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 mathematical modeling continues to be done today.

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 technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

More generally, if a linear operator on a vector space ( possibly infinite-dimensional ) has finite-dimensional range ( e. g., a finite-rank operator ), then the rank of the operator is defined as the dimension of the range.

More generally yet, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.

More formally, a multivariate random variable is a column vector X = ( X < sub > 1 </ sub >, ..., X < sub > n </ sub >)< sup > T </ sup > ( or its transpose, which is a row vector ) whose components are scalar-valued random variables on the same probability space ( Ω,, P ), where Ω is the sample space, is the sigma-algebra ( the collection of all events ), and P is the probability measure ( a function returning every event's probability ).

More generally, a norm on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.

After a basis for the vector space is chosen, this ring is naturally identified with the ring of n-by-n matrices with entries in K. More generally, the endomorphism algebra of the free module is naturally n-by-n matrices with entries in R.

More generally, every module category over a ring is additive, and so in particular, the category of vector spaces over a field is additive.

More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

More formally, a support vector machine constructs a hyperplane or set of hyperplanes in a high-or infinite-dimensional space, which can be used for classification, regression, or other tasks.

More precisely, in order to define the covariant derivative of it is necessary first to extend to a continuously differentiable vector field in an open set.

More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle.

More generally, let V be a vector space over a field K, and let

More generally still, the general linear group of a vector space GL ( V ) is the abstract automorphism group, not necessarily written as matrices.

More detailed discussion of wave functions as elements of vector spaces is below, following further definitions.

More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface.

More precisely, if V is a vector space over &# x211D ;, the inner product is a function

More recently, a number of loop-free distance vector protocols have been developed — notable examples are EIGRP, DSDV and Babel.

More precisely, the vorticity of a flow is a vector field, equal to the curl ( rotational ) of its velocity field v. It can be expressed by the vector analysis formula, where is the nabla operator.