By C.B. Wang

The eigenvalue densities in numerous matrix versions in quantum chromodynamics (QCD) are finally unified during this e-book through a unified version derived from the integrable structures. Many new density versions and unfastened strength features are therefore solved and awarded. The part transition versions together with serious phenomena with fractional power-law for the discontinuities of the unfastened energies within the matrix types are systematically categorized by way of a transparent and rigorous mathematical demonstration. The equipment the following will stimulate new examine instructions equivalent to the real Seiberg-Witten differential in Seiberg-Witten conception for fixing the mass hole challenge in quantum Yang-Mills thought. The formulations and effects will profit researchers and scholars within the fields of section transitions, integrable structures, matrix types and Seiberg-Witten theory.

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**Example text**

L − 1) is the largest integer such that s + lms ≤ 2m − 1. 88) for η in the outside of the cuts to be discussed in the following. Then there is ωl (η) = 1 2 W (η) + y(η), where −y(η) is equal to ms l−1 ηs s=0 glq+s+1 q=0 q + r=[q/2]+1 q r μq 2 2[q/2] (l) q−2[q/2] q b(l) α [q/2] 2r b(l) α (l) q−2r . 90) 40 2 Densities in Hermitian Matrix Models Fig. 91) for s = 0, 1, . . 92) as η → ∞. 2 −1 Now, consider the cuts for ωl (η), determined by α (l) − b(l) α (l) which is equal 2 2 to Λ2 − 4b(l) . Equation Λ2 − 4b(l) = 0 has 2l roots, real or complex.

4 If the parameters as , bs (s = 1, . . , l), and gj (j = 1, . . 6). 4 Density Models 41 Proof Let Γ be a large counterclockwise circle of radius R, and Ω ∗ be the union of closed counterclockwise contours around the upper and lower edges of all the intervals in Ω. 85), Ω ∗ W (η)dη = 0, and Is are real. 5). Change the Ω − and Ω + discussed above just at η ∈ Ω as semicircles of ε radius. 92) and Γ ∗ Wλ−η dλ = 0, there is s 1 2πi Ω∗ ωl (λ) − 12 W (λ) 1 dλ + λ−η 2πi l2 s=1 1 Iˆs = 2πi Γ ωl (λ) − 12 W (λ) dλ → 0, λ−η as R → ∞.

We can transform gj (j = 1, . . , 2m) into a new set of parameters gj (j = 1, . . , 2m) ms s q by a linear transformation so that W (η) = l−1 s=0 η q=0 glq+s+1 Λ , where each ms (s = 0, . . , l − 1) is the largest integer such that s + lms ≤ 2m − 1. 88) for η in the outside of the cuts to be discussed in the following. Then there is ωl (η) = 1 2 W (η) + y(η), where −y(η) is equal to ms l−1 ηs s=0 glq+s+1 q=0 q + r=[q/2]+1 q r μq 2 2[q/2] (l) q−2[q/2] q b(l) α [q/2] 2r b(l) α (l) q−2r . 90) 40 2 Densities in Hermitian Matrix Models Fig.