Ω, ω: omega, last letter of the Greek alphabet.

Ω

ω

The last letter in the Greek alphabet.
A symbol of the end.
The lowest member of a social group, especially in a wolf pack.

As of now, we don't have the slightest clue as to what Omega is because it depends on entirely Universal quantities that may be impossible to measure. Some theorize that 90% or more of the Universe's mass is made up of dark matter, which does not reflect light and is thus not observable using telescopes of any form. Many think of dark matter as some kind of mysterious pseudo-substance, but it could be as simple as billions and billions of burnt out cinder-cores from pathetic star deaths. Some stars explode into supernovae at death, others collapse into black holes. But the largest majority simply burn out, potentially (eventually) reaching a state of death, where all that remains is the fused metal center.

The Omega Constant was appropriately named for the purpose of understanding the fate of the Universe. If the Universe is relatively dense, then Omega is greater than one (even if it only equals 1.00000{billion 0s}1), and the force of gravity will eventually overcome the force(s) that is/are expanding the Universe. As of now (and for the next thousand, million, billion?) years the Universe is actually accelerating apart, theoretically by a cumulative force exerted by virtual particles that occupy the "empty space" between galaxies, a volume that grows exponentially in "emptiness" as the galaxies move apart from each other in very large, quantized amounts.

If Omega equals one (meaning 1.{infinite zeros}), meaning the Universe has the exact amount of matter density to equal its critical density, then it is just dense enough to continually expand, approaching forever in time (at a decelerating rate), asymptotically seeking a state of infinite volume. This is "the heat death of the Universe," see this node as well for details. Note the language used in describing this – "asymptotically" and "approaching forever at a decelerating rate." The next situation is far different, although it initially may sound the same. If Omega is one, then heat death should never actually be reached.

If Omega is less than one, the Universe will expand forever. But this time there's no gradual deceleration, no "approaching forever" -- if Omega is less than zero than the Universe will expand forever, and will reach infinite volume, and will reach a state of heat death. In other words, it will continually accelerate apart, rather than slowing down. No "approaching zero," or "approaching infinity." It defines infinity, and redefines it continually, until it has nothing left to burn. As of now, it appears that this is the Universe we live in. This could change, but as of now, 10-20 billion years from the flash that created all of this, it still seems to be exploding. The big bang isn’t a single event. <opinion>The big bang is the Universe in which entropy is always increasing, due to exponential increases in volume.</opinion>


What I Think (factual part ends here)

I personally believe that Omega is either less than one or greater than one, as either of these situations supports my philosophical position of "eternal recurrence," the idea that this same exact life will be lived again infinite times, even if only occurring one time out of every five googolplex ^ googolplex Universes. If Omega is less than one, the Universe will collapse, and will probably expand again and repeat this cycle forever. Collapsing allows it to conserve all of its energy, probably forming the all-massive true singularity that makes the big bang possible.

But if it expands forever, constantly accelerating apart, reaching infinite volume, then at some arbitrary point we should reach a state of "nothingness," akin to the nothingness that the big bang supposedly came from and destroyed (more likely, transformed, or infused). If every galaxy is constantly pushing away from every other galaxy, and every star is eventually going to die, then we should reach some place where every bit of matter and energy has been converted into entropy, into the effect behind the forces that stretch the cosmic fabric of space, as if warping by the hot energy constantly emanating from every massive point in the Universe!

I believe that a collapsing Universe is not necessary in order to have a repeating Universe. And if I’ve existed in one iteration, than there have to be others, including one that is completely identical in every way to this one (thanks in part to Chaos Theory). And even if Omega does equal one, who’s to say the Universe doesn’t have the patience? As human beings we’ve never witnessed infinity or true nothingness. But the entire mathematics of Calculus is about how to manipulate limits to "approach infinity" and find answers to problems that would otherwise result in division by zero, "0/0", or a "wrong zero" that should actually cancel out to one. This is what unifying the forces is all about; finding Calculus equations that remove the impossible elements from the problem, in order to solve what appears to be impossible and even nonsensical.

We can get around the problem with infinity with a little imagination and practice, so why can’t the ever-expanding-all-massive-all-powerful Universe? I think it’s even a little arrogant to think that this is the only cycle – that we are just lucky in being around to witness the one-time-only event that is the expanding Universe. Stars (although open systems) may represent the most accurate microcosm of the Universe – some of them explode, some of them collapse, but most of them burn out, reaching what is similar to a state of "heat death." Perhaps this Universe is merely a point on a grid of infinite other possibilities, all occurring simultaneously, but as a whole, the entire grid is truly "the Universe," and in order to conserve energyFor matter and energy to truly be conserved, there can be no "heat death" of everything. But that’s just my opinion (and first Nietzsche’s), I could be wrong.

In Modern Greek, the letters omega and omicron are pronounced the same, as a mid-open O vowel. In Ancient Greek, they were different: omega was always a long vowel and omicron was always short (mega = big, micron = small); the capital Ω came from an underlined O (omicron), and the lower-case ω from two o's side by side.

It is clear that omega had a mid-open sound, as in English 'caw': the same noise is indicated onomatopoeically in Greek with omega, for example: κω

They are written in HTML as &Omega; and &omega;

In English the name is pronounced with the stress on the O, which is either long (like ohm) or short (omelette). In Greek the -meg- is the accented syllable.


In mathematics, omega has two important uses, one of which I understand well, and the other not at all. They are unrelated. The first one is lower-case; I think the second one is upper-case.

Lower-case omega is the smallest infinite ordinal. It is the order type of the natural numbers in their natural ordering 0, 1, 2, 3, ... . As it is the smallest ordinal of its cardinality, it is equal to the smallest infinite cardinal, aleph-nought (aleph-null).

From it are constructed other ordinals, such as omega+1, omega·2, omega·omega+1, and so forth. The concept and hierarchy of ordinals are well covered elsewhere in E2.


The other Omega is new to me. I've seen references to it in recent years, but it postdates my formal study of maths, so I'm relying on a recent New Scientist article for my rudimentary understanding. (They are strikingly hopeless at expounding mathematics. I never learn anything useful from their maths articles.)

It is a totally random irrational, uncomputable number between 0 and 1, discovered by Gregory Chaitin, and is the probability that a universal Turing machine will halt on a program chosen at random.

The halting problem for a Turing machine can be translated into a diophantine equation. Chaitin constructed a great big diophantine equation in which successive substitutions of N = 1, 2, 3, ... ask whether there are a finite or an infinite number of solutions to the equation, and the truth value of the answer gives the N-th digit of Omega. But because the digits are incalaculably random, so in general are the solutions of most simple equations in number theory. It says here.

If someone knows something about this, an explanation would be appreciated. A /msg to me for very small corrections, but really we need a proper write-up.

A few words more about Omega (the roguelike):

The version 0.80.2 was "official" for a pretty long time. I remember a thing or two about it; actually, it was the source code from which I drew my first tastes of C programming. Thus, the game has some significance to me. Not that I got it to compile with DJGPP, but...

Anyway, the game was fairly nice, save a few drawbacks:

But apart of these, the game was fairly enjoyable - at least until you grew bored to the HIGH challenge it offered. I mean, this game was hard...

Since then, some efforts have been made to improve the game.

What made it enjoyable? It was almost like Nethack in sense that The DevTeam thinks of everything. For example, the character's sexual orientation was implemented (allows bisexuality and abstaining from sex, too!), all "impossible" states of items were also named...

...bank teller has a programmed-in bug in guard alert code - "Yo mama. Core dumped." =)

What was cool was that it allowed you to play! It asked your physical and mental status, and made character based on your choices.

All in all, a great game if you can appreciate the humor

The following is the game's official "advertisement".


(ESTABLISHING SHOT) A weary ADVENTURER, wearing battered armor and 10 glowing rings, clutching a potion bottle, and laden with all manner of weapons, magical devices, and sacks of gold, lies panting on the ground outside of the seedy-looking entrance to a grimy Dungeon. The nearby scenery is uniformly grey and uninteresting.

Bluff Male Voice: Retrieved the Amulet of Yendor too many times to count?

Sultry Female Voice: Can't see anything in the Eye of Larn?

BMV: Eaten one too many Zombie corpses?

SFV: Run out of Greater Gods to kill?

CLOSE-UP: The ADVENTURER's sweat-streaked face, which is nicked and bruised. He has a black eye.

The ADVENTURER nods wearily, and, it seems, with some boredom.

MBV+SFV: Then take the ultimate challenge ... The final quest ... OMEGA!

(PAN VERTICALLY TOWARD SKY)

The Mormon Tabernacle Choir: Ooooooo -- mega!

A shaft of brilliant sunlight pierces the overcast sky, revealing a Mystic Portal in the sky. A rainbow bridge lances from the portal toward the ADVENTURER. As the ADVENTURER hesitantly sets foot on the bridge, he (with the viewer) is swept through the M.P. in a masterpiece of computer animation. There is a flash of light, and a TRANSFORMED ADVENTURER, in newly polished and chromed armor, wielding a flaming sword, strides confidently toward an edifice that makes the Castle of Ultimate Darkness look like a sandcastle. The landscape is vibrantly colored, and we feel that there are new challenges awaiting just over the horizon.

TMTC: Magnificat! Magnificat! Magnificat!

BMV: Coming Soon to a site near you!

SFV: Challenge Omega -- The Final Quest!

As the ADVENTURER passes through the entrance to the AWESOME CASTLE, a giant portcullis slams shut behind him with the force of a Death Star bulkhead, and we hear a muffled scream, soon cut off.

Satanic Male Voice: If you dare! <laughs insanely>

In computer science, specifically complexity theory, the capital Omega is used to represent a particular complexity class of functions. Specifically:

f(n)=Ω(g(n)) iff there exists constants c>0 and N>0 such that
  f(n) >= c*g(n)  for all n>N

That is, f(n) is "at least as big" as g(n), up to a constant factor, for sufficiently large values of n. (There are some slightly different definitions as well; some researchers require only that there there be infinitely many values for which the relationship holds, not all n>N, and so forth. The various versions are not equivalent, but they're similar enough for most purposes). Thus, Big-Omega is the opposite of Big Oh. The latter states that g(n) is at least as big as f(n) up to a constant factor. Between the two is Big-Theta, which basically means that the two functions can bound each other for sufficiently large values of n and suitably chosen constants.

There is also little-omega notation, which is analogous to little-oh notation: f(n) is ω(g(n)) iff the limit as n->infinity of g(n)/f(n) is zero.

Lower bounds are usually expressed in big-Omega notation, e.g. sorting by comparisons is provably Ω(n log n).

In physics, the Greek letter omega is used to denote several things. The lowercase omega is generally used to denote angular quantities, specifically:

Capital omega is much less frequently utilised, being uncommon as a symbol for a quantity. The most common use of capital omega is as the symbol for the SI unit of resistance, the ohm.

Also, being the twenty-fourth letter in the Greek alphabet, stellar astronomers use it (along with the genitive of the constellation) to indicate the 24-th brightest star in a constellation. Thus we get Omega Centauri, Omega Orionis, Omega Cygni, etc, etc...

In marine navigation, Omega refers to a Very Low Frequency (VLF) system used for radio navigation up until September, 1997 when it ceased operation in lieu of the Global Positioning System (GPS). Omega operated in the 10.2 kilohertz (khz) range with a wavelength of 16 nautical miles (nm). Navigation was accomplished by using a receiver which measured the phase-difference contours of pairs of Omega stations. Nautical charts had "Omega lanes" printed on them consisting of two lines of adjacent zero phase-difference contours. An Omega radio receiver would increment and decrement lane counters as a vessel moved through the water. A line of position (LOP) is established by comparing the intersection of at least two Omega lanes. Accuracy of the system was within 1 to 2 nm which was useable for deep sea navigation, but not for coastal piloting.

Original Omega Station Locations

Letter       Name    Latitude   Longitude
Designation

   A         Norway   66 deg N   13 deg E
   B         Trinidad 11 deg N   62 deg W
   C         Hawaii   21 deg N  158 deg W
   D         N Dakota 46 deg N   95 deg W
   H         Japan    35 deg N  129 deg E

Source:

Quartermaster 3 & 2: Rate Training Manual. NAVEDTRA 10149-F. SuDoc Number D207.208/2:Q2/2 (the SuDoc Number is the number you use if you want to get this from a Federal Depository Library)

O*me"ga (?), n. [NL., fr. Gr. , i.e., the great or long o. Cf. Mickle.]

1.

The last letter of the Greek alphabet. See Alpha.

2.

The last; the end; hence, death.

"Omega! thou art Lord," they said. Tennyson.

Alpha and Omega, the beginning and the ending; hence, the chief, the whole.

Rev. i. 8.

The alpha and omega of science. Sir J. Herschel.

 

© Webster 1913.

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