Consider first-order linear ODEs of the general form:

Suppose that we had a general decision algorithm for statements in a first-order language.

Lenat's quest, in the Cyc project, is to build the basis of a general artificial intelligence by manually representing knowledge in the formal language, CycL, based on extensions to first-order predicate calculus.

Moreover, for the general first-order linear inhomogeneous recurrence relation with variable coefficient ( s ),, there is also a nice method to solve it:

1936-1937: Alonzo Church and Alan Turing, respectively, published independent papers showing that a general solution to the Entscheidungsproblem is impossible: the universal validity of statements in first-order logic is not decidable ( it is only semi-decidable as given by the completeness theorem ).

Arithmetic, mereology, and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with Gödel and Skolem's adherence to first-order logic, led to a general decline in work in second ( or any higher ) order logic.

At that time what were called Pfaffian systems ( i. e. first-order differential equations given as 1-forms ) were in general use ; by the introduction of fresh variables for derivatives, and extra forms, they allowed for the formulation of quite general PDE systems.

In general, let D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M. If

The signatures that are considered in general model theory are often infinite, while a single first-order sentence contains only finitely many symbols.

" For example, a general coupled set of first-order SDEs is often written in the form:

If not, Kalman filter based methods are a first-order approximation ( EKF ) or a second-order approximation ( UKF in general, but if probability distribution is Gaussian a third-order approximation is possible ).

In general, Hendrik Lorentz ( 1892, 1895 ) introduced several new auxiliary variables for moving observers, demonstrating why all first-order optical and electrostatic experiments have produced null results.

He argues ( contra Hintikka ) that while satisfiability might be a first-order matter, the question of whether there is a winning strategy for Verifier over all structures in general " lands us squarely in full second order logic " ( emphasis Feferman's ).

Pattern-directed invocation is a commonly used and more practical alternative to first-order theorem proving in automated theorem proving, and is the primary method of implementing a blackboard system for automated reasoning in general.

In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by introducing a new functional symbol:

Affine arithmetic is meant to be an improvement on interval arithmetic ( IA ), and is similar to generalized interval arithmetic, first-order Taylor arithmetic, the center-slope model, and ellipsoid calculus — in the sense that it is an automatic method to derive first-order guaranteed approximations to general formulas.

In general, these complex maps are referred to as second-order representations of the visual field, as opposed to first-order ( continuous ) representations such as V1 ( Rosa, 2002 ).

Since the set of valid first-order formulas is recursively enumerable but not recursive, there exists no general algorithm to solve this problem.

And it is clearly argued by Lord Percy of Newcastle, in his remarkable long essay, The Heresy Of Democracy, and in a more general way by Voegelin, in his New Science Of Politics, that this same Rousseauan idea, descending through European democracy, is the source of Marx's theory of the dictatorship of the proletariat.

There is little evidence that they are giving any systematic thought to a general theory of the optimum scope and nature of their part in government.

The general theory of resonance shifts is used to derive a general expression for the second moment Af of a polycrystalline paramagnetic sample and is specialized to Af.

We should expect that general phonologic theory should be as adequate for tone as for consonants and vowels, but it has not been.

This can only be for one of two reasons: either the two are quite different and will require totally different theory ( and hence techniques ), or our existing theories are insufficiently general.

-- The theory of elasticity of Gaussian networks has been developed on a more general basis and the equations of state relating variables of pressure, volume, temperature, stress and strain have been precisely formulated.

Dr. Bonnor begins with a discussion of the relativistic theories of the universe, based on the general theory of relativity.

The exact number and placement of Endosymbiotic theory | endosymbiotic events is currently unknown, so this diagram can be taken only as a general guide It represents the most parsimonious way of explaining the three types of endosymbiotic origins of plastids.

He is remembered for developing the theory of general semantics.

*: The article on Whorf states " Drawing on Nietzsche's ideas of perspectivism Alfred Korzybski developed the theory of general semantics which has been compared to Whorf's notions of linguistic relativity.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

The theory of special relativity predicts that any such device would allow communication from the future to the past, which raises problems of causality, unless the device used general relativistic curved spacetimes as an integral part.

The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis ( the symmetrical axis of the system ) are infinitely small, i. e. with infinitesimal objects, images and lenses ; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.

Thus, the Big Bang theory cannot and does not provide any explanation for such an initial condition ; rather, it describes and explains the general evolution of the universe going forward from that point on.

Also, general relativity has passed stringent tests on the scale of the Solar System and binary stars while extrapolation to cosmological scales has been validated by the empirical successes of various aspects of the Big Bang theory.

Bootstrapping is using very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles.

The theory that they were defensive military structures is not accepted by many modern archaeologists ( see the ' general references ' below ), while the alternative notion that they were farmhouses is dismissed by some others.

MacKie's theory has fallen from favour too, mainly because from the 1970s there was a general move away from ' diffusionist ' explanations in archaeology towards those involving exclusively indigenous development.

The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole.

In 1915, Albert Einstein developed his theory of general relativity, having earlier shown that gravity does influence light's motion.

Naveh states, " The striking feature of the blitzkrieg concept is the complete absence of a coherent theory which should have served as the general cognitive basis for the actual conduct of operations ".

Unital Banach algebras over the complex field provide a general setting to study spectral theory.