☽ ☾☼☀☁ ☂ ☔ ☕ ☘☃ ☄ ⚫⚪☉☸■ ☎ ☏ ☐☑☒☓⚀⚁⚂⚃⚄⚅ Greek Alphabets
Other Mathematical Symbols
For a more comprehensive list please refer to wikipedia List of mathematical symbols chexed.com/.../asciicodes.php wikipedia signs http://www.fileformat.info/info/unicode/block/mathematical_operators/images.htm A necessary condition of a statement must be satisfied for the statement to be true. In formal terms, a statement N is a necessary condition of a statement S if S implies N (S ⇒ N). A sufficient condition is one that, if satisfied, assures the statement's truth. In formal terms, a statement S is a sufficient condition of a statement N if S implies N (S ⇒ N). Closedform expression: If the expression can be expressed analytically in terms of a bounded number of certain "wellknown" functions. Typically, these wellknown functions are defined to be elementary functions—constants, one variable x, elementary operations of arithmetic (+ − × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions). RingIt consists of a set together with two binary operations usually called addition and multiplication, which satisfy the following set of axioms: the addition is associative and commutative, has an identity and each element in the set has an additive inverse; the multiplication is associative, is not necessarily commutative, has an identitya and distributes over addition.
FieldA ring whose nonzero elements form a commutative group under multiplication
Hilbert SpaceGeneralizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured.
Complete Space (Cauchy Space)
Orthonormal Basis A subset {v1,...,vk} of a vector space V, with the inner product <,>, is called orthonormal if <vi,vj>=0 when i≠j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: <vi,vi>=1.
General Notes
in relation to population growth. It can model the "Sshaped" curve of growth of some population P. The initial stage of growth is approximately exponential; then, as
saturation begins, the growth slows, and at maturity, growth stops.
 the rate of reproduction is proportional to both the existing
population and the amount of available resources, all else being equal.
 to describe the selflimiting growth of a biological population. up to "all other things being equal"
It indicates that its grammatical object is some equivalence class, to be regarded as a single entity, or disregarded as a single entity. If this object is a class of transformations (such as "isomorphism" or "permutation"), it implies the equivalence of objects one of which is the image of the other under such a transformation. e.g. up to isomorphism "there are seven reflecting tetrominos, up to rotations", which makes reference to the seven possible contiguous arrangements of tetrominoes (collections of four unit squares arranged to connect on at least one side) which are frequently thought of as the seven Tetris pieces (box, I, L, J, T, S, Z.) This could also be written "there are five tetrominos, up to reflections and rotations", which would take account of the perspective that L and J could be thought of as the same piece, reflected, as well as that S and Z could be seen as the same. Morphism Any structurepreserving mappings between two mathematical structures.
Homomorphisma general morphism is called homomorphism. A structurepreserving map between two algebraic structures. The word homomorphism meaning "same" + "shape". Isomorphisms, automorphisms, and endomorphisms are all types of homomorphism.
EXAMPLE 1 The real numbers are a ring, having both addition and multiplication. The set of all 2 × 2 matrices is also a ring, under matrix addition and matrix multiplication. If we define a function between these rings as follows: where r is a real number. Then ƒ is a homomorphism of rings, since ƒ preserves both addition: and multiplication: EXAMPLE 2 For another example, the nonzero complex numbers form a group under the operation of multiplication, as do the nonzero real numbers. (Zero must be excluded from both groups since it does not have a multiplicative inverse, which is required for elements of a group.) Define a function ƒ from the nonzero complex numbers to the nonzero real numbers by f(z) = z, That is, ƒ(z) is the absolute value (or modulus) of the complex number z. Then ƒ is a homomorphism of groups, since it preserves multiplication: Note that ƒ cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:
Automorphism: symmetry  map obj to itself. An automorphism of a set X is an arbitrary permutation of the elements of X
The set of integers, Z has a unique nontrivial automorphism: negation. Endomorphism: An invertible endomorphism of X is called an automorphism [so endomorphism is a one way automorphism???!!!]. an automorphism is an endomorphism (i.e. a morphism from an object to itself) which is also an isomorphism
Isomorphismfrom the Greek adverb endon ("inside") and morphosis ("to form" or "to shape"). meaning "equal," and morphosis,
meaning "to form" or "to shape."
A map that preserves sets and relations among elements. an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f ^{−1}: Y → X, with the property that both f ^{−1}f = id_{X} and f f ^{−1} = id_{Y}.[3]
EXAMPLE Consider the group (Z_{6}, +), the integers from 0 to 5 with addition modulo 6. Also consider the group (Z_{2} × Z_{3}, +), the ordered pairs where the x coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the xcoordinate is modulo 2 and addition in the ycoordinate is modulo 3. These structures are isomorphic under addition, if you identify them using the following scheme:
Hence: isomorphism is to realize that two groups are structurally the same even though the names and notation for the elements are different. We say that groups G and H are isomorphic if there is an isomorphism between them. Another way to think of an isomorphism is as a renaming of elements. For example, the set of complex numbers {1, i, i, 1} under complex
multiplication, the set of integers {0, 1, 2, 3} under addition modulo 4,
and the subgroup {1, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} of S_{4}
look different but are structurally the same.
They are all of order 4 (but that's
not what makes them isomorphic) and are cyclic groups. The maps i ↦ 1 (for the first pair of groups) and 1 ↦ (1 2 3 4) (for the second and third of the groups) provide the necessary
isomorphisms.
