In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting, labeling the objects with distinct whole numbers, Ordinal numbers are thus the labels needed to arrange collections of objects in order. An ordinal number is used to describe the type of a well ordered set. Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, although the distinction between ordinals and cardinals is not always apparent in finite sets, different infinite ordinals can describe the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, a natural number can be used for two purposes, to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is one way to put a finite set into a linear sequence.
This is because any set has only one size, there are many nonisomorphic well-orderings of any infinite set. Whereas the notion of number is associated with a set with no particular structure on it. A well-ordered set is an ordered set in which there is no infinite decreasing sequence, equivalently. Ordinals may be used to label the elements of any given well-ordered set and this length is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it, in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the type of the ordinals less than it, i. e. the ordinals from 0 to 41. Conversely, any set of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is in S—is an ordinal. There are infinite ordinals as well, the smallest infinite ordinal is ω, which is the type of the natural numbers. After all of these come ω·2, ω·2+1, ω·2+2, and so on, ω·3, now the set of ordinals formed in this way must itself have an ordinal associated with it, and that is ω2.
Further on, there will be ω3, ω4, and so on, and ωω, ωωω, later ωωωω and this can be continued indefinitely far. The smallest uncountable ordinal is the set of all countable ordinals, in a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is ordered and there is no infinite decreasing sequence
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is a set. It is necessary for the construction of certain infinite sets in ZF, the axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is small enough to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, suppose P is a definable binary relation such that for every set x there is a unique set y such that P holds. There is a definable function F P, where F P = Y if. Consider the class B defined such for every set y, y ∈ B if, B is called the image of A under F P, and denoted F P or. The axiom schema of replacement states that if F is a class function, as above.
This can be seen as a principle of smallness, the states that if A is small enough to be a set. It is implied by the axiom of limitation of size. In the formal language of set theory, the schema is, ∀ w 1, …, w n ∀ A The axiom schema of collection is closely related to. While replacement says that the image itself is a set, collection merely says that some superclass of the image is a set, in other words, the resulting set, B, is not required to be minimal. This version of collection lacks the requirement on ϕ. Suppose that the variables of ϕ are among w 1, …, w n, x, y. Then the axiom schema is, ∀ w 1, …, w n That is, the relation defined by ϕ is not required to be a function--some x ∈ A may correspond to many y s B. In this case, the image set B whose existence is asserted must contain at least one such y for x in the original set. However, the schema as stated requires that, if an element x of A is associated with at least one set y. The resulting axiom schema is called the axiom schema of boundedness, the axiom schema of collection is equivalent to the axiom schema of replacement over the remainder of the ZF axioms.
However, this is not so in the absence of the Power Set Axiom or constructive counterpart of ZF, the ordinal number ω·2 = ω + ω is the first ordinal that cannot be constructed without replacement
Complement (set theory)
In set theory, the complement of a set A refers to elements not in A. The relative complement of A with respect to a set B, termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A. If A and B are sets, the complement of A in B, termed the set-theoretic difference of B and A, is the set of elements in B. The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard, if R is the set of real numbers and Q is the set of rational numbers, R ∖ Q is the set of irrational numbers. Let A, B, and C be three sets, the following identities capture notable properties of relative complements, C ∖ = ∪. C ∖ = ∪, with the important special case C ∖ = demonstrating that intersection can be expressed using only the relative complement operation. If A is a set, the complement of A is the set of elements not in A.
Formally. The absolute complement of A is usually denoted by A ∁, other notations include A c, A ¯, A ′, ∁ U A, and ∁ A. Assume that the universe is the set of integers, if A is the set of odd numbers, the complement of A is the set of even numbers. If B is the set of multiples of 3, the complement of B is the set of numbers congruent to 1 or 2 modulo 3, assume that the universe is the standard 52-card deck. If the set A is the suit of spades, the complement of A is the union of the suits of clubs and hearts. If the set B is the union of the suits of clubs and diamonds, the complement of B is the union of the suits of hearts, let A and B be two sets in a universe U. The following identities capture important properties of complements, De Morgans laws. Complement laws, A ∪ A ∁ = U, if A ⊂ B, B ∁ ⊂ A ∁. Involution or double complement law, ∁ = A, relationships between relative and absolute complements, A ∖ B = A ∩ B ∁. Relationship with set difference, A ∁ ∖ B ∁ = B ∖ A, the first two complement laws above show that if A is a non-empty, proper subset of U, is a partition of U.
In the LaTeX typesetting language, the command \setminus is usually used for rendering a set difference symbol, when rendered, the \setminus command looks identical to \backslash except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin
International Standard Book Number
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay.
The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces.
Separating the parts of a 10-digit ISBN is done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the worlds oldest publishing house and it holds letters patent as the Queens Printer. The Presss mission is To further the Universitys mission by disseminating knowledge in the pursuit of education, Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global presence, publishing hubs, and offices in more than 40 countries. Its publishing includes journals, reference works, textbooks. Cambridge University Press is an enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press and it originated from Letters Patent granted to the University of Cambridge by Henry VIII in 1534, and has been producing books continuously since the first University Press book was printed.
Cambridge is one of the two privileged presses, authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, and Stephen Hawking. In 1591, Thomass successor, John Legate, printed the first Cambridge Bible, the London Stationers objected strenuously, claiming that they had the monopoly on Bible printing. The universitys response was to point out the provision in its charter to print all manner of books. In July 1697 the Duke of Somerset made a loan of £200 to the university towards the house and presse and James Halman, Registrary of the University. It was in Bentleys time, in 1698, that a body of scholars was appointed to be responsible to the university for the Presss affairs. The Press Syndicates publishing committee still meets regularly, and its role still includes the review, John Baskerville became University Printer in the mid-eighteenth century. Baskervilles concern was the production of the finest possible books using his own type-design, a technological breakthrough was badly needed, and it came when Lord Stanhope perfected the making of stereotype plates.
This involved making a mould of the surface of a page of type. The Press was the first to use this technique, and in 1805 produced the technically successful, under the stewardship of C. J. Clay, who was University Printer from 1854 to 1882, the Press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks, during Clays administration, the Press undertook a sizable co-publishing venture with Oxford, the Revised Version of the Bible, which was begun in 1870 and completed in 1885. It was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories, the Cambridge Modern History was published between 1902 and 1912
In mathematics, the power set of any set S is the set of all subsets of S, including the empty set and S itself. The power set of a set S is variously denoted as P, ℘, P, ℙ, or, in axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. Any subset of P is called a family of sets over S, if S is the set, the subsets of S are, and hence the power set of S is. If S is a set with |S| = n elements. This fact, which is the motivation for the notation 2S, may be demonstrated simply as follows, First and we write any subset of S in the format where γi,1 ≤ i ≤ n, can take the value of 0 or 1. If γi =1, the element of S is in the subset, otherwise. Clearly the number of subsets that can be constructed this way is 2n as γi ∈. Cantors diagonal argument shows that the set of a set always has strictly higher cardinality than the set itself. In particular, Cantors theorem shows that the set of a countably infinite set is uncountably infinite. The power set of the set of numbers can be put in a one-to-one correspondence with the set of real numbers.
The power set of a set S, together with the operations of union, intersection, in fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras this is no true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra. The power set of a set S forms a group when considered with the operation of symmetric difference. It can hence be shown that the power set considered together with both of these forms a Boolean ring. In set theory, XY is the set of all functions from Y to X, as 2 can be defined as, 2S is the set of all functions from S to. Hence 2S and P could be considered identical set-theoretically and this notion can be applied to the example above in which S = to see the isomorphism with the binary numbers from 0 to 2n −1 with n being the number of elements in the set. In S, a 1 in the corresponding to the location in the set indicates the presence of the element.
The number of subsets with k elements in the set of a set with n elements is given by the number of combinations, C
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory, is used to provide an interpretation or motivation of the axioms of ZFC. The rank of a set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, the sets in V are divided into a transfinite hierarchy, called the cumulative hierarchy, based on their rank. The cumulative hierarchy is a collection of sets Vα indexed by the class of numbers, in particular. Thus there is one set Vα for each ordinal number α, Vα may be defined by transfinite recursion as follows, Let V0 be the empty set, V0, = ∅. For any ordinal number β, let Vβ+1 be the set of Vβ. For any limit ordinal λ, let Vλ be the union of all the V-stages so far, a crucial fact about this definition is that there is a single formula φ in the language of ZFC that defines the set x is in Vα.
The sets Vα are called stages or ranks, the class V is defined to be the union of all the V-stages, V, = ⋃ α V α. An equivalent definition sets V α, = ⋃ β < α P for each ordinal α, the rank of a set S is the smallest α such that S ⊆ V α. The first five von Neumann stages V0 to V4 may be visualized as follows, the set V5 contains 216=65536 elements. The set V6 contains 265536 elements, which very substantially exceeds the number of atoms in the known universe, so the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set Vω has the same cardinality as ω, the set Vω+1 has the same cardinality as the set of real numbers. If ω is the set of numbers, Vω is the set of hereditarily finite sets. Vω+ω is the universe of ordinary mathematics, and is a model of Zermelo set theory, if κ is an inaccessible cardinal, Vκ is a model of Zermelo-Fraenkel set theory itself, and Vκ+1 is a model of Morse–Kelley set theory. V is not the set of all sets for two reasons, first, it is not a set, although each individual stage Vα is a set, their union V is a proper class.
Second, the sets in V are only the well-founded sets, the axiom of foundation demands that every set is well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation and these non-well-founded set theories are not commonly employed, but are still possible to study
Union (set theory)
In set theory, the union of a collection of sets is the set of all elements in the collection. It is one of the operations through which sets can be combined and related to each other. For explanation of the used in this article, refer to the table of mathematical symbols. The union of two sets A and B is the set of elements which are in A, in B, for example, if A = and B = A ∪ B =. Sets cannot have duplicate elements, so the union of the sets and is, multiple occurrences of identical elements have no effect on the cardinality of a set or its contents. Binary union is an operation, that is, A ∪ = ∪ C. The operations can be performed in any order, and the parentheses may be omitted without ambiguity, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union and that is, A ∪ ∅ = A, for any set A. This follows from analogous facts about logical disjunction, since sets with unions and intersections form a Boolean algebra, intersection distributes over union A ∩ = ∪ and union distributes over intersection A ∪ = ∩.
One can take the union of several sets simultaneously, for example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. In mathematics a finite union means any union carried out on a number of sets. The most general notion is the union of a collection of sets. If M is a set whose elements are themselves sets, x is an element of the union of M if, in symbols, x ∈ ⋃ M ⟺ ∃ A ∈ M, x ∈ A. This idea subsumes the preceding sections, in that A ∪ B ∪ C is the union of the collection, also, if M is the empty collection, the union of M is the empty set. The notation for the concept can vary considerably. For a finite union of sets S1, S2, S3, …, S n one often writes S1 ∪ S2 ∪ S3 ∪ ⋯ ∪ S n or ⋃ i =1 n S i. In the case that the index set I is the set of natural numbers, whenever the symbol ∪ is placed before other symbols instead of between them, it is of a larger size
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals, Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson and he wrote, the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the stages of the Differential and Integral Calculus. Robinson continued, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort, as a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. The key to our method is provided by the analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.
In 1973, intuitionist Arend Heyting praised non-standard analysis as a model of important mathematical research. A non-zero element of an ordered field F is infinitesimal if and only if its value is smaller than any element of F of the form 1 n, for n. Ordered fields that have infinitesimal elements are called non-Archimedean, more generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the principle for real numbers is a hyperreal field. Robinsons original approach was based on these models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print, on page 88, Robinson writes, The existence of non-standard models of arithmetic was discovered by Thoralf Skolem. Skolems method foreshadows the ultrapower construction Several technical issues must be addressed to develop a calculus of infinitesimals, for example, it is not enough to construct an ordered field with infinitesimals.
See the article on numbers for a discussion of some of the relevant ideas. In this section we outline one of the simplest approaches to defining a hyperreal field ∗ R, let R be the field of real numbers, and let N be the semiring of natural numbers. Denote by R N the set of sequences of real numbers, a field ∗ R is defined as a suitable quotient of R N, as follows. Take a nonprincipal ultrafilter F ⊂ P, in particular, F contains the Fréchet filter. There are at least three reasons to consider non-standard analysis, historical and technical, much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves, a Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. In Venn diagrams the curves are overlapped in every possible way and they are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn and they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, statistics and computer science. A Venn diagram in which in addition the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram and this example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged, the blue circle, set B, represents the living creatures that can fly.
Each separate type of creature can be imagined as a point somewhere in the diagram, living creatures that both can fly and have two legs—for example, parrots—are in both sets, so they correspond to points in the region where the blue and orange circles overlap. That region contains all such and only living creatures. Humans and penguins are bipedal, and so are in the circle, but since they cannot fly they appear in the left part of the orange circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles, the combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly, the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles.
They are rightly associated with Venn, because he comprehensively surveyed and formalized their usage, Venn himself did not use the term Venn diagram and referred to his invention as Eulerian Circles. Of these schemes one only, viz. that commonly called Eulerian circles, has met with any general acceptance, the first to use the term Venn diagram was Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic. Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler and she observes even earlier Euler-like diagrams by Ramon Lull in the 13th Century. In the 20th century, Venn diagrams were further developed, D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers.
B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way.
Ellipses may be used where sets have infinitely many members, thus the set of positive even numbers can be written as
Springer Science+Business Media
Springer hosts a number of scientific databases, including SpringerLink, Springer Protocols, and SpringerImages. Book publications include major works, textbooks and book series. Springer has major offices in Berlin, Dordrecht, on 15 January 2015, Holtzbrinck Publishing Group / Nature Publishing Group and Springer Science+Business Media announced a merger. In 1964, Springer expanded its business internationally, opening an office in New York City, offices in Tokyo, Milan, Hong Kong, and Delhi soon followed. The academic publishing company BertelsmannSpringer was formed after Bertelsmann bought a majority stake in Springer-Verlag in 1999, the British investment groups Cinven and Candover bought BertelsmannSpringer from Bertelsmann in 2003. They merged the company in 2004 with the Dutch publisher Kluwer Academic Publishers which they bought from Wolters Kluwer in 2002, Springer acquired the open-access publisher BioMed Central in October 2008 for an undisclosed amount. In 2009, Cinven and Candover sold Springer to two private equity firms, EQT Partners and Government of Singapore Investment Corporation, the closing of the sale was confirmed in February 2010 after the competition authorities in the USA and in Europe approved the transfer.
In 2011, Springer acquired Pharma Marketing and Publishing Services from Wolters Kluwer, in 2013, the London-based private equity firm BC Partners acquired a majority stake in Springer from EQT and GIC for $4.4 billion. In 2014, it was revealed that Springer had published 16 fake papers in its journals that had been computer-generated using SCIgen, Springer subsequently removed all the papers from these journals. IEEE had done the thing by removing more than 100 fake papers from its conference proceedings. In 2015, Springer retracted 64 of the papers it had published after it was found that they had gone through a fraudulent peer review process, Springer provides its electronic book and journal content on its SpringerLink site, which launched in 1996. SpringerProtocols is home to a collection of protocols, recipes which provide step-by-step instructions for conducting experiments in research labs, SpringerImages was launched in 2008 and offers a collection of currently 1.8 million images spanning science and medicine.
SpringerMaterials was launched in 2009 and is a platform for accessing the Landolt-Börnstein database of research and information on materials, authorMapper is a free online tool for visualizing scientific research that enables document discovery based on author locations and geographic maps. The tool helps users explore patterns in scientific research, identify trends, discover collaborative relationships. While open-access publishing typically requires the author to pay a fee for copyright retention, for example, a national institution in Poland allows authors to publish in open-access journals without incurring any personal cost - but using public funds. Springer is a member of the Open Access Scholarly Publishers Association, the Academic Publishing Industry, A Story of Merger and Acquisition – via Northern Illinois University