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In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single quaternary cubic polynomial which is homogeneous of degree 3. Cubic surfaces are del Pezzo surfaces.
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class.
In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise.
Yuri Ivanovitch Manin is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book "Computable and Uncomputable".
In mathematics, the Hasse–Witt matrixH of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping with respect to a basis for the differentials of the first kind. It is a g × g matrix where C has genus g. The rank of the Hasse–Witt matrix is the Hasse or Hasse–Witt invariant.
In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve.
In mathematics, Arakelov theory is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction is attached to a variety X over a global field, which measures the failure of the Hasse principle for X. If the value of the obstruction is non-trivial, then X may have points over all local fields but not over the global field. The Manin obstruction is sometimes called the Brauer–Manin obstruction, as Manin used the Brauer group of X to define it.
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group. The case with A an elliptic curve E and K the rational number field Q is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922.
In mathematics, a Manin triple consists of a Lie algebra g with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras p and q such that g is the direct sum of p and q as a vector space.
In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004.
In the mathematical discipline of graph theory, Petersen's theorem, named after Julius Petersen, is one of the earliest results in graph theory and can be stated as follows:
Petersen's Theorem. Every cubic, bridgeless graph contains a perfect matching.
Michael David Plummer is a retired mathematics professor from Vanderbilt University. His field of work is in graph theory in which he has produced over a hundred papers and publications. He has also spoken at over a hundred and fifty guest lectures around the world.
The Brouwer Medal is a triennial award presented by the Royal Dutch Mathematical Society and the Royal Netherlands Academy of Sciences. The Brouwer Metal gets its name from Dutch mathematician, L. E. J. Brouwer and is The Netherlands’ most prestigious award in mathematics.
Mark Norman Ellingham is a professor of mathematics at Vanderbilt University whose research concerns graph theory. With Joseph D. Horton, he is the discoverer and namesake of the Ellingham–Horton graphs, two cubic 3-vertex-connected bipartite graphs that have no Hamiltonian cycle.
Belousov Valentin Danilovich was a Moldavian Soviet mathematician and a correspondent member of the Academy of Pedagogical Sciences of the USSR (1968).
References
- Manin, Yuri Ivanovich (1986) [1972], Cubic forms, North-Holland Mathematical Library, 4 (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-444-87823-6, MR 0833513
The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.
Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of Mathematical Reviews and additionally contains citation information for over 3.5 million items as of 2018.