Mathematics and Statistics
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Item Approximate conservation laws of partial differential equations with a small parameter(The Royal Society Publishing, 2025-03) Cheviakov, Alexei; Tarayrah, Mahmood Rajih; Yang, ZhengzhengPartial differential equations (PDEs) involving perturbation terms with a small parameter often have less analytical structure, in particular, fewer symmetries and conservation laws, compared to the unperturbed PDEs. For such perturbed PDEs, approximate conservation laws can be consistently defined. The set of approximate conservation laws comprises equivalence classes where members of each class differ by a trivial approximate conservation law. Similar to exact ones, approximate conservation laws can be systematically constructed using the characteristic approach with approximate multipliers. Examples of new approximate conservation laws are presented for perturbed nonlinear heat and wave equations. For approximately variational problems, an analogue of the first Noether’s theorem relates approximate multipliers to evolutionary components of approximate local Lie symmetry generators. The multiplier method used to obtain approximate conservation laws includes the Noether approach and generalizes it to a non-variational system. The procedure to use approximate local symmetries to obtain new approximate conservation laws from known ones, in terms of fluxes and multipliers, is established and illustrated. It is shown that approximate conservation laws lead to potential systems that can be used to obtain new approximate potential symmetries of the given PDE system with a small parameter.Item Internal Lagrangians and spatial-gauge symmetries(International Press of Boston, 2024) Druzhkov, KostyaA direct reformulation of the Hamiltonian formalism in terms of the intrinsic geometry of infinitely prolonged differential equations is obtained. Concepts of spatial equation and spatial-gauge symmetry of a Lagrangian system of equations are introduced. A noncovariant canonical variational principle is proposed and demonstrated using the Maxwell equations as an example. A covariant canonical variational principle is formulated. The results obtained are applicable to any variational equations, including those that do not originate in physics.Item All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces(AMS: Journal of Algebraic Geometry, 2024) Bellamy, Gwyn; Craw, Alastair; Rayan, Steven; Schedler, Travis; Weiß, Hartmut