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Computational Complexity
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
Computational complexity theory presumes that the problem to be analyzed is well-posed. But in practice, it is often the fuzziness and open-endedness of a problem that makes it hard. If you can categorize or formalize all the possibilities, you may be able in principle to capture this hardness as computational hardness.VT1 Summary: Computational complexity provides other classifications besides NP-hardness. Some problems, including program and model verification, are unsolvable. This partly explains the scarcity of research results and tools to aid in formal modeling. The P-space-hard problems are a step up in complexity from the NP-complete. These include several fundamental problems involving uncertainty, such as stochastic scheduling, queueing networks, and Markov decision processes. The class co-NP helps explain why optimization software spends so much time verifying optimality after it has found the optimal solution. Finally, a problem may be hard in a non-computational sense.
An Introductory Review of Quantum Mechanics
Published in Ramaswamy Jagannathan, Sameen Ahmed Khan, Quantum Mechanics of Charged Particle Beam Optics, 2019
Ramaswamy Jagannathan, Sameen Ahmed Khan
acting on functions in P→-space. Note that the Heisenberg commutation relations are valid in the same form (3.27) in this representation also. One can work in any representation of the operators which preserves the Heisenberg canonical commutation relations. In the momentum representation, the wave function would be a function of momentum of the particle, say Ψ˜(p→,t) such that |Ψ˜(p→,t)|2 gives the probability of finding the particle to have the momentum p→, at time t, if its momentum is measured. It would be useful to note down the following commutation relations:
SRv6 Reliability
Published in Zhenbin Li, Zhibo Hu, Cheng Li, SRv6 Network Programming, 2021
Zhenbin Li, Zhibo Hu, Cheng Li
Figure 6.7 illustrates how SRv6 TI-LFA works. In this figure, the shortest path from node A to node F is A→B→E→F, and node B needs to compute a backup path to node F as follows: 1. Excludes the primary next hop (B–E link) and computes the post-convergence shortest path B→C→D→E→F.Computes the P-space. In this case, nodes B and C are in the P-space.Computes the Q-space. In this case, nodes D, E, and F are in the Q-space.Computes a backup path expressed using a repair segment list. Any path can be represented using the source node→P node→Q node→destination node format. The path from the source node to the P node and the path from the Q node to the destination node are both loop-free. If a PQ node exists, traffic can be directly forwarded to the PQ node, and the entire path is loop-free. In this case, the repair segment list can be composed of the PQ node’s End SID. If no PQ node exists, a loop-free forwarding path from the P node to the Q node needs to be specified, and the repair segment list from the P node to the Q node may be a combination of End and End.X SIDs. In this example, the repair segment list from node B’s furthest P node (C) to its nearest Q node (D) can be End.X SID 3::1.
An extended modified cam clay model for improved accuracy at low and high-end stress levels
Published in Marine Georesources & Geotechnology, 2020
M. D. Liu, D.W. Airey, B. Indraratna, Z. Zhuang, S. Horpibulsuk
The simulations of 31 tests, including the 19 tests reported in Sections 3.1, 3.2 and 3.3, are made with one set of values for soil parameters as listed in Table 1. The soil paths in the e–lnp′ space for all the simulations are shown in Figure 16. Horizontal paths indicate undrained tests. The tests with isotropic virgin yielding states start from ICL. All the tests end at a critical state of deformation, thus on the CSL. The CSL thus obtained is shown in Figure 16. CSL controls the final strength of soil, irrespective of the initial states of the soil, that is, values of OCR or soil structures, and thus it also controls the stiffness of soil deformation because the stiffness of soil deformation is strongly dependent on the degree of mobilization of its strength. Therefore, an improvement in predicting the CSL of the soil has great practical value. The following features of CSL in the e–lnp′ space are observed.
High order parallelisation of an unstructured grid, discontinuous-Galerkin finite element solver for the Boltzmann–BGK equation
Published in International Journal of Computational Fluid Dynamics, 2019
B. Evans, M. Hanna, M. Dawson, M. Mesiti
The condition that must be enforced at a solid wall is zero mass flux across the boundary. In a kinetic theory description, this is expressed as where and is the p-space domain boundary. This condition is ensured by an appropriate modelling of molecular collisions with the wall. We make the assumption that a certain fraction, α, of molecules are adsorbed by the wall and remitted in thermodynamic equilibrium with the wall (diffuse reflection). The remaining fraction, , are not adsorbed by the wall and simply reflect directly back into the domain (specular reflection). The term α is known as the ‘adsorption coefficient’. The distribution function of the net reflected flux of molecules is, therefore, constructed as where , and is the outward facing unit normal at the wall. If is the wall temperature, then is determined as The parameter η is used to enforce the condition in Equation (8), i.e. it is used to ensure conservation of mass at the wall, which implies that For further details on the application of this boundary condition and for a proof of its mass conserving properties, the reader is referred to Evans, Morgan, and Hassan (2011).
Dilatancy behaviour of rockfill materials and its description
Published in European Journal of Environmental and Civil Engineering, 2022
Wan-Li Guo, Zheng-Yin Cai, Ying-Li Wu, Chen Zhang, Jun-Jie Wang
The gradient of the critical state line (CSL) in the q-p space is denoted as the critical state stress ratio Mc, and Mc will be regarded as a constant when the CSL in q-p space is linear. The CSL of HPR in the p-q space (as shown in Figure 3(a)) can be expressed by the linear function as qc = Mcpc, where the constant Mc = 1.722, and the fitting correlation coefficient R2 is 0.997.