More importantly, several of the more advanced of the less developed countries have found through experience that they must plan their own complex investment programs for at least 5 years forward and tentatively for considerably more than that if they are to be sure that the various interdependent activities involved are all to take place in the proper sequence.

More complex techniques use

More complex assistive technology devices have been developed over time, and as a result, sports for people with disabilities " have changed from being a clinical therapeutic tool to an increasingly competition-oriented activity ".

More complex alkenes may be named using the E-Z notation, used to describe molecules having three or four different substituents ( side groups ).

More specialized bicycle components may require more complex tools, including proprietary tools specific for a given manufacturer.

More complex computations may involve many operations and data elements together.

More complex transformations of object properties such as size, shape, lighting effects often require calculations and computer rendering instead of simple re-drawing or duplication.

More complex formulas and procedures apply for such cases, but in its simplest form the critical buckling load formula is given as Equation ( 3 ),

More complex forms of monist dualism also exist, for instance in Hermeticism, where Nous " thought "-that is described to have created man-brings forth both good and evil, dependent on interpretation, whether it receives prompting from the God or from the Demon.

More complex circuits can be analyzed numerically with software such as SPICE or GNUCAP, or symbolically using software such as SapWin.

More complex engines using human power, animal power, water power, wind power and even steam power date back to antiquity.

" More complex variants use computer modeling of actual games based on statistical input generated by professional sports.

More complex to achieve is a display of radiation dose rate, displayed in a unit such as the sievert.

More complex protein hormones bear carbohydrate side-chains and are called glycoprotein hormones.

More recent research has shown the situation to be more complex.

More complex multivariate analyses are sometimes used when multiple characteristics are involved and a univariate analysis could produce confounded results.

More complex molecules have many bonds, and their vibrational spectra are correspondingly more complex, i. e. big molecules have many peaks in their IR spectra.

More complex rewritings involving modus tollens are often seen, for instance in set theory:

More than one standard can be applied to any case based on multiple jurisdictions and numerous other factors in relation to complex state and federal law ; dependant on which jurisdiction and country one is in.

More problematic, and never properly addressed by Cumont or his successors, is how real-life Roman Mithraists subsequently maintained a quite complex and sophisticated Iranian theology behind an occidental facade.

More complex systems will have more layers of neurons with some having increased layers of input neurons and output neurons.

More complex experiments, such as those involving stochastic processes defined in continuous-time, may demand the use of more general probability measures.

* More formal and complex written statements of ownership that permit the easier assumption of shared risk and ownership in companies, and insurance against risk ;

More formally, it is " the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.

More readably and idiomatically, one might use the following equivalent, which demonstrates a simple procedure definition and the use of the mathematical operators and:

More generally, square roots can be considered in any context in which a notion of " squaring " of some mathematical objects is defined ( including algebras of matrices, endomorphism rings, etc.

More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time.

More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related to metamathematics.

More recently ( 2001 and 2002 ), Bailey and Richard Crandall showed that the existence of this and similar formulas has implications for the long-standing question of " normality " – whether and why the digits of certain mathematical constants ( including pi ) appear " random " in a particular sense.

More complex systems arising in biology, medicine, the humanities, management sciences, and similar fields often remained intractable to conventional mathematical and analytical methods.

# More generally, an assignment of elements from a countable mathematical object, such as a countable group, to natural numbers to allow algorithmic manipulation of the mathematical object.

More than 180 named mathematical facts have so benefited from formal codification.

More recent work has restored rigor to infinitesimals, making them once more a legitimate mathematical topic ( see non-standard analysis ).

More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

More recently, the mathematician Underwood Dudley has written a series of books on mathematical cranks, including The Trisectors, Mathematical Cranks, and Numerology: Or, What Pythagoras Wrought.

More recently, Yaneer Bar-Yam has claimed that the gene-centered view ( and thus Ronald Fisher's treatment of evolution ) relies upon a mathematical approximation that is not generally valid.

More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than Planck's constant ħ, so the " deformation parameter " ħ / S can be effectively taken to be zero.

" More formal mathematical descriptions of this relation are often called the term structure of interest rates.

More general affine connections were then studied around 1920, by Hermann Weyl, who developed a detailed mathematical foundation for general relativity, and Élie Cartan, who made the link with the geometrical ideas coming from surface theory.

More generally a stack refers to any category acting more or less like a moduli space with a universal family ( analogous to a classifying space ) parametrizing a family of related mathematical objects such as schemes or topological spaces, especially when the members of these families have nontrivial automorphisms.

More recently, Christopher Isham, Jeremy Butterfield, and their students have continued to develop quantum geometrodynamics to take account of recent work toward a quantum theory of gravity and further developments in the very extensive mathematical theory of initial value formulations of general relativity.

More importantly, his mathematical and graphical calculation methods improved turbine design and engineering.

More recently Paul Ernest has proposed a social constructivist account of mathematical knowledge, drawing on the works of both of these sociologists.

More recently, attempts have been made to construct mathematical models of the stereolithography process and design algorithms to determine whether a proposed object may be constructed by the process.

More significantly, he became interested in developing a linguistic theory using a non-taxonomic approach and based on mathematical formalism, and this line of inquiry represented a decisive break with the Bloomfieldian taxonomic structuralist tradition of linguistic analysis.

More critical or optimized designs now routinely use mathematical modeling on a computer.