Thus a statistical unit, which would be the prime item of concern, is made of discrete components like organs or machine parts.

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, 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 the complexity of a digital filter and the computing time involved, grows inversely with, placing a higher cost on filter functions which better approximate the desired behavior.

Thus, when a differential amplifier is used to extract this difference, its output constitutes the low pass filter section.

Thus, given a linear filter evaluated through, for example, reverse correlation, one can rearrange the two spatial dimensions into one dimension, thus yielding a two dimensional filter ( space, time ) which can be decomposed through SVD.

Thus a blue filter makes the picture look blue.

Thus, the output of the second filter vanishes when the coefficients of the first one are input into it.

Thus, nodule mining could cause habitat alteration, direct mortality of benthic creatures, or suspension of sediment, which can smother filter feeders.

Thus, the circuit behaves as a low-pass filter.

Thus, if a site receives spam advertising " herbal Viagra ", the administrator might place this phrase in the filter configuration.

Thus, applying an edge detection algorithm to an image may significantly reduce the amount of data to be processed and may therefore filter out information that may be regarded as less relevant, while preserving the important structural properties of an image.

Thus a low-cut or rumble filter is used to remove infrasonic energy from a program which may consume undue amplifier power and cause excessive excursions in ( or even damage to ) speakers.

Thus a high-cut or hiss filter may be used to remove annoying white noise at the expense of the crispness of the program material.

Thus in an image processing context the original image ƒ can be recovered from the ' sinogram ' data Rƒ by applying a ramp filter ( in the variable ) and then back-projecting.

Thus a system that can filter images based on their content would provide better indexing and return more accurate results.

Thus, our optimal matched filter is

Thus, in the current context of downsampling, the anti-aliasing filter must be a low-pass filter.

Thus, the circuit behaves as a high-pass filter.

Thus capacitors " filter out " constant pressure differences and slowly-varying, low-frequency pressure differences, while allowing rapid changes in pressure to pass through.

Thus to apply a linear filter, one must first gamma decode the values – and if resampling, one must gamma decode, resample, then gamma encode.

Thus there are many references which filter through into Shakespeare's plays and sonnets derived from this body of thought ; particularly in the description of important characters, and to the power of music above all to ' charm the savage breast ', adjust the elements, and restore the equilibrium and balance, the ' harmony ' of the soul: his characters call for music and are spellbound or restored by it, and in elevated mood, may hear it in the air, or sense its immortal harmonies everywhere.

Thus the Lanczos filter, on its interval, is a product of two sinc functions.

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

Thus, in the immemorial way -- in the way of the right hand that knows and the left that does not -- was the stage set for tragedy at Bari.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

Thus, when the fluid finally does strike the object, it is forced to change its properties -- temperature, density, pressure, and Mach number -- in an extremely violent and irreversible fashion called a shock wave.

Thus, the Big Bang theory cannot and does not provide any explanation for such an initial condition ; rather, it describes and explains the general evolution of the universe going forward from that point on.

Thus, while the book exhibits considerable unity and probably reflects much of the historic Ezekiel, it is the product of a long and complex history and does not necessarily preserve the very words of the prophet.

Thus, the Buddha's silence does not indicate misology or disdain for philosophy.

Thus, one nucleus offers a more tightly bound position to an electron than does another nucleus, with the result that one atom may transfer an electron to the other.

Thus, covalent bonding does not necessarily require the two atoms be of the same elements, only that they be of comparable electronegativity.

Thus, though often de facto the case, Conservative Judaism's halakhic system does not inherently see Orthodox halakhic practice as acceptable and legitimate halakhic practice for a Conservative Jew.

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

Thus, according to Lutz, doublespeak is a form of language that defeats the purpose of inventing language because doublespeak does not communicate the truth but seeks to do the opposite and the doublespeak committee is tasked with correcting this problem that doublespeak has created in the world of language.

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

Thus, interlaced video captures samples the scene motion twice as often as progressive video does, for the same number of frames per second.

Thus, at UV frequencies and higher ( and probably somewhat also in the visible range ), electromagnetic radiation does far more damage to biological systems than simple heating predicts.

Thus, according to Hegel, a great man does not create historical reality himself but only uncovers the inevitable future.

Thus, Annie's strong drive to take revenge on women scholars in general and on Miss de Vine in particular is perfectly comprehensible, and in fact does not in itself prove her to be mentally deranged.

Thus the group of left and right handed 4x4 quaternion matrcies is Sp ( 8 ) xSU ( 2 ) which does include the standard model bosons:

Thus, " DD " does not stand for anything more than " Destroyer ".

Thus it does not stick to the neck and scalp when wet.

Thus he does not scruple to declare that Socrates and Heraclitus were Christians ( Apol., i. 46, ii.

Thus, union density provides a rough picture of union membership only ; it does not account for the circumstance that in some countries, also many persons under education, many unemployed persons, many retired persons and / or many persons who had to leave work due to occupational injuries may also be union members.

Thus in 1992, the Catechism of the Catholic Church, while affirming that " the Church does not know of any means other than Baptism that assures entry into eternal beatitude ", but also stating that " God has bound salvation to the sacrament of Baptism, but he himself is not bound by his sacraments ", stated: " As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them.

Thus, Merleau-Ponty does not postulate that " all consciousness is consciousness of something ", which supposes at the outset a noetic-noematic ground.