Such a position entails the negation of philosophy in its Platonic form as something soaring above and embracing the empirical and mathematical sciences.

With the Prior Analytics, Aristotle is credited with the earliest study of formal logic, and his conception of it was the dominant form of Western logic until 19th century advances in mathematical logic.

The term may also refer to the physical region where the electron can be calculated to be, as defined by the particular mathematical form of the orbital.

The use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simulations form the scientific basis for heavier-than-air flight and a number of other technologies.

The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

The application of the catenary to the construction of arches is attributed to Robert Hooke, whose " true mathematical and mechanical form " in the context of the rebuilding of St Paul's Cathedral alluded to a catenary.

where he wrote that he had found " a true mathematical and mechanical form of all manner of Arches for Building.

He eventually graduated to the Grey Coat mathematical school and was introduced to basic navigation skills which would form the basis of his future career.

The proportions of the spheres of the planets and stars ( which at the time were still thought to revolve around the earth ) were perceived as a form of music, without necessarily implying that any sound would be heard — music refers strictly to the mathematical proportions.

In its mathematical form it is analogous to the description of a physical wave, but its " crests " and " troughs " indicate levels of probability for the occurrence of certain phenomena ( e. g., a spark of light at a certain point on a detector screen ) that can be observed in the macro world of ordinary human experience.

They influenced French physicist André-Marie Ampère's developments of a single mathematical form to represent the magnetic forces between current-carrying conductors.

In mathematical form, this can be stated as:

The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra.

An often-repeated mathematical joke is that topologists can't tell their coffee cup from their donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

The origins of mathematical thought lie in the concepts of number, magnitude, and form.

* Hatch mark, a form of mathematical notation

In fact, mathematical induction is a form of rigorous deductive reasoning.

Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group.

The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects.

In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational functions are defined in terms of each other.

In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form ( they had already been recast as quaternions ) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations.

In whatever mathematical form Pytheas knew the location, he could only have determined when he was there by taking periodic readings of the elevation of the pole ( eksarma tou polou in Strabo and others ).

In its axiomatic form, mathematical statements about probability theory carry the same sort of epistemological confidence shared by other mathematical statements in the philosophy of mathematics.

Lorentz noticed that it was necessary to change the space-time variables when changing frames and introduced concepts like physical length contraction ( 1892 ) to explain the Michelson-Morley experiment, and the mathematical concept of local time ( 1895 ) to explain the aberration of light and the Fizeau experiment.

The Aharonov – Bohm effect illustrates the physicality of electromagnetic potentials, Φ and A, whereas previously it was possible to argue that only the electromagnetic fields, E and B, were physical and that the electromagnetic potentials, Φ and A, were purely mathematical constructs ( Φ and A being non-unique, in addition to not appearing in the Lorentz Force formula ).

** Lorentz transformation, the formula that provides the mathematical backbone for Einstein's theory of special relativity, though Einstein re-derived the transform in a different and much more elegant way

When the acceleration component of the Lorentz mathematical model is small compared to the other components of the equation, then the Debye model is applied.

After Einstein derived special relativity formally from the ( at first sight counter-intuitive ) assumption that the speed of light is the same to all observers, Hermann Minkowski built on mathematical approaches used in non-euclidean geometry and on the mathematical work of Lorentz and Poincaré.

Independently of Larmor, also Lorentz ( 1899 ) extended his transformation for second order terms and noted a ( mathematical ) Time Dilation effect as well.

While for Lorentz length contraction was a real physical effect, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation to simplify the calculation from the resting to a " fictitious " moving system.

Contrary to Lorentz, Poincaré saw more than a mathematical trick in the definition of local time, which he called Lorentz's " most ingenious idea ".

The mathematical tool for that is the so-called Lorentz transformation.

Although recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave Coriolis, in connection with the theory of water wheels, Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.

His notebooks, unfortunately not fully published until the 20th century, reveal a coherent mechanical philosophy of nature with incipient atomism, a force of inertia, and mathematical interpretations of natural philosophy are present.

Wren's challenge to Halley and Hooke, for the reward of a book worth thirty shillings, was to provide, within the context of Hooke's hypothesis, a mathematical theory linking the Kepler's laws with a specific force law.

Certain types of encryption, by their mathematical properties, cannot be defeated by brute force.

See Shannon number for details .</ ref >), it is hard to rule out with mathematical certainty the possibility that the initial position allows either side to force a mate or a threefold repetition after relatively few moves, in which case the search tree might encompass only a very small subset of the set of possible positions.

Note that d < sup > 2 </ sup > R / dt < sup > 2 </ sup > and F / m are both equal to the gravitational acceleration g ( equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass ).

This is the mathematical form of Newton's second law of motion: force equals mass times acceleration, for such situations.

* Proof by exhaustion or brute force method, a method of mathematical proof

Topics of MQPs done in the recent past include the design of the MIR 2 space station life support system module, a study of the effects of stress and nicotine on ADHD, the design of a research rocket, a mathematical viscoelastic cell motility model, experimental research of liquid crystals using atomic force microscopy, and the design of polymers for medicine delivery.

The description of the electrons in the subsection above as little clocks is in effect a statement of the mathematical rules according to which the phases of electrons are to be added and subtracted: they are to be treated as ordinary numbers, except that in the case where the result of the calculation falls outside the range of 0 ≤ θ < 360 °, we force it to " wrap around " into the allowed range, which covers a circle.

These examples are not only intended to force the students into developing mathematical models on their own, but may also be used to promote mathematical interest and understanding by relating the subject to real-life situations.

He notes that the mechanism of " bouncing particles " reproduces the inverse-square force law and that " the strangeness of the mathematical relation will be very much reduced ", but then remarks that the scheme " does not work ", because of the drag it predicts would be experienced by moving bodies, " so that is the end of that theory ".

Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds.

Fortunately, JCSP does not force its users to adopt a mathematical approach themselves, but allows them to benefit from the mathematics that underpins the library.

An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically ( i. e., non-permanently ) when a force is applied to it.

In mathematical notation, an electrical force deriving from a potential will always be divergenceless ( satisfy Laplace's equation ):

The scale of events being simulated by computer simulations has far exceeded anything possible ( or perhaps even imaginable ) using the traditional paper-and-pencil mathematical modeling: over 10 years ago, a desert-battle simulation, of one force invading another, involved the modeling of 66, 239 tanks, trucks and other vehicles on simulated terrain around Kuwait, using multiple supercomputers in the DoD High Performance Computer Modernization Program.

He was the first scientist to explain mathematically the origin of aerodynamic lift, through his circulation hypothesis, the first to dimension the lift force generated by a body moving through an ideal fluid as proportional to the velocity and the circulation around the body, and through a mathematical conformal transformation the first to define the shape of the aerodynamic profile having as essential elements a rounded nose ( leading edge ), double surface ( finite thickness ), cambered or symmetrical, and a sharp tail ( trailing edge ).

The subject in the mathematical calculation of shock wave cancellation around a flat wing in a supersonic gas flow, where they develop a method to solve the Navier-Stokes equations within an MHD force field by the method of characteristics.

Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of Mechanics, which is concerned with the motion of particles or objects when subjected to a force.

In the physical world no truly instantaneous change in velocity is possible as this would require an " infinite force " applied during an " infinitely short time " but as a mathematical model it in most cases describes the effect of a maneuver on the orbit very well.

The brute force approach to balancing is the mathematical analysis of game session results.