For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

Thus, the total applied torque about the origin is given by

Thus, the sum of all applied forces and torques ( with respect to the origin of the coordinate system ) in the body can be given by

Thus, the word myth is of Greek origin, while pith is a Germanic word.

Thus, one cannot expect to find a clear origin.

Thus the total revenue curve for a monopoly is a parabola that begins at the origin and reaches a maximum value then continuously decreases until total revenue is again zero.

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 the origin of the original space can be found at ( 0, 0, ... 0, 1 ).

Thus, it is not correct to establish a strict common origin, but a common cultural identity.

That word came from imbarrare, " to block " or " to bar ", which is a combination of in -, " in ", with barra, " bar " ( from the Vulgar Latin barra, which is of unknown origin ).< sup > 11 </ sup > The problem with this theory is that the first known usage of the word in Italian was by Bernardo Davanzati ( 1529 – 1606 ), long after the word had entered Spanish .< sup > 12 </ sup > Thus, modern scholars believe that the Italian word actually came from the Spanish one .< sup > 13 </ sup >

Thus the spelling of a word is dependent principally upon its structure, its relationship to other words, and its language or origin.

Thus the streamlines are circles that are tangent to the x-axis at the origin.

Thus, the major peculiarities of the ancient Greek political system were ; firstly, its fragmentary nature, and that this does not particularly seem to have tribal origin ; and secondly the particular focus on urban centres within otherwise tiny states.

Thus they express the intermediate filament protein vimentin, a feature used as a marker to distinguish their mesodermal origin.

Thus the word Guacara was pronounced Wakala by the Seminoles who are Muskhogean in their origin and language.

Thus Pennsylvania Dutch, the term, includes residents which historically lived near the " German " origin Pennsylvania Dutch of Germany, in both France and Switzerland, whose borders over time had been traded around to be included in one country and then another, and the Pennsylvania Dutch were not then technically JUST from Germany, although they did share common bloodlines and ancestries, living in close locale.

Thus, the origin and content of the concept of " electron " is questionable.

Thus the origin of the word ' Pohja ' in its use as " North ".

Thus the origin of terrestrial planets is now considered to be an almost solved problem.

Thus, he suggested that they are biological in origin and can therefore be safely and correctly categorised.

Thus the carpets produced became typical of the Indian origin and gradually the industry began to diversify and spread all over the subcontinent.

Thus the length of the spiral measured from the origin can be expressed as:

Thus, by 1928 basic physical understanding of the origin of CFE from bulk metals had been achieved, and the original Fowler-Nordheim-type equation had been derived.

Thus, a is a real Lorentz four-vector representing the change of space-time origin x ↦ x − a where x is in the Minkowski space M < sup > 4 </ sup > and L is a Lorentz transformation, which can be defined as a linear transformation of four-dimensional space-time which preserves the Lorentz distance c²t² − x ⋅ x of every vector ( ct, x ).

Thus far the only people who have been attested with a high level of genetic, historical, linguistic and cultural research to be the descendants of the ancient Mesopotamians are the Assyrian Christians of Iraq and its surrounding areas in north west Iran, north east Syria and south eastern Turkey.

Thus, medical practitioners would select treatment options for specific cases based on the best research for each patient they treat.

Thus, the quality of evidence to support a clinical decision is a combination of the quality of research data and the clinical ' directness ' of the data.

Thus, light extraction in LEDs is an important aspect of LED production, subject to much research and development.

Thus their definition of " liberal " includes the RAND Corporation, a nonprofit research organization with strong ties to the Defense Department.

Thus, the theory ‘ is continually evolving as it is brought to bear on solving problems of a research or practical nature ’.

Thus, the specific processes that can be probed with PET are virtually limitless, and radiotracers for new target molecules and processes are continuing to be synthesized ; as of this writing there are already dozens in clinical use and hundreds applied in research.

Thus, accurately fitting and validating power-law models is an active area of research in statistics.

Thus a conservative classification of Sino-Tibetan / Tibeto-Burman would posit several dozen small coordinate families and isolates ; attempts at subgrouping are either geographic conveniences or hypotheses for further research.

Thus, despite ongoing research, a complete understanding of the causes of mental disability in Graves ’ disease awaits a full description of the effects on neural tissue of thyroid hormones, as well as of the underlying autoimmune process.

Thus, research originally focused on it.

Thus there came about the research for " Le problème de Cauchy pour les équations différentielles d ' un fluide général "; the idea that Prof. Hironaka called " the Nash blowing-up transformation "; and those of " Arc Structure of Singularities " and " Analyticity of Solutions of Implicit Function Problems with Analytic Data ".

Thus began a field of research which is still quite active today.

Thus, considerable research has been applied to speeding up the problem.

Thus far, there have been no comprehensive, systematic research studies of either the strokes, or the stroke clusters.

Thus, marketing research may also be described as the systematic and objective identification, collection, analysis, and dissemination of information for the purpose of assisting management in decision making related to the identification and solution of problems and opportunities in marketing.

Thus systematic planning is required at all the stages of the marketing research process.

Thus, there is a validity issue with much of the research into flashbulb memory in general, as well as any apparent gender differences found therein.

Thus, a lower rating may reflect the lack of research rather than the ineffectiveness of biofeedback for the problem.

Thus, an important industrial requirement is " robustness in poor vacuum conditions "; this needs to be taken into account in research on new emitter materials.

Thus there is scope for future research into cultivation of safer strains.

Thus far, the approach of Cultural Linguistics has been adopted in several areas of applied linguistic research, including intercultural communication, second language learning, and World Englishes.

Thus, considerable concern has arisen over the reading and writing research in L2 due to the booming of Internet.

Thus, as Holmwood states, “ Merton explicitly made power and conflict central issues for research within a functionalist paradigm ”.

Thus, most routine histological and microanatomical illustrations in textbooks and research papers were camera lucida drawings rather than photomicrographs.