Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

Thus, the molecular formula of ethanol may be written in a paper in conformational, three-dimensional, full two-dimensional ( indicating every bond with no three-dimensional directions ), compressed two-dimensional ( CH < sub > 3 </ sub >– CH < sub > 2 </ sub >– OH ), separating the functional group from another part of the molecule ( C < sub > 2 </ sub > H < sub > 5 </ sub > OH ), or by its atomic constituents ( C < sub > 2 </ sub > H < sub > 6 </ sub > O ), according to what is discussed.

Thus, Marcel waving produced a two-dimensional wave, by thermal means only and the change was produced by plastic flow of the hair, rather than by any chemical means.

Thus, the two-dimensional Dirichlet problem is essentially the same problem as that of finding a meromorphic extension of a function defined on a boundary.

Thus, any periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:

Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function .< ref group =" note "> We may also note that:

Thus we have the important result that when a discrete data sequence, s, is proportional to samples of an underlying continuous function, s ( t ), one can deduce something about the continuous Fourier transform, S ( ƒ ).

Thus Fourier analysis might be used to decompose a sound into a unique combination of pure tones of various pitches.

Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1 / T.

Thus, E ( k ) dk represents the contribution to the kinetic energy from all the Fourier modes with k < | k | < k + dk, and therefore,

Thus for normalized cuspidal Hecke eigenforms of integer weight, their Fourier coefficients coincide with their Hecke eigenvalues.

Thus one can think of multiplier operators as operators which are diagonalized by the Fourier transform.

Thus rotation of the Fourier operator gives rise to the fractional Fourier transform, which is related to the chirplet transform.

Thus they can also be thought of as a generalized Fourier series in which the basis functions are the normal modes of an atmosphere at rest.

Thus, using just truncation in the time domain, or equivalently low-pass filters in the frequency domain, one can cut out any convex set in time-frequency space ; just using time domain or frequency domain methods without fractional Fourier transforms only allow cutting out rectangles parallel to the axes.

Thus Dirichlet's theorem says in particular that the Fourier series for f converges and is equal to f wherever f is continuous.

Thus, an expanded definition for data warehousing includes business intelligence tools, tools to extract, transform and load data into the repository, and tools to manage and retrieve metadata.

Thus a 2 × 2 matrix with determinant − 2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation.

Thus there is a three-parameter family of rotations which transform the sphere onto itself, this is the rotation group SO ( 3 ).

Thus, the heat used to transform the working body of molecules from one state to the next cannot be used to do external work, e. g., to push the piston.

Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation.

Thus, in the scaleogram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point.

Thus, as the Pharisees argued that all Israel should act as priests, the Rabbis argued that all Israel should act as rabbis: " The rabbis furthermore want to transform the entire Jewish community into an academy where the whole Torah is studied and kept .... redemption depends on the " rabbinization " of all Israel, that is, upon the attainment of all Jewry of a full and complete embodiment of revelation or Torah, thus achieving a perfect replica of heaven.

Thus these basis qubits and will transform in the following way:

Thus, there is a chiral symmetry of these fermions, e. g., SU ( N < sub > f </ sub >)< sub > L </ sub > ⊗ SU ( N < sub > f </ sub >)< sub > R </ sub >, if they all transform according the same complex representation of G < sub > TC </ sub >.

where α ∈ S < sup > n − 1 </ sup > is a unit vector and s ∈ R. Thus the n-dimensional Radon transform may be rewritten as a function on S < sup > n − 1 </ sup >× R via

Thus the field and its derivative transform in the same way.

Thus hair cells transform the pressure waves of the sound into receptor potentials to initiate sensory transduction.

Thus, using Sebastian Ballesteros ( the latest Cheetah ) as a lure, Dr. Psycho and Circe used periodical mental conditioning and physical implants to gradually transform Vanessa into a new Silver Swan and set her against her former friend.

Thus the wavelet transform of a signal may be represented in terms of both time and frequency.

Thus, our sampling of the DTFT causes the inverse transform to become periodic.

Thus we can transform any minor into any other minor by adding rows and columns, switching them, and multiplying a row or a column by − 1.

Thus for an ion which traverses a 1 m flight path, across a time of 2000 ns, given an initial accelerating voltage of 5000 V and noting that one amu is 1 × 10 < sup >− 27 </ sup > kg, the mass-to-charge ratio ( more correctly the mass-to-ionisation value ratio ) becomes Z amu / charge.

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, the universe would have been even more special before the thermalization than after .” The problem of specific or “ fine-tuned ” initial conditions would not have been solved ; it would have gotten worse.

Thus, the initial step corresponds to k =

Thus, the initial stage of Ebola is not very contagious since its victims experience only internal hemorrhaging.

Thus the initial assumption that p and q − 1 are relatively prime is untenable.

Thus the initial input change circulates along the feedback loop and grows in an avalanche-like manner until finally Q1 switches off and Q2 switches on.

Thus for a transition, K represents the core level hole, the relaxing electron's initial state, and the emitted electron's initial energy state.

Thus it is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven.

Thus, we might naively reason that one can entirely convert the initial combustion heat of the chemical reaction into the work of pushing the piston.

Thus, it was at the core of the initial development of the city.

Thus, the system can be analyzed with Schrödinger's equation and the guiding equation, with an initial distribution for the particles in the system ( see the section on the conditional wave function of a subsystem for details ).

Thus, with succeeding observations, the initial conditions become more and more restricted.

Thus the answer to the initial question ' What is knowledge ' would be heavily circuitous-correct judgement accompanied by ' knowledge ' of the differentness, which Socrates admits is ' silly ' ( 210a ).

Thus, in the above example, after an increase and decrease of percent, the final amount, $ 198, was 10 % of 10 %, or 1 %, less than the initial amount of $ 200.

Thus the time control " 20 minutes + 15 minutes byoyomi " on IGS means that after the initial 20 minutes of thinking time are over, a player is granted 15 additional minutes, which may be spent however he chooses.

Thus the full dictionary does not need be sent with the encoded data ; just the initial dictionary containing the single-character strings is sufficient ( and is typically defined beforehand within the encoder and decoder rather than being explicitly sent with the encoded data.

Thus the second law of thermodynamics is thought to originate in the initial conditions in the universe.

Thus it came as no real surprise when his Man at the Crossroads became controversial, as it contained Moscow May Day scenes and a clear portrait of Lenin, not apparent in initial sketches.

Thus, a biopsy is routinely performed prior to the treatment in the initial stages of papilledema to detect whether a brain tumor is present.

Thus we see an extremely luminous supernovae of extended duration even though the initial explosion energy was the same as that of a normal supernova.

Thus the exchange requires both parties to put up an initial amount of cash, the margin.

Thus a combination must be at least three moves long, but the longer it takes to recoup the initial sacrifice, the more impressive the combination.

Thus, for example, after the initial upbeat Beethoven places the dynamic indication sforzando ( sf ) on each of the next five beats.