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In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's Last Theorem, have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.
The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics.
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he contributed to several areas, including geometrical theory of dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, symplectic topology, differential equations, classical mechanics, differential geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics—KAM theory, and topological Galois theory.
Andrey Nikolaevich Kolmogorov was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.
In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges.
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof, which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Pavel Sergeyevich Alexandrov, sometimes romanized Paul Alexandroff, was a Soviet mathematician. He wrote about three hundred papers, making important contributions to set theory and topology. In topology, the Alexandroff compactification and the Alexandrov topology are named after him.
Hugo Hadwiger was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.
In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three colors. In order to avoid trivial cases, snarks are often restricted to have additional requirements on their connectivity and on the length of their cycles. Infinitely many snarks exist.
The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.
Paul D. Seymour is a British mathematician known for his work in discrete mathematics, especially graph theory. He was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website.
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.
In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks if a similar result holds globally. Precisely, the conjecture states that if a continuously differentiable map on an -dimensional real vector space has a fixed point, and its Jacobian matrix is everywhere Hurwitz, then the fixed point is globally stable.
The Journal of Combinatorial Theory, Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. Series A is concerned primarily with structures, designs, and applications of combinatorics. Series B is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as JCTA and JCTB.
In queueing theory, a discipline within the mathematical theory of probability, Burke's theorem is a theorem asserting that, for the M/M/1 queue, M/M/c queue or M/M/∞ queue in the steady state with arrivals is a Poisson process with rate parameter λ:
- The departure process is a Poisson process with rate parameter λ.
- At time t the number of customers in the queue is independent of the departure process prior to time t.
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.
Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L2 functions, proved by Lennart Carleson (1966). The name is also often used to refer to the extension of the result by Richard Hunt (1968) to Lp functions for p ∈ (1, ∞] and the analogous results for pointwise almost everywhere convergence of Fourier integrals, which can be shown to be equivalent by transference methods.
In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem states that every multivariate continuous function can be represented as a superposition of the two-argument addition of continuous functions of one variable. It solved a more constrained, yet more general form of Hilbert's thirteenth problem.
References
- ↑ Lutz D. Schmadel (2012-06-10). Dictionary of Minor Planet Names. Springer Science & Business Media. p. 717. ISBN 978-3-642-29718-2.
- ↑ Editorial (2015), "Journal Description Arnold Mathematical Journal", Arnold Mathematical Journal, 1 (1): 1–3, doi: 10.1007/s40598-015-0006-6 .