Methods of Mathematical Physics

1947

Citations in the corpus

• Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems (2018, background): “Often such tools are precisely known, usually coming in the form of differential equations that are derived from first physical principles, such as the conservation of energy, mass, and momentum [1].”
• Nodal Domains of Maass Forms I (2012, methodology): “According to Courant’s general nodal domain theorem [CH53] and (1), one has asymptotically”
• Regularization Networks and Support Vector Machines (2000, background): “The mathematical details (such as the convergence or not of certain series) can be found in the theory of integral equations [45,20,23].”
• Analogue Models of and for Gravity (2001, background): “Roughly speaking: in any hyperbolic system of differential equations (no matter how derived) there are by definition wave-like solutions [32, 33].”
• A polynomial chaos expansion in dependent random variables (2018, unknown): “known as the Parseval identity [19] for a multivariate orthogonal system, for every random variable y(X) ∈ L2(Ω,F , P).”
• Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation (2014, unknown): “[13]) and the condition on γ , we see that ∑∞ j=1 λ d 2 +2ε−2γ−2 j < ∞.”
• The hot spots problem in planar domains with one hole (2004, methodology): “By the Courant Nodal Line Theorem ([CH]), the nodal line divides D into two connected compo- nents.”
• Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems (2005, background): “The first term on the right hand side of the hierarchy describes the kinetic energy of the first k particles; the second term is associated with the interactions among the first k particles, and the last term corresponds to interactions between the first k particles and the other N − k particles.”
• ADVANCED SPECTRAL METHODS FOR CLIMATIC TIME SERIES (2002, methodology): “[ 221 ] The rigorous mathematical foundation for the spectral representation of random fields is analogous to that of orthonormal bases for the solutions of the classical equations of mathematical physics [ Courant and Hilbert , 1953, 1962].”
• Coupled oscillators as models of quintom dark energy (2008, unknown): “…described by metric (1), with the φ and σ fields as material sources, the equations of motion are the following: H2 + k a2 = κ 6 [ σ̇2 − φ̇2 + 2V (φ, σ) ] (4) Ḣ + H2 = κ 3 [ −σ̇2 + φ̇2 + V (φ, σ) ] (5) φ̈ + 3Hφ̇ − ∂V ∂φ = 0 (6) σ̈ + 3Hσ̇ + ∂V ∂σ = 0, (7) where κ = 8πG and dots…”
• Observation and Control for Operator Semigroups (2009, unknown): “If the norm of a Banach space is induced by an inner product, then the space is also called a Hilbert space.”
• Optimization formulations for the maximum nonlinear buckling load of composite structures (2011, background): “…load factor sensitivities may be determined by dλj dai = φTj ( dK0 dai + λj dKσ dai ) φj (31) where the eigenvalue problem in (2) has been differentiatedwith respect to any design variable,ai, i = 1, . . . , I, assuming thatλj is simple, see e.g. Courant and Hilbert (1953); Wittrick (1962).”