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

Federigo Enriques

Abramo Giulio Umberto Federigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, other contributions in algebraic geometry. Enriques was born in Livorno, brought up in Pisa, in a Sephardi Jewish family of Portuguese descent, he became a student of Guido Castelnuovo, became an important member of the Italian school of algebraic geometry. He worked on differential geometry, he collaborated with Corrado Segre and Francesco Severi. He had positions at the University of Bologna, the University of Rome La Sapienza, he lost his position in 1938, when the Fascist government enacted the "leggi razziali", which in particular banned Jews from holding professorships in Universities. The Enriques classification, of complex algebraic surfaces up to birational equivalence, was into five main classes, was background to further work until Kunihiko Kodaira reconsidered the matter in the 1950s; the largest class, in some sense, was that of surfaces of general type: those for which the consideration of differential forms provides linear systems that are large enough to make all the geometry visible.

The work of the Italian school had provided enough insight to recognise the other main birational classes. Rational surfaces and more ruled surfaces have the simplest geometry. Quartic surfaces in 3-spaces are now classified as cases of K3 surfaces. Abelian surfaces give rise to Kummer surfaces as quotients. There remains the class of elliptic surfaces, which are fiber bundles over a curve with elliptic curves as fiber, having a finite number of modifications; the question of classification is to show that any surface, lying in projective space of any dimension, is in the birational sense accounted for by the models mentioned. No more than other work in the Italian school would the proofs by Enriques now be counted as complete and rigorous. Not enough was known about some of the technical issues: the geometers worked by a mixture of inspired guesswork and close familiarity with examples. Oscar Zariski started to work in the 1930s on a more refined theory of birational mappings, incorporating commutative algebra methods.

He began work on the question of the classification for characteristic p, where new phenomena arise. The schools of Kunihiko Kodaira and Igor Shafarevich had put Enriques' work on a sound footing by about 1960. Enriques F. Lezioni di geometria descrittiva. Bologna, 1920. Enriques F. Lezioni di geometria proiettiva. Italian ed. 1898 and German ed. 1903. Enriques F. & Chisini, O. Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. Bologna, 1915-1934. Volume 1, Volume 2, Vol. 3, 1924. Severi F. Lezioni di geometria algebrica: geometria sopra una curva, superficie di Riemann-integrali abeliani. Italian ed. 1908 and Enriques F. Problems of Science. Chicago, 1914. Enriques F. Zur Geschichte der Logik. Leipzig, 1927. Castelnouvo G. Enriques F. Die algebraischen Flaechen// Encyklopädie der mathematischen Wissenschaften, III C 6 Enriques F. Le superficie algebriche. Bologna, 1949 On Scientia. Eredità ed evoluzione I numeri e l'infinito Il pragmatismo Il principio di ragion sufficiente nel pensiero greco Il problema della realtà Il significato della critica dei principii nello sviluppo delle matematiche Importanza della storia del pensiero scientifico nella cultura nazionale L'infini dans la pensee des grecs L'infinito nella storia del pensiero L'oeuvre mathematique de Klein La connaissance historique et la connaissance scientifique dans la critique de Enrico De Michelis La filosofia positiva e la classificazione delle scienze I motivi della filosofia di Eugenio Rignano Works by or about Federigo Enriques at Internet Archive O'Connor, John J..

Reviews of the works of Federigo Enriques, MacTutor History PRISTEM page Official home page of center for Enriques studies Federigo Enriques at the Mathematics Genealogy Project

Vito Volterra

Prof Vito Volterra KBE FRS HFRSE was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Born in Ancona part of the Papal States, into a poor Jewish family: his father was Abramo Volterra and mother, Angelica Almagia. Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, where he became professor of rational mechanics in 1883, he started work developing his theory of functionals which led to his interest and contributions in integral and integro-differential equations. His work is summarised in his book Theory of functionals and of Integral and Integro-Differential Equations. In 1892, he became professor of mechanics at the University of Turin and in 1900, professor of mathematical physics at the University of Rome La Sapienza. Volterra had grown up during the final stages of the Risorgimento when the Papal States were annexed by Italy and, like his mentor Betti, he was an enthusiastic patriot, being named by the king Victor Emmanuel III as a senator of the Kingdom of Italy in 1905.

In the same year, he began to develop the theory of dislocations in crystals, to become important in the understanding of the behaviour of ductile materials. On the outbreak of World War I well into his 50s, he joined the Italian Army and worked on the development of airships under Giulio Douhet, he originated the idea of using inert helium rather than flammable hydrogen and made use of his leadership abilities in organising its manufacture. After World War I, Volterra turned his attention to the application of his mathematical ideas to biology, principally reiterating and developing the work of Pierre François Verhulst. An outcome of this period is the Lotka–Volterra equations. Volterra is the only person, a plenary speaker in the International Congress of Mathematicians four times. In 1922, he joined the opposition to the Fascist regime of Benito Mussolini and in 1931 he was one of only 12 out of 1,250 professors who refused to take a mandatory oath of loyalty, his political philosophy can be seen from a postcard he sent in the 1930s, on which he wrote what can be seen as an epitaph for Mussolini’s Italy: Empires die, but Euclid’s theorems keep their youth forever.

However, Volterra was no radical firebrand. As a result of his refusal to sign the oath of allegiance to the fascist government he was compelled to resign his university post and his membership of scientific academies, during the following years, he lived abroad, returning to Rome just before his death. In 1936, he had been appointed a member of the Pontifical Academy of Sciences, on the initiative of founder Agostino Gemelli, he died in Rome on 11 October 1940. He is buried in the Ariccia Cemetery; the Academy organised his funeral. In 1900 he married a cousin, their son Edoardo Volterra was a famous historian of Roman law. 1910. Leçons sur les fonctions de lignes. Paris: Gauthier-Villars. 1912. The theory of permutable functions. Princeton University Press. 1913. Leçons sur les équations intégrales et les équations intégro-différentielles. Paris: Gauthier-Villars. 1926, "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi," Mem. R. Accad. Naz. dei Lincei 2: 31–113. 1926, "Fluctuations in the abundance of a species considered mathematically," Nature 118: 558–60.

1960. Sur les Distorsions des corps élastiques. Paris: Gauthier-Villars. 1930. Theory of functionals and of integral and integro-differential equations. Blackie & Son. 1931. Leçons sur. Paris: Gauthier-Villars. Reissued 1990, Gabay, J. ed. 1954-1962. Opere matematiche. Memorie e note. Vol. 1, 1954. Volterra Volterra's function Lotka–Volterra equation Smith–Volterra–Cantor set Volterra integral equation Volterra series Product integral Volterra operator Volterra space Volterra Semiconductor Poincaré lemma Castelnuovo, G. "Vito Volterra", Rendiconti della Accademia Nazionale delle Scienze detta dei XL, Memorie di Matematica e Applicazioni, Serie 3, XXV: 87–95, MR 0021530, Zbl 0061.00605, archived from the original on 5 March 2016, retrieved 23 June 2014. Fichera, Gaetano, "La figura di Vito Volterra a cinquanta anni dalla morte", in Amaldi, E.. "Vito Volterra fifty years after his death" is detailed biographical survey paper on Vito Volterra, dealing with scientific and moral aspects of his personality.

Gemelli, Agostino, "La relazione del presidente", Acta Pontificia Academia Scientarum, 6: XI–XXIV. The commemorative address pronounced by Agostino Gemelli on the occasion of the first seance of the fourth academic year of Pontificial Academy of Sciences: it includes his commemoration of various deceased members. Goodstein, Judith R; the Volterra Chronicles: The Life and Times of an Extraordinary Mathematician 1860–1940, History of Mathematics, 31, Providence, RI-London: American Mathematical Society/London Ma

Integrated Authority File

The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used for documentation in libraries and also by archives and museums; the GND is managed by the German National Library in cooperation with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero licence; the GND specification provides a hierarchy of high-level entities and sub-classes, useful in library classification, an approach to unambiguous identification of single elements. It comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format; the Integrated Authority File became operational in April 2012 and integrates the content of the following authority files, which have since been discontinued: Name Authority File Corporate Bodies Authority File Subject Headings Authority File Uniform Title File of the Deutsches Musikarchiv At the time of its introduction on 5 April 2012, the GND held 9,493,860 files, including 2,650,000 personalised names.

There are seven main types of GND entities: LIBRIS Virtual International Authority File Information pages about the GND from the German National Library Search via OGND Bereitstellung des ersten GND-Grundbestandes DNB, 19 April 2012 From Authority Control to Linked Authority Data Presentation given by Reinhold Heuvelmann to the ALA MARC Formats Interest Group, June 2012

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