Its first-order Fresnel lens can be seen for about 19 miles ( 30 kilometers ), in good conditions.

* November 1 – The current Cape Lookout, North Carolina, lighthouse is lighted for the first time ( its first-order Fresnel lens can be seen for 19 miles ).

Cape Meares Lighthouse ; first-order Fresnel lens

The Ayres Davies Lens Exhibit Building houses one of the finest collections of restored Fresnel lenses in the world, including the rotating first order Fresnel lens from the Cape Canaveral lighthouse and the restored original Ponce Inlet lighthouse first-order Fresnel lens.

The station includes a first-order Fresnel lens that has a focal plane above ground ( above sea level ).

The light was a first-order Fresnel lens which stood 134 feet above the water.

The light station was replaced by a " floating lighthouse " buoy and decommissioned in 1975, and its-high first-order Fresnel lens was removed in 1983 for display at the Del Norte County Historical Museum in Crescent City.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

No first-order theory, however, has the strength to describe fully and categorically structures with an infinite domain, such as the natural numbers or the real line.

In Gödel's paper, he uses a version of first-order predicate calculus which has no function or constant symbols to begin with.

Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of relations, represented as facts and rules.

Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties.

It has also been shown by L. Berman in 1980 that the problem of verifying / falsifying any first-order statement about real numbers that involves only addition and comparison ( but no multiplication ) is in EXPSPACE.

If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies ; such an effect is caused for example by a little bit of the input leaking around the one-pole filter ; this one-pole – one-zero filter is still a first-order low-pass.

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.

Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model.

# A real closed field has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, *, and ≤.

For an instrument whose payment stream is described by f ( t ), the value V ( t ) satisfies the inhomogeneous first-order ODE (" inhomogeneous " is because one has f rather than 0, and " first-order " is because one has first derivatives but no higher derivatives ) – this encodes the fact that when any cash flow occurs, the value of the instrument changes by the value of the cash flow ( if you receive a $ 10 coupon, the remaining value decreases by exactly $ 10 ).

It is more expressive than propositional logic but has more efficient decision problems than first-order predicate logic.

For example, while it was once assumed that catagenetic processes were first-order reactions, some research has shown that this may not be the case.

There are no first-order administrative divisions as defined by the United States Government, but Puerto Rico has 78 municipalities or " municipios " at the secondary order.

A second-order low-pass filter with a very low quality factor has a nearly first-order step response ; the system's output responds to a step input by slowly rising toward an asymptote.

The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k ≠ n. Each term is proportional to the matrix element, which is a measure of how much the perturbation mixes eigenstate n with eigenstate k ; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies.

Second-and third-order winning percentage has been shown to predict future actual team winning percentage better than both actual winning percentage and first-order winning percentage.

Erwin Schrödinger discussed at length the Stark effect in his third paper on quantum theory ( in which he introduced his perturbation theory ), once in the manner of the 1916 work of Epstein ( but generalized from the old to the new quantum theory ) and once by his ( first-order ) perturbation approach.

Besides those optical experiments, also electrodynamic first-order experiments were conducted, which should lead to positive results according to Fresnel.

It was later shown that all first-order optical experiments must give a negative result due to this coefficient.

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.

In neutron scattering, the first-order Born approximation is almost always adequate, except for neutron optical phenomena like internal total reflection in a neutron guide, or grazing-incidence small-angle scattering.

In addition, a uniform function over a circular area ( in one FT domain ) corresponds to the Airy function, J < sub > 1 </ sub >( x )/ x in the other FT domain, where J < sub > 1 </ sub >( x ) is the first-order Bessel function of the first kind.

Among these, there are 8 which are designated as first-order key disciplines by the Chinese government, and 37 which are designated as second – order key disciplines.

In the second-order theory, one of the normal stress differences can be calculated from the first-order stress relaxation function.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

That is, every consistent set of first-order sentences can be extended to a maximal consistent set.

The Peano axioms are the most widely used axiomatization of first-order arithmetic.

The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger.

In 1920, Thoralf Skolem simplified a previous result by Leopold Löwenheim, leading to the Löwenheim – Skolem theorem and, in 1930, to the notion of a Herbrand universe and a Herbrand interpretation that allowed ( un ) satisfiability of first-order formulas ( and hence the validity of a theorem ) to be reduced to ( potentially infinitely many ) propositional satisfiability problems.

In contrast, other, more systematic algorithms achieved, at least theoretically, completeness for first-order logic.

Initial approaches relied on the results of Herbrand and Skolem to convert a first-order formula into successively larger sets of propositional formulae by instantiating variables with terms from the Herbrand universe.

The traditional test for the presence of first-order autocorrelation is the Durbin – Watson statistic or, if the explanatory variables include a lagged dependent variable, Durbin's h statistic.

The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z-plane to the s-plane.

Here k is first-order rate constant having dimension 1 / time, ( t ) is concentration at a time t and < sub > 0 </ sub > is the initial concentration.

The rate of a first-order reaction depends only on the concentration and the properties of the involved substance, and the reaction itself can be described with the characteristic half-life.

Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.

The cotangent space will then consist of all the possible first-order behaviors of a function near x.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

For example, if we can enumerate all such definable numbers by the Gödel numbers of their defining formulas then we can use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language.

A basic objective of the first normal form defined by Codd in 1970 was to permit data to be queried and manipulated using a " universal data sub-language " grounded in first-order logic.

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

Here the left-hand side is equal to, namely the first-order polygamma function defined through ; the gamma function is equal to if is a positive integer.