⊕

It looks like a Quake-fonted O, but it's not. It's direct sum notation. Given two sets of integers A and B, A ⊕ B consists of the set {a + b: a ∈ B; b ∈ B}. Example:

A = { 4, 5, 7 }
B = { 1, 2 }

A ⊕ B = { 5, 6, 6, 7, 8, 9 }

Both more commonly used and less standardized than ⊗, this symbol has several uses, all somehow related to saying "like + only different":

• In middle schools in some parts of the world, used to signify modulo addition: After setting the modulus (say 12), you can say "7⊕6=1" instead of the more formidable "7+6=1 (mod 12)".

  Computer science makes up new names for everything, but here they steal an idea from middle schools. "⊕" is often used as a symbol for the XOR operation. XOR is just modulo 2 addition. You'll also see "⊕" as a "vectorized" XOR, usually bitwise XOR on some register.

• In abstract algebra, signifies the direct sum of two objects (typically modules). It's really just a finite version of cartesian product (denoted by ×, just to confuse). So A₁⊕...⊕A_n is to all intents and purposes A₁×...×A_n. The infinite version, however, considers only elements that are 0 in all but finitely many positions. So

  A₁⊕A₂⊕... = {(a₁,a₂,...)∈A₁×A₂×.. | ∃N.∀n>N.a_n=0}

Your browser and/or font may not be up to the task of displaying it! It looks like a "+" (plus) in a circle...

In astronomy and astrodynamics (and I presume many other physical sciences), ⊕ denotes "Earth", or "of the Earth". For example:

• R_⊕ = 6378.1363 km (radius)
• M_⊕ = 5.9743 x 10²⁴ kg (mass)
• ω_⊕ = .000072921158553 rad/sec (rotation)
• e_⊕ = .081819221456 (eccentricity)

This usage probably has its roots in astrology - much as the circle with a cross on the bottom (female) is the symbol for Venus, circle with an arrow (male) for Mars, and circle with a dot in the middle for the Sun.

Here's what one source (http://www.awarenessshop.com/KnowledgeBase/stellarsymbolsth.html) had to say about the symbolic Earth in a magical and astrological context:

"EARTH - Though this symbol is not often used in magickal work, its meaning should not be overlooked. Our global home is often represented with a cross circumscribed by a circle: matter surrounded by spirit."

Also, in Susan Cooper's The Dark is Rising sequence of novels, the Earth symbol is used as a powerful talisman for harnessing ancient Welsh magic.

This symbol is regularly used to denote XOR in logic. When using the usual two-valued logic, and using the symbol + for OR, ' for NOT, and * (or just concatenation) for AND, then

• X ⊕ Y = (XY') + (X'Y)

This symbol then represents the familiar either-or concept: either X or Y, but not both. The AND/OR form reads X and not Y or not X and Y.

In old Japanese heraldry, this was the insignia on the standard for the Shimazu family and the feudal Edo period Satsuma domain. The crest is a circle with the chinese character for "ten", which is like a cross or plus sign, in the middle.

The Shimazu clan ruled Satsuma province (present day western half of Kagoshima prefecture) in Kyushu from the Kamakura period to the establishment of the modern Japanese state in the Meiji.

Highly respected warriors throughout Japanes history, they were instrumental in Ashikaga Takauji's recapture of Kyoto and the establishment of the Muromachi shogunate and for being major supporters of the Meiji emperor in the establishment of the modern Japanese state. During the warring states "Sengoku-Jidai" period they controlled almost all of Kyushu before their subjugation by Toyotomi Hideyoshi who would eventually use them as cannon fodder during the infamous Keicho and Bunroku invasions of Korea. It is said that Shimazu Yoshihiro's Satsuma army defeated a Ming army 20 times its size at the battle of Shisen, earning him the name "oni" Shimazu from the Chinese. These Shimazu/Satsuma bearers saw that this insignia flew at major battles from Sekigahara in 1600 (with another one of Yoshihiro's miracles) to the Boshin War in 1868. Though their loss at Sekigahara with Ishida Mitsunari led to Tokugawa Ieyasu's establishment of the Edo shogunate, their victory in the Boshin War spelled the bakufu's destruction and the coming of the Meiji Restoration.