|
Regular Hadamard Matrices
In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order be a square number. The excess, denoted ''E''(''H''), of a Hadamard matrix ''H'' of order ''n'' is defined to be the sum of the entries of ''H''. The excess satisfies the bound , ''E''(''H''), ≤ ''n''3/2. A Hadamard matrix attains this bound if and only if it is regular. Parameters If ''n'' = 4''u''2 is the order of a regular Hadamard matrix, then the excess is ±8''u''3 and the row and column sums all equal ±2''u''. It follows that each row has 2''u''2 ± ''u'' positive entries and 2''u''2 ∓ ''u'' negative entries. The orthogonality of rows implies that any two distinct rows have exactly ''u''2 ± ''u'' positive entries in common. If ''H'' is interpreted as the incidence m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The ''n''-dimensional parallelotope spanned by the rows of an ''n''×''n'' Hadamard matrix has the maximum possible ''n''-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element of ''X'' and one column for each element of ''Y''. The entry in row ''x'' and column ''y'' is 1 if ''x'' and ''y'' are related (called ''incident'' in this context) and 0 if they are not. There are variations; see below. Graph theory Incidence matrix is a common graph representation in graph theory. It is different to an adjacency matrix, which encodes the relation of vertex-vertex pairs. Undirected and directed graphs In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented. The ''unoriented incidence matrix'' (or simply ''incidence matrix'') of an undirected graph is a n\times m matrix ''B'', where ''n'' and ''m'' are the numbers of vertices and edges respectively, such that :B_=\left\{\begin{a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of blocks exhibit symmetry (balance). They have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry. Without further specifications the term ''block design'' usually refers to a balanced incomplete block design (BIBD), specifically (and also synonymously) a 2-design, which has been the most intensely studied type historically due to its application in the design of experiments. Its generalization is known as a t-design. Overview A design is said to be ''balanced'' (up to ''t'') if all ''t''-subsets of the original set occur in equally many (i.e., ''λ'') blocks. When ''t'' is unspecified, it can usually be assumed to be 2, which means that ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel 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 at exactly 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, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard. He obtained his doctorate in 1892 and in the same year was awarded the for his essay on the Riemann zeta function. In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, where he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P Kesava Menon
Puliyakot Keshava Menon (1917 – 22 October 1979) was an Indian mathematician best known as Director of the Joint Cipher Bureau. His sudden demise on 22 October 1979, ended active research in the areas of number theory, combinatorics, algebra and cryptography. Early life P. Kesava Menon was born (1917) in Alathur, which is now part of the Palakkad District of Kerala state in India. His mother, Devaky Amma, hailed from the Kunissery Puliyakot family and, as per custom, Kesava Menon took his family name from his mother. His father, A K Krishnan Unni Kartha, hailed from Aiyiloor in the Palghat district. Menon grew up on Alathur under his uncle's supervision and hence his primary and high school education was conducted in modest surrounding at Alathur itself. Higher education As was the custom for bright students from landed families those days, Menon had to travel to Madras city and join the Madras Christian College for his higher studies. There, he completed his MA in Math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Colbourn
Charles Joseph Colbourn (born October 24, 1953) is a Canadian computer scientist and mathematician, whose research concerns graph algorithms, combinatorial designs, and their applications. From 1996 to 2001 he was the Dorothean Professor of Computer Science at the University of Vermont; since then he has been a professor of Computer Science and Engineering at Arizona State University.Curriculum vitae from Colbourn's web site, retrieved 2011-03-26. Colbourn was born on October 24, 1953, in , ; despite working in the United States since 1996 he retains his Canadian citizenship. He did his und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeff Dinitz
Jeffrey Howard Dinitz (born 1952) is an American mathematician who taught combinatorics at the University of Vermont. He is best known for proposing the Dinitz conjecture, which became a major theorem. Dinitz is married to Susan Dinitz and has three children, Mike, Amy, and Tom. XFL scheduling Dinitz is also well known for scheduling the first season of the now-defunct XFL XFL may refer to: Sports * XFL (2001), a defunct American football league that played its only season in 2001 * XFL (2020), a professional American football league Vehicles * Bell XFL Airabonita, a 1940 U.S. Navy experimental interceptor aircra ... football league. He and a colleague from the Czech Republic, Dalibor Froncek, offered the then-brand-new XFL league their expertise to draft complicated schedules. The XFL administration quickly agreed, which "surprised" Dinitz greatly. After some time on the computer, Dinitz and Froncek sent the XFL a draft schedule, and the new league gratefully accepted. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anne Penfold Street
Anne Penfold Street (1932–2016) was one of Australia's leading mathematicians, specialising in combinatorics. She was the third woman to become a mathematics professor in Australia, following Hanna Neumann and Cheryl Praeger. She was the author of several textbooks, and her work on sum-free sets became a standard reference for its subject matter. She helped found several important organizations in combinatorics, developed a researcher network, and supported young students with interest in mathematics. Early life and education Street was born on 11 October 1932 in Melbourne, the daughter of a medical researcher. She earned a bachelor's degree in chemistry from the University of Melbourne in 1954, while working there as a tutor in chemistry and also studying mathematics. She finished a master's degree in chemistry at Melbourne in 1956. During this time she married another Melbourne chemist, Norman Street, and in 1957 the Streets and their young daughter moved to the Univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jennifer Seberry
Jennifer Roma Seberry (also published as Jennifer Seberry Wallis, born 13 February 1944 in Sydney) is an Australian cryptographer, mathematician, and computer scientist, currently a professor at the University of Wollongong, Australia. She was formerly the head of the Department of Computer Science and director of the Centre for Computer Security Research at the university. Education and career Seberry attended Parramatta High School and got her BSc at University of New South Wales, 1966; MSc at La Trobe University, 1969; PhD at La Trobe University, 1971 (Computational Mathematics); B.Ec. with two years completed at University of Sydney. Her doctoral advisor was Bertram Mond. Seberry was the first person to teach cryptology at an Australian University (University of Sydney). She was also the first woman Professor of Computer Science in Australia. She was the first woman Reader in Combinatorial Mathematics in Australia. she had supervised 30 doctorates and had 71 academic desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
