Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to Markov random fields, which again find broad applicability ; for example in Hopfield networks.

Thus, a statistical filter does not reflect the software author's or administrator's biases as to content, but rather the user's biases.

Thus, in the applied use of loss functions, selecting which statistical method to use to model an applied problem depends on knowing the losses that will be experienced from being wrong under the problem's particular circumstances, which results in the introduction of an element of teleology into problems of scientific decision-making.

Thus, " construct validity " can be evaluated by statistical methods that show whether a common factor can be shown to exist underlying several measurements using different observable indicators.

Thus, by coarsening the variable, we increase sample sizes and achieve better statistical estimation.

Assuming such a statistical ensemble consists of an overall collection of N microscopic states, the ensemble is constructed so that the proportion p < sub > i </ sub >/ N of members of the ensemble which are in microscopic state i is proportional to the probability, over time, of finding the real-world system in that microscopic state i. Thus the ensemble is an imaginary static collection of microscopic states created to mirror the statistics of the successive fluctuations of the macroscopic physical system which is being modeled.

Thus, transfer operators typically appear in physics problems, such as quantum chaos and statistical mechanics, where attention is focused on the time evolution of smooth functions.

Thus, ' brand penetration ' or ' brand share ' reflects only a statistical chance that the majority of customers will buy that brand next time as part of a portfolio of brands they favour.

Thus, we can probe the measurement apparatus with a statistical mixture:

Thus, the R chart is examined before the chart ; if the R chart indicates the sample variability is in statistical control, then the chart is examined to determine if the sample mean is also in statistical control.

Thus, there may be " external incentives " for the production of more comprehensive statistical information which affect some countries, but are much less evident in others, where the information is required by some international body.

Thus there is a minor conflict of terminology since the values within a univariate time series may be treated using certain types of multivariate statistical analyses and may be represented using multivariate distributions.

Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space.

Thus, for example, the CGS unit of pressure, barye, is related to the CGS base units of length, mass, and time in the same way as the SI unit of pressure, pascal, is related to the SI base units of length, mass, and time:

" Thus, any activity of production ( which distributism holds to be the most important part of any economy ) ought to be performed by the smallest possible unit.

In the field of high-energy particle physics, the fundamental velocity unit is the speed of light c. Thus, dividing energy in eV by the speed of light in vacuum, one can describe the momentum of an electron in units of eV / c.

Thus understood, Fascism is totalitarian, and the Fascist State — a synthesis and a unit inclusive of all values — interprets, develops, and potentiates the whole life of a people.

Thus, although the joule has the same dimensions as the newton-metre ( 1 J = 1 N · m = 1 kg · m < sup > 2 </ sup >· s < sup >− 2 </ sup >), these units are not interchangeable as the newton-metre ( N · m ) is the unit of torque and joule the unit of energy.

However, a second definition and usage has historically been in practice in many fields of computer science and information technology, which defines the prefix kilo when used with byte or bit units of data as 1024 ( 2 < sup > 10 </ sup >); this is due to the mathematical coincidence that Thus, in these fields 1 kilobyte is equal to 1 kibibyte, a new unit standardized as part of the binary prefixes to resolve the ambiguity.

Thus for each unit the seller tries to set the price equal to the consumer ’ s reservation price.

Thus, all parts share the same rate of rotation, or the same number of rotations or revolutions per unit of time.

Thus, control unit X was either 1st or 5th, Y was either 2nd or 4th, and Z was 3rd in line.

Thus, it is correctly written either way, and is only incorrect when specification is first made that the word is being used as a unit, or else a symbol of the unit, and then the incorrect letter case for the specified use is employed.

Thus the bound vector represented by ( 1, 0, 0 ) is a vector of unit length pointing from the origin along the positive x-axis.

Thus cutting a cake, where taking a larger piece reduces the amount of cake available for others, is a zero – sum game if all participants value each unit of cake equally ( see marginal utility ).

Thus the shield was symbolic of the individual soldier's subordination to his unit, his integral part in its success, and his solemn responsibility to his comrades in arms – messmates and friends, often close blood relations.

Thus the amount of the test substance required to produce the same biological effect as a given quantity the unit of a standard preparation is compared and the potency of the unknown is expressed as a % of that of the standard by employing a simple formula.

Thus on 16 January 1968, the Zambian pound was replaced by the kwacha with the new official rate equal to one half the old unit, or US $ 1.

Thus if γ ( s ) is the arclength parametrization of C then the unit tangent vector T ( s ) is given by

Thus, as with all heat pumps, the Coefficient of Performance ( amount of thermal energy moved per unit of input work required ) decreases with increasing temperature difference.

Thus polypeptides after treatment become rod-like structures possessing a uniform charge density, that is same net negative charge per unit length.

Thus the topology of the group SL ( n, C ) is the product of the topology of SU ( n ) and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues.

Thus, as long as he receives an appropriate amount of goods, such as Qc, then he will be willing to pay his entire surplus ( ABC ) in addition to the cost per unit under perfect competition ( Pc by Qc )-i. e.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus with regard to the loss of tradition, in the change from order to disorder the metaphysics of change works itself out as a disruption of the individual soul, a change in which man continues as an objective ontological existent, but no longer as a man.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus Trevelyan repeats the story which pictured Victor Emmanuel as refusing to abandon the famous Statuto at the insistence of General Radetzky.

Thus science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus the government simultaneously undertook the vast burden of social security which had traditionally been privately discharged, and created a national scarcity which has engendered calamitous problems of social security.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

Thus, there has come into being a situation in which the state must raise all of its own revenues and, in addition, must give assistance to its local governments.

Thus, in the last few years, a number of programs which looked very promising at the time their development was commenced have since been completely eliminated.

Thus was invented the single thread sewing machine, which Mr. Gibbs in partnership with Mr. Willcox decided to bring to Brown & Sharpe with the proposal that the small company undertake its manufacture.

Thus, direct comparisons can be drawn with free burning arcs which have been studied in detail during the past years and decades by numerous investigators ( Ref. 3 ).

Thus the Af are projections which correspond to some direct-sum decomposition of the space V.

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.

Thus in a context in which there has been discussion of snow but mention of local conditions is new, dominant stress will probably be on here in it rarely snows here, but in a context in which there has been discussion of local weather but no mention of snow, dominant stress will probably be on snows.

Thus in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.

Thus, besides the training provided to youth in school, the existence of the school program can have supplementary benefits to industry which make it an asset to industrial development efforts.

Thus far, the cases which have come before the courts have involved only the issue of referral where the job is vacant due to a strike -- condition ( 1 ) in the Regulation of the Secretary.

Thus, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.

Thus, when Dartmouth's Winter Carnival -- widely recognized as the greatest, wildest, roaringest college weekend anywhere, any time -- was broadcast over a national television hookup, Prexy John Sloan Dickey appeared on the screen in rugged winter garb, topped off by a tam-o'-shanter which he confessed had been acquired from a Smith girl.

Thus some provinces contained tribes which detested each other, and to them independence meant an opportunity for war.

Thus the image of God is that which makes a man a man and not an oyster ; ;

Thus, a well-planned adobe wall of the appropriate thickness is very effective at controlling inside temperature through the wide daily fluctuations typical of desert climates, a factor which has contributed to its longevity as a building material.