Instruct: Given a question, retrieve passages that can help answer the question.
Query: Who introduced the concept of complex numbers?
The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545,[16] though his understanding was rudimentary.
The concept of incidence coloring was introduced by Brualdi and Massey in 1993. They bounded it in terms of "Δ(G)", the maximum degree of a graph "G". Initially, the incidence chromatic number of trees, complete bipartite graphs and complete graphs was found out. They also conjectured that all graphs can have an incidence coloring using "Δ(G) + 2" colors (Incidence coloring conjecture - ICC). This conjecture was disproved by Guiduli, who showed that incidence coloring concept is a directed star arboricity case, introduced by Alon and Algor. His counter example showed that incidence chromatic number is at most "Δ(G) + O(log Δ(G))".
Self-complexity is a person’s perceived knowledge of herself or himself, based upon the number of distinct cognitive structures, or self-aspects, they believe themselves to possess. These self-aspects can include context-dependent social roles, relationships, activities, superordinate traits, and goals of the individual, which combine to form the larger, associative network of their self-concept. According to self-complexity theory, an individual who has a number of self-aspects that are unique in their attributes will have greater self-complexity than one who has only a few self-aspects, or whose self-aspects are closely associated to one another. In other words, self-complexity may invoke the question, “How full is the self-concept”?
An ate complex in chemistry is a salt formed by the reaction of a Lewis acid with a Lewis base whereby the central atom (from the Lewis acid) increases its valence and gains a negative formal charge. (Note that in this definition the meaning of valence is equivalent to coordination number). Often in chemical nomenclature the phrase "ate" is suffixed to the element in question. For example, the ate complex of a boron compound is called a borate. Thus trimethylborane and methyllithium react to form the ate compound MeBLi, lithium tetramethylborate(1-). This concept was introduced by Georg Wittig in 1958. The term is usually reserved for the complexes of metals in groups 2, 11 and 12.
Avanessian’s work has come to be situated primarily outside the academy. It includes widely discussed statements on current affairs such as the refugee crisis, numerous lectures in the international art and culture scene, as well as a large number of interviews and intellectual portraits. Beyond the classical academic mainstream, Avanessian has repeatedly managed to introduce new concepts and theoretical constellations into public discourse. These consist partly of neologisms, partly of concepts used in smaller circles, which Avanessian’s work as editor re-produces for the German-language context. Examples include “speculative realism,” “acceleration,” “Xenofeminism,” “hyperstition,” and, most recently, the concept of the “time complex: postcontemporary.” Hyperstition, moreover, is not just another concept, it also designates a method, namely the actualization in the present, from out of the future, of ideas or fictions.
"Common auxiliaries" are aspect-free concepts that can be used in combination with any other UDC code from the main classes or with other common auxiliaries. They have unique notational representations that makes them stand out in complex expressions. Common auxiliary numbers always begin with a certain symbol known as a facet indicator, e.g. = (equal sign) always introduces concepts representing the language of a document; (0...) numbers enclosed in parentheses starting with zero always represent a concept designating document form. Thus (075) Textbook and =111 English can be combined to express, e.g.(075)=111 Textbooks in English, and when combined with numbers from the main UDC tables they can be used as follows: 2(075)=111 Religion textbooks in English, 51(075)=111 Mathematics textbooks in English etc.In order to preserve the precise meaning and enable accurate parsing of complex UDC expressions, a number of connecting symbols are made available to relate and extend UDC numbers. These are:
This concept was introduced by Rajagopalan and Vazirani who used it to provide a (3/2 + ε) approximation algorithm for the Steiner tree problem on such instances. Subsequently the ε factor was removed by Rizzi and a 4/3 approximation algorithm was obtained by Chakrabarty et al.
The same concept has been used by subsequent authors on the Steiner tree problem, e.g. Robins and Zelikovsky
proposed an approximation algorithm for Steiner tree problem which on quasi-bipartite graphs has approximation ratio 1.28. The complexity of Robins and Zelikovsky's algorithm is "O("m" "n")", where "m" and "n" are the numbers of terminals and non-terminals in the graph, respectively. Furthermore, Gröpl et al. gave a 1.217-approximation algorithm for the special case of "uniformly quasi-bipartite" instances.
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 (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.
In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.
The complex numbers give rise to the fundamental theorem of algebra: every non-constant polynomial equation with complex coefficients has a complex solution. This property is true of the complex numbers, but not the reals. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.[4]
The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates—the magnitude or "modulus" of the product is the product of the two absolute values, or moduli, and the angle or "argument" of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
Split-complex numbers "z" and "w" are said to be hyperbolic-orthogonal if . While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime.
As with other Van Vogt works, the novel introduces a psychological element similar to general semantics. Here the Silkies use the so-called "Logic of Levels". Instead of the symbol/reality dichotomy of General Semantics, the Logic of Levels substitutes the concept of behaviors and instincts as complex neurological structures and feedback loops, simpler levels being built on more complex ones. By manipulating these complexes with their powers the Silkies can defeat adversaries, who find themselves caught in self-amplifying feedback from their own instincts and desires.
The latter approach leads to concepts such as VC dimension and Rademacher complexity which control the complexity of the space formula_8. A smaller hypothesis space introduces more bias into the inference process, meaning that formula_55 may be greater than the best possible risk in a larger space. However, by restricting the complexity of the hypothesis space it becomes possible for an algorithm to produce more uniformly consistent functions. This trade-off leads to the concept of regularization.
The circuit rank can be explained in terms of algebraic graph theory as the dimension of the cycle space of a graph, in terms of matroid theory as the corank of a graphic matroid, and in terms of topology as one of the Betti numbers of a topological space derived from the graph. It counts the ears in an ear decomposition of the graph, forms the basis of parameterized complexity on almost-trees, and has been applied in software metrics as part of the definition of cyclomatic complexity of a piece of code. Under the name of cyclomatic number, the concept was introduced by Gustav Kirchhoff.
Once the basics of electrical network analysis began to be established, it was not long before the ideas of complex impedance and filter design theories were carried over into mechanics by analogy. Kennelly, who was also responsible for introducing complex impedance, and Webster were the first to extend the concept of impedance into mechanical systems in 1920. Mechanical admittance and the associated mobility analogy came much later and are due to Firestone in 1932.
The Oedipus complex (also spelled Œdipus complex) is a concept of psychoanalytic theory. Sigmund Freud introduced the concept in his "Interpretation of Dreams" (1899) and coined the expression in his "A Special Type of Choice of Object made by Men" (1910). The positive Oedipus complex refers to a child's unconscious sexual desire for the opposite-sex parent and hatred for the same-sex parent. The negative Oedipus complex refers to a child's unconscious sexual desire for the same-sex parent and hatred for the opposite-sex parent. Freud considered that the child's identification with the same-sex parent is the successful outcome of the complex and that unsuccessful outcome of the complex might lead to neurosis, pedophilia, and homosexuality.
The third section is perhaps the most complex, and attempts to convey the concepts and language necessary to describe behavior and conversation between individuals. It uses examples to introduce actors speaking to each other, asking questions, disapproving, quoting other people, knowing and wanting things, promising, and playing. The first steps (since he has introduced sets of numbers and questions) are to introduce some new symbols (distinctive patterns of pulses), say they are NOT numbers, and transmit sequences showing two of these new symbols separated by the word "Inq," and followed by a question about an equation, then the symbols reversed, followed by the answer (example below). He expects that after many repetitions, the recipient will decide these new symbols are entities asking and answering the questions, rather than some other context for the questions.
MIT physicist César Hidalgo and Harvard economist Ricardo Hausmann introduced a spectral method to measure the complexity of a country's economy by inferring it from the structure of the network connecting countries to the products that they export. The measure combines information of a country's diversity, which is positively correlated with a country's productive knowledge, with measures of a product ubiquity (number of countries that produce or export the product). This concept, known as the "Product Space", has been further developed by MIT's Observatory of Economic Complexity, and in The Atlas of Economic Complexity in 2011.
Muses also envisioned a mathematical number concept, Musean hypernumbers, that includes hypercomplex number algebras such as complex numbers and split-complex numbers as primitive types. He assigned them levels based on certain arithmetical properties they may possess. While many open questions remain, in particular about defining relations of these levels, Muses pictured a wide range of applicability for this concept. Some of these are based on properties of magic squares, and even related to religious belief. He believed that these hypernumbers were central to issues of consciousness.
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 + 3<i>i is a complex number.[5]
However, with regard to the second sense, which Plotnisky describes by stating that "all imaginary and complex numbers are, by definition, irrational," mathematicians agree with Sokal and Bricmont in not taking complex numbers as irrational. Indeed, the concept of rational numbers has been extended into the complex domain to include Gaussian integers and Gaussian rationals.
Biologist PZ Myers, who works in the field of evolutionary developmental biology, dismissed the concept, stating in 2010 that "Nelson is a creationist who made up this wacky claim of "ontogenetic depth", saying he had a way of objectively measuring the complexity of the developmental process in organisms with a number that described the distance from egg to adult. Unfortunately, he forgot to tell us how one calculated this number, or how it actually accounted for the complexity of a network, or even how we'd get a number that was different for a sponge and a cat. But he did say he'd get back to us with the details tomorrow...six years ago." Myers concludes that "ontogenetic depth is a sloppily-defined concept with no theoretical support for its validity and no apparent operational utility."
The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates — the magnitude or "modulus" of the product is the product of the two absolute values, or moduli, and the angle or "argument" of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
The EER model includes all of the concepts introduced by the ER model. Additionally it includes the concepts of a subclass and superclass (Is-a), along with the concepts of specialization and generalization. Furthermore, it introduces the concept of a union type or category, which is used to represent a collection of objects that is the union of objects of different entity types. EER model also includes EER diagrams that are conceptual models that accurately represent the requirements of complex databases.