For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses, which are systems with the usual spin degrees of freedom for i = 1 ,..., N, with the special fixed " random " couplings Here the ε < sub > i </ sub > and ε < sub > k </ sub > quantities can independently and " randomly " take the values ± 1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e. g. of the energy are invariant.

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 ε is known as the relative permittivity tensor or dielectric tensor.

Here we have another unwanted error ε < sub > b </ sub > with value:

Here h is the zero-field height ( or unreduced height ) of the barrier, e is the elementary positive charge, F is the barrier field, and ε < sub > 0 </ sub > is the electric constant.

Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit.

( Here ε < sub > i </ sub > is the ith error, and is the ith residual.

Here, is the special regular expression denoting the empty set and ε the special one denoting the empty word ;

Here ε < sub > 0 </ sub > is the electric permittivity.

Here ε is the error term ( sometimes called noise ).

Here N is the number of electrons of the molecule under consideration, H is the usual electronic Hamiltonian, is the one-electron Fock operator, and ε < sub > i </ sub > is the orbital energy belonging to the doubly occupied spatial orbital φ < sub > i </ sub >.

Here Z < sub > 0 </ sub > is the impedance of free space, ε < sub > r </ sub > is the relative permittivity of substrate, w is the width of the strip, h is the thickness (" height ") of substrate, and t is the thickness of the strip metallization.

Here | u | and | v | denote the length of the strings respectively.

Here we denote with the Bernoulli number of the second kind ( only because the historical reason of formation of this article ) which differ from the first kind only for the index 1.

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

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

Here we denote a pair of corresponding lower and upper adjoints by f < sup > ∗</ sup > and f < sub > ∗</ sub >, respectively.

Here a and l denote radius and length of the wire, and Y is a constant that depends on the distribution of the current in the wire: Y =

Here, ran and dom denote the range and the domain, respectively.

Here the chain rule has been applied, and the subscripts denote partial derivatives (, ).

Here, the term energy is used in the generalized sense of signal processing to denote a variance of the signal.

Here the vertical bars denote the absolute value.

Here FG: D → D and GF: C → C, denote the respective compositions of F and G, and I < sub > C </ sub >: C → C and I < sub > D </ sub >: D → D denote the identity functors on C and D, assigning each object and morphism to itself.

Here d < sub > X </ sub > and d < sub > Y </ sub > denote the metrics on X and Y respectively.

Here, let R < sup > n + 1 </ sup > denote the real coordinate space of n + 1 dimensions, and regard the landscape to be painted as a hyperplane in this space.

Here we denote transpose by an apostrophe.

Here " T +" or " T −" denote that the result of the test is positive or negative, respectively.

Here, it has to be noted that the military manuals of Asclepiodotus and Aelian use the term lochos to denote a file in the phalanx.

Here, we use to denote the braiding map associated to the permutation ( represented as a product of disjoint cyclic permutations ).

Here let denote the Riemann tensor, regarded as a tensor in.

Here,, and will be used to denote the initial velocity, the velocity along the direction of < var > x </ var > and the velocity along the direction of < var > y </ var >, respectively.

Here, overdots denote differentiation with respect to proper time, counted off along our timelike congruence ( i. e. we take the covariant derivative with respect to the vector field X ).

Here is the film thickness, and are the electric fields in the film and electrode at the interface, is the spontaneous polarization,, and & are the dielectric constants of the film and the metal electrode.

Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of Boltzmann's constant as, and the fourth equality follows from Stokes ' formula for the mobility.

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 as well, the determinants of supply, such as price of substitutes, cost of production, technology applied and various factors inputs of production are all taken to be constant for a specific time period of evaluation of supply.

Here ΔG is change in Gibbs free energy, T is absolute temperature, R is the gas constant and Q is reaction quotient.

Here n is the number of electrons / mole product, F is the Faraday constant ( coulombs / mole ), and ΔE is cell potential.

Here C < sub > p </ sub > is the heat capacity at constant pressure and α is the coefficient of ( cubic ) thermal expansion

Here constant α quantifies the strength of the interaction between adjacent molecules.

Here the disciples and women wait and they gave themselves up to constant prayer: " And when they were come in, they went up into an upper room, where abode both Peter, and James, and John, and Andrew, Philip, and Thomas, Bartholomew, and Matthew, James the son of Alphaeus, and Simon Zelotes, and Judas the brother of James.

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.

Here, the " constant " of integration is no longer a constant, but instead a function of all the variables of the original function except x.

Here a is constant acceleration, or in the case of bodies moving under the influence of gravity, the standard gravity g is used.

Here it is assumed that application of the force causes the spring to expand and, having derived the force constant by least squares fitting, the extension can be predicted from Hooke's law.

Here Δk is the field-dependent difference between LCP and RCP absorption, α is the electric permeability, n is the index of refraction, H is the applied magnetic field, k is the Boltzmann constant and T is the temperature.

Here are the unknowns, are the coefficients of the system, and are the constant terms.

Here V is the molar volume of a substance, T < sub > C </ sub > is the critical temperature and k is a constant valid for almost all substances.

Here Q is the molar activation energy, R is the ideal gas constant, T is absolute temperature, and K < sub > 0 </ sub > is a material dependent factor.

Here p is the momentum of the particle, m is the mass of the particle, E is the kinetic energy of the particle, and ħ is the reduced Planck's constant.

Here he became private chaplain to and benefited from the hospitality of Richard Vaughan, 2nd Earl of Carbery, whose mansion, Golden Grove, is immortalized in the title of Taylor's still popular manual of devotion, and whose first wife was a constant friend of Taylor.

Here is called a spherical harmonic function of degree ℓ and order m, is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively.

Here, just as in Southern Europe, the demands of more intensive agriculture and the invention of the plough, trading, mining and smelting, tanning, building and construction in the growing towns and constant warfare, including the demands of naval shipbuilding, were more important forces behind the destruction of the forests than was shifting cultivation.

Here U is internal energy, H is enthalpy and the entropy, S, is taken to be constant ( see # History ).