A. M. Lyapunov proved that if the system of the first approximation is regular ( e. g., all systems with constant and periodic coefficients are regular ) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable.

The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques ; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.

For instance, if all of the coefficients were equal to unity, a so-called boxcar function, then it would implement a low-pass filter with a low frequency gain of N + 1 and a frequency response given by the sinc function.

They have fairly low coefficients of thermal expansion ( 7740 Pyrex CTE is 3. 25 /° C as compared to about 9 /° C for a typical soda-lime glass ), making them more dimensionally stable.

When the solubility of the salt is very low the activity coefficients of the ions in solution are nearly equal to one.

If a significant amount of the light passed through the sample contains wavelengths that have much lower extinction coefficients than the nominal one, the instrument will report an incorrectly low absorbance.

( Since activity coefficients tend to unity at low concentrations, activities in the Nernst equation are frequently replaced by simple concentrations.

Most polar molecules have low solubility in the hydrocarbon core of a lipid bilayer and consequently have low permeability coefficients across the bilayer.

Since this tends to be a rather large area compared to the projected frontal area, the resulting drag coefficients tend to be low: much lower than for a car with the same drag and frontal area, and at the same speed.

Recent research showed that a-Si: H PV with low temperature coefficients allow the PVT to be operated at high temperatures, creating a more symbiotic PVT system and improving performance of the a-Si: H PV by about 10 %.

The closed-cell structure foams have higher dimensional stability, low moisture absorption coefficients, and higher strength compared to open-cell-structured foams.

In the low temperature limit, the limitations of the Debye model mentioned above do not apply, and it gives a correct relationship between ( phononic ) heat capacity, temperature, the elastic coefficients, and the volume per atom ( the latter quantities being contained in the Debye temperature ).

This extraction is highly controversial among scientists in Australia, regarding the agriculture industry's high water use in a region extremely short of water ( as much due to exceptionally low run-off coefficients as to low rainfall ).

Hydrophobic drugs with high octanol / water partition coefficients are preferentially distributed to hydrophobic compartments such as lipid bilayers of cells while hydrophilic drugs ( low octanol / water partition coefficients ) preferentially are found in hydrophilic compartments such as blood serum.

This decomposition is repeated to further increase the frequency resolution and the approximation coefficients decomposed with high and low pass filters and then down-sampled.

Rainwater harvesting can ensure an independent water supply during mains water restrictions and are effective in ' green droughts ', capturing low rainfall events where runoff coefficients are not sufficient to cause flow into dam storages.

Other alloys with similarly low temperature coefficients are known, such as manganin ( Cu < sub > 86 </ sub > Mn < sub > 12 </ sub > Ni < sub > 2 </ sub >).

Borosilicate glasses are known for having very low coefficients of thermal expansion (~ 3 × 10 < sup >− 6 </ sup > /° C at 20 ° C ), making them resistant to thermal shock, more so than any other common glass.

For the same reason, the output of a DAC requires a low-pass analog filter, called a reconstruction filter, as the output signal must be bandlimited, to prevent aliasing ( here meaning Fourier coefficients being reconstructed as low frequency waves, not as higher frequency aliases ), as in the Whittaker – Shannon interpolation formula.

Today's composite resins have low polymerization shrinkage and low coefficients of thermal shrinkage, which allows them to be placed in bulk while maintaining good adaptation to cavity walls.

In the DWT, each level is calculated by passing only the previous wavelet approximation coefficients ( cA < sub > j </ sub >) through discrete-time low and high pass quadrature mirror filters.

Here, the coefficients χ < sup >( n )</ sup > are the n-th order susceptibilities of the medium and the presence of such a term is generally referred to as an n-th order nonlinearity.

It was there that he and Debye developed their theory ( the Debye – Hückel theory, in 1923 ) of electrolytic solutions, elucidating the behavior of strong electrolytes by considering interionic forces, in order to account for their electrical conductivity and their thermodynamic activity coefficients.

If we now restrict our attention to that portion of the cos ( ωt ) coefficient which varies linearly with V, and then ask ourselves, at what input voltage level, V, will the coefficients of the first and third order terms have equal magnitudes ( i. e., where the magnitudes intersect ), we find that this happens when

The original concept of Reed – Solomon coding describes encoding of k message symbols by viewing them as coefficients of a polynomial p ( x ) of maximum degree k − 1 over a finite field of order N, and evaluating the polynomial at n > k distinct input points.

When a DCT has been used, the coefficients are typically scanned using a zig-zag scan order, and the entropy coding typically combines a number of consecutive zero-valued quantized coefficients with the value of the next non-zero quantized coefficient into a single symbol, and also has special ways of indicating when all of the remaining quantized coefficient values are equal to zero.

In this case is zero unless y is either x + 1, x, or x − 1, when y is x + 1 or x − 1, the R matrix has value c, and in order for the sum of the R matrix coefficients to equal zero, the value of must be − 2c.

An order d linear homogeneous recurrence relation with constant coefficients is an equation of the form

where the coefficients k < sub > i </ sub > are determined in order to fit the initial conditions of the recurrence.

Linear recursive sequences are precisely the sequences whose generating function is a rational function: the denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence.

The denominator is a transform of the auxiliary polynomial ( equivalently, reversing the order of coefficients ); one could also use any multiple of this, but this normalization is chosen both because of the simple relation to the auxiliary polynomial, and so that.

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

The coefficients for a filter of order are defined as

Here the user specifies a desired frequency response, a weighting function for errors from this response, and a filter order N. The algorithm then finds the set of N coefficients that minimize the maximum deviation from the ideal.

where and are geocentric ( spherical ) latitude and longitude respectively, are the fully normalized associated Legendre polynomials of degree and order, and and are the numerical coefficients of the model based on measured data.

EGM96 contains a full set of coefficients to degree and order 360 ( i. e. ), describing details in the global geoid as small as 55 km ( or 110 km, depending on your definition of resolution ).

For example, many of the authors of EGM96 are working on an updated model that should incorporate much of the new satellite gravity data ( see, e. g., GRACE ), and should support up to degree and order 2160 ( 1 / 6 of a degree, requiring over 4 million coefficients ).

< font style =" vertical-align :- 25 %;"> </ font > NGA has announced the availability of EGM2008, complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159.

Where the associated time series flips the order of the coefficients because the linear filter is a convolution, and so both have the same index in this sum.

The rate equation does not simply reflect the reactants stoichiometric coefficients in the overall reaction: It is first order in H < sub > 2 </ sub >, although the stoichiometric coefficient is 2 and it is second order in NO.

The idea is to write the solution of the differential equation as a sum of certain " basis functions " ( for example, as a Fourier series which is a sum of sinusoids ) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial ( written in decreasing order of exponents ), we obtain the f-polynomial of Δ.