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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
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 prohibit some observations, that is
Falsifiability
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
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
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
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
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
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
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
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
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
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
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
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
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
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 associative learning process for behavioral modification
Operant conditioning, also called instrumental conditioning, is a learning process in which voluntary behaviors are modified by association with the addition
Operant_conditioning
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
{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
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
Type of functional equation (mathematics)
desolve(y'=k*y,y) Exact differential equation Functional differential equation Initial condition Integral equations Numerical methods for ordinary differential equations
Differential_equation
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
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)
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
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)
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
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
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
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
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
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
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
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
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
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
Property of a dynamical system where solutions near an equilibrium point remain so
t\geq t_{0}} , with state v ∈ R {\displaystyle v\in \mathbb {R} } and initial condition v ( t 0 ) ≥ 0 {\displaystyle v(t_{0})\geq 0} . The function β ∈ C
Lyapunov_stability
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
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)
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
Interpretation of quantum mechanics
the integral condition and the initial condition w + ( x , t 0 ) = w 0 ( x ) {\displaystyle w_{+}(x,t_{0})=w_{0}(x)} or terminal condition w − ( x , t
Stochastic_quantum_mechanics
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 function, denoted exp(x) or e^x
solution to the differential equation dfx/dt(t) = x fx(t), with initial condition fx(0) = 1; it follows that fx(t) = etx for every t in R. Given a Lie
Exponential_function
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
Mathematical concept
guarantee f ( x ) = e x {\displaystyle f(x)=e^{x}} . However, the initial value condition f ( 1 ) = e {\displaystyle f(1)=e} together with the other regularity
Characterizations of the exponential function
Characterizations_of_the_exponential_function
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
Polygonal curve made from right triangles
{\displaystyle f(x+1)=\left(1+{\frac {i}{\sqrt {x+1}}}\right)\cdot f(x),} the initial condition f ( 0 ) = 1 , {\displaystyle f(0)=1,} and monotonicity in both argument
Spiral_of_Theodorus
Type of MOSFET where the gate is electrically isolated
initial condition for the FG voltage. Among the many solutions proposed for the computer simulation, one of the most promising methods is an Initial Transient
Floating-gate_MOSFET
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)
Physical law relating heat loss to temperature difference
The solution of this differential equation, by integration from the initial condition, is Δ T ( t ) = Δ T ( 0 ) e − t / τ . {\displaystyle \Delta T(t)=\Delta
Newton's_law_of_cooling
Physics of the cause–effect relation
needed] The butterfly effect theory proposes: "Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the
Causality_(physics)
Quantum-mechanical framework for simulating molecules and solids
according to the time-dependent Schrödinger equation under a single initial condition, H ^ ( t ) | Ψ ( t ) ⟩ = i ℏ ∂ ∂ t | Ψ ( t ) ⟩ , | Ψ ( 0 ) ⟩
Time-dependent density functional theory
Time-dependent_density_functional_theory
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)
Pattern defining an infinite sequence of numbers
{\displaystyle n!=n\cdot (n-1)!\quad {\text{for}}\quad n>0,} and the initial condition 0 ! = 1. {\displaystyle 0!=1.} This is an example of a linear recurrence
Recurrence_relation
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
Concept in mathematics
for the initial condition, one can add the plot of several solutions % solve the initial value problem symbolically % for different initial conditions
Autonomous system (mathematics)
Autonomous_system_(mathematics)
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
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
Behavioral paradigm in which organisms learn to predict aversive events
the stress hormone, following the initial stressor. Classical conditioning Extinction (psychology) Eyeblink conditioning Fear processing in the brain Infralimbic
Fear_conditioning
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
Type of differential equation
x(t)=ke^{-At}+{\frac {C}{A}}} where k {\displaystyle k} is a constant encoding the initial condition. Eliasmith, Anderson, Chris, Charles (2003). Neural Engineering. Cambridge
Leaky_integrator
Method in Itô calculus
X_{t}=a(X_{t},t)\,\mathrm {d} t+b(X_{t},t)\,\mathrm {d} W_{t},} with initial condition X0 = x0, where Wt denotes the Wiener process, and suppose that we
Euler–Maruyama_method
Partial differential equation in mathematical finance
x^{2}}}} The terminal condition C ( S , T ) = max { S − K , 0 } {\displaystyle C(S,T)=\max\{S-K,0\}} now becomes an initial condition u ( x , 0 ) = u 0 (
Black–Scholes_equation
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
INITIAL CONDITION
INITIAL CONDITION
Boy/Male
American, British, English
Phonetic Name Based on Initials
Boy/Male
English
Phonetic name based on initials.
Girl/Female
Hindu, Indian
Joy; Win
Girl/Female
Hindu, Indian, Tamil
Sweet
Boy/Male
American, Australian
From the Initials J C
Surname or Lastname
English
English : variant of Earl, with the addition of an inorganic initial H-.
Boy/Male
American, British, English
Phonetic Name Based on Initials
Girl/Female
American, British, English
Initials J and C Combined; Based on the Initials J C or an Abbreviation of Jacinda
Girl/Female
Indian
A Planets of Jupiter
Girl/Female
American, Australian, British, English
Initials J and C Combined; Based on the Initials J C or an Abbreviation of Jacinda
Boy/Male
American, British, English
Attractive; From the Initials J C
Boy/Male
American, Australian, British, English
Phonetic Name Based on Initials; Combination of Initials J and D
Girl/Female
Hindu
There is no ending. ne-no tal-ending, The forehead
Girl/Female
Tamil
The initial reality
Girl/Female
Hebrew, Indian, Spanish
Ann
Boy/Male
Hindu, Indian
The Sprout; Initial
Boy/Male
English
Phonetic name based on initials.
Surname or Lastname
English
English : variant of Osmer with an inorganic initial H-.
Girl/Female
Spanish
Grace.
Girl/Female
Indian
The initial reality
INITIAL CONDITION
INITIAL CONDITION
Girl/Female
Arabic, Muslim
To Become Quiet
Girl/Female
Hindu
Snow
Boy/Male
Tamil
Lord Murugan
Boy/Male
Indian
Heaven, Sky
Girl/Female
Biblical
Sitting together.
Boy/Male
Tamil
Well bred
Male
German
Contracted form of Old High German Willahelm, WILHELM means "will-helmet."Â
Boy/Male
Hindu, Indian, Telugu
Power of Sun Rise
Surname or Lastname
English
English : variant of Hutchens.
Boy/Male
Hebrew
Lofty; exalted; high mountain.
INITIAL CONDITION
INITIAL CONDITION
INITIAL CONDITION
INITIAL CONDITION
INITIAL CONDITION
a.
Hostile; inimical.
a.
Pertaining to an embryo, or the initial state of any organ; embryonic.
v. t.
To initiate; to introduce favorably.
p. pr. & vb. n.
of Initial
a.
Of or pertaining to the beginning; marking the commencement; incipient; commencing; as, the initial symptoms of a disease.
p. pr. & vb. n.
of Instill
v. t.
To print in capital letters, or with an initial capital.
a.
Inimical; hurtful.
imp. & p. p.
of Instill
a.
Beginning; predisposing; exciting; initial.
a.
Inimical; unfriendly.
p. pr. & vb. n.
of Initiate
adv.
In an inimical manner.
a.
Placed at the beginning; standing at the head, as of a list or series; as, the initial letters of a name.
imp. & p. p.
of Initial
n.
The first letter of a word or a name.
v. t.
To begin; to initiate.
adv.
In an initial or incipient manner or degree; at the beginning.
imp. & p. p.
of Initiate
v. t.
To put an initial to; to mark with an initial of initials.