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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.
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Mathematically and conjecture
Mathematically, the conjecture states that, for generic initial data, the maximal Cauchy development possesses a complete future null infinity.
Mathematically and states
Mathematically, the theorem states
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.
David Sklansky, in his book The Theory of Poker, states " Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting.
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, Hooke's law states that
Mathematically, the axiom states that the probability of selecting item i from a pool of j items is given by:
Mathematically speaking, two quantum states and are separated by a selection rule if for any given Hamiltonian, while they are separated by a superselection rule if for all physical observables.
Mathematically, the Pythagorean identity states:
Such omissions were criticized by such groups as Mathematically Correct, and some states and districts have since abandoned this experiment, though it remains widely used.
Mathematically and generic
Mathematically, convection can be described by the convection – diffusion equation or the generic scalar transport equation.
Mathematically one can pose a generic problem as follows:
Mathematically and asymptotically
Mathematically, an object approaches its terminal velocity asymptotically.
Mathematically, this means that they are asymptotically locally Euclidean ( or perhaps asymptotically locally flat ) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds.
Mathematically and flat
Mathematically, the difference between artificial perspective projection ( perspective projection onto a flat surface ) and natural perspective projection ( perspective projection onto a spherical surface ) is a distortion resulting from the small-angle approximation which results from projecting onto a flat surface.
Mathematically, the tension creates a ' flat ' section in the spring's ' torque curve ' ( see graph at right ) and only this flat section is used.
Mathematically and initial
Mathematically, if is the initial shape, and is the shape at time t,
Mathematically, each cell can be modeled as a dissipative, nonlinear dynamical system where information is encoded via its initial state, inputs, and variables used to define its behavior.
Mathematically and data
Mathematically, the climatological annual cycle is commonly estimated from observational data or model output by taking the average of all Januaries, all Februaries, and so forth.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations, where the best approximation is defined as that which minimizes the sum of squared differences between the data values and their corresponding modeled values.
Mathematically, the first vector is the orthogonal, or least-squares, projection of the data vector onto the subspace spanned by the vector of 1's.
Mathematically and is
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, 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 ).
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