jedzenie: Theorem

Evidence: 100 • authority: 72

In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreo…

Evidence: 38 • authority: 92

Tree diet: reducing the treewidth to unlock FPT algorithms in RNA bioinformatics.. Hard graph problems are ubiquitous in Bioinformatics, inspiring the design of specialized Fixed-Parameter Tractable algorithms, many of which rely on a combination of tree-decomposition and dynamic programming. The time/space complexities of such approaches hinge critically on low values for the treewidth tw of the input graph. In order to extend their scope of applicability, we introduce the TREE-DIET problem, i.e. the removal of a minimal set of edges such that a given tree-decomposition can be slimmed down to a prescribed treewidth [Formula: see text]. Our rationale is that the time gained thanks to a smal…

Evidence: 38 • authority: 92

The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion.. In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half d…

Evidence: 38 • authority: 72

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in philosophy of mathematics. The theorems are interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will alwa…

Evidence: — • authority: 80

Too many partners in root–shoot signals. Does hydraulics qualify as the only signal that feeds back over time for reliable stomatal control?. This article is a Commentary on Visentin et al. , 212: 954–963 .

Evidence: — • authority: 80

Figure 6: (A) More authors with few publications. Many authors only contribute one or a small number of papers. (B) Few authors with more publications.

Evidence: — • authority: 88

Tarski’s Theorem for Arithmetic. In the last chapter, we dealt with mathematical languages in considerable generality. We shall now turn to some particular mathematical languages. One of our goals is to reach Gödel’s incompleteness theorem for the particular system known as Peano Arithmetic. We shall give several proofs of this important result; the simplest one is based partly on Tarski’s theorem, to which we first turn. The first concrete language that we will study is the language of first order arithmetic based on addition, multiplication and exponentiation. [We also take as primitive the successor function and the less than or equal to relation, but these are inessential.] We shall for…

Evidence: — • authority: 72

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a , b , c , n {\displaystyle a,b,c,n} with n > 2 {\displaystyle n>2} such that a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} . The cases n = 1 {\displaystyle n=1} and n = 2 {\displaystyle n=2} have been known since antiquity to have infinitely many solutions. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat wrote that "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Although other statements claimed by Fermat without p…

Evidence: — • authority: 72

In game theory, the prisoner's dilemma is a thought experiment involving two rational agents, each of whom can either cooperate for mutual benefit or betray their partner ("defect") for individual gain. The dilemma arises from the fact that while defecting is rational for each agent, cooperation yields a higher payoff for each. The puzzle was designed by Merrill Flood and Melvin Dresher in 1950 during their work at the RAND Corporation. They invited economist Armen Alchian and mathematician John Williams to play a hundred rounds of the game, observing that Alchian and Williams often chose to cooperate. When asked about the results, John Nash remarked that rational behavior in the iterated v…

Evidence: — • authority: 92

Liquid-Vapor Transition in a Model of a Continuum Particle System with Finite-Range Modified Kac Pair Potential.. We prove the existence of a phase transition in dimension d>1 in a continuum particle system interacting with a pair potential containing a modified attractive Kac potential of range γ-1 , with γ>0 . This transition is "close", for small positive γ , to the one proved previously by Lebowitz and Penrose in the van der Waals limit γ↓0 . It is of the type of the liquid-vapor transition observed when a fluid, like water, heated at constant pressure, boils at a given temperature. Previous results on phase transitions in continuum systems with stable potentials required the use of unp…