Fermionic quantum gravity

Quasi-linear Stokes phenomenon for the Painlevé first equation

“…For instance, equation P 1 describes certain solutions to KdV and Bussinesq equations [4,5], bifurcations in some non-integrable nonlinear models [6], continuous limits in matrix models of quantum gravity [7,8,9,10]; Ψ-function associated with P 1 appears in n-large asymptotics of semi-classical orthogonal and bi-orthogonal polynomials [11,12] and thus becomes a primary object in the problem of Laplacian growth [13].…”

“…In context of the string theory, "physical" solutions of P 1 are distinguished from "non-physical" ones by the monotonic asymptotic behavior as x → −∞ [7,8,10,14,15]. There are two kinds of such monotonic boundary conditions, i.e.…”

Grassmann matrix quantum mechanics

“…Ordinary integrals over Grassmann matrices were studied extensively in [9][10][11]. There, it was shown how the problem of Grassmann matrix integrals at large N , M can be expressed as an eigenvalue problem for the composite N × N matrix Φ ij = Aψ iA ψ Aj , which is effectively bosonic.…”

Comparison of three unsupervised neural network models for first Painlevé Transcendent

“…The same start point is given to each network, and other parameter settings for SQP method are given in Table 1 Step 2 Fitness evaluation: MATLAB built-in library includes formulation of SQP method which has been employed for the calculation of fitness values from each start point using the fitness function given in Eqs. (7), (8) and (9) The predefined values of parameters are also provided in Table 1.…”

“…These functions had no physical applications over a long period of time, but now these equations are used extensively to describe different physical processes, due to which research community has shown special interest in these transcendents in the fields of both pure mathematics [1] and applied mathematical physics [2], as well as in theoretical physics [3]. For example, Painlevé equations have arisen for solving the Korteweg-de Vries (KdV) along with Boussinesq equations [4,5], bifurcations in nonlinear non-integrable models [6], quantum gravity matrix models with continuous limits [7,8], reducing to the KdV type equations and its cylindrical variants [9] and so on. d 2 yðxÞ dx 2 ¼ 6yðxÞ 2 þ x;…”