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Journal articles on the topic 'Riemannův problém'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Zeitsch, Peter. "On the Riemann Function." Mathematics 6, no. 12 (2018): 316. http://dx.doi.org/10.3390/math6120316.

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Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in two variables. The first review of Riemann’s method was published by E.T. Copson in 1958. This study extends that work. Firstly, three solution methods were overlooked in Copson’s original paper. Secondly, several new approaches for finding Riemann functions have been developed since 1958. Those techniques are included here and placed in the context of Copson’s original study. There are also numerous equivalences between Riemann functions that have not previously been identified in the literature. Those links are clarified here by showing that many known Riemann functions are often equivalent due to the governing equation admitting a symmetry algebra isomorphic to S L ( 2 , R ) . Alternatively, the equation admits a Lie-Bäcklund symmetry algebra. Combining the results from several methods, a new class of Riemann functions is then derived which admits no symmetries whatsoever.

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2

Zhixin, L., and B. A. Kats. "Riemann boundary-value problem on Cassini spirals." Issues of Analysis 28, no. 1 (2021): 101–10. http://dx.doi.org/10.15393/j3.art.2021.9770.

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3

Han, Xinli, and Lijun Pan. "The Perturbed Riemann Problem with Delta Shock for a Hyperbolic System." Advances in Mathematical Physics 2018 (September 5, 2018): 1–11. http://dx.doi.org/10.1155/2018/4925957.

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In this paper, we study the perturbed Riemann problem with delta shock for a hyperbolic system. The problem is different from the previous perturbed Riemann problems which have no delta shock. The solutions to the problem are obtained constructively. From the solutions, we see that a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity under a perturbation of the Riemann initial data. This shows the instability and the internal mechanism of a delta shock. Furthermore, we find that the Riemann solution of the hyperbolic system is instable under this perturbation, which is also quite different from the previous perturbed Riemann problems.

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4

Durmagambetov, Asset. "Reduction of modern problems of mathematics to the classical Riemann-Poincare-Hilbert problem." European Journal of Pure and Applied Mathematics 11, no. 4 (2018): 1143–76. http://dx.doi.org/10.29020/nybg.ejpam.v11i4.3328.

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Using the example of a complicated problem such as the Cauchy problem for the Navier--Stokes equation, we show how the Poincar\'e--Riemann--Hilbert boundary-value problem enables us to construct effective estimates of solutions for this case. The apparatus of the three-dimensional inverse problem of quantum scattering theory is developed for this. It is shown that the unitary scattering operator can be studied as a solution of the Poincar\'e--Riemann--Hilbert boundary-value problem. This allows us to go on to study the potential in the Schr\"odinger equation, which we consider as a velocity component in the Navier--Stokes equation. The same scheme of reduction of Riemann integral equations for the zeta function to the Poincar\'e--Riemann--Hilbert boundary-value problem allows us to construct effective estimates that describe the behaviour of the zeros of the zeta function very well.

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5

Kurganov, Alexander, and Eitan Tadmor. "Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers." Numerical Methods for Partial Differential Equations 18, no. 5 (2002): 584–608. http://dx.doi.org/10.1002/num.10025.

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6

Chen, Yang, and Alexander R. Its. "A Riemann–Hilbert approach to the Akhiezer polynomials." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1867 (2007): 973–1003. http://dx.doi.org/10.1098/rsta.2007.2058.

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In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the union of disjoint intervals, first introduced by N. I. Akhiezer, via a reformulation as a matrix factorization or Riemann–Hilbert problem. This approach complements the method proposed in a previous paper, which involves the construction of a certain meromorphic function on a hyperelliptic Riemann surface. The method described here is based on the general Riemann–Hilbert scheme of the theory of integrable systems and will enable us to derive, in a very straightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantities involving the corresponding recurrence coefficients. Both of these equations were obtained earlier by A. Magnus. In our approach, however, we are able to go beyond Magnus' results by actually solving the equations in terms of the Riemanni Θ -functions. We also show that the related Hankel determinant can be interpreted as the relevant τ -function.

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7

Bolottn, L. B. "THE SECOND BOUNDARY VALUE PROBLEM OF RIEMANN'S TYPE FOR BIANALYTICAL FUNCTIONS WITH DISCONTINUOUS COEFFICIENTS." Mathematical Modelling and Analysis 9, no. 3 (2004): 193–200. http://dx.doi.org/10.3846/13926292.2004.9637252.

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The paper is devoted to the investigation of one of the basic boundary value problems of Riemann's type for bianalytical functions with discontinuous coefficients. In the course of work there was made out a constructive method for solution of the problem in a unit circle. There was also found out that the solution of the problem under consideration consists in consequent solutions of two Riemann's boundary value problems for analytical functions in a unit circle. Besides, the example is constructed.

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8

Vyugin, Il'ya V. "Riemann-Hilbert problem for scalar Fuchsian equations and related problems." Russian Mathematical Surveys 66, no. 1 (2011): 35–62. http://dx.doi.org/10.1070/rm2011v066n01abeh004727.

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9

Minkevich, A. V. "Towards the theory of regular accelerating Universe in Riemann-Cartan space-time." International Journal of Modern Physics A 31, no. 02n03 (2016): 1641011. http://dx.doi.org/10.1142/s0217751x16410116.

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The problems of modern cosmology and attempts of its solution are considered. The applying of gravitation theory in the Riemann-Cartan space-time as the most natural generalization of Einstein gravitation theory in order to solve the principal cosmological problems is discussed. The gravitation theory in the Riemann-Cartan space-time leading to the solution of the problem of cosmological singularity and dark energy problem is analyzed.

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10

Fatykhov, A. Kh, and P. L. Shabalin. "Solvability homogeneous Riemann-Hilbert boundary value problem with several points of turbulence." Issues of Analysis 25 (September 2018): 31–39. http://dx.doi.org/10.15393/j3.art.2018.5530.

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11

吕, 志新. "Half-Periodic Riemann Boundary Value Problem." Advances in Applied Mathematics 07, no. 07 (2018): 883–89. http://dx.doi.org/10.12677/aam.2018.77106.

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12

Shilin, Andrei P. "Riemann’s diﬀerential boundary-value problem and its application to integro-diﬀerential equations". Doklady of the National Academy of Sciences of Belarus 63, № 4 (2019): 391–97. http://dx.doi.org/10.29235/1561-8323-2019-63-4-391-397.

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The boundary-value problem for analytical functions is investigated. The boundary condition is placed on a closed curve located on the complex plane. The problem belongs to the type of the generalized Riemann boundary-value problems. The boundary condition contains derivatives of the required functions. The problem is reduced to the usual Riemann problem and linear diﬀerential equations. The solution is built in closed form. The application of the solved problem to integro-diﬀerential equations is indicated.

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13

Xu, Yongzhi. "Riemann problem and inverse Riemann problem of ( , 1)bi-analytic functions." Complex Variables and Elliptic Equations 52, no. 10-11 (2007): 853–64. http://dx.doi.org/10.1080/17476930701483809.

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14

Townsend, Jamie F., László Könözsy, and Karl W. Jenkins. "On the development of a rotated-hybrid HLL/HLLC approximate Riemann solver for relativistic hydrodynamics." Monthly Notices of the Royal Astronomical Society 496, no. 2 (2020): 2493–505. http://dx.doi.org/10.1093/mnras/staa1648.

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ABSTRACT This work presents the development of a rotated-hybrid Riemann solver for solving relativistic hydrodynamics (RHD) problems with the hybridization of the HLL and HLLC (or Rusanov and HLLC) approximate Riemann solvers. A standalone application of the HLLC Riemann solver can produce spurious numerical artefacts when it is employed in conjunction with Godunov-type high-order methods in the presence of discontinuities. It has been found that a rotated-hybrid Riemann solver with the proposed HLL/HLLC (Rusanov/HLLC) scheme could overcome the difficulty of the spurious numerical artefacts and presents a robust solution for the Carbuncle problem. The proposed rotated-hybrid Riemann solver provides sufficient numerical dissipation to capture the behaviour of strong shock waves for RHD. Therefore, in this work, we focus on two benchmark test cases (odd–even decoupling and double-Mach reflection problems) and investigate two astrophysical phenomena, the relativistic Richtmyer–Meshkov instability and the propagation of a relativistic jet. In all presented test cases, the Carbuncle problem is shown to be eliminated by employing the proposed rotated-hybrid Riemann solver. This strategy is problem-independent, straightforward to implement and provides a consistent robust numerical solution when combined with Godunov-type high-order schemes for RHD.

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15

Isaacson, E., D. Marchesin, B. Plohr, and B. Temple. "The Riemann Problem Near a Hyperbolic Singularity: The Classification of Solutions of Quadratic Riemann Problems I." SIAM Journal on Applied Mathematics 48, no. 5 (1988): 1009–32. http://dx.doi.org/10.1137/0148059.

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16

Bolotin, I. B., and K. M. Rasulov. "THE FIRST BASIC BOUNDARY VALUE PROBLEM OF RIEMANN'S TYPE FOR BIANALYTICAL FUNCTIONS IN A PLANE WITH SLOTS." Mathematical Modelling and Analysis 9, no. 2 (2005): 91–98. http://dx.doi.org/10.3846/13926292.2004.9637244.

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The paper is devoted to the investigation of one of the basic boundary value problems of Riemann's type for bianalytical functions. In the course of work there was made out a constructive method for solution of the problem given in a plane with slots. There was also found out that the solution of the problem under consideration consists of consequent solutions of two Riemann's boundary value problems for analytical functions in a plane with slots. Besides, a picture of solvability of the problem is being searched and its Noether is identified. Šiame darbe tyrinejamas uždavinys, kai ieškoma dalimis bianaliziniu funkciju, nykstančiu begalybeje, apribotu greta kontūro trūkio tašku ir šiame kontūre tenkinančiu dvi kraštines salygas. Parodoma, kad sprendžiamas uždavinys suvedamas i sprendima dvieju Rimano uždaviniu analizinems funkcijoms.

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17

Ion, Stelian, Dorin Marinescu, and Stefan-Gicu Cruceanu. "Riemann Problem for Shallow Water Equation with Vegetation." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 2 (2018): 145–73. http://dx.doi.org/10.2478/auom-2018-0023.

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Abstract We investigate the existence of the solution of the Riemann Problem for a simplified water ow model on a vegetated surface - system of shallow water type equations. It is known that the system with discontinuous topography is non-conservative even if the porosity is absent. A system with continuous topography and discontinuous porosity is also non-conservative. In order to define Riemann solution for such systems, it is necessary to introduce a family of paths that connects the states defining the Riemann Problem. We focus our attention towards choosing such a family based on physical arguments. We provide the structure of the solution for such Riemann Problems.

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18

Bolibrukh, A. A. "The Riemann-Hilbert problem." Russian Mathematical Surveys 45, no. 2 (1990): 1–58. http://dx.doi.org/10.1070/rm1990v045n02abeh002350.

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19

Gao, Si, and Tiegang Liu. "1D Exact Elastic-Perfectly Plastic Solid Riemann Solver and Its Multi-Material Application." Advances in Applied Mathematics and Mechanics 9, no. 3 (2017): 621–50. http://dx.doi.org/10.4208/aamm.2015.m1340.

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AbstractThe equation of state (EOS) plays a crucial role in hyperbolic conservation laws for the compressible fluid. Whereas, the solid constitutive model with elastic-plastic phase transition makes the analysis of the solid Riemann problem more difficult. In this paper, one-dimensional elastic-perfectly plastic solid Riemann problem is investigated and its exact Riemann solver is proposed. Different from previous works treating the elastic and plastic phases integrally, we resolve the elastic wave and plastic wave separately to understand the complicate nonlinear waves within the solid and then assemble them together to construct the exact Riemann solver for the elastic-perfectly plastic solid. After that, the exact solid Riemann solver is associated with the fluid Riemann solver to decouple the fluid-solid multi-material interaction. Numerical tests, including gas-solid, water-solid high-speed impact problems are simulated by utilizing the modified ghost fluid method (MGFM).

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20

Cerne, Miran. "Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces." American Journal of Mathematics 126, no. 1 (2004): 65–87. http://dx.doi.org/10.1353/ajm.2004.0002.

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21

Kamvissis, Spyridon. "A riemann-hilbert problem in a riemann surface." Acta Mathematica Scientia 31, no. 6 (2011): 2233–46. http://dx.doi.org/10.1016/s0252-9602(11)60396-2.

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22

Kotlyarov, V. P., and E. A. Moskovchenko. "Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening." Zurnal matematiceskoj fiziki, analiza, geometrii 10, no. 3 (2014): 328–49. http://dx.doi.org/10.15407/mag10.03.328.

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23

Takahashi, K., and S. Yamada. "Exact Riemann solver for ideal magnetohydrodynamics that can handle all types of intermediate shocks and switch-on/off waves." Journal of Plasma Physics 80, no. 2 (2013): 255–87. http://dx.doi.org/10.1017/s0022377813001268.

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AbstractWe have built a code to obtain the exact solutions of Riemann problems in ideal magnetohydrodynamics (MHD) for an arbitrary initial condition. The code can handle not only regular waves but also switch-on/off rarefactions and all types of non-regular shocks: intermediate shocks and switch-on/off shocks. Furthermore, the initial conditions with vanishing normal or transverse magnetic fields can be handled, although the code is partly based on the algorithm proposed by Torrilhon (2002) (Torrilhon, M. 2002 Exact solver and uniqueness condition for Riemann problems of ideal magnetohydrodynamics. Research report 2002-06, Seminar for Applied Mathematics, ETH, Zurich, Switzerland), which cannot deal with all types of non-regular waves nor the initial conditions without normal or transverse magnetic fields. Our solver can find all the solutions for a given Riemann problem, and hence, as demonstrated in this paper, one can investigate the structure of the solution space in detail. Therefore, the solver is a powerful instrument to solve the outstanding problem of existence and uniqueness of solutions of MHD Riemann problems. Moreover, the solver may be applied to numerical MHD schemes like the Godunov scheme in the future.

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24

Cheng, Hongjun, and Shiwei Li. "A Deposition Model: Riemann Problem and Flux-Function Limits of Solutions." Abstract and Applied Analysis 2018 (2018): 1–14. http://dx.doi.org/10.1155/2018/8569435.

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The Riemann solutions of a deposition model are shown. A singular flux-function limit of the obtained Riemann solutions is considered. As a result, it is shown that the Riemann solutions of the deposition model just converge to the Riemann solutions of the limit system, the scalar conservation law with a linear flux function involving discontinuous coefficient. Especially, for some initial data, the two-shock Riemann solution of the deposition model tends to the delta-shock Riemann solution of the limit system; by contrast, for some initial data, the two-rarefaction-wave Riemann solution of the deposition model tends to the vacuum Riemann solution of the limit system. Some numerical results exhibiting the formation processes of delta-shocks and vacuum states are presented.

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25

Taghizadeh, N., and V. Soltani Mohammadi. "SOME BOUNDARY VALUE PROBLEMS FOR THE CAUCHY-RIEMANN EQUATION IN HALF LENS." Eurasian Mathematical Journal 9, no. 3 (2018): 73–84. http://dx.doi.org/10.32523/2077-9879-2018-9-3-73-84.

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26

Chern, Shiing-Shen, and Shanyu Ji. "Projective geometry and Riemann's mapping problem." Mathematische Annalen 302, no. 1 (1995): 581–600. http://dx.doi.org/10.1007/bf01444509.

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27

Hildebrandt, Stefan. "Plateau’s Problem and Riemann’s Mapping Theorem." Milan Journal of Mathematics 79, no. 1 (2011): 67–79. http://dx.doi.org/10.1007/s00032-011-0142-y.

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28

Roesler, Friedrich. "Riemann's hypothesis as an eigenvalue problem." Linear Algebra and its Applications 81 (September 1986): 153–98. http://dx.doi.org/10.1016/0024-3795(86)90255-7.

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29

Dai, Wenlong, and Paul R. Woodward. "Structures of reconnection layers based on the ideal magnetohydrodynamic equations." Journal of Plasma Physics 51, no. 3 (1994): 381–98. http://dx.doi.org/10.1017/s0022377800017645.

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A Riemann solver is used, and a set of numerical simulations are performed, to study the structures of reconnection layers in the approximation of the one- dimensional ideal MHD equations. Since the Riemann solver may solve general Riemarin problems, the model used in this paper is more general than those in previous investigations on this problem. Under the conditions used in the previous investigations, the structures we obtained are the same. Our numerical simulations show quantitative agreement with those obtained through the Riemann solver.

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30

Ji, Pengpeng, and Chun Shen. "Construction of the Global Solutions to the Perturbed Riemann Problem for the Leroux System." Advances in Mathematical Physics 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/4808610.

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The global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states. The wave interaction problems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

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31

Kuijlaars, Arno. "The Tacnode Riemann–Hilbert Problem." Constructive Approximation 39, no. 1 (2013): 197–222. http://dx.doi.org/10.1007/s00365-013-9225-z.

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32

Zhang, Tong, and Yuxi Zheng. "Riemann problem for gasdynamic combustion." Journal of Differential Equations 77, no. 2 (1989): 203–30. http://dx.doi.org/10.1016/0022-0396(89)90142-3.

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33

Mohammed, Alip. "The torus related Riemann problem." Journal of Mathematical Analysis and Applications 326, no. 1 (2007): 533–55. http://dx.doi.org/10.1016/j.jmaa.2006.03.011.

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34

Pinelis, Iosif. "A problem concerning Riemann sums." Journal of Classical Analysis, no. 2 (2020): 59–63. http://dx.doi.org/10.7153/jca-2020-16-07.

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35

Rajagopalan, Ganesh, Vadali Mahadev, and Timothy S. Cale. "Surface Evolution During Semiconductor Processing." VLSI Design 6, no. 1-4 (1998): 379–84. http://dx.doi.org/10.1155/1998/62125.

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We discuss our approach to using the Riemann problem to compute surface profile evolution during the simulation of deposition, etch and reflow processes. Each pair of segments which represents the surface is processed sequentially. For cases in which both segments are the same material, the Riemann problem is solved. For cases in which the two segments are different materials, two Riemann problems are solved. The material boundary is treated as the right segment for the left material and as the left segment for the right material. The critical equations for the analyses are the characteristics of the Riemann problem and the ‘jump conditions’ which represent continuity of the surface. Examples are presented to demonstrate selected situations. One limitation of the approach is that the velocity of the surface is not known as a function of the surface angle. Rather, it is known for the angles of the left and right segments. The rate as a function of angle must be assumed for the explicit integration procedure used. Numerical implementation is briefly discussed.

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36

OSTAPENKO, VICTOR A., and JAN AWREJCEWICZ. "A NOVEL ASYMPTOTIC SOLUTION OF ONE NONLINEAR PROBLEM OF FILTERING." International Journal of Modern Physics B 22, no. 15 (2008): 2383–98. http://dx.doi.org/10.1142/s0217979208039538.

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A boundary value problem associated with filtering governed by differential equations with time dependent coefficients and exhibiting a weak nonlinearity is solved. Namely, the boundary value problem is split into two independent boundary problems, i.e., that of Goursat for the concentration of a sorbate and that absorbed by a sorbent. Then each of the formulated problems is solved separately applying the Riemann transformation.

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37

Patanarapeelert, Nichaphat, and Thanin Sitthiwirattham. "Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems." Discrete Dynamics in Nature and Society 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/7895186.

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The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixed-point theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

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38

Shanin, A. V., and A. I. Korolkov. "Diffraction by an impedance strip I. Reducing diffraction problem to Riemann–Hilbert problems." Quarterly Journal of Mechanics and Applied Mathematics 68, no. 3 (2015): 321–39. http://dx.doi.org/10.1093/qjmam/hbv010.

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39

Colţoiu, M., and C. Joiţa. "Some problems related to the Levi problem for Riemann domains over Stein spaces." Complex Variables and Elliptic Equations 65, no. 4 (2019): 713–16. http://dx.doi.org/10.1080/17476933.2019.1608974.

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40

Mamchuev, Murat. "Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order." Mathematics 8, no. 9 (2020): 1475. http://dx.doi.org/10.3390/math8091475.

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We investigate the initial problem for a linear system of ordinary differential equations with constant coefficients and with the Dzhrbashyan–Nersesyan fractional differentiation operator. The existence and uniqueness theorems of the solution of the boundary value problem under the study are proved. The solution is constructed explicitly in terms of the Mittag–Leffler function of the matrix argument. The Dzhrbashyan–Nersesyan operator is a generalization of the Riemann–Liouville, Caputo and Miller–Ross fractional differentiation operators. The obtained results as particular cases contain the results related to the study of initial problems for the systems of ordinary differential equations with Riemann–Liouville, Caputo and Miller–Ross derivatives and the investigated initial problem that generalizes them.

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41

Paliokas, Eugenijus. "MULTIDIMENSIONAL ANALOGUES OF THE RIEMANN–HILBERT BOUNDARY VALUE PROBLEM." Mathematical Modelling and Analysis 12, no. 2 (2007): 205–14. http://dx.doi.org/10.3846/1392-6292.2007.12.205-214.

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42

Ben-Artzi, Matania, and Jiequan Li. "Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem." Numerische Mathematik 106, no. 3 (2007): 369–425. http://dx.doi.org/10.1007/s00211-007-0069-y.

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43

Mohammed, Alip. "The Riemann–Hilbert problem for certain poly domains and its connection to the Riemann problem." Journal of Mathematical Analysis and Applications 343, no. 2 (2008): 706–23. http://dx.doi.org/10.1016/j.jmaa.2008.01.053.

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44

Bachar, Imed, and Hassan Eltayeb. "Hartman-Type and Lyapunov-Type Inequalities for a Fractional Differential Equation with Fractional Boundary Conditions." Discrete Dynamics in Nature and Society 2020 (May 18, 2020): 1–6. http://dx.doi.org/10.1155/2020/8234892.

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We prove Hartman-type and Lyapunov-type inequalities for a class of Riemann–Liouville fractional boundary value problems with fractional boundary conditions. Some applications including a lower bound for the corresponding eigenvalue problem are obtained.

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45

Fokas, A. S., and M. L. Glasser. "The Laplace equation in the exterior of the Hankel contour and novel identities for hypergeometric functions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2157 (2013): 20130081. http://dx.doi.org/10.1098/rspa.2013.0081.

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By using conformal mappings, it is possible to express the solution of certain boundary-value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identity for special functions. A convenient tool for deriving this type of identity is the so-called global relation , which has appeared recently in a wide range of boundary-value problems. As a concrete application, we analyse the Neumann boundary-value problem for the Laplace equation in the exterior of the Hankel contour, which appears in the definition of both the gamma and the Riemann zeta functions. By using the explicit solution of this problem, we derive a number of novel identities involving the hypergeometric function. Also, we point out an interesting connection between the solution of the above Neumann boundary-value problem for a particular set of Neumann data and the Riemann hypothesis.

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46

Zhang, Yu, and Yanyan Zhang. "Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations." Mathematical Modelling of Natural Phenomena 16 (2021): 19. http://dx.doi.org/10.1051/mmnp/2021011.

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The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

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47

TOURIGNY, DAVID S. "RIEMANN ZEROS AND THE INVERSE PHASE PROBLEM." Modern Physics Letters B 27, no. 26 (2013): 1350187. http://dx.doi.org/10.1142/s021798491350187x.

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Finding a universal method of crystal structure solution and proving the Riemann hypothesis are two outstanding challenges in apparently unrelated fields. For centro-symmetric crystals however, a connection arises as the result of a statistical approach to the inverse phase problem. It is shown that parameters of the phase distribution are related to the non-trivial Riemann zeros by a Mellin transform.

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48

Zayko, Yuriy. "Riemann Hypothesis from the Physicist’s Point of View." Transactions on Machine Learning and Artificial Intelligence 8, no. 3 (2020): 01–10. http://dx.doi.org/10.14738/tmlai.83.8217.

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This article presents an attempt to comprehend the evolution of the ideas underlying the physical approach to the proof of one of the problems of the century - the Riemann hypothesis regarding the location of non-trivial zeros of the Riemann zeta function. Various formulations of this hypothesis are presented, which make it possible to clarify its connection with the distribution of primes in the set of natural numbers. A brief overview of the main directions of this approach is given. The probable cause of their failures is indicated - the solution of the problem within the framework of the classical Turing paradigm. A successful proof of the Riemann hypothesis based on the use of a relativistic computation model that allows one to overcome the Turing barrier is presented. This model has been previously applied to solve another problem not computable on the classical Turing machine - the calculation of the sums of divergent series for the Riemann zeta function of the real argument. The possibility of using relativistic computing for the development of artificial intelligence systems is noted.

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Liu, Yujin, and Wenhua Sun. "Analysis of the Stability of the Riemann Problem for a Simplified Model in Magnetogasdynamics." Advances in Mathematical Physics 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/7526413.

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The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin(t>0)in the(x,t)plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. We find that, for some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. For most cases, the Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. While, for some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data.

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L�tkebohmert, W. "Riemann's existence problem for ap-adic field." Inventiones Mathematicae 111, no. 1 (1993): 309–30. http://dx.doi.org/10.1007/bf01231290.

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