Riemann hypothesis is proven by reducing the vanishing of Riemann Zeta to an orthogonality condition for the eigenfunctions of a nonHermitian operator having the zeros of Riemann Zeta as its eigenvalues. Eigenfunctions are analogous to the so called coherent states and in general not orthogonal to each other. The construction of the operator is inspired by the conviction that Riemann Zeta is as...

Riemann approximately proved that every compact, connected Riemann surface is a projective algebraic curve, but the Dirichlet (minimum) principle he invoked was literally incorrect. The literal falsity amounts to the fact that a convex, closed subset of a Banach space need not have a unique element of least norm: it may have none, or infinitely-many. This minimum principle is true in a Hilbert ...

We derive the soliton matrices corresponding to an arbitrary number of higherorder normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order solitons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix R...

The diffraction of a uniform unit-amplitude E-polarized plane wave is considered in the case of its normal incidence on a strip periodic metal grating placed on the anisotropic hyrotropic ferromagnetic half-space boundary. The Dirichlet boundary conditions on the grating strips, the medium interface conjugation conditions, the Meixner condition that the energy is finite in any confined volume a...

We associate to the global field K a Lax–Phillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian L–functions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The adelic aspects are related to previous work by Tate, Iwasawa and Connes. Université de NiceSophia Ant...

In the 20th century, many great mathematicians were attracted by this conjecture. In 1900, D. Hilbert gave a famous speech in an international mathematical conference, in which he proposed 23 problems to mathematicians. The Goldbach conjecture is a part of his 8th problem and the other part is the Riemann hypothesis. G. H. Hardy said that the Goldbach conjecture is one of the most difficult pro...

We continue the study of the Hermitian random matrix ensemble with external source 1 Zn e−nTr( 1 2M −AM)dM, where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a = 1, since the eigenvalues of M accumulate on two intervals for a > 1, and on one interval for 0 < a < 1. The case a > 1 was treated in Part I, where it was proved that lo...

We develop an inverse scattering transform formalism for the good Boussinesq equation on line. Assuming that solution exists, we show it can be expressed in terms of a $3 \times 3$ matrix Riemann-Hilbert problem. The problem is formulated two reflection coefficients whose definitions involve only initial data, and has form which makes suitable evaluation long-time asymptotics via Deift-Zhou ste...

This is a sequel to papers by the last two authors making the Riemann–Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann–Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto–Painl...

We study the complex Sharma–Tasso–Olver equation using Riemann–Hilbert approach. The associated problem for this integrable can be naturally constructed by considering spectral of Lax pair. Subsequently, in case that is irregular, N -soliton solutions deduced. In addition, three-dimensional graphic soliton and wave propagation image are graphically depicted further discussed.