solutions (ψ > 0) exist whenever 1/ √ 2 < h < hc3 ≈ 1.7κ; (3) As h ↓ 1/√2, the limit of any converging subsequence satisfies A = 0, ψ = 1 at infinity.

In this article, we study a system of Hilfer (k,ψ)-fractional differential equations, subject to nonlocal boundary conditions involving (k,ψ)-derivatives and (k,ψ)-integrals. The results for the mentioned are established by using Mönch’s fixed point theorem, then Ulam–Hyers technique is used verify stability solution proposed system. general, symmetry fractional equations related each other. Wh...