Thus a more detailed analysis of the bubble's motion is needed beyond Rayleigh-Plesset to explore the additional energy focusing that an internally formed shock wave might produce.

Thus, before 10 years ago when more detailed studies started, it was considered that there could be anywhere from 2-4 genera within the Actinidiaceae.

Thus it provides an intense, detailed exploration of its subject, providing to some degree both the concentrated focus of the short story and the broad scope of the novel.

Thus, the folklorist Anatole Le Braz gives a detailed account of one ballad singer, Yann Ar Minouz, who wrote and performed songs traveling through Brittany in the late nineteenth century and selling printed versions.

Thus, although Gelukpa scholars give detailed intepretations of the systems of breath meditation set forth in Vasubandu's and Asaṅga's texts, they may not fully account for the higher stages of breath meditation set forth in those texts.

Thus detailed moving graphics caused large ugly fringes of rapidly changing colors to follow them around.

Thus for a more detailed view, the crater must be viewed from orbit.

Thus, from the view-point of the meaning too, this word can be a suitable title for this Surah, for in it the detailed story of the Prophet Moses has been related.

Thus the safety factor dramatically impacts company fortunes and is often worth the additional engineering expense required for detailed analysis and implementation.

Thus, in order to reduce the time required for analysis, less detailed systems ( like MTM-B and MaxiMOST ) are usually used when possible.

( See the most detailed explanation and exposition of this term in Epshtein's articles " Theses on Metarealism and Conceptualism ", " A Catalogue of the New Poetries " ( Russian version ) and in its English version Thus, " meta " means both " through " and " beyond " the reality that we all can see ; hence, " metarealism " is the realism of the hyperphysical nature of things.

Thus, the principle of detailed balance is a sufficient but not necessary condition for the entropy growth in the Boltzmann kinetics.

Thus, a more detailed study would examine changes to the whole of the marginal distribution.

Thus all differences were leveled, and all contrasts erased, in a realm of no distinction, and the harmonious balance of the Lo Shu square could effectively symbolize the world in balanced harmony around a powerful central axis.

Thus, a good discus thrower needs to maintain balance within the circle.

Thus education would play an important role in creating a balance through rigorous exercise of beneficial organs while repressing baser ones.

Thus, Rajaratnam believed that maintaining a balance of power, rather than becoming a de-facto vassal of some larger power, would provide Singapore with freedom to pursue an independent foreign policy.

Thus fiscal federalism provides the tools for " application of the federal approach to governance which lies in its ability to balance the contrasting forces of centralization and decentralization " ( Sharma, 2005b: 177 ).

Thus, God produces the natural evils that follow from simple laws not because he wills those particular effects, but because he wills a world that best reflects his wisdom by achieving the best possible balance between the intrinsic perfection of the work and the simplicity and generality of its laws.

Thus, world conscience is a concept that overlaps with the Gaia hypothesis in advocating a balance of moral, legal, scientific and economic solutions to modern transnational problems such as global poverty and climate change, through strategies such as environmental ethics, climate ethics, natural conservation, ecology, cosmopolitanism, sustainability and sustainable development, biosequestration and legal protection of the biosphere and biodiversity.

Thus a balance must be reached that attempts to achieve both goals whilst not compromising either too much by doing so.

Thus RNA editing by the cellular enzymes is critical to the virus ’ life cycle because it regulates the balance between viral replication and virion assembly.

Thus correct stirrup length creates a balance between control and mobility that fits the discipline performed.

Thus maintaining the balance, of votes if one or the other is unable to attend.

Thus, should it be deemed necessary the NLS Committee will order one or more Midlands or East Anglian-based clubs in the sixth tier to switch divisions ( to move " horizontally " through the leagues so to speak ) so as to maintain numerical balance between North and South.

Thus we have a favourable balance compared with the average production in America or in Europe.

Thus, the crepuscular habit may both reduce predation pressure, thereby increasing the crepuscular populations, and in consequence offer better foraging opportunities to predators that increasingly focus their attention on crepuscular prey until a new balance is struck.

Thus, maintenance of neutral buoyancy in scuba must be a continuous and active procedure — the diving equivalent of balance, in a positive feedback environment.

Thus, the British market played a vital role in helping Canada and Australia stabilize their balance of payments in the immensely difficult economic conditions of the 1930s.

Thus the balance sheet consists of Asset

Thus the centi-ampere balance ranges from 1 to 100 centi-amperes, the deci-ampere balance from 1 to 100 deci-amperes, the ampere balance from 1 to 100 amperes, etc.

Thus, a stock refers to the value of an asset at a balance date ( or point in time ), while a flow refers to the total value of transactions ( sales or purchases, incomes or expenditures ) during an accounting period.

Thus, there is a possible trade-off to be made in an attempt to balance these possibly opposing tendencies .”

Thus the object would balance on any line through the centroid, including any median.

" Thus, directly and indirectly, both strength and weakness can upset the balance of security in international relations.

Thus, the algorithm can be written as a sequence of equations

Thus,,, etc .</ ref > which are particular cases of Diophantine equations.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Thus one can understand equations by a pure understanding of abstract topology or geometry — this idea is of importance in algebraic geometry.

Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found.

Thus, we use equations that have the same form as above.

Thus one can solve many recurrence relations by rephrasing them as difference equations, and then solving the difference equation, analogously to how one solves ordinary differential equations.

Thus there is no fixed point g satisfying these recursion equations.

Thus, the `` square root " may be omitted from these equations with no expected difference in the results of optimization.

in a ring R. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

Thus, a physicist might say that these equations are covariant.

Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube.

Thus, the equations work the same way if you put inputs with consistent units into them, even if you don't know what the unit in question happens to be.

Thus gravity currents may be simulated by the shallow water equations, with special dispensation for the leading edge which behaves as a discontinuity.

Thus, the Lorentz transformation equations take the form

Thus Volterra equations of the first and second types, respectively, would appear as:

Thus, linear momentum equations can be extended to rigid bodies by denoting that they describe the motion of the center of mass of the body.

Thus, several approaches to defining these equations exist, with some the result of curve-fitting experimental data and others based on a particular rheological model.

Thus, these models grew to include hundreds or thousands of equations describing the evolution of hundreds or thousands of prices and quantities over time, making computers essential for their solution.

Thus, Jagiellonian became the center of traditional analysis of differential equations and analytical functions.

Thus the competitive Lotka – Volterra equations are: