According to the spin-statistics theorem, bosonic fields commute while fermionic fields anticommute.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

According to (), the first historical mention of the statement of this theorem appears in ().

According to Birkhoff's theorem, it is the only vacuum solution that is spherically symmetric.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

According to the theorem, it is possible to expand any power of x + y into a sum of the form

According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.

According to the theorem, within every sufficiently powerful logical system ( such as Principia ), there exists a statement G that essentially reads, " The statement G cannot be proved.

* According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely-generated profinite group ( that is, a profinite group that has a dense finitely-generated subgroup ) the subgroups of finite index are open.

According to Rice's theorem, if there is at least one computable function in a particular class C of computable functions and another computable function not in C then the problem of deciding whether a particular program computes a function in C is undecidable.

According to the Rouché – Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.

According to the optical Raychaudhuri's theorem, this requires a violation of the averaged null energy condition.

According to the theorem, the reconstructed signal will match the original signal provided that the original signal contains no frequencies at or above this limit.

According to the Nyquist theorem, error-free detection of the line code requires a channel bandwidth of at least the Nyquist rate, which is half the line code pulse rate.

According to Shannon's source coding theorem, the optimal code length for a symbol is − log < sub > b </ sub > P, where b is the number of symbols used to make output codes and P is the probability of the input symbol.

According to the CPT theorem of particle physics, antihydrogen atoms should have many of the characteristics regular hydrogen atoms have ; i. e., they should have the same mass, magnetic moment, and transition frequencies ( see atomic spectroscopy ) between their atomic quantum states.

According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant.

According to Allen Weiss, in Mirrors of Infinity, this optical effect is a result of the use of the tenth theorem of Euclid ’ s Optics which asserts that “ the most distant parts of planes situated below the eye appear to be the most elevated .” In Fouquet ’ s time, interested parties could cross the canal in a boat, but walking around the canal provides a view of the woods that mark what is no longer the garden and shows the distortion of the grottos previously seen as sculptural.

According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy.

According to the correspondence principle and Ehrenfest's theorem as a system becomes larger or more massive ( action >> Planck's constant ) the classical dynamics tends to emerge, with some exceptions, such as superfluidity.

According to the Kutta – Joukowski theorem, lift is the product of circulation, airspeed, and air density.

According to the Shannon – Hartley theorem, the channel capacity of a properly-encoded signal is proportional to the bandwidth of the channel and the logarithm of the signal-to-noise ratio ( SNR ) ( assuming the noise is additive white Gaussian noise ).

According to Ptolemy's second theorem,

According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the inverse function.

According to one leading historian of statistics, he may have been the earliest discoverer of Bayes theorem.

According to the Standard Model, all elementary particles are either bosons or fermions ( depending on their spin ).

According to the study, any surface can be modified to this effect by application of a suspension of rose-like V < sub > 2 </ sub > O < sub > 5 </ sub > particles, for instance with an inkjet printer.

According to Maijastina Kahlos " monotheism was pervasive in the educated circles in Late Antiquity " and " all divinities were interpreted as aspects, particles or epithets of one supreme God ".

According to quantum theory, the particles do not possess definite positions during the periods between measurements.

According to this view, all that exists is matter and void, and all phenomena result from different motions and conglomerations of base material particles called " atoms " ( literally: " indivisibles ").

* According to the laws of thermodynamics all particles of matter are in constant random motion as long as the temperature is above absolute zero.

According to the theory, as the Universe cooled after the big bang it eventually became possible for common subatomic particles as we know them ( neutrons, protons and electrons ) to exist.

According to the correspondence principle between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is just an approximation for large systems of objects ( or a statistical quantum mechanics of a large collection of particles ).

According to this vacuum energy explanation, since quantum theory holds that vacuum contains virtual particles, the rapidly moving interface between water and gas converts virtual photons into real photons.

According to a recent study conducted in Queensland, Australia, some printers emit sub-micrometre particles which some suspect may be associated with respiratory diseases.

According to Professor Morawska of Queensland University, one printer emitted as many particles as a burning cigarette:

According to Brown & Wallace the de Broglie-Bohm particles play no role in the solution of the measurement problem.

According to Peter R. Holland, in a wider Hamiltonian framework, theories can be formulated in which particles do act back on the wave function.

According to Holland, failure to appreciate key points of the de Broglie – Bohm theory has led to confusion, the key point being " that the trajectories of a many-body quantum system are correlated not because the particles exert a direct force on one another ( à la Coulomb ) but because all are acted upon by an entity – mathematically described by the wavefunction or functions of it – that lies beyond them.

According to Dirac, it was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions.

According to English folk understanding of how witches accomplished affliction, when the dog ate the cake, the witch herself would be hurt because invisible particles she had sent to afflict the girls remained in the girls ' urine, and her cries of pain when the dog ate the cake would identify her as the witch.

According to calculations, the comoving distance ( current proper distance ) to particles from the CMBR, which represent the radius of the visible universe, is about 14. 0 billion parsecs ( about 45. 7 billion light years ), while the comoving distance to the edge of the observable universe is about 14. 3 billion parsecs ( about 46. 6 billion light years ), about 2 % larger.

According to Alfvén, such an ambiplasma would be relatively long-lived as the component particles and antiparticles would be too hot and too low-density to annihilate each other rapidly.

According to the principles of quantum mechanics, particles do not actually have trajectories through space.

According to the researchers the particles self-replicated in microbiological culture, and the researchers further reported having identified DNA in these structures by staining.

According to the kinetic theory of gases, the temperature of a gas is just a measure of the average kinetic energy of the particles in that gas.

According to recent research, some laser printers emit submicrometer particles which have been associated in other environmental studies with respiratory diseases.

According to field theory we account for the Coulomb ( electrostatic ) interaction between charged particles through the fact that charges produce around themselves an electric field, which can be felt by other charges as a force.