Using quantum theory Dirac showed that if magnetic monopoles exist, then one could explain the quantization of electric charge --- that is, why the observed elementary particles carry charges that are multiples of the charge of the electron.

Using all of the above information to plug into the Dirac equation results in

Using the Dirac comb distribution and its Fourier series:

Using the Dirac adjoint, the conserved probability four-current density for a spin-1 / 2 particle field

Using summation notation, it can be written as

Using base-10 ( the familiar decimal notation ) as an example, the number 152853. 5047, which has ten decimal digits of precision, is represented as the significand 1528535047 together with an exponent of 5 ( if the implied position of the radix point is after the first most significant digit, here 1 ).

Using asymptotic notation this result can be restated as

Using comparative analysis of all the material available to him, Vollaerts was able to show the internal logic and coherence in the neumatic notation with regard to a proper articulation of the verbal-melodic line.

Using the above given notation, suppose that some infinite cardinal.

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write:

Using this notation, P ( t ) takes the form,

Using big-O notation, the performance of the interpolation algorithm on a data set of size N is O ( N ); however under the assumption of a uniform distribution of the data on the linear scale used for interpolation, the performance can be shown to be O ( log log N ).

Using Leibniz notation: the substitution yields and thus, formally,, which is the required substitution for.

Using matrix notation, the first fundamental form becomes

Using Knuth's up-arrow notation, Graham's number G ( as defined in Gardner's Scientific American article ) is

Using this sigma notation the above summation is written as:

Using IUPAC notation in which the brackets contain the natural variables ( other than the main four ), we have:

Using the other notation:

Using index notation, the Hodge dual is obtained by contracting the indices of a k-form with the n-dimensional completely antisymmetric Levi-Civita tensor.

Using the limit notation of calculus, the volume may be expressed as:

Using the notation in the diagram on the right, the sides are ( AB ), ( BC ), ( CD ), ( DA ).

Using Lagrange's notation, we can write:

Using Newton's notation, we can write:

Using this notation, Z-X-Z would mean extrinsic composition.

Using hexadecimal notation, which includes the digits < tt > 0123456789ABCDEF </ tt >, it is possible to spell several words.

Using this notation and some knowledge about mathematical functions and probability theory we formalise different versions of the binding and hiding properties of commitments.

Using this notation, we can write Z = O < sub > Q </ sub > since Z is the ring of integers of the field Q of rational numbers.

Using index notation, an eigenvalue equation can be written as

Using a mathematical formulation, let ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub >) be the input sequence of n distinct values over the finite field F ; then the codebook C created from the tuplets of values obtained by evaluating every polynomial ( over F ) of degree less than k at each x < sub > i </ sub > is

Using the avante-garde magazine Playboy as his model, Arledge convinced his superiors at WRCA to let him film a pilot of a show he called " For Men Only.

Using the bottleneck technique most of the time, Elmore really let his guitar sound as I had never heard a guitar sound before.

Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X.

* Using indicator variables, let x < sub > i </ sub > equal 1 if x is in A < sub > i </ sub > and 0 otherwise.

Using the standard formalism of probability, let and be two random variables defined on probability spaces and.

Using the abstract index notation, let a, b, c, ... refer to and μ, ν, ... refer to the tangent bundle.

Using the aforementioned glitch with the Oak's Parcel will let you warp to the map with the number of steps you've walked until the door you walk in.

Using the parts from all the captured Autobots she creates a giant Autobot with herself in control, striking a deal with them that if they would co-operate she would let them go.

* Using " rent " and " let " interchangeably.

Using the parts from all the captured Autobots she created a giant Autobot with herself in control, striking a deal with them that if they would co-operate she would let them go.

Using the parts from all the captured Autobots she created a giant Autobot with herself in control, striking a deal with them that if they would co-operate she would let them go.

Using the parts from all the captured Autobots she created a giant Autobot with herself in control, striking a deal with them that if they would co-operate she would let them go.