Alphabeta MathNumerical Differential Equations and Stochastic Dynamics
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
13 pagesThis page supplies exact dyadic Brownian-increment refinement, the definition of a predictable stopping-time grid, an adaptive Euler–Maruyama update with an explicit…
2 definitions, 2 algorithms, 5 lemmas, 3 theoremsExamples & counterexamples →This page separates ideal independent fair bits from deterministic finite-state streams.
6 algorithms, 3 lemmas, 8 theorems, 1 corollaryExamples & counterexamples →This page uses column probability vectors: Q ij is the rate from source j to destination i, off-diagonal entries are nonnegative, and columns sum to zero.
4 definitions, 2 algorithms, 3 lemmas, 4 theoremsExamples & counterexamples →This page establishes the weak-kernel power integral, defines the Caputo derivative of order 0<α<1 on absolutely continuous functions under Countable Choice (which supplies…
1 definition, 2 algorithms, 9 lemmas, 4 theoremsExamples & counterexamples →- Geometric numerical integration13 results
This page extends the published global leapfrog maps by stating their local admitted domain when a potential is defined on a proper open configuration set.
1 definition, 2 algorithms, 6 lemmas, 1 proposition, 3 theoremsExamples & counterexamples → The page defines underdamped Langevin dynamics and the exact BAOAB B/2–A/2–O–A/2–B/2 update, proves the OU momentum covariance and Gibbs stationarity under explicit…
1 definition, 1 algorithm, 4 lemmas, 4 theoremsExamples & counterexamples →This page defines constant-step linear multistep formulas, their characteristic polynomials, local residual, order, consistency, and the root condition.
1 definition, 2 algorithms, 5 lemmas, 2 theoremsExamples & counterexamples →This page treats the one-dimensional problem -u''=g on (0,1) with exact homogeneous Dirichlet values at both endpoints.
1 algorithm, 2 lemmas, 1 theoremExamples & counterexamples →This page develops the scalar Milstein method from its iterated Itᵒ integral and proves strong order one using conditional local-defect bounds and a stochastic local-to-global recurrence.
2 definitions, 6 lemmas, 3 theoremsExamples & counterexamples →- Nonlinear Boundary-Value Solvers18 results
This page defines square nonlinear two-point boundary-value problems and exact shooting residuals, then separates exact flow sensitivity from the derivative and error contract of a numerical IVP evaluator.
4 definitions, 3 algorithms, 2 lemmas, 5 propositions, 2 theorems, 2 corollariesExamples & counterexamples → Define the local defect and uniform order of a one-step method, together with the one-step Lipschitz bound used to compare inputs in a compact time-state set.
1 definition, 1 lemma, 1 theoremExamples & counterexamples →This page develops finite-budget pseudo-arclength continuation for F:ℝⁿ⁺¹→ℝⁿ on an explicit C² tube, with C³ regularity for the simple-fold curvature expansion.
1 definition, 1 algorithm, 4 lemmas, 4 theoremsExamples & counterexamples →Define finite-channel reaction propensity systems and their marked-event convention, then give complete event laws for Gillespie direct simulation, internal Poisson clocks, and bounded-rate thinning.
1 definition, 4 algorithms, 1 lemma, 5 theoremsExamples & counterexamples →
Level 1
4 pagesThis page certifies finite-state stationary laws and transient hitting statistics.
1 definition, 2 algorithms, 3 lemmas, 8 theorems, 1 corollaryExamples & counterexamples →For a fixed finite interval [0,T] and uniform step h=T/N, the global ODE bound propagates a uniform one-step defect through a stability factor 1+Lh.
1 lemma, 4 theorems, 1 corollaryExamples & counterexamples →This page develops the local equilibrium geometry for scalar folds, transcritical points, and symmetry-preserving pitchforks, then proves the simple Hopf branch and its…
3 definitions, 1 algorithm, 5 lemmas, 5 theoremsExamples & counterexamples →Explicit Poisson tau leaping freezes each channel propensity over a step and draws independent Poisson counts.
1 definition, 2 algorithms, 4 lemmas, 3 theorems, 1 corollaryExamples & counterexamples →
Level 2
9 pagesThis page defines finite discrete coagulation-fragmentation events, conservative fixed-pivot redistribution, and a reported overflow-mass reservoir.
6 definitions, 2 algorithms, 14 lemmas, 12 theoremsExamples & counterexamples →For a fixed delay τ 0, this page defines retarded equations with continuous initial histories and globally state-Lipschitz right-hand sides.
3 definitions, 3 algorithms, 7 lemmas, 4 theoremsExamples & counterexamples →For a semilinear system y'=Ay+g(t,y) with a fixed finite matrix A, these methods integrate the linear part exactly and approximate only the nonlinear forcing.
1 definition, 1 algorithm, 7 lemmas, 1 theoremExamples & counterexamples →This page defines backward Euler and implicit Runge–Kutta residual conventions, local stage branches, and finite Newton or chord solves with explicit factorization costs.
6 definitions, 2 algorithms, 2 lemmas, 2 propositions, 3 theorems, 1 corollaryExamples & counterexamples →The frozen-slow explicit Euler scheme advances a designated slow block once per macro step and advances the fast block through explicit microsteps while holding the slow value fixed.
2 algorithms, 3 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →This page fixes an autonomous phase by adding the unknown period and one scalar phase condition to the periodic endpoint equations.
4 definitions, 4 algorithms, 7 lemmas, 1 proposition, 7 theoremsExamples & counterexamples →For a finite closed system of nonnegative compartments, each production rate is paired with the equal destruction rate of its donor.
2 definitions, 3 algorithms, 3 lemmas, 6 theoremsExamples & counterexamples →This page develops two related but distinct structured-population models.
3 definitions, 4 algorithms, 9 theoremsExamples & counterexamples →- Runge–Kutta Methods and Order13 results
An explicit Runge–Kutta tableau defines ordered stage evaluations and a one-step output, with explicit domain and evaluation-failure status.
2 definitions, 4 algorithms, 2 lemmas, 4 propositions, 1 theoremExamples & counterexamples →
Level 3
9 pagesAn embedded explicit Runge–Kutta pair shares stages and reports the difference between its high and low outputs.
4 definitions, 2 algorithms, 2 lemmas, 1 proposition, 7 theorems, 1 remarkExamples & counterexamples →- Constrained dynamics shake and rattle13 results
This page treats a constant SPD mass matrix and a smooth holonomic constraint on compact regular configuration charts.
2 definitions, 1 algorithm, 4 lemmas, 6 theoremsExamples & counterexamples → For distinct nodes, the page defines the collocation tableau from cardinal polynomials and derives an explicit-left-Euler spectral deferred-correction sweep.
1 definition, 1 algorithm, 9 lemmas, 1 proposition, 8 theoremsExamples & counterexamples →This page treats semi-explicit DAEs x'=f(t,x,z), g(t,x,z)=0 on a selected regular index-one branch.
2 definitions, 3 algorithms, 3 lemmas, 4 theoremsExamples & counterexamples →- Imex and operator splitting methods17 results
This page studies finite-dimensional autonomous additive systems y'=a(y)+b(y).
4 definitions, 1 algorithm, 6 lemmas, 1 proposition, 5 theoremsExamples & counterexamples → - Parallel in time parareal and mgrit14 results
Parareal corrects a serial coarse trajectory with parallel fine propagations.
2 definitions, 3 algorithms, 5 lemmas, 3 theorems, 1 corollaryExamples & counterexamples → - Rosenbrock and linearly implicit methods17 results
The one-stage method solves (I-γ hJ̃)k=f(t,y) and updates y+hk.
1 definition, 3 algorithms, 6 lemmas, 2 propositions, 5 theoremsExamples & counterexamples → This page defines scalar absolute-stability regions for one-step and constant-step multistep methods.
2 definitions, 1 lemma, 1 proposition, 4 theoremsExamples & counterexamples →- Stable sde schemes for superlinear drift13 results
This page develops three specific methods for SDEs outside the globally Lipschitz drift setting.
1 definition, 3 algorithms, 3 lemmas, 6 theoremsExamples & counterexamples →
Level 4
1 pageA hybrid trajectory follows a smooth ODE in each mode, changes mode at an oriented guard crossing, and applies a reset.
2 definitions, 1 algorithm, 4 lemmas, 3 theoremsExamples & counterexamples →
Level 5
1 pageThis page treats two distinct finite-dimensional contact models.
1 definition, 3 algorithms, 7 lemmas, 4 theoremsExamples & counterexamples →