Mathematically it is defined as follows:

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

Mathematically, the catenary curve is the graph of the hyperbolic cosine function.

Mathematically, the conjecture states that the maximal Cauchy development of generic compact or asymptotically flat initial data is locally inextendible as a regular Lorentzian manifold.

Mathematically, density is defined as mass divided by volume:

Mathematically this is described as the convolution of a time-varying input signal x ( t ) with the filter's impulse response h, defined as:

Mathematically, the unification is accomplished under an SU ( 2 ) × U ( 1 ) gauge group.

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 because in reversible processes, the change in entropy of the cold reservoir is the negative of that of the hot reservoir ( i. e., ), keeping the overall change of entropy zero.

Mathematically, this says that the state vector is confined to one of the two eigenspaces of P, and is not allowed to range over the entire Hilbert space.

Mathematically, a sequence of coin flips ( fair or not ) is an example of a Bernoulli process, and its entropy is given by the binary entropy function.

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

Mathematically, the logistic map is written

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 total electromagnetic energy density ( radiation energy density ) in thermodynamic equilibrium from Planck's law is

Mathematically, the light received from stars as a function of star distance in a hypothetical fractal cosmos is:

Mathematically, the formula above entails that all incomes are at least the lower bound x < sub > m </ sub >, which is positive.

Mathematically, this correspondendence is supported by the second term, on the r. h. s.

Mathematically, QED is an abelian gauge theory with the symmetry group U ( 1 ).

Mathematically, this tunneling current is given by

Mathematically, a Yang – Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory.

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, this condition is also approached exponentially ; in theory it takes infinite time, but in practice it is over, for all intents and purposes, in a much shorter period.

Mathematically, the theory is modelled after Bernhard Riemann's metric geometry, but the Lorentz group of spacetime symmetries ( an essential ingredient of Einstein's own theory of special relativity ) replaces the group of rotational symmetries of space.

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

However, in what newspapers such as the Wall Street Journal later call the " math wars ", organizations such as Mathematically Correct complain that some elementary texts based on the standards, including Mathland, have almost entirely abandoned any instruction of traditional arithmetic in favor of cutting, coloring, pasting, and writing.

" Mathematically, the Marshallian PED was based on a point-price definition, using differential calculus to calculate elasticities.

Mathematically, division by zero within reals is explicitly undefined if the problem is zero divided by zero, and is defined as either positive or negative infinity depending on the signs of the numbers involved, or is undefined.

Mathematically a bijective function is used on the characters ' positions to encrypt and an inverse function to decrypt.

Mathematically such a symmetry is represented as a flow, φ, which acts on the variables as follows

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, defining down to be positive, the net force acting on an object falling near the surface of Earth is ( according to the drag equation ):

Mathematically, the ( real valued ) amplitude of one wave component is represented by a scalar wave function u that depends on both space and time:

Mathematically, this corresponds to the fact that spherical coordinates do not define a coordinate chart on the sphere at zenith and nadir.

McDuffie argued that the 40 % tariff on cotton finished goods meant that “ the manufacturer actually invades your barns, and plunders you of 40 out of every 100 bales that you produce .” Mathematically incorrect, this argument still struck a nerve with his constituency.

Mathematically, the parallel projection of the point on the-plane gives.

Mathematically oriented quantitative analysts tend to have more of a reliance on numerical analysis, and less of a reliance on statistics and econometrics.

Mathematically, how the impulse is described depends on whether the system is modeled in discrete or continuous time.

Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that

Mathematically, texture filtering is a type of anti-aliasing, but it filters out high frequencies from the texture fill whereas other AA techniques generally focus on visual edges.

Mathematically it can be explained by considering the normal modes of electromagnetic field which explicitly depend on the boundary ( or matching ) conditions on the surfaces of the interacting bodies.

Mathematically, he is most well known for his treatise Logarithmo-technica on logarithms, published in 1668.

Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-rank, symmetric tensor on M, conventionally denoted by g. Moreover the metric is required to be nondegenerate with signature (< tt >-+++</ tt >).