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 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, QCD is a non-Abelian gauge theory based on a local ( gauge ) symmetry group called SU ( 3 ).

Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding bases are Fourier transforms of one another ( i. e., position and momentum are conjugate variables ).

Mathematically, the extremely wideband excitation provided by the spark gap was bandpass filtered by the self-oscillating antenna side circuit, which because of its simple construction, also had a relatively broad and badly controlled filter characteristic.

Mathematically the switching mixer is not much different from a multiplying mixer, just because instead of the LO sine wave term we would use the signum function.

Mathematically ideal drums with membranes of these two different shapes ( but otherwise identical ) would sound the same, because the eigenfrequency | eigenfrequencies are all equal, so the Timbre # Spectra | timbral spectra would contain the same overtones.

Mathematically, nominal numbering is a one-to-one function from a set of objects being named to the set of numerals: it is a function because each object is assigned a numeral, and it is one-to-one because different objects are assigned different numerals.

Mathematically, fading is usually modeled as a time-varying random change in the amplitude and phase of the transmitted signal.

Mathematically, the rate of change of X ( its derivative with respect to time ) vanishes,

Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times Boltzmann's constant k < sub > B </ sub >.

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, the return from a short position is equivalent to that of owning ( being " long ") a negative amount of the instrument.

Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: left is negative, right is positive.

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, the LM curve is defined by the equation, where the supply of money is represented as the real amount M / P ( as opposed to the nominal amount M ), with P representing the price level, and L being the real demand for money, which is some function of the interest rate i and the level Y of real income.

Mathematically, if is the loss of a portfolio, then is the level-quantile, i. e.

Mathematically, corrosion rates of glasses are characterized by normalized corrosion rates of elements NR < sub > i </ sub > ( g / cm < sup > 2 </ sup >· d ) which are determined as the ratio of total amount of released species into the water M < sub > i </ sub > ( g ) to the water-contacting surface area S ( cm < sup > 2 </ sup >), time of contact t ( days ) and weight fraction content of the element in the glass f < sub > i </ sub >:

Mathematically, the axiom states that the probability of selecting item i from a pool of j items is given by:

Mathematically, the number of points in the inner circle is equal to the number of points in the outer circle, i. e. there is a bijection between them.

Mathematically, we shall not distinguish the experiment of three dice tossed once from that of one die tossed three times.

Mathematically, this means the coefficient for the projection of onto.

Mathematically, the conjecture states that, for generic initial data, the maximal Cauchy development possesses a complete future null infinity.

Mathematically speaking, magnetic fields and electric fields are convertible with relative motion as a 2nd-order tensor or bivector.

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