Functions and mappings rsFrom Wikipedia, the free encyclopediaContents1 2 2 real matrices 11.1 Prole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Equi-areal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Functions of 2 2 real matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 2 2 real matrices as complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Range (mathematics) 52.1 Distinguishing between the two uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Rayleigh dissipation function 73.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Reection (mathematics) 84.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Reection across a line in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Reection through a hyperplane in n dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Rigid transformation 135.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Distance formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 Translations and linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Rotation of axes 16iii CONTENTS6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 Examples in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4 Rotation of conic sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4.1 Identifying rotated conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.5 Generalization to several dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.6 Examples in several dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.6.1 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Sammon mapping 217.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Second derivative 238.1 Second derivative power rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.4 Relation to the graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4.1 Concavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4.2 Inection points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4.3 Second derivative test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.5 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.6 Quadratic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.7 Eigenvalues and eigenvectors of the second derivative . . . . . . . . . . . . . . . . . . . . . . . . 268.8 Generalization to higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.8.1 The Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.8.2 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.9.1 Print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.9.2 Online books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Semilinear transformation 299.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3 General semilinear group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31CONTENTS iii9.4 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.4.1 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.4.2 Mathieu group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110Set function 3210.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.3Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211Shear mapping 3311.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.1.1 Horizontal and vertical shear of the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.1.2 General shear mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.2Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612Signomial 3712.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813Similarity invariance 3913.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914Splitting lemma (functions) 4014.1Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015Squeeze mapping 4115.1Logarithm and hyperbolic angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.2Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.3Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3.1 Corner ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3.2 Relativistic spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.3.3 Bridge to transcendentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516Squeeze theorem 4616.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.2.1 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47iv CONTENTS16.2.2 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.2.3 Third example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.2.4 Fourth example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5016.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5117Steiners problem 5217.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318Superfunction 5418.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.2Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.3Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.4.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.4.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.4.3 Quadratic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.4.4 Algebraic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.4.5 Rational function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.4.6 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.5Abel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.6Applications of superfunctions and Abel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5919Surjective function 6019.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.3.1 Surjections as right invertible functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.3.4 Cardinality of the domain of a surjection. . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520Swirl function 6620.1Symmmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.3Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 68CONTENTS v20.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Chapter 12 2 real matricesIn mathematics, the set of 22 real matrices is denoted by M(2, R). Two matrices p and q in M(2, R) have a sum p+ q given by matrix addition. The product matrix p q is formed from the dot product of the rows and columns of itsfactors through matrix multiplication. Forq=_a bc d_,letq=_d bc a_.Then q q* = q* q = (ad bc) I, where I is the 22 identity matrix. The real number ad bc is called the determinantof q. When ad bc 0, q is an invertible matrix, and thenq1= q / (ad bc).The collection of all such invertible matrices constitutes the general linear group GL(2, R). In terms of abstractalgebra, M(2, R) with the associated addition and multiplication operations forms a ring, and GL(2, R) is its group ofunits. M(2, R) is also a four-dimensional vector space, so it is considered an associative algebra. It is ring-isomorphicto the coquaternions, but has a dierent prole.The 22 real matrices are in one-one correspondence with the linear mappings of the two-dimensional Cartesiancoordinate system into itself by the rule_xy__a bc d__xy_=_ax +bycx +dy_.1.1 ProleWithin M(2, R), the multiples by real numbers of the identity matrix I may be considered a real line. This real lineis the place where all commutative subrings come together:Let Pm = {xI + ym : x, y R} where m2 { I, 0, I }. Then Pm is a commutative subring and M(2, R) = Pmwhere the union is over all m such that m2 { I, 0, I }.To identify such m, rst square the generic matrix:_aa +bc ab +bdac +cd bc +dd_.12 CHAPTER 1. 2 2 REAL MATRICESWhen a + d = 0 this square is a diagonal matrix. Thus one assumes d = a when looking for m to form commutativesubrings. When mm= I, then bc = 1 aa, an equation describing a hyperbolic paraboloid in the space of parameters(a, b, c). Such an m serves as an imaginary unit. In this case Pm is isomorphic to the eld of (ordinary) complexnumbers.When mm = +I, m is an involutory matrix. Then bc = +1 aa, also giving a hyperbolic paraboloid. If a matrix is anidempotent matrix, it must lie in such a Pm and in this case Pm is isomorphic to the ring of split-complex numbers.The case of a nilpotent matrix, mm = 0, arises when only one of b or c is non-zero, and the commutative subring Pmis then a copy of the dual number plane.When M(2, R) is recongured with a change of basis, this prole changes to the prole of split-quaternions wherethe sets of square roots of I and I take a symmetrical shape as hyperboloids.1.2 Equi-areal mappingMain article: Equiareal mapFirst transform one dierential vector into another:_dudv_=_p rq s__dxdy_=_p dx +r dyq dx +s dy_.Areas are measured with densitydx dy , a dierential 2-form which involves the use of exterior algebra. Thetransformed density isdu dv= 0 +ps dx dy +qr dy dx + 0= (ps qr) dx dy= (det g) dx dy.Thus the equi-areal mappings are identied with SL(2, R) = {g M(2, R) : det(g) = 1}, the special linear group. Giventhe prole above, every such g lies in a commutative subring Pm representing a type of complex plane according tothe square of m. Since g g* = I, one of the following three alternatives occurs:mm = I and g is on a circle of Euclidean rotations; ormm = I and g is on an hyperbola of squeeze mappings; ormm = 0 and g is on a line of shear mappings.Writing about planar ane mapping, Rafael Artzy made a similar trichotomy of planar, linear mapping in his bookLinear Geometry (1965).1.3 Functions of 2 2 real matricesThe commutative subrings of M(2, R) determine the function theory; in particular the three types of subplanes havetheir own algebraic structures which set the value of algebraic expressions. Consideration of the square root functionand the logarithmfunction serves to illustrate the constraints implied by the special properties of each type of subplanePm described in the above prole. The concept of identity component of the group of units of Pm leads to the polardecomposition of elements of the group of units:If mm = I, then z = exp(m).If mm = 0, then z = exp(s m) or z = exp(s m).If mm = I, then z = exp(a m) or z = exp(a m) or z = m exp(a m) or z = m exp(a m).1.4. 2 2 REAL MATRICES AS COMPLEX NUMBERS 3In the rst case exp( m) = cos() + m sin(). In the case of the dual numbers exp(s m) = 1 + s m. Finally, in the caseof split complex numbers there are four components in the group of units. The identity component is parameterizedby and exp(a m) = cosh a + m sinh a.Now exp(am) = exp(am/2) regardless of the subplane Pm, but the argument of the function must betaken from the identity component of its group of units. Half the plane is lost in the case of the dual number structure;three-quarters of the plane must be excluded in the case of the split-complex number structure.Similarly, if exp(a m) is an element of the identity component of the group of units of a plane associated with 22matrix m, then the logarithm function results in a value log + a m. The domain of the logarithm function suers thesame constraints as does the square root function described above: half or three-quarters of Pm must be excluded inthe cases mm = 0 or mm = I.Further function theory can be seen in the article complex functions for the Cstructure, or in the article motor variablefor the split-complex structure.1.4 2 2 real matrices as complex numbersEvery 22 real matrix can be interpreted as one of three types of (generalized[1]) complex numbers: standard complexnumbers, dual numbers, and split-complex numbers. Above, the algebra of 22 matrices is proled as a union ofcomplex planes, all sharing the same real axis. These planes are presented as commutative subrings Pm. We candetermine to which complex plane a given 22 matrix belongs as follows and classify which kind of complex numberthat plane represents.Consider the 22 matrixz=_a bc d_.We seek the complex plane Pm containing z.As noted above, the square of the matrix z is diagonal when a + d = 0. The matrix z must be expressed as the sumof a multiple of the identity matrix I and a matrix in the hyperplane a + d = 0.Projecting z alternately onto thesesubspaces of R4yieldsz= xI +n, x =a +d2, n = z xI.Furthermore,n2= pI where p =(ad)24+bc .Now z is one of three types of complex number:If p < 0, then it is an ordinary complex number:Let q= 1/p, m = qn . Then m2= I, z= xI +mp .If p = 0, then it is the dual number:z= xI +nIf p > 0, then z is a split-complex number:Let q= 1/p, m = qn . Then m2= +I, z= xI +mp .Similarly, a 22 matrix can also be expressed in polar coordinates with the caveat that there are two connectedcomponents of the group of units in the dual number plane, and four components in the split-complex number plane.4 CHAPTER 1. 2 2 REAL MATRICES1.5 References[1] Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine77(2):11829Rafael Artzy (1965) Linear Geometry, Chapter 2-6 Subgroups of the Plane Ane Group over the Real Field,p. 94, Addison-Wesley.Helmut Karzel & Gunter Kist (1985) Kinematic Algebras and their Geometries, found inRings and Geometry, R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437509, esp 449,50, D.Reidel ISBN 90-277-2112-2 .Svetlana Katok (1992) Fuchsian groups, pp. 113, University of Chicago Press ISBN 0-226-42582-7 .Garret Sobczyk (2012). Chapter 2: Complex and Hyperbolic Numbers. New Foundations in Mathematics:The Geometric Concept of Number. Birkhuser. ISBN 978-0-8176-8384-9.Chapter 2Range (mathematics)For other uses, see Range.In mathematics, and more specically in naive set theory, the range of a function refers to either the codomain orXYf(x)f : X Yxf is a function from domain X to codomain Y. The smaller oval inside Y is the image of f . Sometimes range refers to the imageand sometimes to the codomain.the image of the function, depending upon usage. Modern usage almost always uses range to mean image.The codomain of a function is some arbitrary set. In real analysis, it is the real numbers. In complex analysis, it isthe complex numbers.The image of a function is the set of all outputs of the function. The image is always a subset of the codomain.56 CHAPTER 2. RANGE (MATHEMATICS)2.1 Distinguishing between the two usesAs the term range can have dierent meanings, it is considered a good practice to dene it the rst time it is usedin a textbook or article.Older books, when they use the word range, tend to use it to mean what is now called the codomain.[1][2] Moremodern books, if they use the word range at all, generally use it to mean what is now called the image.[3] To avoidany confusion, a number of modern books don't use the word range at all.[4]As an example of the two dierent usages, consider the function f(x) = x2as it is used in real analysis, that is, as afunction that inputs a real number and outputs its square. In this case, its codomain is the set of real numbers R , butits image is the set of non-negative real numbers R+, since x2is never negative if x is real. For this function, if weuse range to mean codomain, it refers to R . When we use range to mean image, it refers to R+.As an example where the range equals the codomain, consider the function f(x) = 2x , which inputs a real numberand outputs its double. For this function, the codomain and the image are the same (the function is a surjection), sothe word range is unambiguous; it is the set of all real numbers.2.2 Formal denitionWhen range is used to mean codomain, the range of a function must be specied. It is often assumed to be theset of all real numbers, and {y | there exists an x in the domain of f such that y = f(x)} is called the image of f.When range is used to mean image, the range of a function f is {y | there exists an x in the domain of f such thaty = f(x)}. In this case, the codomain of f must be specied, but is often assumed to be the set of all real numbers.In both cases, image f range f codomain f, with at least one of the containments being equality.2.3 See alsoBijection, injection and surjectionCodomainImage (mathematics)Naive set theory2.4 Notes[1] Hungerford 1974, page 3.[2] Childs 1990, page 140.[3] Dummit and Foote 2004, page 2.[4] Rudin 1991, page 99.2.5 ReferencesChilds (2009). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.).Springer. ISBN 978-0-387-74527-5. OCLC 173498962.Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7.OCLC 52559229.Hungerford, Thomas W. (1974). Algebra. Graduate Texts in Mathematics 73. Springer. ISBN 0-387-90518-9. OCLC 703268.Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill. ISBN 0-07-054236-8.Chapter 3Rayleigh dissipation functionIn physics, the Rayleigh dissipation function, named for Lord Rayleigh, is a function used to handle the eects ofvelocity-proportional frictional forces in Lagrangian mechanics. It is dened for a system of N particles asF=12Ni=1(kxv2i,x +kyv2i,y +kzv2i,z).The force of friction is negative the velocity gradient of the dissipation function, Ff= vF . The function is halfthe rate at which energy is being dissipated by the system through friction.3.1 ReferencesGoldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. p. 24. ISBN0-201-02918-9.7Chapter 4Reection (mathematics)This article is about reection in geometry. For reexivity of binary relations, see reexive relation.In mathematics, areection (also spelledreexion)[1] is a mapping from a Euclidean space to itself that is anisometry with a hyperplane as a set of xed points; this set is called the axis (in dimension 2) or plane (in dimension3) of reection. The image of a gure by a reection is its mirror image in the axis or plane of reection. For examplethe mirror image of the small Latin letter p for a reection with respect to a vertical axis would look like q. Its imageby reection in a horizontal axis would look like b. A reection is an involution: when applied twice in succession,every point returns to its original location, and every geometrical object is restored to its original state.The term reection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely thenon-identity isometries that are involutions. Such isometries have a set of xed points (the mirror) that is an anesubspace, but is possibly smaller than a hyperplane. For instance a reection through a point is an involutive isometrywith just one xed point; the image of the letter p under it would look like a d. This operation is also known as a centralinversion (Coxeter 1969, 7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, thereection in the point situated at the origin is the same as vector negation. Other examples include reections in aline in three-dimensional space. Typically, however, unqualied use of the term reection means reection in ahyperplane.A gure which does not change upon undergoing a reection is said to have reectional symmetry.Some mathematicians use "ip" as a synonym for reection.[2]4.1 ConstructionIn plane (or 3-dimensional) geometry, to nd the reection of a point one drops a perpendicular from the point ontothe line (plane) used for reection, and continues it to the same distance on the other side. To nd the reection of agure, one reects each point in the gure.To reect point P in the line AB using compass and straightedge, proceed as follows (see gure):Step 1 (red): construct a circle with center at P and some xed radius r to create points A' and B' on the lineAB, which are equidistant from P.Step 2 (green): construct circles centered at A' and B' having radius r. P and Q will be the points of intersectionof these two circles.Point Q is then the reection of point P in line AB.4.2 PropertiesThe matrix for a reection is orthogonal with determinant 1 and eigenvalues (1, 1, 1, ... 1, 1). The product of twosuch matrices is a special orthogonal matrix which represents a rotation. Every rotation is the result of reecting in aneven number of reections in hyperplanes through the origin, and every improper rotation is the result of reecting84.3. REFLECTION ACROSS A LINE IN THE PLANE 9MC BAAB CM/2A reection through an axis followed by a reection across a second axis parallel to the rst one results in a total motion which is atranslation.in an odd number. Thus reections generate the orthogonal group, and this result is known as the CartanDieudonntheorem.Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reections in anehyperplanes.In general, a group generated by reections in ane hyperplanes is known as a reection group.Thenite groups generated in this way are examples of Coxeter groups.4.3 Reection across a line in the planeFor more details on reection of light rays, see Specular reection Direction of reection.Reection across a line through the origin in two dimensions can be described by the following formulaRefl(v) = 2v ll l l v10 CHAPTER 4. REFLECTION (MATHEMATICS)A BPA' B'QOPoint Q is reection of point P in the line ABWhere v denotes the vector being reected, l denotes any vector in the line being reected in, and vl denotes the dotproduct of v with l. Note the formula above can also be described asRefl(v) = 2Projl(v) vWhere the reection of line l on a is equal to 2 times the projection of v on line l minus v. Reections in a line havethe eigenvalues of 1, and 1.4.4 Reection through a hyperplane in n dimensionsGiven a vector a in Euclidean space Rn, the formula for the reection in the hyperplane through the origin, orthogonalto a, is given byRefa(v) = v 2v aa aawhere va denotes the dot product of v with a. Note that the second term in the above equation is just twice the vectorprojection of v onto a. One can easily check thatRefa(v) = v, if v is parallel to a, andRefa(v) = v, if v is perpendicular to a.4.4. REFLECTION THROUGH A HYPERPLANE IN N DIMENSIONS 11BCAA'B'C'/2A reection across an axis followed by a reection in a second axis not parallel to the rst one results in a total motion that is arotation around the point of intersection of the axes.Using the geometric product the formula is a little simplerRefa(v) = avaa2Since these reections are isometries of Euclidean space xing the origin they may be represented by orthogonalmatrices. The orthogonal matrix corresponding to the above reection is the matrix whose entries areRij= ij 2aiaja2where ij is the Kronecker delta.The formula for the reection in the ane hyperplane v a = c not through the origin isRefa,c(v) = v 2v a ca aa.12 CHAPTER 4. REFLECTION (MATHEMATICS)4.5 See alsoCoordinate rotations and reectionsHouseholder transformationInversive geometryPoint reectionPlane of rotationReection mappingReection groupSpecular reection4.6 Notes[1] Reexion is an archaic spelling.[2] Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media, p.251Gallian, Joseph (2012), Contemporary Abstract Algebra (8th ed.), Cengage Learning, p. 32Isaacs, I. Martin (1994), Algebra: A Graduate Course, American Mathematical Society, p. 64.7 ReferencesCoxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons,ISBN 978-0-471-50458-0, MR 123930Popov, V.L. (2001), Reection, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Weisstein, Eric W., Reection, MathWorld.4.8 External linksReection in Line at cut-the-knotUnderstanding 2DReection and Understanding 3DReection by Roger Germundsson, The WolframDemon-strations Project.Chapter 5Rigid transformationIn mathematics, arigidtransformation (isometry) of a vector space preserves distances between every pair ofpoints.[1][2] Rigid transformations of the plane R2, space R3, or real n-dimensional space Rn are termed a Euclideantransformation because they form the basis of Euclidean geometry.[3]The rigid transformations include rotations, translations, reections, or their combination. Sometimes reectionsare excluded from the denition of a rigid transformation by imposing that the transformation also preserve thehandedness of gures in the Euclidean space (a reection would not preserve handedness; for instance, it wouldtransform a left hand into a right hand). To avoid ambiguity, this smaller class of transformations is known as properrigid transformations (informally, also known as roto-translations).In general, any proper rigid transformationcan be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as animproper rotation followed by a translation (or as a sequence of reections).Any object will keep the same shape and size after a proper rigid transformation.All rigid transformations are examples of ane transformations. The set of all (proper and improper) rigid transfor-mations is a group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of properrigid transformation is called special Euclidean group, denoted SE(n).In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to representthe linear and angular displacement of rigid bodies. According to Chasles theorem, every rigid transformation canbe expressed as a screw displacement.5.1 Formal denitionA rigid transformation is formally dened as a transformation that, when acting on any vector v, produces a trans-formed vector T(v) of the formT(v) = R v + twhere RT = R1 (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin.A proper rigid transformation has, in addition,det(R) = 1which means that R does not produce a reection, and hence it represents a rotation (an orientation-preserving or-thogonal transformation).Indeed, when an orthogonal transformation matrix produces a reection, its determinantis 1.5.2 Distance formulaA measure of distance between points, or metric, is needed in order to conrm that a transformation is rigid. TheEuclidean distance formula for Rn is the generalization of the Pythagorean theorem. The formula gives the distance1314 CHAPTER 5. RIGID TRANSFORMATIONsquared between two points X and Y as the sum of the squares of the distances along the coordinate axes, that isd(X, Y)2= (X1Y1)2+ (X2Y2)2+. . . + (XnYn)2= (X Y) (X Y).where X=(X1, X2, ..., X) and Y=(Y1, Y2, ..., Y), and the dot denotes the scalar product.Using this distance formula, a rigid transformation g:RnRn has the property,d(g(X), g(Y))2= d(X, Y)2.5.3 Translations and linear transformationsA translation of a vector space adds a vector d to every vector in the space, which means it is the transformationg(v):vv+d. It is easy to show that this is a rigid transformation by computing,d(v + d, w + d)2= (v + d w d) (v + d w d) = (v w) (v w) = d(v, w)2.A linear transformation of a vector space, L:Rn Rn, has the property that the transformation of a vector, V=av+bw,is the sum of the transformations of its components, that is,L(V) = L(av +bw) = aL(v) +bL(w).Each linear transformation L can be formulated as a matrix operation, which means L:v[L]v, where [L] is an nxnmatrix.A linear transformation is a rigid transformation if it satises the condition,d([L]v, [L]w)2= d(v, w)2,that isd([L]v, [L]w)2= ([L]v [L]w) ([L]v [L]w) = ([L](v w)) ([L](v w)).Now use the fact that the scalar product of two vectors v.w can be written as the matrix operation vTw, where the Tdenotes the matrix transpose, we haved([L]v, [L]w)2= (v w)T[L]T[L](v w).Thus, the linear transformation L is rigid if its matrix satises the condition[L]T[L] = [I],where [I] is the identity matrix. Matrices that satisfy this condition are called orthogonal matrices. This conditionactually requires the columns of these matrices to be orthogonal unit vectors.Matrices that satisfy this condition form a mathematical group under the operation of matrix multiplication called theorthogonal group of nxn matrices and denoted O(n).Compute the determinant of the condition for an orthogonal matrix to obtaindet([L]T[L]) = det[L]2= det[I] = 1,5.4. REFERENCES 15which shows that the matrix [L] can have a determinant of either +1 or 1. Orthogonal matrices with determinant1 are reections, and those with determinant +1 are rotations. Notice that the set of orthogonal matrices can beviewed as consisting of two manifolds in Rnxn separated by the set of singular matrices.The set of rotation matrices is called the special orthogonal group, and denoted SO(n). It is an example of a Lie groupbecause it has the structure of a manifold.5.4 References[1] O. Bottema & B. Roth (1990). Theoretical Kinematics. Dover Publications. reface. ISBN 0-486-66346-9.[2] J. M. McCarthy (2013). Introduction to Theoretical Kinematics. MDA Press. reface.[3] Galarza, Ana Irene Ramrez; Seade, Jos (2007), Introduction to classical geometries, BirkhauserChapter 6Rotation of axesAn xy-Cartesian coordinate system rotated through an angle to an x'y'-Cartesian coordinate systemIn mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to anx'y'-Cartesian coordinate systemin which the origin is kept xed and the x'- and y'-axes are obtained by rotating the x-and y-axes counterclockwise through an angle . A point P has coordinates (x, y) with respect to the original systemand coordinates (x', y') with respect to the new system.[1] In the new coordinate system, the point P will appear tohave been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more thantwo dimensions is dened similarly.[2][3] A rotation of axes is a linear map[4][5] and a rigid transformation.6.1 MotivationWhen we want to study the equations of curves and when we wish to use the methods of analytic geometry, coor-dinate systems become essential. When we use the method of coordinate geometry we place the axes at a positionconvenient with respect to the curve under consideration. For example, when we study the equations of ellipses andhyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin.But now suppose that we have a problem in which the curve (hyperbola, parabola, ellipse, etc.) is not situated so166.2. DERIVATION 17conveniently with respect to the axes. We would then like to change the coordinate system in order to have the curveat a convenient and familiar location and orientation. The process of making this change is called a transformationof coordinates.[6]The solutions to many problems can be simplied by rotating the coordinate axes to obtain new axes through the sameorigin.6.2 DerivationThe equations dening the transformation in two dimensions, which rotates the xy-axes counterclockwise through anangle into the x'y'-axes, are derived as follows.In the xy-system, let the point P have polar coordinates (r, ) . Then, in the x'y'-system, P will have polar coordinates(r, ) .We haveandSubstituting equations (1) and (2) into equations (3) and (4), we obtainEquations (5) and (6) can be represented in matrix form as_xy_=_ cos sin sin cos __xy_,which is the standard matrix equation of a rotation of axes in two dimensions.[8]The inverse transformation isor_xy_=_cos sin sin cos __xy_.18 CHAPTER 6. ROTATION OF AXES6.3 Examples in two dimensions6.3.1 Example 1Find the coordinates of the point P1= (x, y) = (3, 1) after the axes have been rotated through the angle 1= /6, or 30.Solution:x=3 cos(/6) + 1 sin(/6) = (3)(3/2) + (1)(1/2) = 2y= 1 cos(/6) 3 sin(/6) = (1)(3/2) (3)(1/2) = 0.The axes have been rotated counterclockwise through an angle of1=/6 and the new coordinates areP1=(x, y) = (2, 0) . Note that the point appears to have been rotated clockwise through /6 , that is, it now coincideswith the (new) x'-axis.6.3.2 Example 2Find the coordinates of the point P2= (x, y) = (7, 7) after the axes have been rotated clockwise 90, that is, throughthe angle 2= /2 , or 90.Solution:_xy_=_ cos(/2) sin(/2)sin(/2) cos(/2)__77_=_0 11 0__77_=_77_.The axes have been rotated through an angle of2= /2 , which is in the clockwise direction and the newcoordinates are P2= (x, y) = (7, 7) . Again, note that the point appears to have been rotated counterclockwisethrough /2 .6.4 Rotation of conic sectionsMain article: Conic sectionThe most general equation of the second degree has the formThrough a change of coordinates (a rotation of axes and a translation of axes), equation (9) can be put into a standardform, which is usually easier to work with. It is always possible to rotate the coordinates in such a way that in the newsystem there is no x'y' term. Substituting equations (7) and (8) into equation (9), we obtainwhereIf we select so that cot 2 = (AC)/B we will have B= 0 and the x'y' term in equation (10) will vanish.[11]When a problem arises with B, D and E all dierent from zero, we may eliminate them by performing in successiona rotation (eliminating B) and a translation (eliminating the D and E terms).[12]6.5. GENERALIZATION TO SEVERAL DIMENSIONS 196.4.1 Identifying rotated conic sectionsA non-degenerate conic section given by equation (9) can be identied by evaluating B24AC . The conic sectionis:___an ellipse or a circle, if B24AC0.[13]6.5 Generalization to several dimensionsSuppose a rectangular xyz-coordinate system is rotated around its z-axis counterclockwise (looking down the positivez-axis) through an angle , that is, the positive x-axis is rotated immediately into the positive y-axis. The z-coordinateof each point is unchanged and the x- and y-coordinates transform as above. The old coordinates (x, y, z) of a pointQ are related to its new coordinates (x', y', z') by__xyz__=__cos sin 0sin cos 00 0 1____xyz__. [14]Generalizing to any nite number of dimensions, a rotation matrix A is an orthogonal matrix that diers from theidentity matrix in at most four elements. These four elements are of the formaii= ajj= cos and aij= aji= sin ,for some and some i j.[15]6.6 Examples in several dimensions6.6.1 Example 3Find the coordinates of the point P3= (w, x, y, z) = (1, 1, 1, 1) after the positive w-axis has been rotated throughthe angle 3= /12 , or 15, into the positive z-axis.Solution:____wxyz____=____cos(/12) 0 0 sin(/12)0 1 0 00 0 1 0sin(/12) 0 0 cos(/12)________wxyz________0.96593 0.0 0.0 0.258820.0 1.0 0.0 0.00.0 0.0 1.0 0.00.25882 0.0 0.0 0.96593________1.01.01.01.0____=____1.224751.000001.000000.70711____.6.7 See alsoRotation20 CHAPTER 6. ROTATION OF AXES6.8 Notes[1] Protter & Morrey (1970, p. 320)[2] Anton (1987, p. 231)[3] Burden & Faires (1993, p. 532)[4] Anton (1987, p. 247)[5] Beauregard & Fraleigh (1973, p. 266)[6] Protter & Morrey (1970, pp. 314-315)[7] Protter & Morrey (1970, pp. 320-321)[8] Anton (1987, p. 230)[9] Protter & Morrey (1970, p. 320)[10] Protter & Morrey (1970, p. 316)[11] Protter & Morrey (1970, pp. 321-322)[12] Protter & Morrey (1970, p. 324)[13] Protter & Morrey (1970, p. 326)[14] Anton (1987, p. 231)[15] Burden & Faires (1993, p. 532)6.9 ReferencesAnton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduc-tion to Groups, Rings, and Fields, Boston: Houghton Miin Co., ISBN 0-395-14017-XBurden, Richard L.; Faires, J. Douglas (1993),Numerical Analysis (5th ed.), Boston: Prindle, Weber andSchmidt, ISBN 0-534-93219-3Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading:Addison-Wesley, LCCN 76087042Chapter 7Sammon mappingSammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lowerdimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection. It is particularly suited for use in exploratory data analysis. Themethod was proposed by John W. Sammon in 1969.[1] It is considered a non-linear approach as the mapping cannotbe represented as a linear combination of the original variables as possible in techniques such as principal componentanalysis, which also makes it more dicult to use for classication applications.[2]Denote the distance between ith and jth objects in the original space by dij , and the distance between their projectionsbydij . Sammons mapping aims to minimize the following error function, which is often referred to as Sammonsstress or Sammons error:E=1i 0.The sign function is not continuous at zero and therefore the second derivative for x = 0 does not exist. But the abovelimit exists for x = 0 :limh0sgn(0 +h) 2 sgn(0) + sgn(0 h)h2= limh01 2 0 + (1)h2= limh00h2= 08.6 Quadratic approximationJust as the rst derivative is related to linear approximations, the second derivative is related to the best quadraticapproximation for a function f. This is the quadratic function whose rst and second derivatives are the same as thoseof f at a given point. The formula for the best quadratic approximation to a function f around the point x = a isf(x) f(a) +f(a)(x a) +12f(a)(x a)2.This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.8.7 Eigenvalues and eigenvectors of the second derivativeFor many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of the secondderivative can be obtained. For example, assuming x [0, L] and homogeneous Dirichlet boundary conditions, i.e.,8.8. GENERALIZATION TO HIGHER DIMENSIONS 27v(0) = v(L) = 0 , the eigenvalues are j= j22L2and the corresponding eigenvectors (also called eigenfunctions)are vj(x) =2L sin_jxL_. Here, vj (x) = jvj(x),j= 1, . . . , .For other well-known cases, see the main article eigenvalues and eigenvectors of the second derivative.8.8 Generalization to higher dimensions8.8.1 The HessianMain article: Hessian matrixThe second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a functionf:R3 R, these include the three second-order partials2fx2,2fy2, and2fz2and the mixed partials2fxy,2fxz, and2fy z.If the functions image and domain both have a potential, then these t together into a symmetric matrix known as theHessian. The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivativetest. (See also the second partial derivative test.)8.8.2 The LaplacianMain article: Laplace operatorAnother common generalization of the second derivative is the Laplacian. This is the dierential operator 2denedby2f=2fx2+2fy2+2fz2.The Laplacian of a function is equal to the divergence of the gradient.8.9 References[1] A. Zygmund (2002). Trigonometric Series. Cambridge University Press. pp. 2223. ISBN 978-0-521-89053-3.[2] Thomson, Brian S. (1994). Symmetric Properties of Real Functions. Marcel Dekker. p. 1. ISBN 0-8247-9230-0.8.9.1 PrintAnton, Howard; Bivens, Irl; Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single andMultivariable (8th ed.), New York: Wiley, ISBN 978-0-471-47244-5Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra1 (2nd ed.), Wiley, ISBN 978-0-471-00005-128 CHAPTER 8. SECOND DERIVATIVEApostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications1 (2nd ed.), Wiley, ISBN 978-0-471-00007-5Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN978-0-03-029558-4Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early TranscendentalFunctions (4th ed.), Houghton Miin Company, ISBN 978-0-618-60624-5Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0-914098-89-8Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), NewYork: St. Martins Press, ISBN 978-0-312-18548-08.9.2 Online booksCrowell, Benjamin (2003), CalculusGarrett, Paul (2004), Notes on First-Year CalculusHussain, Faraz (2006), Understanding CalculusKeisler, H. Jerome (2000), Elementary Calculus: An Approach Using InnitesimalsMauch, Sean (2004), Unabridged Version of Seans Applied Math BookSloughter, Dan (2000), Dierence Equations to Dierential EquationsStrang, Gilbert (1991), CalculusStroyan, Keith D. (1997), A Brief Introduction to Innitesimal CalculusWikibooks, Calculus8.10 External linksDiscrete Second Derivative from Unevenly Spaced PointsChapter 9Semilinear transformationIn linear algebra, particularly projective geometry, a semilinear transformation between vector spaces V and Wover a eld K is a function that is a linear transformation up to a twist, hence semi-linear, where twist means "eldautomorphism of K". Explicitly, it is a function T :V W that is:linear with respect to vector addition:T(v +v) = T(v) +T(v)semilinear with respect to scalar multiplication:T(v) = T(v), where is a eld automorphism of K, andmeans the image of the scalar under the automorphism. There must be a single automorphism for T,in which case T is called -semilinear.The invertible semilinear transforms of a given vector space V (for all choices of eld automorphism) form a group,called the general semilinear group and denoted L(V ), by analogy with and extending the general linear group.Similar notation (replacing Latin characters with Greek) are used for semilinear analogs of more restricted lineartransform; formally, the semidirect product of a linear group with the Galois group of eld automorphism. Forexample, PU is used for the semilinear analogs of the projective special unitary group PSU. Note however, that it isonly recently noticed that these generalized semilinear groups are not well-dened, as pointed out in (Bray, Holt &Roney-Dougal 2009) isomorphic classical groups G and H (subgroups of SL) may have non-isomorphic semilinearextensions. At the level of semidirect products, this corresponds to dierent actions of the Galois group on a givenabstract group, a semidirect product depending on two groups and an action. If the extension is non-unique, there areexactly two semilinear extensions; for example, symplectic groups have a unique semilinear extension, while SU(n,q)has two extension if n is even and q is odd, and likewise for PSU.9.1 DenitionLet K be a eld and k its prime subeld. For example, if K is C then k is Q, and if K is the nite eld of order q= pi,then k is Z/pZ.Given a eld automorphism of K, a function f :V W between two K vector spaces V and W is -semilinear,or simply semilinear, if for all x, y in V and l in K it follows:1. f(x +y) = f(x) +f(y),2. f(lx) = lf(x),where ldenotes the image of l under .Note that must be a eld automorphism for f to remain additive, for example, must x the prime subeld asnf(x) = f(nx) = f(x + +x) = nf(x)Also2930 CHAPTER 9. SEMILINEAR TRANSFORMATION(l1 +l2)f(x) = f((l1 +l2)x) = f(l1x) +f(l2x) = (l1 +l2)f(x)so (l1 +l2)= l1 +l2. Finally,(l1l2)f(x) = f(l1l2x) = l1f(l2x) = l1l2f(x)Every linear transformation is semilinear, but the converse is generally not true. If we treat V and W as vector spacesover k, (by considering K as vector space over k rst) then every -semilinear map is a k-linear map, where k is theprime subeld of K.9.2 ExamplesLet K= C, V= Cn, with standard basis e1, . . . , en. Dene the map f :V Vbyf (ni=1ziei) =ni=1 zieif is semilinear (with respect to the complex conjugation eld automorphism) but not linear.Let K= GF(q) the Galois eld of order q= pi, p the characteristic. Let l= lp. By the Freshmans dreamit is known that this is a eld automorphism. To every linear map f :VW between vector spaces V andW over K we can establish a -semilinear mapf_ni=1liei_:= f_ni=1liei_Indeed every linear map can be converted into a semilinear map in such a way. This is part of a general observationcollected into the following result.9.3 General semilinear groupGiven a vector space V, the set of all invertible semilinear maps (over all eld automorphisms) is the group L(V ).Given a vector space V over K, and k the prime subeld of K, then L(V ) decomposes as the semidirect productL(V ) = GL(V ) Gal(K/k)where Gal(K/k) is the Galois group of K/k. Similarly, semilinear transforms of other linear groups can be dened asthe semidirect product with the Galois group, or more intrinsically as the group of semilinear maps of a vector spacepreserving some properties.We identify Gal(K/k) with a subgroup of L(V ) by xing a basis B for V and dening the semilinear maps:bBlbb bBlbbfor any Gal(K/k). We shall denoted this subgroup by Gal(K/k)B. We also see these complements to GL(V) inL(V ) are acted on regularly by GL(V) as they correspond to a change of basis.9.4. APPLICATIONS 319.3.1 ProofEvery linear map is semilinear, thus GL(V ) L(V ). Fix a basis B of V. Now given any semilinear map f withrespect to a eld automorphism Gal(K/k), then dene g :V Vbyg_bBlbb_:=bBf_l1bb_=bBlbf(b)As f(B) is also a basis of V, it follows that g is simply a basis exchange of V and so linear and invertible:g GL(V ).Set h := fg1. For every v=bB lbb in V,hv= fg1v=bBlbbthus h is in the Gal(K/k) subgroup relative to the xed basis B. This factorization is unique to the xed basis B.Furthermore, GL(V) is normalized by the action of Gal(K/k)B, so L(V ) = GL(V ) Gal(K/k).9.4 Applications9.4.1 Projective geometryThe L(V ) groups extend the typical classical groups in GL(V). The importance in considering such maps followsfrom the consideration of projective geometry. The induced action of L(V ) on the associated vector space P(V)yields the projective semilinear group, denoted PL(V ), extending the projective linear group, PGL(V).The projective geometry of a vector space V, denoted PG(V), is the lattice of all subspaces of V. Although the typicalsemilinear map is not a linear map, it does follow that every semilinear map f :V W induces an order-preservingmap f :PG(V ) PG(W). That is, every semilinear map induces a projectivity. The converse of this observation(except for the projective line) is the fundamental theorem of projective geometry. Thus semilinear maps are usefulbecause they dene the automorphism group of the projective geometry of a vector space.9.4.2 Mathieu groupMain article: Mathieu groupThe group PL(3,4) can be used to construct the Mathieu group M24, which is one of the sporadic simple groups;PL(3,4) is a maximal subgroup of M24, and there are many ways to extend it to the full Mathieu group.9.5 ReferencesGruenberg, K. W. and Weir, A.J. Linear Geometry 2nd Ed. (English) Graduate Texts in Mathematics. 49.New York Heidelberg Berlin: Springer-Verlag. X, 198 pp. (1977).Bray, John N.; Holt, Derek F.; Roney-Dougal, Colva M. (2009), Certain classical groups are not well-dened,Journal of Group Theory 12 (2): 171180, doi:10.1515/jgt.2008.069, ISSN 1433-5883, MR 2502211This article incorporates material from semilinear transformation on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.Chapter 10Set functionIn mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input isa set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.10.1 ExamplesExamples of set functions include:The function that assigns to each set its cardinality, i.e. the number of members of the set, is a set function.The functiond(A) =limn|A {1, . . . , n}|n,assigning densities to suciently well-behaved subsets A {1, 2, 3, ...}, is a set function.The Lebesgue measure is a set function that assigns a non-negative real number to each set of real numbers.(Kolmogorov and Fomin 1975)A probability measure assigns a probability to each set in a -algebra. Specically, the probability of the emptyset is zero and the probability of the sample space is 1, with other sets given probabilities between 0 and 1.A possibility measure assigns a number between zero and one to each set in the powerset of some given set.See Possibility theory.A Random set is a set-valued random variable. See Random compact set.10.2 ReferencesA.N. Kolmogorov and S.V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-010.3 Further readingSobolev, V.I. (2001), Set function, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4*Regular set function at Encyclopedia of Mathematics32Chapter 11Shear mappingA horizontal shearing of the plane with coecient m = 1.25, illustrated by its eect (in green) on a rectangular grid and some gures(in blue). The black dot is the origin.In plane geometry, ashearmapping is a linear map that displaces each point in xed direction, by an amountproportional to its signed distance from a line that is parallel to that direction.[1] This type of mapping is also calledshear transformation, transvection, or just shearing.An example is the mapping that takes any point with coordinates (x, y) to the point (x + 2y, y) . In this case, thedisplacement is horizontal, the xed line is the x -axis, and the signed distance is the y coordinate. Note that pointson opposite sides of the reference line are displaced in opposite directions.Shear mappings must not be confused with rotations. Applying a shear map to a set of points of the plane will changeall angles between them (except straight angles), and the length of any line segment that is not parallel to the directionof displacement.Therefore it will usually distort the shape of a geometric gure, for example turning squares intonon-square parallelograms, and circles into ellipses. However a shearing does preserve the area of geometric gures,the alignment and relative distances of collinear points. A shear mapping is the main dierence between the uprightand slanted (or italic) styles of letters.The same denition is used in three-dimensional geometry, except that the distance is measured from a xed plane.A three-dimensional shearing transformation preserves the volume of solid gures, but changes areas of plane gures(except those that are parallel to the displacement).This transformation is used to describe laminar ow of a uidbetween plates, one moving in a plane above and parallel to the rst.In the general n -dimensional Cartesian space Rn, the distance is measured from a xed hyperplane parallel tothe direction of displacement.This geometric transformation is a linear transformation of Rnthat preserves the n-dimensional measure (hypervolume) of any set.3334 CHAPTER 11. SHEAR MAPPINGboundary plate (2D, stationary)velocity, ufluidy dimensionboundary plate (2D, moving)gradient,In uid dynamics a shear mapping depicts uid ow between parallel plates in relative motion.11.1 Denition11.1.1 Horizontal and vertical shear of the planeIn the plane R2= RR , a horizontal shear (or shear parallel to the x axis) is a function that takes a generic pointwith coordinates (x, y) to the point (x +my, y) ; where m is a xed parameter, called the shear factor.The eect of this mapping is to displace every point horizontally by an amount proportionally to its y coordinate.Any point above the x -axis is displaced to the right (increasing x ) if m > 0 , and to the left if m < 0 . Points belowthe x -axis move in the opposite direction, while points on the axis stay xed.Straight lines parallel to the x -axis remain where they are, while all other lines are turned, by various angles, aboutthe point where they cross the x -axis. Vertical lines, in particular, become oblique lines with slope 1/m . Thereforethe shear factor m is the cotangent of the angle by which the vertical lines tilt, called the shear angle.If the coordinates of a point are written as a column vector (a 21 matrix), the shear mapping can be written asmultiplication by a 22 matrix:_xy_=_x +myy_=_1 m0 1__xy_.A vertical shear (or shear parallel to the y -axis) of lines is similar, except that the roles of x and y are swapped. Itcorresponds to multiplying the coordinate vector by the transposed matrix:_xy_=_xmx +y_=_1 0m 1__xy_.The vertical shear displaces points to the right of the y -axis up or down, depending on the sign of m. It leaves verticallines invariant, but tilts all other lines about the point where they meet the y -axis. Horizontal lines, in particular, gettilted by the shear angle to become lines with slope m .11.1. DEFINITION 35transform="skewX(-30)"Through a shear mapping coded in SVG,a rectangle becomes a rhombus.11.1.2 General shear mappingsFor a vector space V and subspace W, a shear xing W translates all vectors parallel to W.To be more precise, if V is the direct sum of W and W, and we write vectors asv = w + wcorrespondingly, the typical shear xing W is L whereL(v) = (w + Mw ) + w where M is a linear mapping from W into W. Therefore in block matrix terms L can be represented as_I M0 I_36 CHAPTER 11. SHEAR MAPPING11.2 ApplicationsThe following applications of shear mapping were noted by William Kingdon Cliord:A succession of shears will enable us to reduce any gure bounded by straight lines to a triangle of equalarea."... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area ofany triangle is half the area of the rectangle on the same base and with height equal to the perpendicularon the base from the opposite angle.[2]The area-preserving property of a shear mapping can be used for results involving area. For instance, the Pythagoreantheorem has been illustrated with shear mapping.[3]An algorithm due to Alan W. Paeth uses a sequence of three shear mappings (horizontal, vertical, then horizontalagain) to rotate a digital image by an arbitrary angle. The algorithm is very simple to implement, and very ecient,since each step processes only one column or one row of pixels at a time.[4]11.3 References[1] Denition according to Weisstein, Eric W. Shear From MathWorld A Wolfram Web Resource[2] William Kingdon Cliord (1885) Common Sense and the Exact Sciences, page 113[3] Mike May S.J. Pythagorean theorem by shear mapping, from Saint Louis University; requires Java and Geogebra.Clickon the Steps slider and observe shears at steps 5 and 6.[4] Alan Paeth (1986), A Fast Algorithm for General Raster Rotation. Proceedings of Graphics Interface '86, pages 7781.Chapter 12SignomialA signomial is an algebraic function of one or more independent variables. It is perhaps most easily thought of asan algebraic extension of multi-dimensional polynomialsan extension that permits exponents to be arbitrary realnumbers (rather than just non-negative integers) while requiring the independent variables to be strictly positive (sothat division by zero and other inappropriate algebraic operations are not encountered).Formally, let X be a vector of real, positive numbers.X= (x1, x2, x3, . . . , xn)TThen a signomial function has the formf(x1, x2, . . . , xn) =Mi=1__cinj=1xaijj__where the coecients ck and the exponents aij are real numbers. Signomials are closed under addition, subtraction,multiplication, and scaling.If we restrict allci to be positive, then the function f is a posynomial. Consequently, each signomial is either aposynomial, the negative of a posynomial, or the dierence of two posynomials. If, in addition, all exponents aij arenon-negative integers, then the signomial becomes a polynomial whose domain is the positive orthant.For example,f(x1, x2, x3) = 2.7x21x1/32x0.732x41x2/53is a signomial.The term signomial was introduced by Richard J. Dun and Elmor L. Peterson in their seminal joint work ongeneral algebraic optimizationpublished in the late 1960s and early 1970s. A recent introductory exposition isoptimization problems.[1] Although nonlinear optimization problems with constraints and/or objectives dened bysignomials are normally harder to solve than those dened by only posynomials (because, unlike posynomials, signo-mials are not guaranteed to be globally convex), signomial optimization problems often provide a much more accuratemathematical representation of real-world nonlinear optimization problems.12.1 References[1] C. Maranas and C. Floudas, Global optimization in generalized geometric programming, pp. 351370, 1997.3738 CHAPTER 12. SIGNOMIAL12.2 External linksS. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A Tutorial on Geometric ProgrammingChapter 13Similarity invarianceIn linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under sim-ilarities of its domain. That is, f is invariant under similarities if f(A)=f(B1AB) where B1AB is a matrixsimilar to A. Examples of such functions include the trace, determinant, and the minimal polynomial.A more colloquial phrase that means the same thing as similarity invariance is basis independence, since a matrixcan be regarded as a linear operator, written in a certain basis, and the same operator in a new base is related to onein the old base by the conjugation B1AB , where B is the transformation matrix to the new base.13.1 See alsoInvariant (mathematics)Gauge invarianceTrace diagram39Chapter 14Splitting lemma (functions)See also splitting lemma in homological algebra.In mathematics, especially in singularity theory thesplittinglemma is a useful result due to Ren Thom whichprovides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degeneratecritical point.14.1 Formal statementLetf:(Rn,0)(R,0) be a smooth function germ, with a critical point at 0 (so(f/xi)(0)=0, (i=1,...,n) ). Let V be asubspace of Rnsuch that the restriction f|V is non-degenerate, and write B for the Hessian matrix of this restriction.Let W be any complementary subspace to V. Then there is a change of coordinates (x, y) of the form (x, y)=((x, y), y) withxV,yW , and a smooth function h on W such thatf (x, y) =12xTBx +h(y).This result is often referred to as the parametrized Morse lemma, which can be seen by viewing y as the parameter.It is the gradient version of the implicit function theorem.14.2 ExtensionsThere are extensions to innite dimensions, to complex analytic functions, to functions invariant under the action ofa compact group, . . .14.3 ReferencesPoston, Tim; Stewart, Ian (1979), Catastrophe Theory and Its Applications, Pitman, ISBN 978-0-273-08429-7.Brocker, Th (1975), Dierentiable Germs and Catastrophes, Cambridge University Press, ISBN 978-0-521-20681-5.40Chapter 15Squeeze mappingr = 3/2 squeeze mappingIn linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesianplane, but is not a rotation or shear mapping.For a xed positive real number a, the mapping(x, y) (ax, y/a)is the squeeze mapping with parameter a. Since{(u, v) : uv= constant}is a hyperbola, if u = ax and v = y/a, then uv = xy and the points of the image of the squeeze mapping are on thesame hyperbola as (x,y) is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, asdid mile Borel in 1914,[1] by analogy with circular rotations which preserve circles.4142 CHAPTER 15. SQUEEZE MAPPING15.1 Logarithm and hyperbolic angleThe squeeze mapping sets the stage for development of the concept of logarithms. The problem of nding the areabounded by a hyperbola (such as xy = 1) is one of quadrature. The solution, found by Grgoire de Saint-Vincentand Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight intologarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area. Thearea of a hyperbolic sector is taken as a measure of a hyperbolic angle associated with the sector. The hyperbolicangle concept is quite independent of the ordinary circular angle, but shares a property of invariance with it: whereascircular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular andhyperbolic angle generate invariant measures but with respect to dierent transformation groups. The hyperbolicfunctions, which take hyperbolic angle as argument, perform the role that circular functions play with the circularangle argument.[2]15.2 Group theoryA squeeze mapping moves one purple hyperbolic sector to another with the same area.It also squeezes blue and green rectangles.If r and s are positive real numbers, the composition of their squeeze mappings is the squeeze mapping of theirproduct. Therefore the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicativegroup of positive real numbers. An additive view of this group arises from consideration of hyperbolic sectors andtheir hyperbolic angles.15.3. APPLICATIONS 43From the point of view of the classical groups, the group of squeeze mappings is SO+(1,1), the identity componentof the indenite orthogonal group of 2 2 real matrices preserving the quadratic form u2 v2. This is equivalent topreserving the form xy via the change of basisx = u +v, y= u v ,and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hy-perbolic rotation is analogous to interpreting the group SO(2) (the connected component of the denite orthogonalgroup) preserving quadratic form x2+ y2) as being circular rotations.Note that the SO+" notation corresponds to the fact that the reectionsu u, v vare not allowed, though they preserve the form (in terms of x and y these are x y, y x and x x, y y); theadditional "+" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity compo-nent because the group O(1,1) has 4 connected components, while the group O(2) has 2 components: SO(1,1) has 2components, while SO(2) only has 1. The fact that the squeeze transforms preserve area and orientation correspondsto the inclusion of subgroups SO SL in this case SO(1,1) SL(2) of the subgroup of hyperbolic rotations inthe special linear group of transforms preserving area and orientation (a volume form).In the language of Mbiustransforms, the squeeze transformations are the hyperbolic elements in the classication of elements.15.3 ApplicationsIn studying linear algebra there are the purely abstract applications such as illustration of the singular-value decom-position or in the important role of the squeeze mapping in the structure of 2 2 real matrices. These applicationsare somewhat bland compared to two physical and a philosophical application.15.3.1 Corner owIn uid dynamics one of the fundamental motions of an incompressible ow involves bifurcation of a ow runningup against an immovable wall. Representing the wall by the axis y = 0 and taking the parameter r = exp(t) where t istime, then the squeeze mapping with parameter r applied to an initial uid state produces a ow with bifurcation leftand right of the axis x = 0. The same model gives uid convergence when time is run backward. Indeed, the areaof any hyperbolic sector is invariant under squeezing.For another approach to a ow with hyperbolic streamlines, see the article potential ow, section Power law with n= 2.In 1989 Ottino[3] described the linear isochoric two-dimensional ow asv1= Gx2v2= KGx1where K lies in the interval [1, 1]. The streamlines follow the curvesx22Kx21= constantso negative K corresponds to an ellipse and positive K to a hyperbola, with the rectangular case of the squeeze mappingcorresponding to K = 1.Stocker and Hosoi[4] described their approach to corner ow as follows:we suggest an alternative formulation to account for the corner-like geometry, based on the use of hy-perbolic coordinates, which allows substantial analytical progress towards determination of the ow ina Plateau border and attached liquid threads. We consider a region of ow forming an angle of /2 anddelimited on the left and bottom by symmetry planes.44 CHAPTER 15. SQUEEZE MAPPINGStocker and Hosoi then recall Moatts[5] consideration of ow in a corner between rigid boundaries, induced by anarbitrary disturbance at a large distance. According to Stocker and Hosoi,For a free uid in a square corner, Moatts (antisymmetric) stream function ... [indicates] that hyper-bolic coordinates are indeed the natural choice to describe these ows.15.3.2 Relativistic spacetimeSelect (0,0) for a here and now in a spacetime. Light radiant left and right through this central event tracks two linesin the spacetime, lines that can be used to give coordinates to events away from (0,0). Trajectories of lesser velocitytrack closer to the original timeline (0,t). Any such velocity can be viewed as a zero velocity under a squeeze mappingcalled a Lorentz boost. This insight follows from a study of split-complex number multiplications and the diagonalbasis which corresponds to the pair of light lines. Formally, a squeeze preserves the hyperbolic metric expressed inthe form xy; in a dierent coordinate system. This application in the theory of relativity was noted in 1912 by Wilsonand Lewis,[6] by Werner Greub,[7] and by Louis Kauman.[8] Furthermore, Wolfgang Rindler, in his popular textbookon relativity, used the squeeze mapping form of Lorentz transformations in his demonstration of their characteristicproperty.[9]15.3.3 Bridge to transcendentalsThe area-preserving property of squeeze mapping has an application in setting the foundation of the transcendentalfunctions natural logarithm and its inverse the exponential function:Denition: Sector(a,b) is the hyperbolic sector obtained with central rays to (a, 1/a) and (b, 1/b).Lemma: If bc = ad, then there is a squeeze mapping that moves the sector(a,b) to sector(c,d).Proof: Take parameter r = c/a so that (u,v) = (rx, y/r) takes (a, 1/a) to (c, 1/c) and (b, 1/b) to (d, 1/d).Theorem (Gregoire de Saint-Vincent 1647) If bc = ad, then the quadrature of the hyperbola xy = 1 against theasymptote has equal areas between a and b compared to between c and d.Proof: An argument adding and subtracting triangles of area , one triangle being {(0,0), (0,1), (1,1)}, shows thehyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma.Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic pro-gression, the projections upon the asymptote increase in geometric sequence. Thus the areas form logarithms of theasymptote index.For instance, for a standard position angle which runs from (1, 1) to (x, 1/x), one may ask When is the hyperbolicangle equal to one?" The answer is the transcendental number x = e.A squeeze with r = e moves the unit angle to one between (e, 1/e) and (ee, 1/ee) which subtends a sector also of areaone. The geometric progressione, e2, e3, ..., en, ...corresponds to the asymptotic index achieved with each sum of areas1,2,3, ..., n,...which is a proto-typical arithmetic progression A + nd where A = 0 and d = 1 .15.4 See alsoEqui-areal mappingIndenite orthogonal groupIsochoric processLorentz transformation15.5. REFERENCES 4515.5 References[1] mile Borel (1914) Introduction Geometrique quelques Thories Physiques, page 29, Gauthier-Villars, link from CornellUniversity Historical Math Monographs[2] Mellon W. Haskell (1895) On the introduction of the notion of hyperbolic functions Bulletin of the American MathematicalSociety 1(6):1559,particularly equation 12, page 159[3] J. M. Ottino (1989) The Kinematics of Mixing: stretching, chaos, transport, page 29, Cambridge University Press[4] Roman Stocker & A.E. Hosoi (2004) Corner ow in free liquid lms, Journal of Engineering Mathematics 50:26788[5] H.K. Moatt (1964) Viscous and resistive eddies near a sharp corner, Journal of Fluid Mechanics 18:118[6] Edwin Bidwell Wilson & Gilbert N. Lewis (1912) The space-time manifold of relativity. The non-Euclidean geometry ofmechanics and electromagnetics, Proceedings of the American Academy of Arts and Sciences 48:387507, footnote p.401[7] W. H. Greub (1967) Linear Algebra, Springer-Verlag. See pages 272 to 274[8] Louis Kauman (1985) Transformations in Special Relativity, International Journal of Theoretical Physics 24:22336[9] Wolfgang Rindler, Essential Relativity, equation 29.5 on page 45 of the 1969 edition, or equation 2.17 on page 37 of the1977 edition, or equation 2.16 on page 52 of the 2001 editionHSM Coxeter & SL Greitzer (1967) Geometry Revisited, Chapter 4 Transformations, A genealogy of transfor-mation.P. S. Modenov and A. S. Parkhomenko (1965) Geometric Transformations, volume one. See pages 104 to 106.Walter, Scott (1999). The non-Euclidean style of Minkowskian relativity (PDF). In J. Gray. The SymbolicUniverse: Geometry and Physics. Oxford University Press. pp. 91127.(see page 9 of e-link)Chapter 16Squeeze theoremSandwich theorem redirects here. For the result in measure theory, see Ham sandwich theorem.In calculus, the squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich ruleand sometimes the squeeze lemma) is a theorem regarding the limit of a function.The squeeze theorem is used in calculus and mathematical analysis. It is typically used to conrm the limit of afunction via comparison with two other functions whose limits are known or easily computed. It was rst usedgeometrically by the mathematicians Archimedes and Eudoxus in an eort to compute , and was formulated inmodern terms by Gauss.In many languages (e.g. French, German and Italian), the squeeze theorem is also known as the two policemen (anda drunk) theorem, or some variation thereof. The story is that if two policemen are escorting a drunk prisonerbetween them, and both ocers go to a cell, then (regardless of the path taken, and the fact that the prisoner may bewobbling about between the policemen) the prisoner must also end up in the cell.16.1 StatementThe squeeze theorem is formally stated as follows.Let I be an interval having the point a as a limit point. Let f, g, and h be functions dened on I,except possibly at a itself. Suppose that for every x in I not equal to a, we have:g(x) f(x) h(x)and also suppose that:limxag(x) = limxah(x) = L.Then limxaf(x) = L.The functions g and h are said to be lower and upper bounds (respectively) of f.Here a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left-or right-hand limits.A similar statement holds for innite intervals: for example, if I = (0, ), then the conclusion holds, taking thelimits as x .4616.2. EXAMPLES 4716.1.1 ProofFrom the above hypotheses we have, taking the limit inferior and superior:L = limxag(x) liminfxaf(x) limsupxaf(x) limxah(x) = L,so all the inequalities are indeed equalities, and the thesis immediately follows.A direct proof, using the (, ) denition of limit, would be to prove that for all real > 0 there exists a real > 0such that for all x with 0 < |x a | < , we have < f(x) L < . Symbolically, > 0 > 0 : x (0 < |x a| < < f(x) L < ).Aslimxag(x) = Lmeans that > 0 1> 0 : x (0 < |x a| < 1 < g(x) L < ). (1)andlimxah(x) = Lmeans that > 0 2> 0 : x (0 < |x a| < 2 < h(x) L < ), (2)then we haveg(x) f(x) h(x)g(x) L f(x) L h(x) LWe can choose := min {1, 2} . Then, if |x a| < , combining (1) and (2), we have < g(x) L f(x) L h(x) L< , < f(x) L < which completes the proof. 16.2 Examples16.2.1 First exampleThe limit48 CHAPTER 16. SQUEEZE THEOREMx2sin(1/ x) being squeezed in the limit as x goes to 0limx0x2sin(1x)cannot be determined through the limit lawlimxa(f(x) g(x)) = limxaf(x) limxag(x),becauselimx0sin(1x)does not exist.However, by the denition of the sine function,1 sin(1x) 1.16.2. EXAMPLES 49It follows thatx2 x2sin(1x) x2Since limx0x2= limx0x2= 0 , by the squeeze theorem, limx0x2sin(1x) must also be 0.16.2.2 Second exampleProbably the best-known examples of nding a limit by squeezing are the proofs of the equalitieslimx0sin xx= 1,limx01 cos xx= 0.The rst follows by means of the squeeze theorem from the fact thatcos x < sin xx< 1for x close enough, but not equal to 0.These two limits are used in proofs of the fact that the derivative of the sine function is the cosine function. That factis relied on in other proofs of derivatives of trigonometric functions.16.2.3 Third exampleIt is possible to show thatdd tan = sec2by squeezing, as follows.In the illustration at right, the area of the smaller of the two shaded sectors of the circle issec2 2,since the radius is sec and the arc on the unit circle has length . Similarly the area of the larger of the two shadedsectors issec2( + ) 2.What is squeezed between them is the triangle whose base is the vertical segment whose endpoints are the two dots.The length of the base of the triangle is tan( + ) tan(), and the height is 1. The area of the triangle is thereforetan( + ) tan()2.From the inequalities50 CHAPTER 16. SQUEEZE THEOREM1tan ( tan + ) sec ) + ( secsec2 2 tan( + ) tan()2 sec2( + ) 2we deduce thatsec2 tan( + ) tan() sec2( + ),provided > 0, and the inequalities are reversed if < 0. Since the rst and third expressions approach sec2 as 0, and the middle expression approaches (d/d) tan , the desired result follows.16.2.4 Fourth exampleThe squeeze theorem can still be used in multivariable calculus but the lower (and upper functions) must be below(and above) the target function not just along a path but around the entire neighborhood of the point of interest andit only works if the function really does have a limit there.It can, therefore, be used to prove that a function has alimit at a point, but it can never be used to prove that a function does not have a limit at a point.[1]lim(x,y)(0,0)x2yx2+y216.3. REFERENCES 51cannot be found by taking any number of limits along paths that pass through the point, but since0 x2x2+y2 1|y| y |y||y| x2yx2+y2 |y|lim(x,y)(0,0)|y| = 0lim(x,y)(0,0)|y| = 00 lim(x,y)(0,0)x2yx2+y2 0therefore, by the squeeze theorem,lim(x,y)(0,0)x2yx2+y2= 016.3 ReferencesWeisstein, Eric W., Squeezing Theorem, MathWorld.[1] Stewart, James (2008). Chapter 15.2 Limits and Continuity. Multivariable Calculus (6th ed.). pp. 909910. ISBN0495011630.16.4 External linksSqueeze Theorem by Bruce Atwood (Beloit College) after work by, Selwyn Hollis (Armstrong Atlantic StateUniversity), the Wolfram Demonstrations Project.Squeeze Theorem proof on Proofs.wiki.Chapter 17Steiners problem-1-0.500.511.520 1 2 3 4 5 6f(x)xSteiner's problemx^(1/x)Steiners problem is the problem of nding the maximum of the functionf(x) = x1/x.[1]It is named after Jakob Steiner.The maximum is at x=e , where e denotes the base of natural logarithms. One can determine that by solving theequivalent problem of maximizingg(x) = ln f(x) = ln xx.5217.1. REFERENCES 53The derivative of g can be calculated to beg(x) =1 ln xx2.It follows that g(x) is positive for 0 < x < e and negative for x > e , which implies that g(x) (and therefore f(x) )increases for 0 < x < e and decreases for x > e. Thus, x = e is the unique global maximum of f(x).17.1 References[1] Eric W. Weisstein. Steiners Problem. MathWorld. Retrieved December 8, 2010.Chapter 18SuperfunctionMain article: Iterated functionMain article: Innite compositions of analytic functionsIn mathematics, superfunction is a nonstandard name for an iterated function for complexied continuous iterationindex. Roughly, for some function f and for some variable x, the superfunction could be dened by the expressionS(z; x) =f_f_. . . f(x) . . .__. .z function the of evaluations f.Then, S(z;x) can be interpreted as the superfunction of the function f(x). Such a denition is valid only for a positiveinteger index z. The variable x is often omitted. Much study and many applications of superfunctions employ variousextensions of these superfunctions to complex and continuous indices; and the analysis of the existence, uniqueness andtheir evaluation. The Ackermann functions and tetration can be interpreted in terms of super-functions.18.1 HistoryAnalysis of superfunctions arose from applications of the evaluation of fractional iterations of functions. Superfunc-tions and their inverses allow evaluation of not only the rst negative power of a function (inverse function), but alsoof any real and even complex iterate of that function. Historically, an early function of this kind considered wasexp; the function! has then been used as the logo of the Physics department of the Moscow State University.[1]At that time, these investigators did not have computational access for the evaluation of such functions, but thefunction exp was luckier than! : at the very least, the existence of the holomorphic function such that((u)) = exp(u) had been demonstrated in 1950 by Hellmuth Kneser.[2]Relying on the elegant functional conjugacy theory of Schrders equation,[3] for his proof, Kneser had constructedthe superfunction of the exponential map through the corresponding Abel function X , satisfying the related AbelequationX(exp(u)) = X(u) + 1.so that X(S(z; u)) = X(u) +z. The inverse function Kneser found,S(z; u) = X1(z +X(u))is an entire super-exponential, although it is not real on the real axis; it cannot be interpreted as tetrational, becausethe condition S(0; x)=x cannot be realized for the entire super-exponential. The real exp can be constructedwith the tetrational (which is also a superexponential); while the real! can be constructed with the superfactorial.5418.2. EXTENSIONS 5518.2 ExtensionsThe recurrence formula of the above preamble can be written asS(z + 1; x) = f(S(z; x)) z N : z> 0S(1) = f(x).Instead of the last equation, one could write the identity function,S(0) = x ,and extend the range of denition of the superfunction S to the non-negative integers. Then, one may positS(1) = f1(x),and extend the range of validity to the integer values larger than 2.The following extension, for example,S(2) = f2(x)is not trivial, because the inverse function may happen to be not dened for some values of x . In particular, tetrationcan be interpreted as super-function of exponential for some real base b ; in this case,f= expb.Then, at x=1,S(1) = logb 1 = 0,butS(2) = logb 0is not dened.For extension to non-integer values of the argument, the superfunction should be dened in a dierent way.For complex numbers aandb, such that abelongs to some connected domain D C , the superfunction (froma to b ) of a holomorphic function f on the domain D is function S , holomorphic on domain D , such thatS(z+1) = f(S(z)) z D : z+1 DS(a) = b.18.3 UniquenessIn general, the superfunction is not unique. For a given base function f , from a given (a d) superfunction S ,another (a d) superfunction G could be constructed as56 CHAPTER 18. SUPERFUNCTIONG(z) = S(z +(z))where is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that (a) = 0 .The modied super-function may have a narrower range of holomorphy. The variety of possible super-functions isespecially large in the limiting case, when the width of the range of holomorphy becomes zero; in this case, one dealswith real-analytic superfunctions.[4]If the range of holomorphy required is large enough, then, the super-function is expected to be unique, at least insome specic base functions H . In particular, the (C, 0 1) super-function of expb , for b > 1 , is called tetrationand is believed to be unique, at least for C= {z C: (z) > 2} ; for the case b > exp(1/e) ,[5] but up to 2009,the uniqueness was more conjecture than a theorem with a formal mathematical proof.18.4 ExamplesThis short collection of elementary superfunctions is illustrated in.[6] Some superfunctions can be expressed throughelementary functions; they are used without mention that they are superfunctions. For example, for the transferfunction "++", which means unity increment, the superfunction is just addition of a constant.18.4.1 AdditionChose a complex number c and dene the function addc as addc(x) = c +x, x C . Further dene the functionmulc as mulc(x) = c x, x C .Then, the function S(z; x) = x + mulc(z) is the superfunction (0 to c) of the functionaddcon C.18.4.2 MultiplicationExponentiation expc is superfunction (from 1 to c ) of function mulc .18.4.3 Quadratic polynomialsThe examples but the last one, below, are essentially from Schrders pioneering 1870 paper.[3]Let f(x) = 2x21 . Then,S(z; x) = cos(2zarccos(x))is a (C, 01) superfunction (iteration orbit) of f.Indeed,S(z + 1; x) = cos(2 2zarccos(x)) = 2 cos(2zarccos(x))21 = f(S(z; x))and S(0; x) = x .In this case, the superfunction S is periodic, with period T=2ln(2)i 9.0647202836543876194i ; and the super-function approaches unity in the negative direction of the real axis,limzS(z) = 1.18.5. ABEL FUNCTION 5718.4.4 Algebraic functionSimilarly,f(x) = 2x1 x2has an iteration orbitS(z; x) = sin(2zarcsin(x)).18.4.5 Rational functionIn general, the transfer (step) function f(x) needs not be an entire function. An example involving a meromorphicfunction f reads,f(x) =2x1x2x D; D = C\{1, 1}.Its iteration orbit (superfunction) isS(z; x) = tan(2zarctan(x))on C, the set of complex numbers except for the singularities of the function S. To see this, recall the double angletrigonometric formulatan(2) =2 tan()1 tan()2 C\{ C : cos() = 0|| sin() = cos()}.18.4.6 ExponentiationLet b > 1 , f(u) = expb(u) , C= {z C : (u) > 2} . The tetration tetb is then a (C, 01) superfunction ofexpb .18.5 Abel functionMain article: Abel equationThe inverse of a superfunction for a suitable argument x can be interpreted as the Abel function, the solution of theAbel equation,X(exp(u)) = X(u) + 1.and henceX(S(z; u)) = X(u) +z.The inverse function when dened, isS(z; u) = X1(z +X(u)),58 CHAPTER 18. SUPERFUNCTIONfor suitable domains and ranges, when they exist. The recurs