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By C.B. Wang

The eigenvalue densities in numerous matrix types in quantum chromodynamics (QCD) are eventually unified during this ebook through a unified version derived from the integrable platforms. Many new density versions and loose strength features are as a result solved and awarded. The section transition versions together with severe phenomena with fractional power-law for the discontinuities of the loose energies within the matrix versions are systematically categorised through a transparent and rigorous mathematical demonstration. The tools right here will stimulate new study instructions resembling the real Seiberg-Witten differential in Seiberg-Witten thought for fixing the mass hole challenge in quantum Yang-Mills thought. The formulations and effects will gain researchers and scholars within the fields of section transitions, integrable structures, matrix versions and Seiberg-Witten theory.

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100) for η ∈ [−2b, 2b], where b2 + 12gb4 = 1. 6), where η− = a − 2b, η+ = a + 2b. The density formula and the conditions coincide with the results (45) and (46) in [1] for the case W (η) = 12 η2 + g3 η3 . , W (η) = g0 + g3 η3 + g4 η4 , the conditions become 3g3 (η + a) + 4g4 η2 + aη + a 2 + 2b2 ≥ 0, g3 = − 8a(a 2 + 6b2 ) 3b2 (5a 4 + 3(a 2 − 4b2 )2 ) 2(a 2 + 2b2 ) g4 = 2 4 . 106) is satisfied if and only if τ = 4b is restricted in the a2 √ interval 0 < τ ≤ τ− or τ+ ≤ τ , where τ+ = 1 + 5, and τ− is uniquely determined 1/2 by the conditions: 0 < τ− < 1/2 and 1 − 2τ− + 34 τ−2 = 0.

In the momentum aspect, the parameters and the corresponding functions such as uN and vN control the wave functions of the random variables, so that the asymptotics of these functions are not directly related to the behaviors of the particles. If there is an asymptotic relation, it should be a “relative” asymptotics. These functions are closely connected to the moments of the eigenvalues. Each reduction from the integrable system is not a necessity, but a case of the probability. And the occurrence of each possible case is not based on certainty principle, but the uncertainty principle.

87) , where l ≥ 1. According to the Cayley-Hamilton theorem for J (l) , choose α (l) = 2 2 (Λ + Λ2 − 4b(l) )/2, where Λ = Λ(η) = tr J (l) , b(l) = det J (l) and b(l) > 0. 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 .

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