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

Thus, an analog signal is one represented by a continuous stream of data, in this case along an electrical circuit in the form of voltage, current or charge changes ( compare with digital signals below ).

Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i. e. acting on every point in it.

Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.

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 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 there is very wide and continuous range of timescales covering the response of such a system to externally applied stress.

Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

Thus, subcapsular sinus is a sinus immediately deep to the capsule, and its endothelium is continuous with that of the afferent lymph vessel.

Thus the normals allow the calculation of the relative surface heights of the line on the object using a line integral, if we assume a continuous surface.

Thus, during the zoom, there is a continuous perspective distortion, the most directly noticeable feature being that the background appears to change size relative to the subject.

Thus for example continuous real-valued functions on a topological space form a presheaf of rings on the space.

Intuitively, the continuous operator A never " lengthens " any vector more than by a factor of c. Thus the image of a bounded set under a continuous operator is also bounded.

A punitive force sent by the Bulgarians was defeated. Thus we see a continuous cycle of dynastic strife amongst Vlastimir ’ s successors, stirred on by the Byzantine and Bulgarians, who were effectively using the Serbs as pawns.

Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ ( g ) is a unitary operator for all g ∈ G ; i. e., for all v, w ∈ H. ( I. e.

Thus, a continuous system such as a plate or shell is modeled as a discrete system with a finite number of elements interconnected at finite number of nodes.

Thus, RCA is often considered to be an iterative process, and is frequently viewed as a tool of continuous improvement.

Thus, in the ideal continuous case, detection of zero-crossings in the second derivative captures local maxima in the gradient.

The category Top of topological spaces has as morphisms the continuous functions, where a function f is continuous if the inverse image f < sup > − 1 </ sup >( O ) of any open set in the codomain of f is open in the domain of f. Thus any continuous function f from a space X to a space Y defines an inverse mapping f < sup > − 1 </ sup > from Ω ( Y ) to Ω ( X ).

Thus, while smaller in terms of total number of planes, the FAE of January 1995 felt qualitatively capable of facing the FAP on more or less equal terms, in striking contrast to the situation during the crisis of 1981, where except for a small number of missions, the FAE had been kept on the ground armed and ready for immediate action, to be committed only in case of a full-fledged war, their positions in the Cordillera del Cóndor was well defended due to tactically placed SAMs by the FAE, and units armed with British-made Blowpipe and Russian-made SA-16 MANPADs, nevertheless, such defences didn ´ t stop the continuous raids of the Peruvian Air Force, which lost several aircraft in the effort.

Thus, sets of complementary wavelets are useful in wavelet based compression / decompression algorithms where it is desirable to recover the original information with minimal loss.

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

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

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 two-dimensional Fourier transform of the initial function is the one variable Fourier transform of the Radon transform of that function.

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 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, 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.