Mathematically, the condition for a vector field to be projective is equivalent to the existence of a one-form satisfying

Mathematically, this is expressed by the following condition:

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

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 talented, Lamarr also co-invented — with composer George Antheil — an early technique for spread spectrum communications and frequency hopping, necessary for wireless communication from the pre-computer age to the present day.

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, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function ; this is also known as a two-dimensional Weierstrass transform.

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 pseudoscalar is an element of the top exterior power of a vector space, or the top power of a Clifford algebra ; see pseudoscalar ( Clifford algebra ).

Mathematically, the number of strands is the greatest common divisor of the number of leads and the number of bights ; the knot may be tied with a single strand if and only if the two numbers are coprime.

The instrument used for assessing environmental thermal comfort in terms of operative temperature is called a eupatheoscope and was invented by A. F. Dufton in 1929 .< ref > Glossary of Meteorology, American Meteorological Society, < http :// amsglossary. allenpress. com / glossary / browse? s = e & p = 42 >, accessed Sept 2010 </ ref > Mathematically, operative temperature can be shown as ;

Mathematically, the Theis equation is the solution of the groundwater flow equation for a step increase in discharge rate at the pumping well ; a slug test is instead an instantaneous pulse at the pumping well.

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, 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.

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.

Mathematically, this takes the form:

Mathematically, Gauss's law takes the form of an integral equation:

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

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

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 speaking, the simplification reduces the state space of the system to a finite number, and the partial differential equations ( PDEs ) of the continuous ( infinite-dimensional ) time and space model of the physical system into ordinary differential equations ( ODEs ) with a finite number of parameters.

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

Mathematically, cardiac output ( blood flow measured in volume per unit time ) is computed by multiplying the heart rate ( in beats per minute ) and the stroke volume ( the volume of blood ejected during ventricular systole ).

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

Mathematically, the permissibility of time travel is represented by the existence of closed timelike curves.

Mathematically, if is the initial shape, and is the shape at time t,