New advances in mathematical and computer sciences are aiding intelligence analysts in more accurately calculating the quantitative probabilities of future occurrences or in some cases, the directions of specific events or actions by targeted individuals. One of the areas of science that have seen phenomenal increase in their applications to intelligence calculations is Differential Equations – a prominent area in calculus.
Differential equations are mathematical equations that unravel an unknown function of one or several variables that relate the values of the function itself and its derivatives of various orders and magnitudes.
Examples of Differential Equations
Simply put, a differential equation is an equation that contains a derivative. A derivative is a measure of how a function changes as its input changes.
In some cases, as in the case of the first example above, the equation may contain just one derivative and little else. In others, as in the last example, it may contain more than one derivative, and the derivative may be first-order (see explanations of “separable first-order linear ordinary differential equations” below), second-order or any other order!
While a differential equation must contain at least one derivative, the equation may contain other terms involving variables, as in the three middle examples given.
Although the second and third equations look fairly similar, the information they contain is very different.
Separable First-Order Linear Ordinary Differential Equations
A separable linear ordinary differential equation of the first order has the general form:
where f(t) is some known function. We may solve this by separation of variables (moving the y terms to one side and the t terms to the other side),
Integrating, we find
Differential equations are mathematical equations that unravel an unknown function of one or several variables that relate the values of the function itself and its derivatives of various orders and magnitudes.
Examples of Differential Equations
Simply put, a differential equation is an equation that contains a derivative. A derivative is a measure of how a function changes as its input changes.
In some cases, as in the case of the first example above, the equation may contain just one derivative and little else. In others, as in the last example, it may contain more than one derivative, and the derivative may be first-order (see explanations of “separable first-order linear ordinary differential equations” below), second-order or any other order!
While a differential equation must contain at least one derivative, the equation may contain other terms involving variables, as in the three middle examples given.
Although the second and third equations look fairly similar, the information they contain is very different.
Separable First-Order Linear Ordinary Differential Equations
A separable linear ordinary differential equation of the first order has the general form:
where f(t) is some known function. We may solve this by separation of variables (moving the y terms to one side and the t terms to the other side),
Integrating, we find
where C is a constant. Then, by exponentiation, we obtain
with A another arbitrary constant. It is easy to confirm that this is a solution by plugging it into the original differential equation:
Some elaboration is needed because ƒ(t) is not necessarily a constant—indeed, it might not even be integrable.
Arguably, one must also assume something about the domains of the functions involved before the equation is fully defined. Are we talking about complex functions, or just real, for example? The usual textbook approach is to discuss forming the equations well before considering how to solve them.
Why Differential Equations Are Becoming More Useful in Intelligence Calculations
Differential equations have often played prominent roles in engineering, physics, economics, and now, intelligence.
In making intelligence calculations, the philosophical bases of most of the modern intelligence analytical instruments (such as the Lockwood Analytical Method for Prediction) are that:
By solving a differential equation, we seek to find the function that satisfies it (the conditions or interactions of the human free will). In other words we apply differential equations in order to find the relationship between the two variables directly, and not necessarily involving any derivatives.
Applications of differential equations to intelligence analyses have produced excellent results for intelligence analysts because of the advancements in database technologies. With richer databases, most occurrences in our modern societies have moved from the indeterminate realm to the deterministic sphere. That is, in predicting human actions, intelligence analysts now depend less guesswork and more on scientific accuracy. In other words, most human actions can now be effectively calculated based on existing data.
with A another arbitrary constant. It is easy to confirm that this is a solution by plugging it into the original differential equation:
Some elaboration is needed because ƒ(t) is not necessarily a constant—indeed, it might not even be integrable.
Arguably, one must also assume something about the domains of the functions involved before the equation is fully defined. Are we talking about complex functions, or just real, for example? The usual textbook approach is to discuss forming the equations well before considering how to solve them.
Why Differential Equations Are Becoming More Useful in Intelligence Calculations
Differential equations have often played prominent roles in engineering, physics, economics, and now, intelligence.
In making intelligence calculations, the philosophical bases of most of the modern intelligence analytical instruments (such as the Lockwood Analytical Method for Prediction) are that:
- The future is not predetermined
- The future is the sum total of all interactions of "free will"
- The future is a spectrum of constantly changing relative probabilities.
By solving a differential equation, we seek to find the function that satisfies it (the conditions or interactions of the human free will). In other words we apply differential equations in order to find the relationship between the two variables directly, and not necessarily involving any derivatives.
Applications of differential equations to intelligence analyses have produced excellent results for intelligence analysts because of the advancements in database technologies. With richer databases, most occurrences in our modern societies have moved from the indeterminate realm to the deterministic sphere. That is, in predicting human actions, intelligence analysts now depend less guesswork and more on scientific accuracy. In other words, most human actions can now be effectively calculated based on existing data.
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