将发育现象正式化为连续时间系统:数学与语言发展之间的关系

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,

Children master language quickly and with relative ease, supported by both biological predisposition and reinforcement. B. F. Skinner (1957) proposed that language is learned through reinforcement, while Noam Chomsky (1965) argued that language acquisition mechanisms are biologically determined.
The critical period for language acquisition suggests that the ability to acquire language is at its peak early in life. As people age, this proficiency decreases. Language development begins very...

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system....

Developmental psychology explores the changes and continuities in human abilities throughout life, encompassing physical, cognitive, linguistic, and social dimensions. Human development is not restricted to growth, but includes aspects of decline, particularly in physical abilities as individuals age. Developmental psychologists seek to understand how people change as they age and how their mental and social skills evolve.
Developmental Milestones
A key concept in developmental psychology is...

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...