Here k is first-order rate constant having dimension 1 / time, ( t ) is concentration at a time t and < sub > 0 </ sub > is the initial concentration.

Here N < sub > x </ sub > is the number of x atoms per volume, λ the electron escape depth, θ the analyzer angle, T the transmission of the analyzer, I ( t ) the electron excitation flux at depth t, dΩ the solid angle, and δt is the thickness of the layer being probed.

This is an approximate equation that is derived from the incompressible Navier-Stokes equations ( written in spherical coordinate system ) and describes the motion of the radius of the bubble R as a function of time t. Here, μ is the viscosity, p the pressure, and γ the surface tension.

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Here u ( x, t ) is the temperature of the rod at position x and time t and α is a constant that depends on how fast heat diffuses through the rod.

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Here A < sub >–</ sub > and A < sub >+</ sub > are the absorbances of LCP or RCP, respectively ; ω is the modulator frequency – usually a high acoustic frequency such as 50 kHz ; t is time ; and δ < sub > 0 </ sub > is the time-dependent wavelength shift.

Here G is final average grain size, G < sub > 0 </ sub > is the initial average grain size, t is time, m is a factor between 2 and 4, and K is a factor given by:

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Here N ( t ) is the quantity at time t, and N < sub > 0 </ sub >

Here dt is the matrix of one-forms obtained by taking the exterior derivative of the components of t.

Here, ∆ t < sub > 1 </ sub > and ∆ t < sub > 2 </ sub > represent the propagation delay s of the logic elements.

Here, the right hand side is clearly not convergent for any non-zero value of t. However, by truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of for sufficiently small t. Substituting and noting that results in the asymptotic expansion given earlier in this article.

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Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

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Here K denotes the field of real numbers or complex numbers, I is a closed and bounded interval b and p, q are real numbers with 1 < p, q < ∞ so that

Here highbit ( S ) denotes the most significant bit of S ; the '< tt >*</ tt >' operator denotes unsigned integer multiplication with lost overflow ; '< tt >^</ tt >' is the bitwise exclusive or operation applied to words ; and P is a suitable fixed word.

# If A is a cartesian product of intervals I < sub > 1 </ sub > × I < sub > 2 </ sub > × ... × I < sub > n </ sub >, then A is Lebesgue measurable and Here, | I | denotes the length of the interval I.

Here is the particle's velocity and × denotes the cross product.

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

Here denotes the maximum-likelihood estimate, and is the sample mean of the observations.

Here, denotes the natural logarithm.

Here C < sub > b </ sub >( X ) denotes the C *- algebra of all continuous bounded functions on X with sup-norm.

Here denotes vertical composition of natural transformations, and denotes horizontal composition.

( Here V0, V1, and V2 denote verbs and Ny denotes a noun.

Here y *( w ) denotes the value of the linear functional y * ( which is an element of the dual space of W ) when evaluated at the element w ∈ W. This scalar in turn is multiplied by x to give as the final result an element of the space V.

Here, || ||< sub > 2 </ sup > is the matrix 2-norm, c < sub > n </ sub > is a small constant depending on n, and ε denotes the unit round-off.

Here 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z / 2Z is given by reducing integers modulo 2.

Here, denotes the radical of J and I ( U ) is the ideal of all polynomials which vanish on the set U.

Here, " prequel " denotes status as a " franchise-renewing original " that depicts events earlier in the ( internally inconsistent ) narrative cycle than those of a previous installment.

Here N < sub > m </ sub > denotes the number of turns in loop m, Φ < sub > m </ sub > the magnetic flux through this loop, and L < sub > m, n </ sub > are some constants.

( Here K ( a ) denotes the smallest subfield of L containing K and a )

Here denotes the standard complex inner product on and.

Here it also denotes a hand gesture, now linked to three other hand mudrās — the action ( karma ), pledge ( samaya ), and dharma mudrās — but also involves " mantra recitations and visualizations that symbolize and help to effect one ’ s complete identification with a deity ’ s divine form or awakening mind ( bodhicitta ).

Here, denotes the resolution limit in arcseconds and is in millimeters.

As explained above, Gaussian elimination writes a given m × n matrix A uniquely as a product of an invertible m × m matrix S and a row-echelon matrix T. Here, S is the product of the matrices corresponding to the row operations performed.

Here a rank 1 tensor ( matrix product of a column vector and a row vector ) is the same thing as a rank 1 matrix of the given size.

Here D is a traceless matrix such that αDe gives the derivative of e with respect to z.

Here ℓ is the rank of the covariance matrix.

Here is a-dimensional vector, is the known-dimensional mean vector, is the known covariance matrix and is the quantile function for probability of the chi-squared distribution with degrees of freedom.

Here a < sub > 1 </ sub >, ..., a < sub > m </ sub > denote the rows of the matrix A.

Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v. This is a reflection in the hyperplane perpendicular to v ( negating any vector component parallel to v ).

* the algebra of all n-by-n matrices over the field ( or commutative ring ) K. Here the multiplication is ordinary matrix multiplication.

Here M < sup > T </ sup > denotes the transpose of the matrix M. One can easily verify that the set of all such matrices forms a group.

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( Here and are matrices the same size as ( is the identity matrix ); and is the square root of − 1 ).

Here I < sub > p </ sub > denotes a p × p identity matrix.

Here, is the Hankel matrix of the sequence.

Here ω < sup > α </ sup >< sub > β </ sub > defines a k × k matrix of one-forms on U. In fact, given any such matrix the above expression defines a connection on E restricted to U. This is because ω < sup > α </ sup >< sub > β </ sub > determines a one-form ω with values in End ( E ) and this expression defines ∇ to be the connection d + ω, where d is the trivial connection on E over U defined by differentiating the components of a section using the local frame.

Here the rank of Q < sub > i </ sub > should be interpreted as meaning the rank of the matrix B < sup >( i )</ sup >, with elements B < sub > j, k </ sub >< sup >( i )</ sup >, in the representation of Q < sub > i </ sub > as a quadratic form:

Here it is assumed that the mixing matrix is square ().

Here π < sub > n </ sub > is a homotopy group, GL ( R ) is the direct limit of the general linear groups over R for the size of the matrix tending to infinity, B is the classifying space construction of homotopy theory, and the < sup >+</ sup > is Quillen's plus construction.

Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix.