Hanfried Lenz was a German mathematician, known for his work in geometry and combinatorics. Hanfried Lenz was the eldest son of Fritz Lenz an influential German geneticist, associated with Eugenics and hence with the Nazi racial policies during the Third Reich, he started to study mathematics and physics at the University of Tübingen, but interrupted his studies from 1935-37 to do his military service. After that he continued to study in Munich and Leipzig. In 1939 when World War II broke out in Europe, he became a soldier in the western front and during a vacation he passed the exams for his teacher certification, he married Helene Ranke in 1943 and 1943-45 he worked on radar technology in a laboratory near Berlin. After World War II Hanfried Lenz was classified as a "follower" by the denazification process, he started to work as a math and physics teacher in Munich and in 1949 he became an assistant at the Technical University of Munich. He received his PhD in 1951 and his Habilitation in 1953.
He worked as a lecturer until he became an associate professor in 1959. In 1969 he became a full professor at the Free University of Berlin and worked there until his retirement in 1984, he was politically active and in connection with his opposition to the rebuilding of the German army in the early 50s, he became a member of the Social Democratic Party in 1954. Due to being alienated by the student movement of the'60s, his leanings became more conservative again and in 1972 he left the SPD to join the Christian Democratic Union. Hanfried Lenz is known for his work on the classification of projective planes and in 1954 he showed how one can introduce affine spaces axiomatically without constructing them from projective spaces or vector spaces; this result is now known as the theorem of Lenz. During his years he worked in the area of combinatorics and published a book on design theory. In 1995 the Institute of Combinatorics and its Applications awarded the Euler Medal to Hanfried Lenz. Christoph Kaiser: Lernen heißt irren dürfem.
Berliner Zeitung, 2002-4-15 Prof. Dr. Hanfried Lenz ist am 1. Juni 2013 gestorben - news at the math department of the Free University of Berlin Walter Benz: Zum mathematischen Werk von Hanfried Lenz, Journal of Geometry 43, 1992 Hanfried Lenz: Mehr Glück als Verstand, Books on Demand 2002, Autobiography "Ich habe halt Schwein gehabt". FU-Nachrichten, number 5,2005 Wikipedia userpage of Hanfried Lenz in the German Wikipedia
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
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect, thus any two distinct lines in a projective plane intersect in only one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic; the archetypical example is the real projective plane known as the extended Euclidean plane. This example, in different guises, is important in algebraic geometry and projective geometry where it may be denoted variously by PG, RP2, or P2, among other notations. There are many other projective planes, both infinite, such as the complex projective plane, finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces.
Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes. A projective plane consists of a set of lines, a set of points, a relation between points and lines called incidence, having the following properties: The second condition means that there are no parallel lines; the last condition excludes the so-called degenerate cases. The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines, thus the expression "point P is incident with line ℓ " is used instead of either "P is on ℓ " or "ℓ passes through P ". To turn the ordinary Euclidean plane into a projective plane proceed as follows: To each set of mutually parallel lines add a single new point; that point is considered incident with each line of this set. The point added; these new points are called points at infinity. Add a new line, considered incident with all the points at infinity; this line is called the line at infinity. The extended structure is a projective plane and is called the extended Euclidean plane or the real projective plane.
The process outlined above, used to obtain it, is called "projective completion" or projectivization. This plane can be constructed by starting from R3 viewed as a vector space, see § Vector space construction below; the points of the Moulton plane are the points of the Euclidean plane, with coordinates in the usual way. To create the Moulton plane from the Euclidean plane some of the lines are redefined; that is, some of their point sets will be changed. Redefine all the lines with negative slopes so that they look like "bent" lines, meaning that these lines keep their points with negative x-coordinates, but the rest of their points are replaced with the points of the line with the same y-intercept but twice the slope wherever their x-coordinate is positive; the Moulton plane is an affine plane. It can be projectivized, as in the previous example. Desargues' theorem is not a valid theorem in either the Moulton plane or the projective Moulton plane; this example has just thirteen points and thirteen lines.
We label the points P1... P13 and the lines m1...m13. The incidence relation can be given by the following incidence matrix; the rows are labelled by the points and the columns are labelled by the lines. A 1 in row i and column j means that the point Pi is on the line mj, while a 0 means that they are not incident; the matrix is in Paige-Wexler normal form. To verify the conditions that make this a projective plane, observe that every two rows have one common column in which 1's appear and that every two columns have one common row in which 1's appear. Among many possibilities, the points P1,P4,P5,and P8, for example, will satisfy the third condition; this example is known as the projective plane of order three. Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace through the origin in a 3-dimensional vector space, a "line" in the projective plane arises from a plane through the origin in the 3-space.
This idea can be made more precise as follows. Let K be any division ring. Let K3 denote the set of all triples x = of elements of K. For any nonzero x in K3, the minimal subspace of K3 containing x is the subset of K3. Let x and y be linearly independent elements of K3, meaning that kx + my = 0 implies that k = m = 0; the minimal subspace of K3 containing x and y is the subset of K3. This 2-dimensional subspace contains various 1-di
R. H. Bruck
Richard Hubert Bruck was an American mathematician best known for his work in the field of algebra in its relation to projective geometry and combinatorics. Bruck studied at the University of Toronto, where he received his doctorate in 1940 under the supervision of Richard Brauer, he spent most his career as a professor at University of Wisconsin–Madison, advising at least 31 doctoral students. He is best known for his 1949 paper coauthored with H. J. Ryser, the results of which became known as the Bruck–Ryser theorem, concerning the possible orders of finite projective planes. In 1946, he was awarded a Guggenheim Fellowship. In 1956, he was awarded the Chauvenet Prize for his article Recent Advances in the Foundations of Euclidean Plane Geometry. In 1962, he was an invited speaker at the International Congress of Mathematicians in Stockholm. In 1963, he was a Fulbright Lecturer at the University of Canberra. In 1965 a Groups and Geometry conference was held at the University of Wisconsin in honor of Bruck's retirement.
Dick Bruck and his wife Helen were supporters of the fine arts. They were patrons of the regional American Players Theatre in Wisconsin. Bruck, R. H. "Contributions to the theory of loops", Transactions of the American Mathematical Society, 60: 245–354, doi:10.2307/1990147 Bruck, R. H.. J.. "The nonexistence of certain finite projective planes". Canadian Journal of Mathematics. 1: 88–92. Doi:10.4153/CJM-1949-009-2. Bruck, R. H.. H. "Finite Nets. I. Numerical invariants", Canadian Journal of Mathematics, 3: 94–106, doi:10.4153/cjm-1951-012-7 Bruck, R. H. "Finite Nets. II. Uniqueness and imbedding", Pacific Journal of Mathematics, 13: 421–457, doi:10.2140/pjm.1963.13.421 Bruck, R. H.. "Recent Advances in the Foundations of Euclidean Geometry". The American Mathematical Monthly. Mathematical Association of America. 62: 2–17. JSTOR 2308175. Bruck, R. H. "Difference sets in a finite group", Transactions of the American Mathematical Society, 78: 464–481, doi:10.1090/s0002-9947-1955-0069791-3 Bruck, R. H. A Survey of Binary Systems, Berlin: Springer-Verlag Bruck, R. H.
"Some theorems on Math. Z. 73: 59–78, doi:10.1007/bf01163269 Bruck, R. H.. C. "The construction of translation planes from projective spaces", Journal of Algebra, 1: 85–102, doi:10.1016/0021-869390010-9 Biography at the University of Texas Bruck–Ryser–Chowla Theorem at Mathworld
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Combinatorics is an area of mathematics concerned with counting, both as a means and an end in obtaining results, certain properties of finite structures. It is related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. To understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon. According to H. J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with the enumeration of specified structures, sometimes referred to as arrangements or configurations in a general sense, associated with finite systems, the existence of such structures that satisfy certain given criteria, the construction of these structures in many ways, optimization, finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge in their objectives, their methods, the degree of coherence they have attained." One way to define combinatorics is to describe its subdivisions with their problems and techniques. This is the approach, used below. However, there are purely historical reasons for including or not including some topics under the combinatorics umbrella. Although concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite but discrete setting. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory and geometry, as well as in its many application areas. Many combinatorial questions have been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the twentieth century, however and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.
One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc. thus computing all 26 − 1 possibilities. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle. In the Middle Ages, combinatorics continued to be studied outside of the European civilization; the Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, these formulas may have been familiar to Indian mathematicians as early as the 6th century CE.
The philosopher and astronomer Rabbi Abraham ibn Ezra established the symmetry of binomial coefficients, while a closed formula was obtained by the talmudist and mathematician Levi ben Gerson, in 1321. The arithmetical triangle— a graphical diagram showing relationships among the binomial coefficients— was presented by mathematicians in treatises dating as far back as the 10th century, would become known as Pascal's triangle. In Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for algebraic combinatorics. Graph theory enjoyed an explosion of interest at the same time in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject.
In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics; the twelvefold way provides a unified framework for counting permutations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theo
Dieter Jungnickel is a German mathematician specializing in combinatorics. In 1971 Jungnickel entered the Free University of Berlin, graduating in 1975, he studied the finite difference method with Hanfried Lenz and completed his Habilitation in 1978. He became a professor of mathematics at University of Giessen in 1980; the uses of finite geometry in coding theory led Jungnickel and Thomas Beth to write "Variations of Seven Points" which used the Fano plane as a starting point. With Thomas Beth and Hanfried Lenz he wrote Design Theory on combinatorial design. Albrecht Beutelspacher reviewed it positively as deserving high estimation, as an impressive work, it was re-published by Cambridge University Press in 1986. He updated the work in 1989 For the second edition Design Theory was split into two volumes and two. In 1990 Jungnickel wrote an article on geometric and graph-theoretical aspects of latin squares of interest in coding theory. In September 1990 University of Vermont was the site of a conference concerned with the mathematics of coding theory, commemorating the contributions of Marshall Hall.
Together with Scott Vanstone, Jungnickel edited the Proceedings. In 1993 Jungnickel joined Augsburg University where he occupies the chair for discrete mathematics and operations research. Jungnickel wrote about finite fields in 1993: Finite fields and Arithmetics. A reviewer notes that "The author does a beautiful job showing and developing the practical applicability of the fascinating area of finite field theory". In 1999 his book Graphs and Algorithms appeared as translation of the 1994 German version. A reviewer calls it a "first class textbook" and indispensable for teachers of combinatorial optimization; the second edition appeared in 2005, the third in 2008, the fourth in 2013