If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

Thus a regular space encountered in practice can usually be assumed to be T < sub > 3 </ sub >, by replacing the space with its Kolmogorov quotient.

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the viscosity () and the rate of energy dissipation ().

Thus the " null distribution " of the test statistic, i. e. its probability distribution assuming the null hypothesis is true, is stochastically smaller than the Kolmogorov – Smirnov distribution.

Thus, two drawbacks to elution mode chromatography, especially at the preparative scale, are operational complexity, due to gradient solvent pumping, and low throughput, due to low column loadings.

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 it has arithmetic complexity of O ( n < sup > 3 </ sup >).

Thus, the eventually attainable perfection and complexity of manufactured products, while they can be calculated in theory, cannot be predicted with confidence.

Thus, the eventually attainable perfection and complexity of

Thus the challenge of making sense of all this complexity is formidable.

Thus, the so-called blub paradox ( after a hypothetical programming language of average complexity called ' Blub ') says that anyone preferentially using some particular programming language will ' know ' that it is more powerful than some, but not that it is less powerful than others.

Thus the division between " hardware " and " software " began with abstraction being used to deal with the complexity of computing.

Thus, complexity is not a special quality of systems, like for instance the concept of " heat ", but simply a label for all systems whose computations are sophisticated.

Thus, two drawbacks to elution mode chromatography, especially at the preparative scale, are operational complexity, due to gradient solvent pumping, and low throughput, due to low column loadings.

Thus Teilhard postulates the Omega Point as this supreme point of complexity and consciousness, which in his view is the actual cause for the universe to grow in complexity and consciousness.

Thus, as God creates, the universe evolves towards higher forms of complexity, consciousness, and finally with humans, personality, because God, who is drawing the universe towards Him, is a person.

Thus, the eventually attainable perfection and complexity of manufactured products, while they can be calculated in theory, cannot be predicted with confidence.

Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.

To understand the full sweep and complexity of cultures across all of human history, anthropology draws and builds upon knowledge from the social and biological sciences as well as the humanities and physical sciences .” Thus, the field is divided into four subareas: Sociocultural Anthropology, Biological ( or Physical ) Anthropology, Archaeology, and Linguistic Anthropology.

Thus the need for services as much larger units of functionality than traditional functions or classes, lest the sheer complexity of thousands of such granular objects overwhelm the application designer.

Thus, the key result of BTW's paper was its discovery of a mechanism by which the emergence of complexity from simple local interactions could be spontaneous — and therefore plausible as a source of natural complexity — rather than something that was only possible in the lab ( or lab computer ) where it was possible to tune control parameters to precise values.

Thus the number of quantifier alternations required to define a set gives a measure of the complexity of the set.

Thus, Dumont ’ s pleasure in the expansion of power, Roux ’ s internal struggle, Nägeli ’ s drive towards complexity, and Rolph ’ s principle of insatiability and assimilation are fused together into the biological side of Nietzsche ’ s theory of will to power, which is developed in a number of places in his published writings.

Thus, this maintenance is desired to be performed infrequently, even when such increased intervals cause increased complexity and cost to the overhaul.

Thus, barring continual conquest of your neighbors ( which is always subject to diminishing returns ), innovation that increases productivity is – in the long run – the only way out of the dismal science dilemma of declining marginal returns on added investments in complexity.

Thus, the period of colonialism brought in sources for raw materials and captive markets for exports, while decolonialization brought new opportunities in the form of dependence.

Thus, he claims in his memoirs that the USSR had become " a raw material appendage to Bulgaria ," something obliquely confirmed by Gorbachev when he wrote in his memoirs that " Bulgaria was a country which had lived beyond its means for a long time.

Thus while increased Soviet fuel and raw materials export prices imposed a severe burden on Czechoslovakia, the cost was substantially less than if the country had imported these materials from noncommunist countries.

Thus color-indicating information in the sensor's raw data requires " aggressive " matrixing ( essentially, removal of common-mode signals ) to produce color data in a standard color space, which can increase color noise in low-light situations.

Thus, from the processing of raw materials into useful goods, the next step was the concept of the assembly line, as introduced by Eli Whitney.

Thus the overall size of motion vector plus prediction error will be greater than the raw encoding.

Thus the newly independent LP had to ' scramble for raw materials ,' particularly timber, as the July 29, 1990 Portland Oregonian explained.

Thus, overall placement in the caption area for each judge is more important than the raw score awarded by the judge.

Thus, intermediate consumption is an accounting flow which consists of the total monetary value of goods and services consumed or used up as inputs in production by enterprises, including raw materials, services and various other operating expenses.

Thus far, however, most such cases against file sharers have been settled out of court.

Thus, one can consider data compression as data differencing with empty source data, the compressed file corresponding to a " difference from nothing ".

Thus a representation that compresses a 10MB file to 2MB has a compression ratio of 2 / 10

Thus a representation that compresses a 10MB file to 2MB would yield a space savings of 1-2 / 10 =

Thus to add a person ( or subject ) to one's kill file is to arrange for that person to be ignored by one's newsreader in the future.

Thus, it is easy to convert from a vector file to a range of bitmap / raster file formats but it is much more difficult to go in the opposite direction, especially if subsequent editing of the vector picture is required.

Thus, rather than describing the outline of the glyph directly, a Metafont file describes the pen paths.

Thus, by changing the value of one of these parameters at one location in the Metafont file, one can produce a consistent change throughout the entire font.

Thus, each square has a unique identification of file letter followed by rank number.

Thus, the minimum memory required for a SAX parser is proportional to the maximum depth of the XML file ( i. e., of the XML tree ) and the maximum data involved in a single XML event ( such as the name and attributes of a single start-tag, or the content of a processing instruction, etc.

Thus the queen's file is named " Q " and the king's file is named " K ".

Thus, crosscut saws have sawteeth that are usually shaped, often with a metal file, in such a way that they form a series of tiny knifelike edges. The wood cells ( straws ) are contacted by the knife-edge of the tooth and cut.

Thus to get an application to support multiple languages one would design the application to select the relevant language resource file at runtime.

Thus many PostScript printer drivers have an option to save as EPS, or to add EPS DSC information to their output which you can " print to file ".

Thus if all the files in an archive are text files, and have ASCII names, then the archive is essentially an ASCII text file ( containing many NUL characters ).

Thus, an MXF file received & unwrapped may reveal SMPTE D10 compliant essence with eight mono AES audio components ; the recipient has no way of knowing whether these components are multiple stereo pairs, 5. 1 or serve some other purpose.

Thus they were not responsible for controlling any specific file downloads.

Thus, since downloading the Bible through peer-to-peer file sharing software is an addition to the list of well-established legitimate methods for obtaining a free Bible, the benefits of this addition are not substantial and the overall use of P2P software should not be considered a non-infringing use.

Thus it was labeled a combined-primary file or a combined-demand file.

Thus, pre-Sixth Edition Unix versions read an executable file into memory and jumped to the first low memory address of the program, relative address zero.

Thus the IDEF1 language was created to allow a neutral description of data structures that could be applied regardless of the storage method or file access method.