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Sublinear Operator
A sublinear function (or functional, as is more often used in functional analysis), in linear algebra and related areas of mathematics, is a function f : V → F displaystyle f:Vrightarrow mathbf F on a vector space V over F, an ordered field (e.g
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Functional Analysis
Functional analysis
Functional analysis
is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject
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Convex Function
In mathematics, a real-valued function defined on an n-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space
Euclidean space
(or more generally a vector space) of at least two dimensions. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992
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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] An ISBN is assigned to each edition and variation (except reprintings) 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, and 10 digits long if assigned before 2007
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Introduction To Algorithms
Introduction to Algorithms
Introduction to Algorithms
is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The first edition of the book was widely used as the textbook for algorithms courses at many universities[1] and is commonly cited as a reference for algorithms in published papers, with over 10000 citations documented on CiteSeerX.[2] The book sold half a million copies during its first 20 years.[3] Its fame has led to the common use of the abbreviation "CLRS" (Cormen, Leiserson, Rivest, Stein), or, in the first edition, "CLR" (Cormen, Leiserson, Rivest).[4] In the preface, the authors write about how the book was written to be comprehensive and useful in both teaching and professional environments. Each chapter focuses on an algorithm, and discusses its design techniques and areas of application. Instead of using a specific programming language, the algorithms are written in Pseudocode
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Clifford Stein
Clifford Seth Stein (born December 14, 1965), a computer scientist, is a professor of industrial engineering and operations research at Columbia University
Columbia University
in New York, NY, where he also holds an appointment in the Department of Computer Science. Stein is chair of the Industrial Engineering and Operations Research Department at Columbia University
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Ronald L. Rivest
Ronald Linn Rivest (/rɪˈvɛst/;[5][6] born May 6, 1947) is a cryptographer and an Institute Professor at MIT.[2] He is a member of MIT's Department of Electrical Engineering and Computer Science
Computer Science
(EECS) and a member of MIT's Computer Science
Computer Science
and Artificial Intelligence Laboratory (CSAIL). He was a member of the Election Assistance Commission's Technical Guidelines Development Committee, tasked with assisting the EAC in drafting the Voluntary Voting
Voting
System Guidelines.[7] Rivest is one of the inventors of the RSA algorithm (along with Adi Shamir and Len Adleman).[1] He is the inventor of the symmetric key encryption algorithms RC2, RC4, RC5, and co-inventor of RC6. The "RC" stands for "Rivest Cipher", or alternatively, "Ron's Code"
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Charles E. Leiserson
Charles Eric Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof. As part of this effort, he developed the Cilk multithreaded language. He invented the fat-tree interconnection network, a hardware-universal interconnection network used in many supercomputers, including the Connection Machine
Connection Machine
CM5, for which he was network architect. He helped pioneer the development of VLSI theory, including the retiming method of digital optimization with James B. Saxe and systolic arrays with H. T. Kung. He conceived of the notion of cache-oblivious algorithms, which are algorithms that have no tuning parameters for cache size or cache-line length, but nevertheless use cache near-optimally
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Thomas H. Cormen
Thomas H. Cormen[1] is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein. In 2013, he published a new book titled Algorithms Unlocked. He is a professor of computer science at Dartmouth College
Dartmouth College
and former Chair of the Dartmouth College
Dartmouth College
Department of Computer Science. Between 2004 and 2008 he directed the Dartmouth College
Dartmouth College
Writing Program.[2] His research interests are algorithm engineering, parallel computing, speeding up computations with high latency.Contents1 Early life and education 2 Honors and awards 3 Bibliography 4 Notes 5 External linksEarly life and education[edit] Thomas H. Cormen was born in New York City
New York City
in 1956
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Norm (mathematics)
In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. A simple example is two dimensional Euclidean space
Euclidean space
R2 equipped with the "Euclidean norm" (see below) Elements in this vector space (e.g., (3, 7)) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude. A vector space on which a norm is defined is called a normed vector space
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Linear Algebra
Linear
Linear
algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , displaystyle a_ 1 x_ 1 +cdots +a_ n x_ n =b, linear functions such as ( x 1 , … , x n ) ↦ a 1 x 1 + … + a n x n , displaystyle (x_ 1 ,ldots ,x_ n )mapsto a_ 1 x_ 1 +ldots +a_ n x_ n , and their representations through matrices and vector spaces.[1][2][3] Linear
Linear
algebra is central to almost all areas of mathematics
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Big O Notation
Big O notation
Big O notation
is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity
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Computer Science
Computer science
Computer science
is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. It is the scientific and practical approach to computation and its applications and the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to, information. An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems.[1] Its fields can be divided into a variety of theoretical and practical disciplines
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Hahn–Banach Theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry. The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C [ a , b ] displaystyle Cleft[a,bright] of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly,[1] and a more general extension theorem, the M
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Functional (mathematics)
In mathematics, the term functional (as a noun) has at least two meanings.In modern linear algebra, it refers to a linear mapping from a vector space V displaystyle V into its field of scalars, i.e., to an element of the dual space V ∗ displaystyle V^ * . In mathematical analysis, more generally and historically, it refers to a mapping from a space X displaystyle X into the real numbers, or sometimes into the complex numbers, for the purpose of establishing a calculus-like structure on X displaystyle X . Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space X displaystyle X .This article is mainly concerned with the second concept, which arose in the early 18th century as part of the calculus of variations
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