Mathematically, this can be represented for n species as:

Mathematically the 0-1-knapsack problem can be formulated as:

Mathematically the bounded knapsack problem can be formulated as:

Mathematically, the potential of the ion cores can be treated by various models, the simplest being the nearly free electron model.

Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by the square root of ; or by keeping time a real quantity and embedding the vectors in a Minkowski space.

Mathematically, a neuron's network function is defined as a composition of other functions, which can further be defined as a composition of other functions.

Mathematically, the modulated wave can be written in the form:

Mathematically ( using the Laplace transform ) the impedance of this network can be written as:

Mathematically, the E field can be thought of as a function that associates a vector with every point in space.

Mathematically, convection can be described by the convection – diffusion equation or the generic scalar transport equation.

Mathematically, the pressure of a mixture of gases can be defined as the summation

Mathematically corrected punch motions can be programmed to simulate any type and model of production tablet press.

Mathematically, this can be expressed by means of the signal to noise ratio ( SNR or S / N ).

Mathematically speaking, there are seventeen basic patterns, described as wallpaper groups, that can be used to tile an infinite plane.

Mathematically, evanescent waves can be characterized by a wave vector where one or more of the vector's components has an imaginary value.

Mathematically, the maximum lift-to-drag ratio can be estimated as:

Mathematically, this can be written as:

Mathematically the problem can be stated like this:

Mathematically, the criterion for whether spontaneous fission can occur in a time short enough to be observed by present methods, is approximately:

Mathematically, we can represent this constraint in terms of a surface integral:

Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers ( the Cartesian coordinates of P ): x = x < sub > 2 </ sub >, ..., x < sub > n </ sub >.

Mathematically, we can describe the reciprocal lattice as the set of all vectors K that satisfy the above identity for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces.

Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.

Mathematically, a basic parametric mixture model can be described as follows:

Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must have negative-real values, i. e. the real part of all the poles are less than zero.

Mathematically, a baseband modulated signal may be approximated by a sinusoidal continuous wave signal with a frequency f < sub > m </ sub >.

Mathematically, quasicrystals have been shown to be derivable from a general method, which treats them as projections of a higher-dimensional lattice.

Mathematically, the equation may be rearranged to compute torque for a given power output.

Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions.

Mathematically, it refers to a property of solutions to the Schrödinger equation ; since the Schrödinger equation is linear, any linear combination of solutions to a particular equation will also be a solution of it.

Mathematically, assuming the terminator lies on a great circle, such a crescent will actually be the figure bounded by a half-ellipse and a half-circle, with the major axis of the ellipse coinciding with a diameter of the semicircle.

Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.

Mathematically, it may be represented as follows:

Mathematically, the Lorentz group may be described as the generalized orthogonal group O ( 1, 3 ), the matrix Lie group which preserves the quadratic form

Mathematically, the motion of a continuum using the Eulerian description is expressed by the mapping function

In 1827, the German scientist Georg Ohm expressed his law in this famous book " Die galvanische Kette, mathematisch bearbeitet " ( The Galvanic Circuit Investigated Mathematically ) in which he gave his complete theory of electricity.

Mathematically, incompressibility is expressed by saying that the density ρ of a fluid parcel does not change as it moves in the flow field, i. e.,

Mathematically, this is expressed by saying that the curl of the flow field is everywhere equal to zero.

Mathematically, this is expressed as 1 minus unavailability.

Mathematically, the D4σ beam width in the x-dimension for the beam profile is expressed as

Mathematically this is expressed using the virial theorem, which states that, to maintain equilibrium, the gravitational potential energy must equal twice the internal thermal energy.

Mathematically this is expressed as:

Mathematically, the FT is expressed as:

If a curve γ represents the path of a particle then the instantaneous velocity of the particle at a given point P is expressed by a vector, called the tangent vector to the curve at P. Mathematically, given a parametrized C < sup > 1 </ sup > curve γ = γ ( t ), for every value t

Mathematically, the function is expressed as the derivative of the consumption function with respect to disposable income.

Mathematically, the image is expressed as:

Mathematically the von Mises yield criterion is expressed as:

Mathematically, it can be expressed as:

Mathematically, each physical law can be expressed with respect to the coordinates given by an inertial frame of reference by a mathematical equation ( for instance, a differential equation ) which relates the various coordinates of the various objects in the spacetime.

Mathematically, this may be expressed as,

Mathematically, this is expressed by the following condition:

Mathematically, the marginal propensity to import ( MPM ) function is expressed as the derivative of the import ( I ) function with respect to disposable income ( Y ).