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Register a new userSquare root meadows
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Added by
Inge Bethke
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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Let Q_0 denote the rational numbers expanded to a meadow by totalizing inversion such that 0^{-1}=0. Q_0 can be expanded by a total sign function s that extracts the sign of a rational number. In this paper we discuss an extension Q_0(s ,sqrt) of the signed rationals in which every number has a unique square root.
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A meadow is a commutative ring with a total inverse operator satisfying 0^{-1}=0. We show that the class of finite meadows is the closure of the class of Galois fields under finite products. As a corollary, we obtain a unique representation of minimal finite meadows in terms of finite prime fields.
Square-root topological states are new topological phases, whose topological property is inherited from the square of the Hamiltonian. We realize the first-order and second-order square-root topological insulators in phononic crystals, by putting additional cavities on connecting tubes in the acoustic Su-Schrieffer-Heeger model and the honeycomb lattice, respectively. Because of the square-root procedure, the bulk gap of the squared Hamiltonian is doubled. In both two bulk gaps, the square-root topological insulators possess multiple localized modes, i.e., the end and corner states, which are evidently confirmed by our calculations and experimental observations. We further propose a second-order square-root topological semimetal by stacking the decorated honeycomb lattice to three dimensions.
A combination of program algebra with the theory of meadows is designed leading to a theory of computation in algebraic structures which use in addition to a zero test and copying instructions the instruction set ${x Leftarrow 0, x Leftarrow 1, xLeftarrow -x, xLeftarrow x^{-1}, xLeftarrow x+y, xLeftarrow xcdot y}$. It is proven that total functions on cancellation meadows can be computed by straight-line programs using at most 5 auxiliary variables. A similar result is obtained for signed meadows.
The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favour of the square root voting system, where the voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic union of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: q simeq 1/2+1/sqrt{{pi} M}. The prefactor 1/sqrt{{pi}} appears here as a result of averaging over the ensemble of unions with random populations.
We prove the Kato conjecture for elliptic operators, $L=- ablacdotleft((mathbf A+mathbf D) abla right)$, with $mathbf A$ a complex measurable bounded coercive matrix and $mathbf D$ a measurable real-valued skew-symmetric matrix in $mathbb{R}^n$ with entries in $BMO(mathbb{R}^n)$;, i.e., the domain of $sqrt{L},$ is the Sobolev space $dot H^1(mathbb{R}^n)$ in any dimension, with the estimate $|sqrt{L}, f|_2lesssim | abla f|_2$.
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