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Parameter in differential equations and dynamical systems
In mathematics and particularly in dynamical systems, an initial condition is the initial value (often at time t = 0 {\displaystyle t=0} ) of a differential
Initial_condition
Idea that small causes can have large effects
reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different
Butterfly_effect
Partial differential equation
hyperbolic equation. The solution to the equation and along with the initial condition ∂ u ∂ t + u ∂ u ∂ x = 0 , u ( x , 0 ) = f ( x ) {\displaystyle {\frac
Burgers'_equation
Type of calculus problem
In calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown
Initial_value_problem
Existence and uniqueness of solutions to initial value problems
is y(t) = 0, which is obtained for the initial condition y(0) = 0. Beginning with any other initial condition y(0) = y0 ≠ 0, the solution y ( t ) = y
Picard–Lindelöf_theorem
Millennium Prize Problem
^{3}} , which needs extra conditions on the growth behavior of the initial condition and the solutions. In order to rule out the problems at infinity,
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Linearization technique for nonlinear differential systems
equilibrium point, referred to as Type A, and linearization around the initial condition, referred to as Type B. Let q {\displaystyle \mathbf {q} } be an equilibrium
Carleman_linearization
Multiple simulation method for weather forecasting
when making an ensemble weather forecast: initial condition uncertainty and model uncertainty. Initial condition uncertainty arises due to errors in the
Ensemble_forecasting
Property of a statement that can be logically contradicted
In the case of a theory, falsifiability requires that, given an initial condition, the theory must theoretically exclude some observations, that is
Falsifiability
Describing pressure difference over an interface in fluid mechanics
by the capillary length. The solution of the equation requires an initial condition for position, and the gradient of the surface at the start point.
Young–Laplace_equation
Set of different models of the climate system
carbon cycle, atmospheric chemistry, land use etc. can be perturbed. Initial condition ensembles involve the same model in terms of the same atmospheric
Climate_ensemble
Field of mathematics and science based on non-linear systems and initial conditions
Tornado in Texas?. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability
Chaos_theory
Proposal concerning the state of the universe prior to the Planck epoch
first proposed it in 1983. The concept can also be considered as an initial condition for models of quantum cosmology. The Hartle-Hawking proposal includes
Hartle–Hawking_proposal
Differential equation containing derivatives with respect to only one variable
any solution that satisfies this initial condition is a restriction of the solution that satisfies this initial condition with domain I max {\displaystyle
Ordinary differential equation
Ordinary_differential_equation
Analog of the continuous Laplace operator
Transform the initial condition into the coordinate system % of the eigenvectors for t = 0:0.05:5 % Loop through times and decay each initial component Phi
Discrete_Laplace_operator
Partial differential equation describing the evolution of temperature in a region
of the heat equation is a solution that corresponds to the initial condition of an initial point source of heat at a known position. These can be used
Heat_equation
African in the 16th-century Portuguese court
received constant racist abuse. His "inferior condition", undoubtedly a result of his initial condition as a slave, was frequently brought up by other
João_de_Sá_Panasco
Equation in mathematical physics
{\displaystyle F(\pm 1)=0} . Usually, one has the following initial condition with the Neumann boundary condition { v ( x , 0 ) = v 0 ( x ) , Ω × { 0 } ∂ n v = 0
Allen–Cahn_equation
Doubling map on the unit interval
on initial conditions—the mapping from the truncated initial condition has deviated exponentially from the mapping from the true initial condition. And
Dyadic_transformation
Statement on solutions to ordinary differential equations
′ ( t ) = f ( t , y ( t ) ) {\displaystyle y'(t)=f(t,y(t))} with initial condition y ( t 0 ) = y 0 , {\displaystyle y(t_{0})=y_{0},} where the function
Carathéodory's existence theorem
Carathéodory's_existence_theorem
Differential equations involving stochastic processes
condition and let F : Ω → U {\displaystyle F:\Omega \to U} be some initial condition, meaning it is a measurable function with respect to the initial
Stochastic differential equation
Stochastic_differential_equation
Type of thermodynamic potential
except such as at the close of the processes are left in their initial condition. The initial state of the body, according to Gibbs, is supposed to be such
Gibbs_free_energy
Aspect of learning procedure
Classical conditioning (also respondent conditioning and Pavlovian conditioning) is a behavioral procedure in which a biologically potent stimulus (e
Classical_conditioning
Pendulum with another pendulum attached to its end
sensitive dependence on initial conditions. The sensitivity of a dynamical system's trajectory at a particular initial condition can be quantified by the
Double_pendulum
Simple polynomial map exhibiting chaotic behavior
for the parameter r and performing calculations starting from an initial condition x 0 {\displaystyle x_{0}} , the sequence x 0 {\displaystyle x_{0}}
Logistic_map
Partial differential equation
x^{2}}},} which is the simplest form of a diffusion equation. If the initial condition is p ( x , 0 ) = δ ( x ) {\displaystyle p(x,0)=\delta (x)} , the solution
Fokker–Planck_equation
Formula relating stochastic processes to partial differential equations
t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-V(x)w} with initial condition w(x, 0) = δ(x). The Feynman–Kac formula can also be used to evaluate
Feynman–Kac_formula
Phenomenon of isolated many-body quantum systems not reaching thermal equilibrium
failing to reach thermal equilibrium, and retaining a memory of its initial condition in local observables for infinite times. Textbook quantum statistical
Many-body_localization
Human disease from a bacterial infection of the skin
prophylactic antibiotics are sometimes used after resolution of the initial condition. The disease prognosis includes: Spread of infection to other areas
Erysipelas
Principle in systems theory
systems, a direct cause-and-effect relationship exists between the initial condition and the final state of the system: When a computer's 'on' switch is
Equifinality
Mathematical condition for convergence
In mathematics, the convergence condition by Courant–Friedrichs–Lewy (CFL) is a necessary condition for convergence while solving certain partial differential
Courant–Friedrichs–Lewy condition
Courant–Friedrichs–Lewy_condition
Markovian quantum master equation for density matrices (mixed states)
property of being trace-preserving and completely positive for any initial condition. The Schrödinger equation or, actually, the von Neumann equation,
Lindbladian
Theorem in electrical circuit analysis
widely used to make circuit analysis simpler and to study a circuit's initial-condition and steady-state response. Thévenin's theorem can be used to convert
Thévenin's_theorem
Technique for solving differential equations
t}{L^{2}}}\right),} where Dn are coefficients determined by initial condition. Given the initial condition we can get f ( x ) = ∑ n = 1 ∞ D n sin n π x L . {\displaystyle
Separation_of_variables
Space of all possible states that a system can take
particular initial condition, located in the full phase space that represents the set of states compatible with starting from any initial condition. As a whole
Phase_space
Type of differential equation
{\displaystyle {\frac {d}{dt}}x(t)=f(x(t),x(t-\tau ))} with given initial condition ϕ : [ − τ , 0 ] → R n {\displaystyle \phi \colon [-\tau ,0]\to \mathbb
Delay_differential_equation
Approaches for approximating solutions to differential equations
{\displaystyle {\frac {dy}{dt}}=-y^{2},\ t\in [0,a]\quad \quad (2)} with the initial condition y ( 0 ) = 1. {\displaystyle y(0)=1.} This problem has the exact solution
Explicit_and_implicit_methods
DC circuit analysis technique
widely used for circuit analysis simplification and to study circuit's initial-condition and steady-state response. Norton's theorem was independently derived
Norton's_theorem
Dimensionless quantity in fluid dynamics
point, then the value of M from the subsonic equation is used as the initial condition for fixed point iteration of the supersonic equation, which usually
Mach_number
Theory in computer science
liveness properties. For example, CTL can specify that when some initial condition is satisfied (e.g., all program variables are positive or no cars
Computation_tree_logic
Existence and uniqueness theorem for certain partial differential equations
f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)} with initial condition f ( x ) = 0 {\displaystyle f(x)=0} on the hypersurface x n = 0 {\displaystyle
Cauchy–Kovalevskaya_theorem
Model-free reinforcement learning algorithm
implicitly assumes an initial condition before the first update occurs. High initial values, also known as "optimistic initial conditions", can encourage
Q-learning
Probability distribution
fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition, and in simplified mathematical models of the behavior of the longest
Tracy–Widom_distribution
Observational basis of thermodynamics
and exactly predictable from the macroscopic specification of the initial condition of the process. This is why entropy increases in natural processes
Laws_of_thermodynamics
Problem in control theory
this minimum number of steps will be at most N (depending on the initial condition), provided that the system is null controllable (that it can be brought
Dead-beat_control
Machine learning algorithm
produces no unique solution, but an aggregate of many solutions. At initial condition, the nodes carry a label that denotes the community they belong to
Label_propagation_algorithm
universality class - the PNG droplet - converges under suitable scaling and initial condition to the Airy2 process and that it is a stationary process with almost
Airy_process
Exponential growth based on a constant rate
equation: d P d t = r P {\displaystyle {\frac {dP}{dt}}=rP} with initial condition: P(0)= P0 This model is often referred to as the exponential law.
Malthusian_growth_model
Alert posture used by the United States Armed Forces
The defense readiness condition (DEFCON) is an alert state used by the United States Armed Forces. The DEFCON system was developed in 1959 by the Joint
DEFCON
Quality of a system having two stable equilibrium states
{\displaystyle y(t)} over time depends on the initial condition y ( 0 ) {\displaystyle y(0)} . If the initial condition is positive ( y ( 0 ) > 0 {\displaystyle
Bistability
Visual representation of an iterated function
a cobweb plot, it is possible to infer the long-term status of an initial condition under repeated application of a map. For a given iterated function
Cobweb_plot
Solution to a stochastic differential equation
with its Borel σ {\displaystyle \sigma } -algebra, such that: 1. (Initial Condition) The process starts at ξ {\displaystyle \xi } at time τ {\displaystyle
Diffusion_process
Systems that do not produce or consume energy
pairs {v(·), i(·)} with the fixed initial state x (e.g., all voltage–current trajectories for a given initial condition of the system). A system is considered
Passivity_(engineering)
Fluid dynamics principle regarding bodies with sharp corners
The Kutta condition is a principle in steady-flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies with sharp corners, such
Kutta_condition
Mathematical series
the ordinary differential equation (1 + x)u′(x) − αu(x) = 0 with initial condition u(0) = 1. The unique solution of this problem is the function u(x)
Binomial_series
Short "burst" or "envelope" of restricted wave action that travels as a unit
Schrödinger equation (with 2Δx, m, and ħ set equal to one) satisfying the initial condition ψ ( x , 0 ) = 2 / π 4 exp ( − x 2 + i k 0 x ) , {\displaystyle \psi
Wave_packet
Chaotic model of atmospheric convection
famously said to resemble a butterfly. The system's extreme sensitivity to initial conditions gave rise to the popular concept of the butterfly effect—the
Lorenz_system
Unchanging property for all objects of a class
array are valid values. The Default_Initial_Condition of the Stack type, by specifying an empty stack, ensures the initial truth of the invariant, and Push
Class_invariant
Principle suggesting that time travel paradoxes are inherently impossible
extensions for the same initial condition—indeed, Echeverria et al. found an infinite number of consistent extensions for every initial trajectory they analyzed—can
Novikov self-consistency principle
Novikov_self-consistency_principle
Patient is a carrier for a disease or infection but experiences no symptoms
(as incidental findings) while treating other diseases. Knowing that a condition is asymptomatic is important because: It may be contagious, and the contribution
Asymptomatic
Type of pattern that returns to its original configuration after a number of steps
number of generations it takes before the pattern returns to its initial condition is called the period of the oscillator. An oscillator with a period
Oscillator (cellular automaton)
Oscillator_(cellular_automaton)
{sgn}(X_{t})\,\mathrm {d} B_{t},} driven by canonical Brownian motion B, with initial condition X0 = 0, where sgn denotes the sign function sgn ( x ) = { + 1 ,
Tanaka_equation
Coarsest topology making certain functions continuous
is the initial topology induced by some family of maps on X . {\displaystyle X.} This section gives a sufficient (but not necessary) condition. A family
Initial_topology
Economic concept
elasticities add to more than 1, which is the Marshall-Lerner condition. If the initial trade surplus is positive so X - eM > 0, the sum of the magnitudes
Marshall–Lerner_condition
Physical theory of the cosmos
where they showed that mathematical singularities were an inevitable initial condition of relativistic models of the Big Bang. Then, from the 1970s to the
Big_Bang
The Initial D (頭文字（イニシャル）D) video game series, known in Japan as Initial D Arcade Stage, is an arcade racing game series developed by Sega, based on the
Initial_D_(video_game_series)
Class of numerical techniques
(boundary condition) U ( x , 0 ) = U 0 ( x ) (initial condition) {\displaystyle {\begin{cases}U_{t}=U_{xx}\\U(0,t)=U(1,t)=0&{\text{(boundary condition)}}\\U(x
Finite_difference_method
Stochastic differential equation
{\displaystyle dX_{t}=\Phi (X_{t-})dL_{t}} exists for each deterministic initial condition x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} and yields a Feller
Infinitesimal generator (stochastic processes)
Infinitesimal_generator_(stochastic_processes)
Boundary-value problem in differential equations
In mathematics, a Cauchy (French: [koʃi]) boundary condition augments an ordinary differential equation or a partial differential equation with conditions
Cauchy_boundary_condition
Mathematical way of attaining a desired output from a dynamic system
}}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),} and the initial condition x ( t 0 ) = x 0 {\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}
Optimal_control
the height function to the KPZ fixed point and only depend on the initial condition. Matetski-Quastel-Remenik constructed the KPZ fixed point for the
KPZ_fixed_point
solutions to the dispersion relation may be complex-valued. Now, an initial condition y ( x , 0 ) {\displaystyle y(x,0)} can be written as a superposition
Briggs–Bers_criterion
Providing boundary conditions for Maxwell's equations uniquely fixes a solution
the source distribution is outside of the volume specified in the initial condition. One example is that the field outside a uniformly charged sphere
Electromagnetism uniqueness theorem
Electromagnetism_uniqueness_theorem
Describes state evolution of a linear system
matrix functions; and x 0 {\displaystyle \mathbf {x} _{0}} is the initial condition at t 0 {\displaystyle t_{0}} . Using the state-transition matrix
State-transition_matrix
Concept in dynamical systems
{\mathcal {O}}(\varepsilon ^{2})} terms. Uniformity with respect to the initial condition x 0 {\displaystyle x_{0}} : if we vary x 0 {\displaystyle x_{0}} this
Method_of_averaging
Machine learning algorithm
implicitly assumes an initial condition before the first update occurs. A high (infinite) initial value, also known as "optimistic initial conditions", can
State–action–reward–state–action
State–action–reward–state–action
Control flow construct for executing code repeatedly
pre-test loop. They are commonly formatted in manner similar to for initialization, condition, increment do body repeat This syntax came from B and was originally
Loop_(statement)
Changing of one's behaviour to assume a role
starts to define "I" and separate him or herself from an adult is the initial condition for and the result of role play. There are also experiments that found
Role-playing
Continuously computed dead reckoning
orientation of the system in the inertial reference frame as the initial condition and integrating the angular displacement, the system's current orientation
Inertial_navigation_system
Mathematics of general relativity
field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of
Energy_condition
Nowhere analytic, infinitely differentiable function
Fabius (1966). This function satisfies the initial condition f ( 0 ) = 0 {\displaystyle f(0)=0} , the symmetry condition f ( 1 − x ) = 1 − f ( x ) {\displaystyle
Fabius_function
Any medical condition that affects the integumentary system
A skin condition, also known as a cutaneous condition, is any medical condition that affects the integumentary system—the organ system that encloses the
Skin_condition
Numerical problem-solving method
for solving initial value problems. This problem, in which an ordinary differential equation is given together with an initial condition, plays a central
One-step_method
c 0 {\displaystyle c_{0}} is an arbitrary shift dependent on the initial condition x ( 0 ) {\displaystyle x(0)} (specifically by the formula c 0 = arctan
Quadratic_integrate_and_fire
{\displaystyle \sigma \in {\mathcal {K_{\infty }}}} so that for any initial condition x 0 ∈ R n {\displaystyle x_{0}\in {\mathbb {R} }^{n}} and for any
Comparison_function
Mathematical method
c , 0 ≤ t ≤ d {\displaystyle b\leq x\leq c,\;0\leq t\leq d} with initial condition u ( x , 0 ) = u 0 ( x ) {\displaystyle u(x,0)=u_{0}(x)\,} and the
Lax–Friedrichs_method
Stochastic process modeling random walk with friction
Fokker–Planck equation with initial condition P ( x , t 0 ) = δ ( x − x 0 ) {\displaystyle P(x,t_{0})=\delta (x-x_{0})} . Conditioned on a particular value
Ornstein–Uhlenbeck_process
System with an infinite-dimensional state-space
formulas as illustrated below on the examples given above. Setting the initial condition w 0 {\displaystyle w_{0}} equal to zero and denoting Laplace transforms
Distributed_parameter_system
Special type of continuous-time Markov process
_{k+1}p_{k+1}(t)-(\lambda _{k}+\mu _{k})p_{k}(t),k=1,2,\ldots ,\,} and the initial condition ∑ k = 0 ∞ p k ( t ) = 1. {\displaystyle \sum _{k=0}^{\infty }p_{k}(t)=1
Birth–death_process
Belgian mathematician
is determined by the initial condition N(0). The solution can also be written as a weighted harmonic mean of the initial condition and the carrying capacity
Pierre_François_Verhulst
Memoryless property of a stochastic process
any process to one which includes the complete history from a given initial condition and it would be made to be Markovian. But the state space would be
Markov_property
Nonlinear time delay differential equation
{k}{c}}+f(y_{0})e^{-ct}} where y 0 {\displaystyle y_{0}} is any initial condition for the initial value problem. However, the above model assumes that variations
Mackey–Glass_equations
Part of mathematics that addresses the stability of solutions
understood, it is natural to ask next whether a small change in the initial condition will lead to similar behavior. Stability theory addresses the following
Stability_theory
System of moving vectors in differential geometry
differential equations, which has a unique solution for any initial condition given by the second condition (for instance, by the Picard–Lindelöf theorem). The
Parallel_transport
Elementary cellular automaton
program, and infinitely repeating, the patterns of 0s and 1s at the initial condition can be represented by an infinitely repeating string. Each production
Rule_110
Type of problem involving ODEs or PDEs
then it is a Cauchy boundary condition. A type 0 boundary condition has no physical boundary. Aside from the boundary condition, boundary value problems are
Boundary_value_problem
Sound change
word to have an allophone conditioned by what the first consonant or vowel is. When this conditioning element is lost in initial dropping, these allophones
Initial_dropping
Method to calculate trajectory calculations for spacecraft
{\displaystyle \mathbf {v} _{\infty }} . This velocity is then used as the initial condition for the heliocentric trajectory: v helio = v planet + v ∞ {\displaystyle
Patched_conic_approximation
Motion of particles in a fluid
and one might say that the variable x depends on the time t and the initial condition x = x0. Examples are given below. In the case of a flow of a vector
Flow_(mathematics)
Type of stochastic process
ergodic properties so that after initially flowing away from its initial condition it would generate samples from the posterior probability model. Jump
Jump_diffusion
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