According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors ( see Eötvös experiment ), there is a universality of free fall ( also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass ): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.

When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA < sub > 0 </ sub > ( a second-order variant of Peano arithmetic restricted to induction over Σ < sup > 0 </ sup >< sub style =" margin-left :- 0. 6em "> 1 </ sub > formulas ).

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder ( weak order ), an equivalence relation, or a relation with any other special properties, if and only if R is.

* Define a b as " not b < a " ( i. e., take the inverse complement of the relation ), which corresponds to defining a ~ b as " neither a < b nor b < a "; these relations and ~ are in general not transitive ; however, if they are, ~ is an equivalence ; in that case "<" is a strict weak order.

* The relation is a strict weak order, where the associated equivalence is equality.

If one reagent is a weak acid or base and the other is a strong acid or base, the titration curve is irregular and the pH shifts less with small additions of titrant near the equivalence point.

Titration curves corresponding to weak bases and strong acids are similarly behaved, with the solution being acidic at the equivalence point and indicators such as methyl orange and bromothymol blue being most appropriate.

The simplest way to test the weak equivalence principle is to drop two objects of different masses or compositions in a vacuum, and see if they hit the ground at the same time.

They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms ( e. g., a weak equivalence of spaces passes to an isomorphism of homology groups ), verified that all existing ( co ) homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.

Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers ( which are not constructively equivalent to the Cauchy real numbers without countable choice < ref > For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman ; 1998 ; < cite > A weak countable choice principle </ cite >; available from .</ ref >).

* 1686 – Isaac Newton uses a fixed length pendulum with weights of varying composition to test the weak equivalence principle to 1 part in 1000

* 1889 – Loránd Eötvös uses a torsion balance to test the weak equivalence principle to 1 part in one billion

* 1962 – Robert Dicke, Peter Roll, and R. Krotkov use a torsion fiber balance to test the weak equivalence principle to 2 parts in 100 billion

* 1968 – Kenneth Nordtvedt studies a possible violation of the weak equivalence principle for self-gravitating bodies and proposes a new test of the weak equivalence principle based on observing the relative motion of the Earth and Moon in the Sun's gravitational field

* Strict weak ordering – a strict partial order in which incomparability is an equivalence relation

Certain axioms, such as stability of fibrations under composition and pullbacks, factorization of every morphism into the composition of an acyclic cofibration followed by a fibration or a cofibration followed by an acyclic fibration, where the word " acyclic " indicates that the corresponding arrow is also a weak equivalence, and other requirements are set up to allow the abstract treatment of homotopy theory.

* Isaac Newton uses a fixed length pendulum with weights of varying composition to test the weak equivalence principle to 1 part in 1000.

Due to this circumstance, a functor with these properties is sometimes called a weak equivalence of categories ( unfortunately this conflicts with terminology from homotopy theory ).

One may therefore view an adjointness relation between two functors as a " very weak form of equivalence ".

Critics of the SAP argue in favor of a weak anthropic principle ( WAP ) similar to the one defined by Brandon Carter, which states that the universe's ostensible fine tuning is the result of selection bias: i. e., only in a universe capable of eventually supporting life will there be living beings capable of observing any such fine tuning, while a universe less compatible with life will go unbeheld.

The anthropic principle is often criticized for lacking falsifiability and therefore critics of the anthropic principle may point out that the anthropic principle is a non-scientific concept, even though the weak anthropic principle, " conditions that are observed in the universe must allow the observer to exist ", is " easy " to support in mathematics and philosophy, i. e. it is a tautology or truism

While the description of strong and weak interactions within the Standard Model is based on gauge symmetries governed by the simple symmetry groups SU ( 3 ) and SU ( 2 ) which allow only discrete charges, the remaining component, the weak hypercharge interaction is described by an abelian symmetry U ( 1 ) which in principle allows for arbitrary charge assignments.

The weak anthropic principle could then be applied to conclude that we ( as conscious beings ) would only exist in one of those few universes that happened to be finely tuned, permitting the existence of life with developed consciousness.

Popularly known as the Sapir – Whorf hypothesis, or Whorfianism, the principle is often defined as having two versions: ( i ) the strong version that language determines thought and that linguistic categories limit and determine cognitive categories and ( ii ) the weak version that linguistic categories and usage influence thought and certain kinds of non-linguistic behaviour.

Also the distinction between a weak and a strong version of the hypothesis is a later invention, as Sapir and Whorf never set up such a dichotomy although in their writings at times their view of this relativity principle are phrased in stronger or weaker terms.

The notion of " weak " and " strong " versions of Whorf's principle of linguistic relativity is a misunderstanding of Whorf promulgated by Stuart Chase, whom Whorf considered " utterly incompetent by training and background to handle such a subject.

* Social Choice and Individual Values for the '( weak ) Pareto principle '

When some strong acid is added to an equilibrium mixture of the weak acid and its conjugate base, the equilibrium is shifted to the left, in accordance with Le Chatelier's principle.

In that sense the behavior of light in this apparatus is deterministic, but there is no way to predict where in the resulting interference pattern any individual photon will make its contribution ( although, there may be ways to use weak measurement to acquire more information without violating the Uncertainty principle ).

Even a whole particle trajectory can be measured by a weak measurement.

Another serious shortcoming was the tank's weak armament, the 2 pounder ( 40 mm ) gun, which was improved by the addition of a 3 inch howitzer in the hull ( the Mk IICS had the howitzer in the turret ) to deliver an HE shell albeit not on a howitzer's usual high trajectory.

Increasingly a transit point for heroin being trafficked to Western Europe ; minor transit point for cannabis ; remains highly vulnerable to money-laundering activity given a primarily cash-based and unregulated economy, weak law enforcement, and instances of corruption.

Austrian Germanist Rudolf Simek says that Bifröst either means " the swaying road to heaven " ( also citing bifa ) or, if Bilröst is the original form of the two ( which Simek says is likely ), " the fleetingly glimpsed rainbow " ( possibly connected to bil, perhaps meaning " moment, weak point ").

The Fifth incarnation of Doctor Who, Peter Davison, was noted for wearing a stalk of celery on his lapel, claiming it at one point to be an excellent restorative, though the human olfactory sense was comparatively weak.

His backside ( which he couldn't reach ) is as vulnerable as any human back, rendering this his weak point.

He ran as a Labour Party candidate for London University in the 1922 and 1923 general elections after the death of his friend W. H. R. Rivers, but at that point his faith in the party was weak or uncertain.

Because of the weak, induced dipole-dipole interaction between helium atoms, their macroscopic physical properties are mainly determined by their zero point energy (" ground state kinetic energy ").

" James Wierzbicki for the St. Louis Post-Dispatch described Adams's score as the weak point in an otherwise well-staged performance, noting the music as " inappropriately placid ," " cliché-ridden in the abstract " and " heavily in Adams's worn-out Minimalist clichés.

Modern tests and contemporary accounts agree therefore that well-made plate armour could protect against longbows, however there are a number of caveats to this point ; not all plate armour was well-made or well looked after, and there were also weak points in the eye and air holes and joints where arrows could penetrate, meaning that even if the armour was proof against nearly all arrows, being shot at by thousands of longbowmen would have been an uncomfortable experience, physically and mentally.

This was almost certainly done to discredit him but perhaps, in doing so, accusers were attacking a perceived weak point in their adversary's character.

Deteriorating weather conditions and weak, unacclimatised ponies affected the initial depot-laying journey, so that the expedition's main supply point, One Ton Depot, was laid north of its planned location at 80 ° S. Lawrence Oates, in charge of the ponies, advised Scott to kill ponies for food and advance the depot to 80 ° S, which Scott refused to do.

The pivoting part that supports the cylinder is called the crane ; it is the weak point of swing-out cylinder designs.

It is almost customary to have at least one instance, usually during the final boss battle, where the Force can be ejected and launched into a boss ' weak point, lodged there, and constantly damage it until the boss falls.

While Sapir never made a point of studying directly how languages affected the thought processes of their speakers, some notion of ( probably " weak ") linguistic relativity lay inherent in his basic understanding of language, and would be taken up by his student Benjamin Lee Whorf.

The part of the blade between the Center of Percussion ( CoP ) and the point is called the foible ( weak ) of the blade, and that between the Center of Balance ( CoB ) and the hilt is the forte ( strong ).

The language's diverse appeal is seen as a strong point, though the consequently wide divergence between implementations is seen as one of the language's weak points.

However, the attacks did reveal a weak point in the fortifications and the combined forces on the inside and the outside almost made a breakthrough.

This raised fuel consumption, however ( already a weak point for the Wankel engine ), at the same time that the oil crisis of 1973 raised the price of gasoline.

From this point of view, the weak topology is the coarsest polar topology ; see weak topology ( polar topology ) for details.

No function has this property, but it can be thought of as a limit of functions whose integrals over space are unity, and whose support ( the region where the function is non-zero ) shrinks to a point ( see weak solution ).