The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.

In states where a quantum mechanical particle is bound, it must be localized as a wave packet, and the existence of the packet and its minimum size implies a spread and minimal value in particle wavelength, and thus also momentum and energy.

In 1790, Immanuel Kant wrote in Kritik der Urtheilskraft ( Critique of Judgement ) that the analogy of animal forms implies a common original type, and thus a common parent.

is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.

The holographic principle thus implies that the subdivisions must stop at some level, and that the fundamental particle is a bit ( 1 or 0 ) of information.

Physicalism thus implies ( through modal realism ) that:

Bell's theorem implies, and it has been proven mathematically, that compatible measurements cannot show Bell-like correlations, and thus entanglement is a fundamentally non-classical phenomenon.

The existence of these " universal " properties implies that superconductivity is a thermodynamic phase, and thus possesses certain distinguishing properties which are largely independent of microscopic details.

Totality implies reflexivity, that is, a ≤ a, thus a total order is also a partial order.

Modern quantum mechanics implies that uncertainty is inescapable, and thus that Laplace's vision needs to be amended: a theory of everything must include gravitation and quantum mechanics.

However, as Plutarch implies, since naval power relied on the mass mobilisation of the common citizens ( thetes ) as rowers, such a policy put more power into the hands of average Athenians — and thus into Themistocles's own hands.

The definition of the fundamental solution thus implies that, if the Laplacian of u is integrated over any volume that encloses the source point, then

That Dasein is thrown into an already existing world and thus into its mortal possibilities does not only mean that Dasein is an essentially temporal being ; it also implies that the description of Dasein can only be carried out in terms inherited from the Western tradition itself.

This definition of depression implies that a recession and a depression will have different ending dates and thus distinct durations.

In the double-sideband suppressed-carrier transmission ( DSB-SC ) modulation, unlike AM, the wave carrier is not transmitted ; thus, a great percentage of power that is dedicated to it is distributed between the sidebands, which implies an increase of the cover in DSB-SC, compared to AM, for the same power used.

" Beth Te pilla " is derived from the cognate verb, " hitpallel " ( פ ּ ליל ) which is also spelled as pa ^ li ^ yl ( pronounced: paleel ) which implies " estimate, judge, render a verdict " thus " hitpallel " means " to pray " or " to seek a judgment for oneself " or " to plead ".

The Greek prefix epi-in epigenetics implies features that are " on top of " or " in addition to " genetics ; thus epigenetic traits exist on top of or in addition to the traditional molecular basis for inheritance.

In reality, this goes much further than simply A and B, as it implies every single cascade or trigger chain related to the events in the transaction, and thus every check on all of the indirectly impacted values as well.

If the rest mass is imaginary this implies that the denominator is imaginary since the total energy is an observable and thus must be real ; therefore the quantity under the square root must be negative, which can only happen if v is greater than c. As noted by Gregory Benford et al., among others, special relativity implies that tachyons, if they existed, could be used to communicate backwards in time ( see Tachyonic antitelephone article ).

According to Herodotus and Pausanias ( vi. 17. 6 ), on the authority of Hesiod, his father was Amythaon, whose name implies the " ineffable " or " unspeakably great "; Melampus and his heirs were thus Amythaides of the " House of Amythaon ".

At least one line in the C. S. Lewis book The Last Battle implies that Lewis learned of Narnia's events-and thus wrote the Narnia books-after the Railway Accident in 1949, when Susan told him the stories in the belief that she was relating mere childhood make-believe.

In Aztec mythology, the god Nanahuatzin or Nanahuatl ( or Nanauatzin, the suffix-tzin implies respect or familiarity ; ), the most humble of the gods, sacrificed himself in fire so that he would continue to shine on Earth as the sun, thus becoming the sun god.

We now show that it is impossible to construct a 20 ° angle ; this implies that a 60 ° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected.

" ( p. 128 ) Rice believes that " the Precession is fundamental to an understanding of what powered the development of Egypt " ( p. 10 ), to the extent that " in a sense Egypt as a nation-state and the king of Egypt as a living god are the products of the realisation by the Egyptians of the astronomical changes effected by the immense apparent movement of the heavenly bodies which the Precession implies.

Thus, we can interpret Boltzmann's constant as being simply the expression ( in terms of more standard physical units ) of the abstract logarithmic unit that is needed to convert the dimensionless pure-number quantity ln W ( which uses an arbitrary choice of base, namely e ) to the more fundamental pure logarithmic quantity Log ( W ), which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy.

Semi-strong-form efficiency implies that neither fundamental analysis nor technical analysis techniques will be able to reliably produce excess returns.

In Western and secular culture, religious education implies a type of education which largely separate from academia, and which ( generally ) regards religious belief as a fundamental tenet and operating modality, as well as a prerequisite condition of attendance.

The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, atoroidal, and has infinite fundamental group.

As the name implies, the fundamental meaning of monodromy comes from ' running round singly '.

Similarly, when ideas about self-organization originate in, say, biology or social science, the farther one tries to take the concept into chemistry, physics or mathematics, the more resistance is encountered, usually on the grounds that it implies direction in fundamental physical processes.

In history, a distinction between eras or periods, implies a major and fundamental change in the system of institutions governing a society.

This theorem implies that flat space-time is stable, a fundamental issue for the theory of general relativity.

It suffices to take for d the degree of the symmetric function, since the part in degree d is of that ring is mapped isomorphically to rings with more indeterminates by φ < sub > n </ sub > for all n ≥ d. This implies that for studying relations between individual elements, there is no fundamental difference between symmetric polynomials and symmetric functions.

And since 31 does not divide and measures 496, the fundamental theorem of arithmetic implies that must divide 16 and be amongst the numbers 1, 2, 4, 8 or 16.

This theorem implies the calculation of the fundamental group of the circle as the group of integers, since the group of integers is obtained from the groupoid by identifying, in the category of groupoids, its two vertices.

also known as the fundamental tensor of F at v. Strong convexity of F < sup > 2 </ sup > implies the subadditivity with a strict inequality if u / F ( u ) ≠ v / F ( v ).

In mathematical finance, a risk-neutral measure, also called an equivalent martingale measure, is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure.

The fundamental requirement for incompressible flow is that the density,, is constant within an infinitesimal volume, dV, which moves at the velocity of the fluid, v. Mathematically, this constraint implies that the material derivative ( discussed below ) of the density must vanish to ensure incompressible flow.

The fundamental theorem of calculus for line integrals implies that if V is defined in this way, then so that V is a scalar potential of the conservative vector field F. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it.

Loschmidt's paradox is equivalent to the question of how it is possible that there could be a thermodynamic arrow of time given time-symmetric fundamental laws, since time-symmetry implies that for any process compatible with these fundamental laws, a reversed version that looked exactly like a film of the first process played backwards would be equally compatible with the same fundamental laws, and would even be equally probable if one were to pick the system's initial state randomly from the phase space of all possible states for that system.

Each type in the tower conceptually " sits on " a more fundamental type, so an integer is a rational number and a number, but the inverse is not necessarily true, i. e. not every number is an integer ; this asymmetry implies that a language can allow implicit coercions of numerical types — without creating semantic problems — in only one direction: coercing an integer to a rational loses no information and does not affect the results of a function, but to coerce most reals to an integer could well result in a problem ( for example, the real 1 / 3 does not equal any integer ).

The fundamental theorem of Galois theory implies that the corresponding fixed field

Otherwise, without free agent's a priori fundamental source, socially essential concepts created from human mind, such as justice, would be undermined ( responsibility implies freedom of choice ) and, in short, civilization and human values would crumble.