It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic.

In mathematical terms, quenched disorder is harder to analyze than its annealed counterpart, since the thermal and the noise averaging play very different roles.

It has become painfully clear that the very attempt to make the language of social research free of values by erecting mathematical and physical models, is itself a conditioned response to a world which pays a premium price for technological manipulation.

The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

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.

Ampère begun developing a mathematical and physical theory to understand the relationship between electricity and magnetism.

(" the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in a realm outside of space-time ") Whilst our knowledge of concrete, physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects.

A more radical defense is to deny the separation of physical world and the platonic world, i. e. the mathematical universe hypothesis.

But these are physical representations of the corresponding mathematical entities ; the line and the curve are idealized concepts whose width is 0 ( see Line ).

The study of condensed matter physics involves measuring various material properties via experimental probes along with using techniques of theoretical physics to develop mathematical models that help in understanding physical behavior.

Caltech received $ 144 million in federal funding for the physical sciences, $ 40. 8 million for the life sciences, $ 33. 5 million for engineering, $ 14. 4 million for environmental sciences, $ 7. 16 million for computer sciences, and $ 1. 97 million for mathematical sciences in 2008.

These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system.

If numerical iterative methods have to be employed, the aim is to iterate until full machine accuracy is obtained ( the best that is possible with a finite word length on the computer, and within the mathematical and / or physical approximations made ).

It seeks to understand physical systems, using mathematical modeling, in terms of inputs, outputs and various components with different behaviors ; use control systems design tools to develop controllers for those systems ; and implement controllers in physical systems employing available technology.

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.

The " space " in cyberspace has more in common with the abstract, mathematical meanings of the term ( see space ) than physical space.

A successful mathematical classification method for physical lattice defects, which works not only with the theory of dislocations and other defects in crystals but also, e. g., for disclinations in liquid crystals and for excitations in superfluid < sup > 3 </ sup > He, is the topological homotopy theory.

The polyphonic organization of different melodies to sound at the same time was still a relatively new invention then, and it is understandable that the mathematical or physical relationships in frequency that give rise to the musical intervals as we hear them, should be foremost among the preoccupations of Medieval musicians.

This is an idealized mathematical model: real physical Diesels do have an increase in pressure during this period, but it is less pronounced than in the Otto cycle.

The Copenhagen interpretation is a consensus among some of the pioneers in the field of quantum mechanics that it is undesirable to posit anything that goes beyond the mathematical formulae and the kinds of physical apparatus and reactions that enable us to gain some knowledge of what goes on at the atomic scale.

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.

" Kant stated that all mathematical and scientific statements are synthetic a priori propositions because they are necessarily true but our knowledge about the attributes of the mathematical or physical subjects we can only get by logical inference.

In short, virtual manipulatives are dynamic visual / pictorial replicas of physical mathematical manipulatives, which have long been used to demonstrate and teach various mathematical concepts.

Armstrong was of the opinion that anyone who had actual contact with the development of radio understood that the radio art was the product of experiment and work based on physical reasoning, rather than on the mathematicians ' calculations and formulae ( known today as part of " mathematical physics ").

In 1951, together with William Cochran and Vladimir Vand, Crick assisted in the development of a mathematical theory of X-ray diffraction by a helical molecule.

Kirchhoff's diffraction formula provides a rigorous mathematical foundation for diffraction, based on the wave equation.

In 1921, he laid the mathematical foundation of fiber diffraction analysis.

As derived below, the electron density within the crystal and the diffraction patterns are related by a simple mathematical method, the Fourier transform, which allows the density to be calculated relatively easily from the patterns.

The recorded series of two-dimensional diffraction patterns, each corresponding to a different crystal orientation, is converted into a three-dimensional model of the electron density ; the conversion uses the mathematical technique of Fourier transforms, which is explained below.

A detailed mathematical treatment of Fraunhofer diffraction is given in this article.

In his Optics ,, Francis Weston Sears offers a mathematical approximation suggested by Fresnel that predicts the main features of diffraction patterns and uses only simple mathematics.

A mathematical formula is nothing more than a pattern for solving a specific problem.

The equation is used for the mathematical process of solving the problem.

However, it is essential that the various mathematical symbols used in the equations be understood so that the mathematical processes can be done properly and in their correct order.

We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

A mathematical block diagram for the leveling system is shown in Fig. 7-2.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The importance of this 5 can largely be explained by the natural mathematical properties of the middle number and its special relationship to all the rest of the numbers -- quite apart from any numerological considerations, which is to say, any symbolic meaning arbitrarily assigned to it.

it is also their mathematical mean, since it is equal to half the sum of every opposing pair, all of which equal 10.

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 study of altruism was the initial impetus behind George R. Price's development of the Price equation, which is a mathematical equation used to study genetic evolution.

The working principle of a yupana is unknown, but in 2001 an explanation of the mathematical basis of these instruments was proposed by Italian mathematician Nicolino De Pasquale.

The abacus teaches mathematical skills that can never be replaced with talking calculators and is an important learning tool for blind students.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

In mathematical notation, this is:

An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom.

Acoustic theory is the field relating to mathematical description of sound waves.

The mathematical equation for an ideal gas undergoing a reversible ( i. e., no entropy generation ) adiabatic process is