Betti number

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces, the sequence of Betti numbers is 0 from some point onward, they are all finite; the nth Betti number represents the rank of the nth homology group, denoted Hn, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology, computer science, digital images, etc; the term "Betti numbers" was coined by Henri Poincaré after Enrico Betti. The modern formulation is due to Emmy Noether. Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface; the first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, 2-dimensional simplicial complexes: b0 is the number of connected components b1 is the number of one-dimensional or "circular" holes b2 is the number of two-dimensional "voids" or "cavities"Thus, for example, a torus has one connected surface component so b0 = 1, a circular hole at each end of the central space so b1 = 2 and a single cavity enclosed within the surface so b2 = 1.

Related to the Betti numbers of a topological surface is the Poincaré polynomial of that surface. The Poincaré polynomial of a surface is defined to be the generating function of its Betti numbers. For example, the Betti numbers of the torus are 1, 2, 1; the same definition applies to any topological space. The two-dimensional Betti numbers are easier to understand because we see the world in 0, 1, 2, 3-dimensions. For a non-negative integer k, the kth Betti number bk of the space X is defined as the rank of the abelian group Hk, the kth homology group of X; the kth homology group is H k = ker δ k / I m δ k + 1, the δ k s are the boundary maps of the simplicial complex and the rank of Hk is the kth Betti number. Equivalently, one can define it as the vector space dimension of Hk since the homology group in this case is a vector space over Q; the universal coefficient theorem, in a simple torsion-free case, shows that these definitions are the same. More given a field F one can define bk, the kth Betti number with coefficients in F, as the vector space dimension of Hk.

Given a topological space which has finitely generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, viz the polynomial where the coefficient of x n is b n. This is a simple example of. We have a simplicial complex with 0-simplices: a, b, c, d, 1-simplices: E, F, G, H and I, the only 2-simplex is J, the shaded region in the figure, it is clear. This means that the rank of H 0 is 1, the rank of H 1 is 1 and the rank of H 2 is 0; the Betti number sequence for this figure is 1,1,0,0.... This may be proved straightforwardly by mathematical induction on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components; the first Betti number is called the cyclomatic number—a term introduced by Gustav Kirchhoff before Betti's paper. See cyclomatic complexity for an application to software engineering; the "zero-th" Betti number of a graph is the number of connected components k. The Betti numbers bk do not take into account any torsion in the homology groups, but they are useful basic topological invariants.

In the most intuitive terms, they allow one to count the number of holes of different dimensions. For a finite CW-complex K we have χ = ∑ i = 0 ∞ i b i, where χ denotes Euler characteristic of K and any field F. For any two spaces X and Y we have P X × Y = P X P Y, where P X den

Surface (mathematics)

In mathematics, a surface is a generalization of a plane which needs not be flat – that is, the curvature is not zero. This is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study; the mathematical concept of surface is an idealization of what is meant by surface in common language and computer graphics. A surface is defined by equations that are satisfied by the coordinates of its points; this is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, called an implicit surface. If the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation x 2 + y 2 + z 2 − 1 = 0. A surface may be defined as the image, in some space of dimension at least 3, of a continuous function of two variables.

In this case, one says that one has a parametric surface, parametrized by these two variables, called parameters. For example, the unit sphere may be parametrized by the Euler angles called longitude u and latitude v by x = cos cos y = sin cos z = sin . Parametric equations of surfaces are irregular at some points. For example, all but two points of the unit sphere, are the image, by the above parametrization, of one pair of Euler angles. For the remaining two points, one has cos v = 0, the longitude u may take any values. There are surfaces for which there cannot exist a single parametrization that covers the whole surface. Therefore, one considers surfaces which are parametrized by several parametric equations, whose images cover the surface; this is formalized by the concept of manifold: in the context of manifolds in topology and differential geometry, a surface is a manifold of dimension two. This allows defining surfaces in spaces of dimension higher than three, abstract surfaces, which are not contained in any other space.

On the other hand, this excludes surfaces that have singularities, such as the vertex of a conical surface or points where a surface crosses itself. In classical geometry, a surface is defined as a locus of a point or a line. For example, a sphere is the locus of a point, at a given distance of a fixed point, called the center. A ruled surface is the locus of a moving line satisfying some constraints. In this article, several kinds of surfaces are compared. An unambiguous terminology is thus necessary to distinguish them. Therefore, we call. We call; every differential surface is a topological surface. For simplicity, unless otherwise stated, "surface" will mean a surface in the Euclidean space of dimension 3 or in R3. A surface, not supposed to be included in another space is called an abstract surface; the graph of a continuous function of two variables, defined over a connected open subset of R2 is a topological surface. If the function is differentiable, the graph is a differential surface. A plane is both a differentiable surface.

It is a ruled surface and a surface of revolution. A circular cylinder is a differential surface. A circular cone is an algebraic surface, not a differential surface. If one removes the apex, the remainder of the cone is the union of two differential surfaces; the surface of a polyhedron is a topological surface, neither a differential surface nor an algebraic surface. A hyperbolic paraboloid is an algebraic surface, it is a ruled surface, for this reason, is used in architecture. A two-sheet hyperboloid is an algebraic surface and the union of two non-intersecting differential surfaces. A parametric surface is the image of an open subset of the Euclidean plane (typically R

International Standard Serial Number

An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.

The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.

The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.

Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the

Complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.

For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia