A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation is used to denote that is an element of a set . The study of sets and their properties is the object of set theory.Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set.Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor.Symbols used to operate on sets include (which means "and" or intersection), and (which means "or" or union). The symbol is used to denote the set containing..
An equivalence relation on a set is a subset of , i.e., a collection of ordered pairs of elements of , satisfying certain properties. Write "" to mean is an element of , and we say " is related to ," then the properties are 1. Reflexive: for all , 2. Symmetric: implies for all 3. Transitive: and imply for all , where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., or .
An equivalence class is defined as a subset of the form , where is an element of and the notation "" is used to mean that there is an equivalence relation between and . It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of . For all , we have iff and belong to the same equivalence class.A set of class representatives is a subset of which contains exactly one element from each equivalence class.For a positive integer, and integers, consider the congruence , then the equivalence classes are the sets , etc. The standard class representatives are taken to be 0, 1, 2, ..., .
A -rational point is a point on an algebraic curve , where and are in a field . For example, rational point in the field of ordinary rational numbers is a point satisfying the given equation such that both and are rational numbers.The rational point may also be a point at infinity.For example, take the elliptic curve(1)and homogenize it by introducing a third variable so that each term has degree 3 as follows:(2)Now, find the points at infinity by setting , obtaining(3)Solving gives , equal to any value, and (by definition) . Despite freedom in the choice of , there is only a single point at infinity because the two triples (, , ), (, , ) are considered to be equivalent (or identified) only if one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a valid point. The triples (, , 1) correspond to the ordinary points (, ), and the triples (, , 0) correspond to the points at infinity, usually called the line at infinity.The rational points on elliptic..
Let be a sequence over a finite alphabet (all the entries are elements of ). Define the block growth function of a sequence to be the number of admissible words of length . For example, in the sequence , the following words are admissible lengthadmissible words1234so , , , , and so on. Notice that , so the block growth function is always nondecreasing. This is because any admissible word of length can be extended rightwards to produce an admissible word of length . Moreover, suppose for some . Then each admissible word of length extends to a unique admissible word of length .For a sequence in which each substring of length uniquely determines the next symbol in the sequence, there are only finitely many strings of length , so the process must eventually cycle and the sequence must be eventually periodic. This gives us the following theorems: 1. If the sequence is eventually periodic, with least period , then is strictly increasing until it reaches , and is..
Order the natural numbers as follows:Now let be a continuous function from the reals to the reals and suppose in the above ordering. Then if has a point of least period , then also has a point of least period .A special case of this general result, also known as Sharkovsky's theorem, states that if a continuous real function has a periodic point with period 3, then there is a periodic point of period for every integer .A converse to Sharkovsky's theorem says that if in the above ordering, then we can find a continuous function which has a point of least period , but does not have any points of least period (Elaydi 1996). For example, there is a continuous function with no points of least period 3 but having points of all other least periods.Sharkovsky's theorem includes the period threetheorem as a special case (Borwein and Bailey 2003, p. 79)...
A generalization of the p-adic norm first proposed by Kürschák in 1913. A valuation on a field is a function from to the real numbers such that the following properties hold for all : 1. , 2. iff , 3. , 4. implies for some constant (independent of ). If (4) is satisfied for , then satisfies the triangle inequality, 4a. for all . If (4) is satisfied for then satisfies the stronger ultrametric inequality 4b. . The simplest valuation is the absolute value for real numbers. A valuation satisfying (4b) is called non-Archimedean valuation; otherwise, it is called Archimedean.If is a valuation on and , then we can define a new valuation by(1)This does indeed give a valuation, but possibly with a different constant in axiom 4. If two valuations are related in this way, they are said to be equivalent, and this gives an equivalence relation on the collection of all valuations on . Any valuation is equivalent to one which satisfies the triangle inequality..
An important result in valuation theory which gives information on finding roots of polynomials. Hensel's lemma is formally stated as follows. Let be a complete non-Archimedean field, and let be the corresponding valuation ring. Let be a polynomial whose coefficients are in and suppose satisfies(1)where is the (formal) derivative of . Then there exists a unique element such that and(2)Less formally, if is a polynomial with "integer" coefficients and is "small" compared to , then the equation has a solution "near" . In addition, there are no other solutions near , although there may be other solutions. The proof of the lemma is based around the Newton-Raphson method and relies on the non-Archimedean nature of the valuation.Consider the following example in which Hensel's lemma is used to determine that the equation is solvable in the 5-adic numbers (and so we can embed the Gaussian integers inside in a nice..
Let be a non-Archimedean field. Its valuation ring is defined to beThe valuation ring has maximal idealand the field is called the residue field, class field, or field of digits. For example, if (p-adic numbers), then (-adic integers), (-adic integers congruent to 0 mod ), and = GF(), the finite field of order .
A function is said to be periodic (or, when emphasizing the presence of a single period instead of multiple periods, singly periodic) with period iffor , 2, .... For example, the sine function , illustrated above, is periodic with least period (often simply called "the" period) (as well as with period , , , etc.).The constant function is periodic with any period for all nonzero real numbers , so there is no concept analogous to the least period for constant functions. The following table summarizes the names given to periodic functions based on the number of independent periods they posses.number of periodsname1singly periodic function2doubly periodic function3triply periodic function
If a sequence has the property that the block growth function for all , then it is said to have minimal block growth, and the sequence is called a Sturmian sequence. An example of this is the sequence arising from the substitution system(1)(2)yielding , which gives us the Sturmian sequence 01001010....Sturm functions are sometimes also said to forma Sturmian sequence.
A sequence , , ... such that the metric satisfiesCauchy sequences in the rationals do not necessarily converge,but they do converge in the reals.Real numbers can be defined using either Dedekindcuts or Cauchy sequences.
An ultrametric is a metric which satisfies the followingstrengthened version of the triangle inequality,for all . At least two of , , and are the same.Let be a set, and let (where N is the set of natural numbers) denote the collection of sequences of elements of (i.e., all the possible sequences , , , ...). For sequences , , let be the number of initial places where the sequences agree, i.e., , , ..., , but . Take if . Then defining gives an ultrametric.The p-adic norm metric is another example ofan ultrametric.
A singular point of an algebraic curve is a point where the curve has "nasty" behavior such as a cusp or a point of self-intersection (when the underlying field is taken as the reals). More formally, a point on a curve is singular if the and partial derivatives of are both zero at the point . (If the field is not the reals or complex numbers, then the partial derivative is computed formally using the usual rules of calculus.)The following table gives some representative named curves that have various types of singular points at their origin.singularitycurveequationacnodecuspcusp curvecrunodecardioidquadruple pointquadrifoliumramphoid cuspkeratoid cusptacnodecapricornoidtriple pointtrifoliumConsider the following two examples. For the curvethe cusp at (0, 0) is a singular point. For the curve is a nonsingular point and this curve is nonsingular.Singular points are sometimes known as singularities,and vice versa...
An isolated singularity is a singularity for which there exists a (small) real number such that there are no other singularities within a neighborhood of radius centered about the singularity. Isolated singularities are also known as conic double points.The types of isolated singularities possible for cubic surfaces have been classified (Schläfli 1863, Cayley 1869, Bruce and Wall 1979) and are summarized in the following table from Fischer (1986).namesymbolnormal formCoxeter-Dynkin diagramconic double pointbiplanar double pointbiplanar double pointbiplanar double pointbiplanar double pointuniplanar double pointuniplanar double pointuniplanar double pointelliptic cone point--
There are (at least) two mathematical objects known as Weierstrass forms. The first is a general form into which an elliptic curve over any field can be transformed, given bywhere , , , , and are elements of .The second is the definition of the gamma functionaswhere is the Euler-Mascheroni constant (Krantz 1999, p. 157).
The group of an ellipticcurve which has been transformed to the formis the set of -rational points, including the single point at infinity. The group law (addition) is defined as follows: Take 2 -rational points and . Now 'draw' a straight line through them and compute the third point of intersection (also a -rational point). Thengives the identity point at infinity. Now find the inverse of , which can be done by setting giving .This remarkable result is only a special case of a more general procedure. Essentially, the reason is that this type of elliptic curve has a single point at infinity which is an inflection point (the line at infinity meets the curve at a single point at infinity, so it must be an intersection of multiplicity three).
Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. The Weierstrass elliptic function describes how to get from this torus to the algebraic form of an elliptic curve.Formally, an elliptic curve over a field is a nonsingular cubic curve in two variables, , with a -rational point (which may be a point at infinity). The field is usually taken to be the complex numbers , reals , rationals , algebraic extensions of , p-adic numbers , or a finite field.By an appropriate change of variables, a general elliptic curve over a field with field characteristic , a general cubic curve(1)where , , ..., are elements of , can be written in the form(2)where the right side of (2) has no repeated factors. Any elliptic curve not of characteristic 2 or 3 can also be written in Legendre normal form(3)(Hartshorne 1999).Elliptic curves are illustrated above for various values of and..
Let be a valuated field. The valuation group is defined to be the setwith the group operation being multiplication. It is a subgroup of the positive real numbers, under multiplication.
Let be a complete non-Archimedean valuated field, with valuation ring , and let be a power series with coefficients in . Suppose at least one of the coefficients is nonzero (so that is not identically zero) and the sequence of coefficients converges to 0 with respect to . Then has only finitely many zeros in .
A collection of equations satisfies the Hasse principle if, whenever one of the equations has solutions in and all the , then the equations have solutions in the rationals . Examples include the set of equationswith , , and integers, and the set of equationsfor rational. The trivial solution is usually not taken into account when deciding if a collection of homogeneous equations satisfies the Hasse principle. The Hasse principle is sometimes called the local-global principle.
The field is called an algebraic closure of if is algebraic over and if every polynomial splits completely over , so that can be said to contain all the elements that are algebraic over .For example, the field of complex numbers is the algebraic closure of the field of reals .
For a field with multiplicative identity 1, consider the numbers , , , etc. Either these numbers are all different, in which case we say that has characteristic 0, or two of them will be equal. In the latter case, it is straightforward to show that, for some number , we have . If is chosen to be as small as possible, then will be a prime, and we say that has characteristic . The characteristic of a field is sometimes denoted .The fields (rationals), (reals), (complex numbers), and the p-adic numbers have characteristic 0. For a prime, the finite field GF() has characteristic .If is a subfield of , then and have the same characteristic.
An algebraic curve over a field is an equation , where is a polynomial in and with coefficients in . A nonsingular algebraic curve is an algebraic curve over which has no singular points over . A point on an algebraic curve is simply a solution of the equation of the curve. A -rational point is a point on the curve, where and are in the field .The following table lists the names of algebraic curves of a given degree.ordercurveexamples2quadratic curvecircle, ellipse, hyperbola, parabola3cubic curvecissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, Mordell curve, Ochoa curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, witch of Agnesi4quartic curveampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid..