Their examples can be described by an inductively defined family of regular expressions over the binary alphabet as follows – cf.

Modern Russian has a singular vs plural number system, but the declension of noun phrases containing numeral expressions follows complex rules.

It follows from this that equality is the smallest equivalence relation on any set S, in the sense that it is a subset of any other equivalence relation on S. An equation is simply an assertion that two expressions are related by equality ( are equal ).

In fact, this is a free Kleene algebra in the sense that any equation among regular expressions follows from the Kleene algebra axioms and is therefore valid in every Kleene algebra.

Such an equality can hold only if in any matrix position the entry that is multiplied by a given power is the same on both sides ; it follows that the constant matrices with coefficient in both expressions must be equal.

His view of arithmetical algebra is as follows: " In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic ; the signs and denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers ; thus in expressions such as we must suppose and to be quantities of the same kind ; in others, like, we must suppose greater than and therefore homogeneous with it ; in products and quotients, like and we must suppose the multiplier and divisor to be abstract numbers ; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science.

The purpose of the YPWW as stated in the COGIC official manual is as follows: “ You are evidently aware of the fact that true religious education consists of instruction and practical expressions of the truths learned.

Some other quantified expressions are constructed as follows,

The rate expressions given in the 1864 paper could not be integrated, so they were simplified as follows.

In the second formula, the inner summation in the right hand side is only extended over indices within bounds, 0 ≤ i ≤ m and 0 ≤ j ≤ n. Alternative expressions of addition and multiplication, without using explicit bounds in the sums, are as follows:

Alternative expressions of the three theorems are as follows:

The dance follows the classical text of Hastha Lakshanadeepika, which has elaborate description of mudras ( gestural expressions by the hand palm and fingers ).

so the identity follows from equating the two expressions we obtained for and dividing out by ( this is allowed if one thinks of the identities as polynomial identities over the ring of polynomials in the indeterminate variables ).

It follows, then, that less complex societies, such as the Australian Aborigines, have less complex religious systems, involving totems associated with particular clans.

While the screenplay closely follows Hornby's novel, it omits scenes involving drug use and has a different ending.

A slow speed-chase follows involving the male flying around the female exposing his expanded under tail coverts to her.

These epic introductory tendencies give way to the main portion of the story, usually involving a battle of some kind ( such as in the Iliad ) that follows this pattern: dressing for battle ( description of Achilles shield, preparation for battle ), altar sacrifice / libation to the gods, some battle change ( perhaps involving drugs ), treachery ( Achilles ankle is told to be his weak spot ), a journey to the Underworld, and the final battle.

In the year Apr 2010 to Mar 2011 there were 11, 227 recorded offences involving firearms, broken down as follows.

As is common for reactions involving solids and solution, initiation follows an induction period during which reactive magnesium becomes exposed to the organic reagents.

It follows Arkady Renko, a chief investigator for the Militsiya, who is assigned to a case involving three corpses found in Gorky Park, an amusement park in Moscow, who have had their faces and fingertips cut off by the murderer to prevent identification.

A ritual involving a honeycomb follows.

After the E6 ( of " these are words "-) there follows an ascent involving different inversions of the D dim chord.

The story follows four former prostitutes on the run following a justifiable homicide and a prison escape, who later encounter difficulties involving bank robbery and Pinkerton detectives.

The Heroes of Dune interquel series ( 2008 – present ) by Brian Herbert and Kevin J. Anderson follows events involving the Atreides before, between, and after Dune, Dune Messiah, and Children of Dune.

In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem.

Other significant accidents involving multiple fatalities were as follows:

Conversely, an argument involving König's infinity lemma shows that any non-surjective rule must have an orphan, and ( letting be the size of a bounding box of this orphan ) a very similar counting argument shows that the number of patterns that fit within an square and do not contain an orphan is too small to provide a distinct successor to every starting pattern within an square, from which it follows that some two of the possible starting patterns are twins.

Based on the popular manga and anime series YuYu Hakusho created by Yoshihiro Togashi, Dark Tournament follows the protagonist Yusuke Urameshi, a rebellious teenager who dies and is brought back to life in order to serve as a “ Spirit Detective ”, solving cases involving apparitions and demons within the living world.

There follows a mad rush to the local hospital-also involving both Sandra's parents, Sandra's boyfriend, and Beryl's Mum-as it turns out Sandra has swallowed the ring in a mouthful of trifle.

Because arguments involving underdetermination involve both a claim about what the evidence is and that such evidence underdetermines a conclusion, it is often useful to separate these two claims within the underdetermination argument as follows:

The video games follow the same storyline, although the Game Boy Advance adaption follows the television show's story more closely, involving the characters who are from a video game appearing on Earth, and Ace ventures into the Carnival of Doom to find the pieces of the Amulet and defeat Lord Fear, eventually sending the villains back to the Sixth Dimension, the homeworld of the characters.

Like other Front Mission titles, Front Mission 4 is part of a serialized storyline that follows the stories of various characters and their struggles involving mecha known as wanzers.

This chapter follows two separate plot lines: one involving Laura at a gymnastics tournament with her friends watching, as well as another involving people of the Alpha Shade universe.

Combining historical fantasy, martial arts, and science-fiction, Tong ’ s film follows archaeologist Jack ( Jackie Chan ) and scientist William ( Tony Leung Ka-Fai ) on their adventures in investigating the veracity of a myth involving immortality, levitation, and a Korean princess-turned-concubine for Emperor Qin towards the end of the Qin Dynasty.

It follows that the solution to the current disunity of the free nations is only to a very limited extent a matter of devising new machinery of consultation and coordination.

Reports already issued on the sampling census, 1955 - 57, in various areas run as follows ( using only the French and omitting corresponding Flemish titles ).

It follows, then, provided the possibilities have been exhausted, that the only real alternative is the general viewpoint of the `` left '', which has been represented on the Continent by Fritz Buri and, to some extent at least, is found in much that is significant in American and English theology.

In French, the period only follows an abbreviation if the last letter in the abbreviation is not the last letter of its antecedent: " M ." is the abbreviation for " monsieur " while " Mme " is that for " madame ".

For Hume, every effect only follows its cause arbitrarily — they are entirely distinct from one another.

F follows the same format but has only 111 cantigas, of which 7 have no text, only miniatures.

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free ( and hence has only one prime factor of two ) will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.

" 2-4-6-8 ", a popular diet of this variety, follows a four-day cycle in which only 200 calories are consumed the first day, 400 the second day, 600 the third day, 800 the fourth day, 1, 000 the fifth day, and then the cycle repeats.

Hong Kong follows British practice in calling physicians " Doctor " even though many of them hold only an MBBS qualification.

The rule is deterministic ; in other words, for a given time interval only one future state follows from the current state.

" The Constitution grants the Governor-General a wide range of powers, but in practice he or she follows the conventions of the Westminster system and ( with rare exceptions ) acts only on the advice of the Prime Minister of Australia or other ministers.

It follows now that we need only prove Theorem 2 for formulas φ in normal form.

As the personal representative of the monarch, the governor follows only the sovereign in the NSW order of precedence.

Logically, it follows that a user should trust an HTTPS connection to a website if and only if all of the following are true:

One may define a partial order on the idempotents of a ring as follows: if a and b are idempotents, we write if and only if.

Up to reversal of orientation, the only homology classes which contain simple-closed curves are as follows: ( 0, 0 ), ( 1, 0 ), ( 1, 1 ), ( 2, 0 ), ( 0, 1 ).

The outer " back " of sapwood, approximately flat, follows the natural growth rings ; modern bowyers often thin the sapwood, while in the Mary Rose bows the back of the bow was the natural surface of the wood, only the bark being removed.

This article will not follow this nomenclature: In what follows, the term " Lorentz force " will refer only to the expression for the total force.

* The degrees of Chief Commander, Commander, Officer, and Legionnaire are awarded only to members of armed forces of foreign nations under the criteria outlined in Army Regulation 672-7 and is based on the relative rank or position of the recipient as follows:

There is a classical analogue to the quantum no-cloning theorem, which we might state as follows: given only the result of one flip of a ( possibly biased ) coin, we cannot simulate a second, independent toss of the same coin.

However, the nuclear force acts only over relatively short ranges ( a few nucleon diameters ), since it follows an exponentially decaying Yukawa potential which makes it insignificant at longer distances.

This is obtained as follows: Initially, the only matching clause-head for the query is the first one, so proving the query is equivalent to proving the body of that clause with the appropriate variable bindings in place, i. e., the conjunction.

People supporting the non-aggression principle claim that the moral prohibition of the use of violence follows from argumentation ethics, which applies only when people are using argumentation to solve disputes.