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He approximated to the Pythagoreans in this, that ( as is clear from his explanation of the soul ) he regarded number as the conditioning principle of consciousness, and consequently of knowledge also ; he thought it necessary, however, to supply what was wanting in the Pythagorean assumption by the more accurate definition, borrowed from Plato, that it is only insofar as number reconciles the opposition between the same and the different, and has raised itself to self-motion, that it is soul.
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The nuclei of these fibers, as is shown in Figures 3 and 4, showed remarkable proliferation and were closely approximated, forming a chainlike structure at either the center or the periphery of the fiber.
UTC is a discontinuous time scale composed from segments that are linear transformations of atomic time, the discontinuities being arranged so that UTC approximated UT2 until the end of 1971, and UT1 thereafter.
Even when the statistical model is nonlinear, it can be approximated by a linear model for which an analysis of variance may be appropriate.
Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach.
Specifically, in quantum mechanics, the state of an atom, i. e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion ( see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions.
) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this independent-particle model of products of single electron wave functions.
Many processes are close to adiabatic and can be easily approximated by using an adiabatic assumption, but there is always some heat loss ; as no perfect insulators exist.
For a Gaussian response system ( or a simple RC roll off ), the rise time is approximated by:
In Geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve.
where L, coastline length, a function of the measurement unit, ε, is approximated by the expression.
F is a constant and D is a parameter that Richardson found depended on the coastline approximated by L. He gave no theoretical explanation but Mandelbrot identified L with a non-integer form of the Hausdorff dimension, later the fractal dimension.
A real number a is said to be computable if it can be approximated by some computable function in the following manner: given any integer, the function produces an integer k such that:
Light interaction is often approximated by the even simpler models: diffuse reflection and specular reflection, although both can be BRDFs.
However, this value is only approximate, as reaction quotient is defined in terms of ion activities which can be approximated with the concentrations as calculated here.
Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.
In the Newtonian limit, i. e. when is sufficiently large compared to the Schwarzschild radius, the redshift can be approximated by a binomial expansion to become
Thus, if is the total number of trials and is the number of trials where the event occurred, the probability of the event occurring will be approximated by the relative frequency as follows:
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
approximated and from
Precipitation can be approximated from coastal weather data and air temperature from water temperatures.
In the quantization stage the representative signal values are approximated by values from a finite set.
The energy of formation of a molecule containing only single bonds then can be approximated from an electronegativity table, and depends on the constituents and sum of squares of differences of electronegativities of all pairs of bonded atoms.
Many diffuse reflectors are described or can be approximated by Lambert's cosine law, which describes surfaces that have equal luminance when viewed from any angle.
Although simple programs can be written in a few hours, programs that use complex mathematical formulas whose solutions can only be approximated or that draw data from many existing systems may require more than a year of work.
The full form factor could then be approximated by adding up the contribution from each of the pixel-like squares.
The average meso sequence length may be approximated from the relative abundance of pentads as follows:
The human auditory system, which processes frequencies in a roughly logarithmic fashion approximated by the Bark scale, does not perceive them with equal sensitivity ; signals in the 2 – 4-kHz octave sound loudest, and the loudness of other frequencies drops increasingly, depending both on the distance from the peak-sensitivity area and on the level.
Further on, he departed from the date of the foundation of the Realms in Exile, calculated at 3320 of the Second Age on the basis of average reigns of the Kings in Númenor ; from the duration of the time of peace before the war of the Last Alliance, approximated at 100 years ; and from the date of the failing of the Kings in Gondor, proposed as T. A.
Gravitation from Earth and Moon dominate the spacecraft's acceleration, and since the spacecraft's own mass is negligible in comparison, the spacecraft's trajectory may be better approximated as a restricted three-body problem.
All model parameters ( i. e., class priors and feature probability distributions ) can be approximated with relative frequencies from the training set.
Since ΔA << 1, this expression can be approximated by expanding the exponentials in a Taylor series to first-order and then discarding terms of ΔA in comparison with unity and converting from radians to degrees:
The Egyptian historian Manetho referred to Memphis as Hut-ka-Ptah ( meaning " Enclosure of the ka of Ptah "), which he approximated in Greek as Aί γυ πτoς ( Ai-gy-ptos ), from which derives the Latin AEGYPTVS and the modern English name of Egypt.
Linguists have approximated that the precursor to the language of the Huastecs diverged from the Proto-Mayan language between 2200 and 1200 BCE.
Settlers began to trickle into the Kittitas Valley with the opening of the Snoqualmie Wagon Road in 1867, which approximated the modern-day route of Interstate 90 past Thorp, from Seattle to Ellensburg.
The results of this model may be approximated, to within one degree, from the following formula:
* The Doppler rotation rate in the Sun ( measured from Doppler-shifted absorption lines ), can be approximated as:
The Globe's actual dimensions are unknown, but its shape and size can be approximated from scholarly inquiry over the last two centuries.
is close to that of a circular one, and the mean orbital speed can be approximated either from observations of the orbital period and the semimajor axis of its orbit, or from knowledge of the masses of the two bodies and the semimajor axis.
approximated and number
Its light curve can be approximated by adding together a number of sine waves, with periods that range between 0. 8 and 1. 5 hours.
The number of dōgu has been approximated at 4, 500.
Meton of Athens approximated the cycle to a whole number ( 6940 ) of days, obtained by 125 long months of 30 days and 110 short months of 29 days.
If the surfaces are approximated by a finite number of planar patches, each of which is taken to have a constant radiosity B < sub > i </ sub > and reflectivity ρ < sub > i </ sub >, the above equation gives the discrete radiosity equation,
Liouville showed that this number is what we now call a Liouville number ; this essentially means that it can be more closely approximated by rational numbers than can any irrational algebraic number.
The value of fully meshed networks is proportional to the exponent of the number of subscribers, assuming that communicating groups of any two endpoints, up to and including all the endpoints, is approximated by Reed's Law.
In some countries, ( particularly in Commonwealth nations ), the REN is better known as the ringer approximated loading number ( RAL ).
The set of all dyadic fractions is dense in the real line: any real number x can be arbitrarily closely approximated by dyadic rationals of the form.
For example, in computer chess, rather than computing the full minimax tree of all possible moves for the remainder of the game, a more limited tree of minimax possibilities is computed, with the tree being pruned at a certain number of moves, and the remainder of the tree being approximated by a static evaluation function.
A number of sub-national and international borders are defined by, or are approximated by, parallels.
The value of the Avogadro constant N < sub > A </ sub > was first approximated by Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas.
This number can be approximated by
The Egyptian Rhind papyrus of 1800BC gives the area of a circle as ( 64 / 81 ) < sup > 2 </ sup >, where is the diameter of the circle, and pi approximated to 256 / 81, a number that appears in the older Moscow Mathematical Papyrus, and used for volume approximations ( i. e. hekat ( volume unit )).
In constructive analysis, this does not hold, because the constructive interpretation of existential quantification (" there exists ") requires one to be able to construct the real number c ( in the sense that it can be approximated to any desired precision by a rational number ).
The first problem was to know how well a real number can be approximated by rational numbers.
It appears that these bounds depends on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger that the lower bound for algebraic numbers, which is itself larger than the lower bound for all real numbers.
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