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 the sum is extended over all ( k, j ) with and with

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 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, Joseph Lynch Martin ( a. k. a. " Greenbrier Joe ") established a trading post called Pensacola.

Here k may be negative.

Here, dq < sub > k </ sub >/ T represents a very small exchange of heat energy at temperature T. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process.

Here we see that the period of 3 < sup > k </ sup > modulo 7 is 6.

Here we show that the size of the ( sub ) tree rooted at any node x of degree d in the heap must have size at least F < sub > d + 2 </ sub >, where F < sub > k </ sub > is the kth Fibonacci number.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Here, S ( P, Ei ) represents the similarity between the prototype and the ith exemplar, k is the subscript for the dimensions ( k

Here the maps i: A ∩ B ↪ A, j: A ∩ B ↪ B, k: A ↪ X, and l: B ↪ X are inclusion maps and denotes the direct sum of abelian groups.

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 Z < sub > l </ sub > denotes the ℓ-adic integers, but the definition is by means of the system of ' constant ' sheaves with the finite coefficients Z / l < sup > k </ sup > Z.

" Here A is a polynomial ring over a field, that is, A = k ..., x < sub > n </ sub >.

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 X < sub > i </ sub > are any vector fields on P. Dϕ is a tensorial ( k + 1 )- form on P.

Here, if we define the sequence p < sup > k </ sup > by

Here to be a global norm means to be an element k of K such that there is an element l of L with ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from L < sub > P </ sub >; here the " prime " p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean.

Here, on the hottest day, it is cool beneath the stone and fresh from the water flowing in the sluices at the bottom of the vaults.

Here in these little rooms -- or stages arched open to the sky and river -- they choose a few lines out of the hundreds they may know and sing them according to one of the modes into which Persian music is divided.

Here, on a desk, is a stack of pamphlets representing the efforts of some of the best men of the day to penetrate these questions.

Here, if anywhere, it is not wholly incontrovertible.

Here we may observe that at least one modern philosophy of history is built on the assumption that ideas are the primary objectives of the historian's research.

Here an important caveat is in order.

Here, then, is what Swift would have called a modest proposal by way of a beginning.

But this we know: Here is a great life that in every area of American politics gives the American people occasion for pride and that has invested the democratic process with the most decent qualities of honor, decency, and self-respect.

Here is a word of advice when you go shopping for your pansy seeds.

Here then is our problem: aircraft are vital to winning a war today because they can perform those missions which a missile is totally incapable of performing ; ;

Here is truly a `` Great Recording of the Century '', and its greatness is by no means diminished by the fact that it is not quite perfect.

Here is an original kedgeree recipe from the Family Club's kitchen:

Here is the promise of a vacation trip they can afford.

Here is where things stand today:

Here the Af distance is 2.44 Aj.

Here the pulmonary vein, as in type 2,, is noted to draw away from the bronchus, and to follow a more direct, independent course to the hilum ( figs. 23, 24 ).

Here the number of trials is a random variable, not a fixed number.

Here there is a specific preventive component which applies in a more generalized sense to any casework situation.

( Here an entry is a form plus the information that pertains to it.

Here again, in the written language it is possible to help the reader get his stresses right by using underlining or italics, but much of the time there is simply reliance on his understanding in the light of context.

Here is the best short explanation of the origins of the Cold War that has been written.

Here the left-hand side is equal to, namely the first-order polygamma function defined through ; the gamma function is equal to if is a positive integer.

( Here, Peano Arithmetic ( PA ) is understood as the first-order theory of arithmetic with symbols of addition and multiplication, and schema of recursion.

Here, the denominator appearing in the n < nowiki >' th </ nowiki > term of the perturbative expansion could become arbitrarily small, causing the n < nowiki >' th </ nowiki > correction to be as large or larger than the first-order correction.