Michael Hartley Freedman is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional Generalized Poincaré conjecture. Freedman and Robion Kirby showed. Freedman was born in Los Angeles, California, U. S, his father, Benedict Freedman, was an American Jewish aeronautical engineer, musician and mathematician. His mother, Nancy Mars Freedman, performed as an actress and trained as an artist, his parents cowrote a series of novels together. He entered the University of California and after two semesters dropped out in the same year he wrote a letter to Ralph Fox a Princeton professor at the time and was admitted to graduate school so in 1968 he continued his studies at Princeton University where he received Ph. D. degree in 1973 for his doctoral dissertation titled Codimension-Two Surgery, written under the supervision of William Browder. After graduating, Freedman was appointed a lecturer in the Department of Mathematics at the University of California, Berkeley.
He held this post from 1973 until 1975, when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed assistant professor in the Department of Mathematics at the University of California San Diego, he spent the year 1980/81 at IAS, returning to UC San Diego, where in 1982 he was promoted to professor. He was appointed the Charles Lee Powell chair of mathematics at UC San Diego in 1985. Freedman has received numerous other awards and honors including Sloan and Guggenheim Fellowships, a MacArthur Fellowship and the National Medal of Science, he is an elected member of the National Academy of Sciences, a fellow of the American Academy of Arts and Sciences and of the American Mathematical Society. He works at Microsoft Station Q at the University of California, Santa Barbara, where his team is involved in the development of the topological quantum computer. Freedman, Michael Hartley, "The topology of four-dimensional manifolds", Journal of Differential Geometry, 17: 357–453, ISSN 0022-040X, MR 0679066 Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol 39, Princeton University Press, New Jersey, 1990.
ISBN 0-691-08577-3 Freedman, Michael H.: Z2-systolic-freedom. Proceedings of the Kirbyfest, 113–123, Geom. Topol. Monogr. 2, Geom. Topol. Publ. Coventry, 1999. Freedman, Michael H.. Mathematics of quantum computation, 287–320, Comput. Math. Ser. Chapman & Hall/CRC, Boca Raton, Florida, 2002. Michael H. Freedman, Microsoft Technical Fellow and Director of Station Q Michael Freedman at the Mathematics Genealogy Project
Eli Biham is an Israeli cryptographer and cryptanalyst a professor at the Technion Israeli Institute of Technology Computer Science department. Starting from October 2008 and till 2013, Biham was the dean of the Technion Computer Science department, after serving for two years as chief of CS graduate school. Biham received his Ph. D. for inventing differential cryptanalysis, while working under Adi Shamir. It had, been invented at least twice before. A team at IBM discovered it during their work on DES, was requested/required to keep their discovery secret by the NSA, who evidently knew about it as well. Among his many contributions to cryptanalysis one can count: differential cryptanalysis - publicly invented during his Ph. D. studies under Adi Shamir Attacking all triple modes of operation. Impossible differential cryptanalysis - joint work with Adi Shamir and Alex Biryukov Breaking the ANSI X9.52 CBCM mode Breaking the GSM security mechanisms Co-invention of related-key attacks. Differential Fault Analysis - joint work with Adi Shamir Biham has taken part in the design of several new cryptographic primitives: Serpent, a block cipher, one of the final five contenders to become the Advanced Encryption Standard Tiger, a hash function fast on 64-bit machines, Py, one of a family of fast stream ciphers.
SHAvite-3, a hash function, one of the 14 semifinalists in the NIST hash function competition. Eli Biham's homepage at Technion
Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large scale. Quantum mechanics differs from classical physics in that energy, angular momentum and other quantities of a bound system are restricted to discrete values. Quantum mechanics arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others; the modern theory is formulated in various specially developed mathematical formalisms.
In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position and other physical properties of a particle. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the laser, the transistor and semiconductors such as the microprocessor and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he described in a paper titled On the nature of light and colours.
This experiment played a major role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays; these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, the 1900 quantum hypothesis of Max Planck. Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics. Following Max Planck's solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect.
Around 1900–1910, the atomic theory and the corpuscular theory of light first came to be accepted as scientific fact. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, for which Niels Bohr developed his theory of the atomic structure, confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept introduced by Arnold Sommerfeld; this phase is known as old quantum theory. According to Planck, each energy element is proportional to its frequency: E = h ν, where h is Planck's constant. Planck cautiously insisted that this was an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.
In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material, he won the 1921 Nobel Prize in Physics for this work. Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle, with a discrete quantum of energy, dependent on its frequency; the foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wi
A conceptual model is a representation of a system, made of the composition of concepts which are used to help people know, understand, or simulate a subject the model represents. It is a set of concepts; some models are physical objects. The term conceptual model may be used to refer to models which are formed after a conceptualization or generalization process. Conceptual models are abstractions of things in the real world whether physical or social. Semantic studies are relevant to various stages of concept formation. Semantics is about concepts, the meaning that thinking beings give to various elements of their experience; the term conceptual model is normal. It could mean "a model of concept" or it could mean "a model, conceptual." A distinction can be made between what models are made of. With the exception of iconic models, such as a scale model of Winchester Cathedral, most models are concepts, but they are intended to be models of real world states of affairs. The value of a model is directly proportional to how well it corresponds to a past, future, actual or potential state of affairs.
A model of a concept is quite different because in order to be a good model it need not have this real world correspondence. In artificial intelligence conceptual models and conceptual graphs are used for building expert systems and knowledge-based systems. Conceptual models range in type from the more concrete, such as the mental image of a familiar physical object, to the formal generality and abstractness of mathematical models which do not appear to the mind as an image. Conceptual models range in terms of the scope of the subject matter that they are taken to represent. A model may, for instance, represent a single thing, whole classes of things, very vast domains of subject matter such as the physical universe; the variety and scope of conceptual models is due to the variety of purposes had by the people using them. Conceptual modeling is the activity of formally describing some aspects of the physical and social world around us for the purposes of understanding and communication." A conceptual model's primary objective is to convey the fundamental principles and basic functionality of the system which it represents.
A conceptual model must be developed in such a way as to provide an understood system interpretation for the model's users. A conceptual model, when implemented properly, should satisfy four fundamental objectives. Enhance an individual's understanding of the representative system Facilitate efficient conveyance of system details between stakeholders Provide a point of reference for system designers to extract system specifications Document the system for future reference and provide a means for collaborationThe conceptual model plays an important role in the overall system development life cycle. Figure 1 below, depicts the role of the conceptual model in a typical system development scheme, it is clear that if the conceptual model is not developed, the execution of fundamental system properties may not be implemented properly, giving way to future problems or system shortfalls. These failures do have been linked to; those weak links in the system design and development process can be traced to improper execution of the fundamental objectives of conceptual modeling.
The importance of conceptual modeling is evident when such systemic failures are mitigated by thorough system development and adherence to proven development objectives/techniques. As systems have become complex, the role of conceptual modelling has expanded. With that expanded presence, the effectiveness of conceptual modeling at capturing the fundamentals of a system is being realized. Building on that realization, numerous conceptual modeling techniques have been created; these techniques can be applied across multiple disciplines to increase the user's understanding of the system to be modeled. A few techniques are described in the following text, many more exist or are being developed; some used conceptual modeling techniques and methods include: workflow modeling, workforce modeling, rapid application development, object-role modeling, the Unified Modeling Language. Data flow modeling is a basic conceptual modeling technique that graphically represents elements of a system. DFM is a simple technique, like many conceptual modeling techniques, it is possible to construct higher and lower level representative diagrams.
The data flow diagram does not convey complex system details such as parallel development considerations or timing information, but rather works to bring the major system functions into context. Data flow modeling is a central technique used in systems development that utilizes the structured systems analysis and design method. Entity-relationship modeling is a conceptual modeling technique used for software system representation. Entity-relationship diagrams, which are a product of executing the ERM technique, are used to represent database models and information systems; the main components of the diagram are the relationships. The entities can represent objects, or events; the relationships are responsible for relating the entities to one another. To form a system process, the relationships are combined with the entities and any attr
Michael Aaron Nielsen is a quantum physicist, science writer, computer programming researcher living in San Francisco. In 2004 Nielsen was characterized as Australia's "youngest academic" and secured a Federation Fellowship at the University of Queensland, he worked at the Los Alamos National Laboratory, as the Richard Chace Tolman Prize Fellow at Caltech, a Senior Faculty Member at the Perimeter Institute for Theoretical Physics. Nielsen obtained his PhD in physics in 1998 at the University of New Mexico. With Isaac Chuang he is the co-author of a popular textbook on quantum computing. In 2007, Nielsen announced a marked shift in his field of research: from quantum information and computation to “the development of new tools for scientific collaboration and publication”; this work includes "massively collaborative mathematics" projects like the Polymath project with Timothy Gowers. Besides writing books and essays, he has given talks about Open Science, he was a member of the Working Group on Open Data in Science at the Open Knowledge Foundation.
In 2015 Nielsen published the online textbook Neural Networks and Deep Learning. The same year he joined the Recurse Center as a Research Fellow. Since 2017 Nielsen works as a Research Fellow at Y Combinator Research.. Nielsen, Michael A. Reinventing Discovery: The New Era of Networked Science. Princeton, N. J: Princeton University Press. ISBN 0-691-14890-2; this book is based on themes that are covered in his essay on the Future of Science. Nielsen, Michael A.. Neural Networks and Deep Learning. Determination Press
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise and in mathematics and physics as infinite-dimensional function spaces; the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, ergodic theory. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications.
The success of Hilbert space methods ushered in a fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes, in analogy with Cartesian coordinates in the plane; when that set of axes is countably infinite, the Hilbert space can be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is in the older literature referred to as the Hilbert space.
Linear operators on a Hilbert space are fairly concrete objects: in good cases, they are transformations that stretch the space by different factors in mutually perpendicular directions in a sense, made precise by the study of their spectrum. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ3, equipped with the dot product; the dot product takes two vectors x and y, produces a real number x · y. If x and y are represented in Cartesian coordinates the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3; the dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: · y = ax1 · y + bx2 · y for any scalars a, b, vectors x1, x2, y, it is positive definite: for all vectors x, x · x ≥ 0, with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a inner product. A vector space equipped with such an inner product is known as a inner product space.
Every finite-dimensional inner product space is a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted ||x||, to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ. Multivariable calculus in Euclidean space relies on the ability to compute limits, to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n consisting of vectors in ℝ3 is convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞. Just as with a series of scalars, a series of vectors that converges also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as N → ∞; this property expresses the completeness of
String (computer science)
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed. A string is considered a data type and is implemented as an array data structure of bytes that stores a sequence of elements characters, using some character encoding. String may denote more general arrays or other sequence data types and structures. Depending on programming language and precise data type used, a variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements; when a string appears in source code, it is known as a string literal or an anonymous string. In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set called an alphabet. Let Σ be a non-empty finite set of symbols, called the alphabet.
No assumption is made about the nature of the symbols. A string over Σ is any finite sequence of symbols from Σ. For example, if Σ = 01011 is a string over Σ; the length of a string s can be any non-negative integer. The empty string is the unique string over Σ of length 0, is denoted ε or λ; the set of all strings over Σ of length n is denoted Σn. For example, if Σ = Σ2 =. Note that Σ0 = for any alphabet Σ; the set of all strings over Σ of any length is the Kleene closure of Σ and is denoted Σ*. In terms of Σn, Σ ∗ = ⋃ n ∈ N ∪ Σ n For example, if Σ = Σ* =. Although the set Σ* itself is countably infinite, each element of Σ* is a string of finite length. A set of strings over Σ is called a formal language over Σ. For example, if Σ =, the set of strings with an number of zeros, is a formal language over Σ. Concatenation is an important binary operation on Σ*. For any two strings s and t in Σ*, their concatenation is defined as the sequence of symbols in s followed by the sequence of characters in t, is denoted st.
For example, if Σ =, s = bear, t = hug st = bearhug and ts = hugbear. String concatenation is an non-commutative operation; the empty string ε serves as the identity element. Therefore, the set Σ* and the concatenation operation form a monoid, the free monoid generated by Σ. In addition, the length function defines a monoid homomorphism from Σ* to the non-negative integers. A string s is said to be a substring or factor of t if there exist strings u and v such that t = usv; the relation "is a substring of" defines a partial order on Σ*, the least element of, the empty string. A string s is said to be a prefix of t if there exists a string u such that t = su. If u is nonempty, s is said to be a proper prefix of t. Symmetrically, a string s is said to be a suffix of t if there exists a string u such that t = us. If u is nonempty, s is said to be a proper suffix of t. Suffixes and prefixes are substrings of t. Both the relations "is a prefix of" and "is a suffix of" are prefix orders. A string s = uv.
For example, if Σ = the string 0011001 is a rotation of 0100110, where u = 00110 and v = 01. The reverse of a string is a string in reverse order. For example, if s = abc the reverse of s is cba. A string, the reverse of itself is called a palindrome, which includes the empty string and all strings of length 1, it is useful to define an ordering on a set of strings. If the alphabet Σ has a total order one can define a total order on Σ* called lexicographical order. For example, if Σ = and 0 < 1 the lexicographical order on Σ* includes the relationships ε < 0 < 00 < 000 <... < 0001 < 001 < 01 < 010 < 011 < 0110 < 01111 < 1 < 10 < 100 < 101 < 111 < 1111 < 11111... The lexicographical order is total if the alphabetical order is, but isn't well-founded for any nontrivial alphabet if the alphabetical order is. See Shortlex for an alternative string ordering that preserves well-foundedness. A number of additional operations on strings occur in the formal theory; these are given in the article on string operations.
Strings admit the following interpretation as nodes on a graph: Fixed-length strings can be viewed as nodes on a hypercube Variable-length strings can be viewed as nodes on the k-ary tree, where k is the number of symbols in Σ Infinite strings can be viewed as i