Archive for February, 2010

I.4: More Extremal Problems and Determining Whether Different Mathematical Properties are Compatible, Pages 65 -66.

Sunday, February 28th, 2010

A very hard problem that only has very inaccurate bounds is how many numbers can one choose between 1 and n if no three of them are allowed to lie in an arithmetic progression?

How many steps are needed to multiply together two n digit numbers?

A similar problem is how to compute the multiplication of two n x n matricies efficiently?

It has been shown that even normalized gaps between two succesive prime numbers can be arbitrarily large, proved by Westzynthius in 1931.  There are also infinitely primes for which p and p + 2 is also a prime, proved by Goldston, Pintz and Yildirim in 2005.

 Given a mathematical structure and a selection of properties that it has then which combinations of properties imply which other ones?

I.4: More Discovering Patterns, Explaining Apparent Coincidences, Counting and Measuring, Exact Counting, Estimates, Averages and Extremal problems. Pages 59 – 64.

Saturday, February 27th, 2010

The triangiular lattice defines the best packing of circles in the plane.  For higher dimensions a lattice, L, is described in Rn.  It has three properties: 1.  If x and y belong to L then so do x + y and x – y,  2.  If x belongs to L then x is isolated in the sense that there is some d > 0 so that the distance between x and any other point of L is at least x,  and 3.  L is not contained in any n – 1 dimensional subset of Rn.  One lattice is the set of all points with integer coordinates in Rn.

 In 24 dimensions one can define the Leech lattice which also has the followint three properties:  1.  There is a 24 by 24 matrix M with determinant equal to one such that L consists of all integer combinations of the columns of M, 2.  if v is an element, point, of L then then the square of the distance from 0 to v is an even integer and 3:  the non-zero vector nearest to zero is at distance two.  The balls of radius one about the points of in L form a packing of R24.  In this lattice there are 196,560 nonzero vectors in L nearest to zero all a distance 2 from one another.   This lattice also has a huge number of rotational symmetries.

The Conway group Co1 is the quotient of the symmetry group and the subgroup consisting of the identity and minus the identity.  Co1 is on of the famous sporadic simple groups.

The Leech group was shown in 2004  by Henry Cohn and Abhinav Kumar to give the densest possible packing in R24 among all packings derived from lattices.

The largest of all sp[oradic finite simple groups is called the Monster group.  It contains 

246 x 320 x 59 x 76 x 112 x 133 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71 elements.  This group is related to a lattice with smallest possible dimension of 196,883.  

 

The elliptic modular function of algebraic numbers is defined as:  j(z) = e-2piz + 744 + 196,884 e2piz   + 21,493,760 e4piz + 864,299,970 e6piz+ … .  That the coefficient on e2piz and the monster group smallest possible lattice number are related is called the monstrous moonshine conjecture.  This relationship was proved by Richard Borcherds in 1992.  The conection was proved with vertex algebra and string theory, a concept from theoretical physics.

 

There is another example of deep connections in mirror symmetry.

 

The j – fucntion leads to another coincidence.   It turns out that e2pÖ163 = 262,537,412,640,768,743.99999999999925 … .  The fact that this number is soo close to an integer is explained by algebraic numbers.

 

There are 60 rotational symmetries of an icosahedron.  This can be seen by a simple counting aurguement.  The group of rotations is called, A5, the alternating group of 5 elements.

 

The number of n-step random walks that start at zero is 2n.

 

The number of walks of length 2n that start and end at zero can also be counted.    This can be done by creating a recurrence relation.  This relation can be evaluated by use of generating functions.  This corresponds to a legal bracketing system that is never negative.  This aurgument gives a way of efficiently counting the number of walks of length 2n that start and end at zero. 

 

The number r(n) of regions that a plane is cut into by n lines if no two lines are parallel and no three concurrent is 1/2 x (n(n+3)).

 

The number s(n) of ways of writing n as a sum of four squares is equal to 8 times the the sum of all divisors of n that are not multiples of 4.

 

The number of lines in space that meet a given 4 lines that are in gneral position is 2.  Generalizations of this are accomplished using Schubert calculus.

 

Let p(n) be the number of ways of espressing a positive integer as a sum of positive integers.  p(n) is called the partition function.  There is an approximation to p(n) called an due to Hardy and Ramanujan that is so accurate that p(n) is always the closest integer to an.

 

Sometimes counting problems cannot be answered exactly.  In this case it may be possible to give upper and lower bounds for the answer.  Examples include: estimating the number of prime numbers less than n, counting the number of self avoiding walks of length n and the number of integer coordinate points in the plane contained in a circle of radius t in the plane.

 

Sometimes when counting is doen for averages.  Examples include: The avergae distance between the start point and end point of a self avoiding walk of length n and the number of distinct prime factors of  a number between m and 2m.

 

There are extremal problems too.  These include: if X is a set with n distinct elements then how many subsets of X are there if none of the subsets is contained in another subset and what is the shape of a heavy chain supported by two spots in a ceiling?   There is a combinatoric formula to answer the first problem and the shape is that of a catenary.

I.4 Proving a More Abstract Result, Identifying Characteristic Properties, Generalization and Reformulation, Higher Dimensions and Several Variables, Discovering Patterns: Pages 56 – 58

Friday, February 26th, 2010

A more abstract result, Lagrange’s theorem,  can be used to prove a more specific result, Fermat’s little theorem.  Fermat’s little theorem states that if p is a prime and if a is not a multiple of p, then ap-1 leaves a remainder of 1 when you divide by p.  Lagrange’s theorem states that the size of a group is always divisible by the size of any of its subgroups.  An aurgument shows that Fermat’s little theorem is just a special case of Lagrange’s theorem.

This process of abstraction has multiple benefits: existence of a more general theorem, revealing other interesting specific cases, one only needs to prove this more general theorem once and it  connects otherwise unconnected results.

Characteristic properties are the defining properties of some objects rather than the object itself.   Examples, are the imaginary number, i, the square root of two and x raised to the power a for x and a real numbers.  Abstraction is also portrayed as the opposit of classification.

It is shown how reformulation of the concept of dimension allows one to generalize the concept of dimension to non-integer dimensions.  Nocommutative geometry uses this same process.  A manifold X is the set, C(X), of all complex valued continuous funtions defined on X.  C(X) is a vector space.  C(X) is an algebra and even a C*-algebra.  This algebra is commutative in the operation of multiplication.  If one now considers non-commutative algebras then this provides a reformulation with an inferred noncommutative geometry.  A third example is given that generalizes the fundamental theorem of arithmetic that uses primes and multiplication to form all integers in exactly one way.  Rings of numbers such as a + i b√5 with a and b integers may have multiple factorizations.  Ideal numbers are formed and defined.  We asssociate the set of all multiples of a number with the number.  A subset of a ring with the closure property is called an ideal.  One can use this ideal to determine a unique factorization. 

A single linear eauations in one variable is much easier to analyze than are polynomial systems of equations in several variables.  Partial differential equations using multiple variables and are much more difficult to analyze than differential equations in just one variable.  It is a fruitful method to generalize an idea to higher dimensions and/or to several variables.

One can discover patterns to solve packing problems in low dimensions.  For example, packing the 2-dimensional plane with circles or 3-dimensions with spheres.  In high dimensions this is an unsolved problem.  One should not just give up then but it may be fruitful to reformulate the problem so that it can be solved.  For example, one can find the most dense known packing in a higher dimension.

I3: Projective, Lorentz, Manifolds and Differential Geometry, Riemannian Metrics, I4: The General Goals of Mathematical Research: Solving Equations, Linear, Polynomial, Diophantine, Differential, Classifying: Identifying Building Blocks and Families, Equivalence, Non-equivalence and Invariants, Generalizing,Weakening Hypothesis and Strengthening Conclusions, Page 43 – 55

Thursday, February 25th, 2010

One view of projective geometry is all sets of lines in R3 that go through the origin with both points of the units sphere regarded as the same.    A transformation takes any invertible linear map and apply it to R3.  Linear maps may be multiples of one another.  The resulting group of transformations is like GL3(R).  Regarding all non-zero multiples of any given matrix are equivalent gives the projective special linear group PSL3(R).  It is the three dimensional equivalent of PSL2(R).  Because PSL3(R) is bigger than PSL2(R) the projective plane comes with a richer set of transformations than the hyperbolic plane.  So fewer geometric properties are preserved and, for example, there is no obvious notion of projective distance.

Lorentz geometry is used in the special theory of reletivity to model four-dimensional spacetime.  This space time is also known as Minkowski space.  The usual notion of distance betwen two points (s,x,y,z) and (t,a,b,c) is replaced by the generalized distance -(s-t)(s-t) + (x-a)(x-a) + (y-b)(y-b) + (z-c)(z-c).   The minus sign on the first term reflects the fundamental difference between space and time.  A Lorentz transformation is a linear map from R4 to R4 that preserves the generalized distance.  Let g be the linear map that sends (t,x,y,z) to (s,a,b,c) and G the corresponding 4 by 4 matrix with elements (-1,1,1,1,) on the diagonal and off diagonal elements equal to zero.   A Lorentz transformation is one whose matrix A satifies AGAT = I, I the 4 by 4 identity matrix and superscript T the transpose of the matrix.  The transpose of matrix A is matrix B with elements Bij = Aji.   

A point is said to be spacelike if -tt+xx+yy+zz > 0, timelike if tt + xx +yy + zz < 0 and in the light cone if tt + xx + yy + zz =0.  Lorentz geography is used in general relativity which may also be thought of as the study of Lorentzian manifolds.

A d-dimensional manifold is any geometrical object M with the property that every point x in M is surrounded by what may appear to be a portion of d-dimensional Euclidean space.  For example, small parts of a sphere, toris or projective plane are very close to plane and thus are 2-manifolds.   One can use an atlas to formally define a manifold.  Calculus is possible for functions defined on manifolds. 

Differentiability may be defined for manifolds.  Manifolds with transitions functions that are differentiable, and equal on different charts, are said to be differentiable manifolds.  Manifolds for which transition functions are continuous but not necessarily differentiable are called topological manifolds.

The manifold ideas generalize where the Euclidean spaces are replaced by manifolds.  The domain of the linear map is called a tangent space.

On any given manifold there may be a more than one metric defined.  The Riemannian Metric allows one to calculate distances of paths on a manifold.

The general goals of mathematical research are listed.  These goals include: solving equations, classifying, generalizing, discovering patterns, explaining apparent coincidences, counting and measuring determining whether different mathematical properties are compatible, working with arguments that are not fully rigorous, finding explicit proofs and algorithms, and addressing what you find in a mathematical paper.

Solving equations involvers three issues: does a given equation have any solutions, if an equation has solutions, then does it have exactly one solution and what is the set in which solutions are required to exist?  The first two questions are known as existence and uniqueness.  The third may provide a generalization.

Linear eauations may be written in matrix notation as A x = b.  Here A is a matrix, x and b are column vectors.  A and b are known and the problem is to solve for x.

Polynomial equations are of the form: anxn +  an-1xn-1 + … + a2x2 +  a1x1 + a0 = 0.  Solving quadratic requires the intoduction of irrational numbers and complex numbers.  There are fomulas for finding solutions of polynomials for degrees n = 1, 2, 3 and 4.  There cannot be a formula for polynomials of degree 5 or higher.  Abel and Galois showed that no formula exists for degress 5 or higher.

The field of algebraic geometry is concerned with solutions of polynomial equations in more than one variables. 

Diophantine equations are those where the solutions are restricted to the field of integers.  The most famous diophantine equation is the Fermat equation xn + yn = zn.  Andrew Wiles proved that this equation has no integer solutions for n > 2.  It does however, have an infinite number of solutions for n = 2.  There is no systematic solution to Diophantine equations.   Diophantine equations are solved in subsets, since there is no systematic approach to solve them all at once.

Differential equations include the simple harmonic motion equation and the heat equation.  The simple harmonic motion equation is:  d2x/dt2 + k2x = 0.  The solutions to these equations are functions.  The general solution to the simple hamonic motion equation is:  x(t) = A sin(kt) + B cos( kt).  These differential equations are called linear because, for example, if we write f(f) = d2x/dt2 + k2x, then f is  alinear map in the sense f(f + g) = f(f) + f(g) and f(af) = af(f).  The heat equation has this same property.  Such differential equations are called linear.  A few differential equations including some nonlinear ones can be solved exactly.

The three body problem is an example where the solution may not be written down explicitly, and the solution may be in fact chaotic.

Regular polytopes in dimensions three and higher fall into three families: the n-dimensional versions of the tetrahedron, the cube and the octahedron,  five exceptional examples, the dododecahedron, the icosahedron and three four dimensional polytopes, which have 120 three dimensional faces, 600 tetrahedra faces and one with verticies of the forms with plus and minus 1’s and plus and minus 2’s with zeroes.

Primes are used as multiplicative building blocks of integers.  In a similar way finite groups are all products of basic groups that are called simple groups.

Two objects are called equivalent if we are not concerned about their differences.  A topologist might consider a sphere and a cube equivalent, as well as a donut and a teacup.  The sphere and the torus are examples of compact orientable surfaces.  Given a compact orientable surface on can find a an equivalent surface built out of triangles that is topologically equivalent. 

One defines the Euler characteristic of one of the surfaces built out of triangles as the number of verticies – the number of edges plus the number of faces.  This gives us a way of showing that a sphere is topologically different from a torus.  One has Euler character istic of 2 and the other of 0, so they are topologically different.  This is because the Euler characteristic is the same for all triangulations of a surface and if two surfaces are continuous deformations of one another then the have the same Euler characteristic. 

The Euler characteristic is an example of an invariant.  Ifg one has a general invariant f, then if two objects X and Y are equivalent then f(X) =  f(Y).  To show that two objects are not equivalent one can show that f(X) and f(Y) are different.

In mathematics onetakes specific cases and tries to generalize them.   This may help in understanding the specific case.

Sometimes one can waken a hypothesis and then prove the conclusion.  For example, is there a number (integer) that can be written as the sum of four cubes in ten different ways?  Attacking this problem diectly is very hard.  However, one can show that there must be such a number by looking at how many ways there are of summing ten different cubes for all number sless than 1,000,000,000.  A combinatoric argument show that there must be such a number.  This argument does not supply the number but only guarantees its existance.

Instead of weakening the hypothesis one may think of this method as strenthening the conclusion.  This uses the augument that P \Rightarrow  Q may also be written v\Rightarrow vQ.

I3. Holomorphic Functions, Geometry, Symmetry Groups, Euclidean, Affine, Topology, Spherical and Hyperbolic, Pages 37 – 42.

Wednesday, February 24th, 2010

Differentiable functions  from the complex numbers to the complex numbers are quite special and are called holomorphic.  Holomorphic functions do satisfy the Cauchy-Riemann equations.  If the holomorphic function f(z) is written f(x + iy) = u(x + iy) + iv(x + iy) then u/x = v/y and u/y = – ¶v/x.  One consequence of this is that 2u/x2 + 2u/y2 = 2v/xy – 2v/yx = 0.  It follows that both u and v satisfy the Laplace equation, Df = 2f/x2 + 2f/y2 so Du = 2u/x2 + 2u/y2 and Dv = 2v/x2 + 2v/y2.   

Complex differentiation is a much stronger condition then real differentiation. 

Let f be the derivative of the holomorphic function F.  The path integral is written as òP f(z) dz.  If P is a path that starts at point u and ends at point w then the path integral is well defined, that is unique whatever path is taken in the complex plane and in fact òP f(z) dz = F(w) – F(u).  If u and w are the same point then the path integral is equal to zero.

It turns out that the path integral is true if we restrict the domain to be a simply connected subset of C.  Simply connected means an open set without holes.  If there are holes then the path integrals may differ depending upon how the paths circumnavigate the holes.  Path integrals have a close connection with the topology of subsets of the plane.

A holomorphic function can be differentiated an infinite number of times.  So for complex functions differentiability implies infinite differentiability. 

Any holomorphic function can be expanded in a power series: f(z) = S¥n=0 an(z-w)n , where f is defined and differentiable on an open disk centered at w.  This series is called the Taylor series expansion of f.  

 The enitire behavior of a holomorphic function is determined by their values in a tiny open region.  The process of analytical continuation determines the values of the function outside the tiny open region.  This process is how the Riemman  Zeta function is defined everywhere.

The therorem of Liouville states that if f is a holomorphic function defined on the entire complex plane and f is bounded then f must be constant.  A counterexample to this in the real line is the function sin(x), which is not constant but is bounded and infinitely differentiable.

Modern geometry has established two basic ideas: the relationship between geometry and symmetry and the notion of a manifold.

Geometry is concerned with the usual geometrical objects, point, line, plane, space, surve, sphere, cube, distance and angle, as well as refelection, rotation, translation, stretch, shear, projection, angle-preserving map, continuous deformation and transformation.  For any group of transformations there is a corresponding notion of geometry, in which one finds phenomena that are unaffected by transformations in that group.  Two objects are equivalent if they can be turned into each other from transformations within a group.  Tus different groups of transformations lead to different geometries.

Euclidean geometry requires the specification of the dimension, n, and the transformations in the group.  The group includes rigid transformations.  These transformations preserve distance.  These transformations are rotations, reflections and translations.  The rotations form a special group called the special orthogonal group, denoted as, SO(n).   The larger orthogonal group, O(n), includes rotations and reflections.  An orthogonal map is a linear map T that preserves distances so that d(x,y) = d(Tx,Ty) for all x and y.  If the determinant of an orthogonal map is equal to 1 then the linear map T is a rotation.  If it is equal to -1 then the linear map T turns space “inside out” as in a reflection. 

The group GLn(R) of all invertible linear transformations of  Rn and the translation transformation forms a larger group of transformations of the form x # Tx + b, b is a fixed vector and T is an inveetible linear map.  The resulting geometry is called affine geometry.  Distance and angle are not included in affine geometry.  Points, lines and planes after an invertible linear map remain as points, lines and planes and so are part of geometry.  There are parallelograms and ellipses are in affine geometry (but not rectangles or circles).

Let G be a group of transformations of Rn, if we are doing G-geometry, then two shapes are equivalent if we can use transformations in G to obtain each other.  In G-geometry basic objects form equivalence classes rather than shapes themselves.   One can use a continuous deformation to move within an equivalence class.  However, the surface of a sphere cannot be deformed into a torus.  One does not have a hole and the other does.  Invariants, algebraic topology and differential topology deal with these issues. 

Spherical geometry occurs on an n-dimensional sphere, Sn, which is the surface of an n+1 dimensional ball of radius 1.  When n = 2 then the appropriate group of transformations is SO(3) the group of all rotations about axes that go through the origin.  These are symmetries of the sphere S2.  On the sphere lines are great circles.  Distances are defined along segments of great circles.  There are angles defined between these great circle segments.  The angles in a spherical triangle add to more than 180 degrees.  Two distinct spherical lines in spherical geometry intersect at two points.

 The group of transformations that defines hyperbolic geometry is called PSL2(R), the projective special linear group in two dimensions.  The special linear group SL2(R) is the set of all 2 by 2 matrices, with row 1 elements of a and b and row two elements of c and d, with determinant ad – bc = 1.  To form the projective group one makes this matrix equivalent to the 2 by 2 matrix with row 1 elements of -a and -b and row two elements of -c and -d. 

The half plane model of hyperbolic geometry uses the group PSL2(R) andthe upper half-plane of complex numbers.  The transformation takes the point z = x + iy and the 2 by 2 matrix above to the point (az + b)/(cz + d).  There is only one distance definition that is allowed by this geometry.  The Euclid parrallel postulate fails to hold.  That is given a line and a point not on the line there more than one line through the point that does not intersect the line.  The other Euclidean geometry axioms hold.  This means that the parallel axiom cannot be inferred from the other Euclidean axioms.  The angles in a hyperbolic triangle always sum to less than 180 degrees.  This reflects the fact that the hyperbolic plane has negative curvature, whereas the Euclidean plan is flat (zero curvature).

There are also disk models and hyperboloid models of hyperbolic geometry.  The disk model is defined on the open unit disk.  The hyperboloid model is the revolution of a hyperbola about the axis.  The geometry transformation involve hyperbolic rotations defined by the 2 by 2 matrix with row 1 elements of cosh(q) and sinh(q) and row 2 elements of sinh(q) and cosh(q).  The three hyperbolic geometry models are all equivalent.

I3. Partial Differential Equations and Integration, Pages 34 – 36.

Tuesday, February 23rd, 2010

The Laplacian is defined as:  Df  = 2f/x2 + 2f/y2 + 2f/z2  The Laplacian may be expanded to more or less than three variables.  The heat equation is written:   T/t = kDT.  This equation describes the temperature of a 3-dimensional point as a function of time and the temperature near the point.  The Laplace equation is  Df  = 0.   The Laplace equation says that that f is equal to the average of the points surrounding the point in question.  The one dimensional wave equation is written: (1/v2)2h/t2 = 2h/x2.  This equation related the displacement of a vibrating string as a function of time. 

The three dimensional wave equation may be written: (1/v2)2h/t2 = Dh.  This equation can be rewritten with the operation p2 where p2h = Dh – (1/v2)2h/t2.  This operation is called the d’Alembertian after d’Alembert who was the first to formulate the wave equation.  With this notation the wave equation becomes: p2h = 0.

Reimann integration examples are provided.  Here a function, f, is approximated or bracketed  by a mesh in a set of points S of step functions.  The mesh is then made finer and finer as the approximation tends to a limit.   If f is the function and S is the set then the total amount of f in S is the integral of f over S written as: òS f(x) dx.  Here x is an element of S.

The fundamental theorem of calculus states that integration and differentiation are inverse functions (up to an additive constant) for functions with reasonable continuity properties.

The Heaviside step function, H(x), does not have the required continuity properties.  H(x) = 0 when x < 0 and H(x) = 1 when x s 0.  The problem occurs at the point zero where the derivative approaches infinity from below.

                             

I.3 Continuity and Differentiation; Pages 31 – 33.

Monday, February 22nd, 2010

One basic definition of a limit of a sequence of numbers an is that : whatever positive delta you choose, there is a number N such that for all bigger numbers n the difference between an and l (ell) is less than d.   This may be written mathematically as:  \forall d, N,  \forall n s N, |an – l| < d.   The idea of limit applies just as well when one has a distance between objects defined rather than the difference between numbers.  In this case there is a limiting object.  

One says that “the limit of the sequence an is l (ell)” or “”an converges to l”.  One can also say that “this happens as n tends to infinity”.  Any sequence that has a limit is called convergent.

We say that a function f is continuous at a if  \forall  > 0,  $ d > 0, (| x – a| < d \Rightarrow  |f(x) – f(a)| < e).  As with limits this idea can be extended to more general cases where distance between objects in X and a possibly different distance between objects in Y is known.  Then we have f is continuous at a if  \forall  > 0,  $ d > 0, ( distance(x,a) < d \Rightarrow  distance(f(x),f(a)) < e).  We also say f is continuous if f is continuous at every a.  It turns out that continuous functions are functions that conserve the structure of convergent sequences.  For a continuous function f, if an converges to x, an z x, then f(an) converges to f(x), f(an) z f(x).  

The derivative of a function f at a value a is the rate of change of f(x) as x passes through a.  One may think of this as the linear approximation to f at a.  This derivative idea of functions may be generalized to linear maps.  If u is a function from Rn to Rm then then u is said to be differentiable at x Î Rn if there is a linear map T: Rn to Rm such that u(x + h) = u(x)  + T(h) + e(h), with e(h) small relative to h.  The linear map T is the derivative of u at h.

A special case is with m=1.  The matrix of T is a row vector of length n which may be denoted as  \nabla \!\,f(x) and is referred to as the gradient of f at x.  This vector points in the direction in which f increases most rapidly and its magnitude is the rate of change in that direction.  

I.3 Quotients, Functions between Algebraic Structures: Homomorphisms, Isomorphisms, Automorphisms, Linear Maps and Matrices, Eigenvalues and Eigenvectors, Basic Concepts of Mathematical Analysis: Limits; Pages 25 – 30

Sunday, February 21st, 2010

Define the set Q[x] as the set of all polynomials in the variable x with rational coefficients.  Q[x] is a commutative ring but not a field.  One regards a unique polynomial to be equivalent to the zero polynomial.  Quotients of a set of polynomials allows one to divide the set of all polynomials into equivalence classes where any polynomial “is equivalent to” a polynomial at most some finite degree.  This converts Q[x] into a field.  This creates a way to transform a polynomial with the difference between the original polynomial and the equivalent one a multiple of the unique polynomial, P(x), used to create the equivalence classes.  In the newly created equivalence field all polynomials that are not equivalent to zero (multiples of the unique polynomial) have inverses.  We decide that equivalent polynomials are equal.  The resulting mathematical structure is denote Q[x]/P(x).  It turns out that Q[x]/P(x) is the smallest field that contains Q and also has a root of the polynomial P(x).  It turns out that this is the same field we described earlier as Q(g).

We regarded to objects as equal when the were equivalent.  Formally this is the notion of equivalence classes and equivalence relations.  An example of this is the rational numbers where we regard 1/2 and 2/4 as equivalent to one another.   This means we must be careful when defining binary operations on the equivalence classes.  One must get equivalence objects out when one puts equivalence objects into the binary operation.

A quotient group is defined with G a group and H a subgroup of G.  g1 and g2 elements of G are said to be equivalent if g1-1g2 belongs to H.  The equivalence class of an element g is the set of all elements gh such that h Î H, this is written gH.  It is called a left coset of H. 

A natural candidate for a binary operation * on the set of left cosets: g1H * g2H = g1g2H.  One must check that if you pick elements from the cosets g1H and g2H that the product of these elements will indeed be in g1g2H.  This may not be true.  If one has the additional assumption that H is a normal subgroup then it is true.  H is a normal subgroup if when if h is an element of H, then ghg-1 is an element of H for every g of G.  Elements of the form ghg-1 are called conjugates of h.  A normal subgroup is one that is closed under conjugation.

If H is a normal subgroup then the set of left cosets forms a group with the new binary operation abd the groupis written G/Hand is called the quotient subgroup of G by H.

It turns out that a torus can be described as a quotient subgroup of R2 by H.  For a H that is an integer translation of points in R2.    Other examples of this are fundamental groups, homology and cohomology groups and a moduli space. 

There are interesting functions between algebraic structures.  Some basic structures are homomorphisms, isomorphisms and automorphisms.  Particular mathematical structures include: groups, fields and vector spaces.  Consider the class of functions between two mathematical structures X and Y.  Functions that preserve structure are interesting.  If there is structure between elements of X, say a and b, and this structure is preserved between the images of these elements, say f(a) and f(b) (elements of Y) then the structure is preserved.

    Examples of this structure might be ab = c and the preserved structure is f(a)f(b) = f(c) with X and Y groups.  Or, this same strucure if X and Y are fields.  Also, if X and Y are fields we may have the additive structure f(a) + f(b) = f(c) whenever a + b = c.  Vector space structure to be preserved is linear combinations, so that, f(av + bw) = af(v) + bf(w).

A function that preserves structure is called a homomorphism.  A homomorphism of vector spaces is called a linear map.  A homomorphism can collapse structure.

An isomporphism is a homomorphism that has an inverse that is also a homomorphism.  For most algebraic structures if f has an inverse g then g is automatically a homomorphism.  In such cases an isommorphism that is also a homomorphism that is also a bijection.If there is an isomorphism between two algebraic structures X and Y then X and Y are said to be isomorphic.  Isomorphic menas “the same in all essential respects”.  The kind of objects in the sets X and Y are not essential.

An automorphism is an isonmorphism from X to itself.  What is important about automorphisms is the algebraic symmetries induced in the group.  The composition of two automophisms is another automporphism.   The automorphisms of a structure X form a group.

An example of automorphisms is given with Q(2).  There are two automorphisms  that form a group consisting of the elements +1, -1 under multiplication.  This is also the group of integers modulo 2, the group of symmetries of a non equilateral isosceles triangle and many more.

Kernels of a homomorphism between tow algebraic structures are the set of elements of X such that the image of the element is the identity of Y.  (Additive if both addition and multiplication are the binary operations.)  If G and K are groups then kernel of a homomorphism form G to K is a normal subgroup of G.  If H is a normal subgroup of G then the quotient map which takes each element of G to the left coset gH is a homomorphism from G to the quotient group G/H with kernel H.  The kernel of any ring homomorphism is an ideal.  every ideal I in a ring R is the kernel of a quotient map from R to R/I. 

Linear maps preserve linear combinations in vector spaces.  These can be written as matrices with the usual definition of matrix multiplication.  The linear maps that can be inverted are automporphisms.  There are also infinite dimension linear maps, such as, differentiation and integration. 

One can determine eigenvalues and eigenvectors for linear maps. 

One of the basic concepts of mathematical analysis is that one can specify an object indirectly as the limit of objects that one can list directly.  One uses better and better approximations.  This is the basis of calculus. 

Limits of sequences can approach a limit. 

             

I.3 More Algebraic Structures: Fields, Vector Spaces, Rings and Creating New Strucures out of Old Ones: Substructures and Products, Pages 20 – 24.

Saturday, February 20th, 2010

A field has two binary operations.  The operations are both commutative, associative and both have identity elements.  Each eleemnt has an inverse except for the identity element of one of the operations (zero if the operations are addition and multiplication).  Also, the distributative law must apply, this ties the two operations together.  If the operations are addition and multiplication then the distributative law is x * (y + z) = x * y + x * z.

The number systems Q, R and C form fields with addition and multiplication.  Also, Fp = the set of integers modulo a prime p, with addition and multiplication also defined modulo p form a field.

One can create a new field by adjoining the roots of polynomials which are not already in F to a field F creating a new extended field F’.  This was done to create the complex numbers from the reals with the roots of the polynomial P(x) =  x2 + 1.  The exact same process can be used with, for example, F5 as the field and appending i to the elements.  We can also append roots to the rationals forming the extended field Q(Ö2) or in general Q(g), where g Î root of a polynomial which is not a rational number.   

Filed extensions can be interesting and are used in automorphisms and vector spaces. 

Vector spaces are formed as linear combinations of some basis vectors.  Given scalers a and b and vectors v and w the element av + bw is also in the vector space.  Addition of vectors must be associative and commutative with an identity and inverse element for each v under addition.  Scaler multiplication must be associative a(bv) = ab(v).  Two distributative laws must be satisfied: (a + b)v = av +bv and a(v + w) = av + aw for any scalers a and b and any vectors v and w.   

A set of vectors forms a basis for V if every vector in V can be written in exactly one way as a linear combination of the set of basis vectors.  A set of vectors spans the space V if there is at least one linear combination of the spanning set that equals each vector in V.  If there is a spanning set where no vector may be formed by more than one linear combination of spanning set then the spanning set vectors are said to be independent.

The number of elements in a basis of V is said to be the dimension of V. This number is unique.  A vector space may have an infinite number of basis vectors, then the vector space is called infinite dimensional. 

The vectors in a vector space may be functions.  The scalars used to define a vector space may be from any field, F.  In this case one is said to form a vector space V over F.

Notice again that a vector space over Cartesian coordinates do not occur in the actual world because there is not Euclidean geometry.

Another algebraic structure is a ring.  A ring has most but not necessarily all of the properties of a field.  Requirements of the multiplication operation are less stringent.  For example, non zero elements of the ring are not required to have multiplicative inverses.  Sometimes the multiplication is not required to be commutative.  Although there are commutative rings.  Examples of commutative rings are the set of integers and the set of all polynomials with coefficients in some field F. 

Structures that we already know may be used to create new structures.  These new structures are important to use as examples and counterexamples.  Examples are important in mathematical thought. 

One possible group is the group G of all symmetries of an icosohedron.

Substructures may also be important.  For example, primes are a subset of the integers and the set a + b i with a and b rational are an interesting subset of the complex numbers.

In general groups have subgroups, vector spaces have subspaces and rings have subrings.  Even the plane has interesting subsets such as the Mandelbrot set.   

Products of groups may be defined.  Let G and H be two groups then the product group G x H.  Using the binary operation from these groups we may form (g1,h1) (g2,h2) = (g1g2,h1h2).  The binary operation from G is used on the first coordinate and that from H is used on the second coordinate.

Products of vector spaces V and W, or V x W, are all pairs (v,w) with v in V and w in W.  Addition and scaler multiplication are defined as  (v1,w1) + (v2,w2) = (v1 + v2 , w1 + w2) and l(v,w) = (lv,lw).   The dimension of the resulting space is the sum of the dimensions of V and W.  The space V x W is usually written V Å W.  It is called the direct sum of V and W.

It is not always possible to define products structures that retain the necesary properties in this way.  Sometimes we can change the multiplication definition to satisfy the necessary properties.

We can define more complicated products of groups than the direct product defined above.  Consider the dihedral group D4 which is the group of all symmetries of a square.  We can create this group as comprised of 90 degree turns and reflections.  D4 is the product of the group {I, T1, T2, T3}, consisting of four rotations or turns and {I,R} consisting of the identity and a reflection.  One must be careful to define multiplication to enforce RTR = T-1.  This is an example of a semi-direct product of two groups.  Semi-direct groups may be formed by defining different binary operations of the set of pairs (g,h).

I.2 3. Some Elementary Logic, I.3 Some Fundamental Mathematical Definitions Pages 13 – 19

Friday, February 19th, 2010

Some elementary logic: logical connectives, quantifiers,negation, free and bound variables and levels of formality followed by some fundamental mathematical definitions: the main number systems, the natural numbers, the integers, the rational numbers,the real numbers, the complex numbers and one of the four important mathematical structures: groups.

Mathematicians use two quantifiers “for all” denoted “ \forall“, and “there exists”, dentoted “$“.  An example is given of the joke that makes use of the two different uses of the word “nothing”.

The logical connective between two mathematical sentences “and” is sometimes written “,“.  The mathematical “or”, “.“, includes the case when both mathematical sentences it ties are true.  A third connective is implies, “\Rightarrow “.  P \Rightarrow  Q means “if P then Q”, this statement is true under all circumstances except one, that is, if P is true and Q is false.  A cute example of this is provided where a young girl says to “put your hand up if you are a girl”.  Her brother puts his hand up on the grounds that this was not eliminated by the statement.  That is the girl did not add “and do not put your hand up if you are not a girl”.

The symbol “¬” is introduced to indicate “not”.  What is the negation of the sentence “Every number in the set A is odd”?  It is some member of A is even.

Free and bound variables are defined.   A variable is any letter used to stand for a mathematical object.  A free variable is free to take on any value.  A bound variable takes on a range of values specified by the mathematical notation.

There are various levels of formality used to communicate mathematical concepts.  Which one is used depends upon the audience.  An example is given as: “Every nonempty set of positive integers has a least element” can be completely formally written as:  \forall A \subset \!\, N  [($ n ÎN, n Î  A) \Rightarrow  ( x ÎA,  \forall y  Î A, (y > x) . (y = x))], where A is a sub-set of positive integers and N the set of Natural numbers, “\subset \!\,” is defined to mean “subset”.  

Numbers are a very basic mathematical concept.   More useful are collections of number with properties called number systems.

The natural numbers are the positive integers.  Sometimes zero is included and sometimes not.  One may add and multiply natural numbers but differences and division are not always possible.

The integers include the natural numbers, zero and the negative of the natural numbers.  The negative integer has the defining property that when you add the negative number to the integer the result is zero.  This sytem of numbers includes the addition, multiplication and subtraction properties.

The rational numbers, Q, include fractions or ratios of integers, except for not allowing for division by the zero element.  So these numbers have the addition, multiplication, and subtraction properties, as well as, division, except by zero properties.

The real numbers, denoted by R, are extensions of the rational numbers to include numbers such as the square root of 2 and pi.

 The complex numbers, denoted by C, are a further extension of the real numbers to include for example, the square root of -1.  Inclusion of this one number and the real and imaginary parts of the complex number allows one to list roots of all polynomial equations as complex numbers.

Complex numbers may be writeen in polar coordinates.  Multiplication of complex numbers in polar coordinates is simply a contraction followed by a rotation of the point in two space or an Argand diagram.

There are four important algebraic strucures: groups, fields, vector spaces and rings.

Groups are important and occur when one considers symmetries.  Symmetries may arise as rigid motions of geometrical objects.  Rigid motions include translations and rotations but not stretching of any kind.  However, it should be noted that this is true in Euclidean space and not the actual space in which we live as well as for idealized objects and not those made up of actual matter.  Because the symmetries can be composed there are underlying groups of symmetires.  The composition operation is associative and has an identity element, it also includes and inverse.  Any set with a binary operation, such as composition of symmetries, that has these properties is called a group.  The composition does not have to be commutative. 

If the group also has the commutative property then it is called Abelian.  Z, Q, R and C are all examples of Abelian groups under the operation of addition.