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ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
THE STRENGTH OF JULLIEN’S INDECOMPOSABILITY THEOREM
"... Abstract. Jullien’s indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis. We identify the strength of ..."
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Abstract. Jullien’s indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis. We identify the strength of the theorem relative to standard reverse mathematics markers. We show that it lies strictly between weak Σ 1 1 choice and ∆11 comprehension. §1. Introduction. A linear order (U;<U) (denoted simply U below) is scattered if it does not contain a copy of the order of rational numbers. A cut in the order U is a pair 〈L,R 〉 so that L ∩ R = ∅, L ∪ R = U, L is closed downward (or leftward) in <U, and R is closed upward (or rightward). The cut is a decomposition of U if U does not embed into either one of L, R. The order U
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"... Dynamics of a classical Hall system driven by a timedependent AharonovBohm flux. (English summary) J. Math. Phys. 48 (2007), no. 5, 052901, 14 pp. Summary: “We study the dynamics of a classical particle moving in a punctured plane under the influence of a homogeneous magnetic field, an electric ba ..."
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Dynamics of a classical Hall system driven by a timedependent AharonovBohm flux. (English summary) J. Math. Phys. 48 (2007), no. 5, 052901, 14 pp. Summary: “We study the dynamics of a classical particle moving in a punctured plane under the influence of a homogeneous magnetic field, an electric background, and driven by a timedependent singular flux tube through the hole. We exhibit a striking (de)localization effect: when the electric background is absent we prove that a linearly timedependent flux tube opposite to the homogeneous flux eventually leads to the usual classical Hall motion. The particle follows a cycloid whose center is drifting orthogonal to the electric field; if the fluxes are additive, the drifting center eventually gets pinned by the flux tube whereas the kinetic energy grows with the additional flux.” References 1. Asch, J., Benguria, R. D., and ˇSt’ovíček, P., ”Asymptotic properties of the differential equation h 3 (h′ ′ + h′) = 1, ” Asymptotic Anal. 41, 23–40 (2005). MR2124892 (2005j:34069) 2. Asch, J., Hradeck´y, I., and ˇSt’ovíček, P., ”Propagators weakly associated to a family of Hamiltonians and the adiabatic theorem for the Landau Hamiltonian with a timedependent AharonovBohm flux, ” J. Math. Phys. 46, 053303 (2005). MR2143006 (2006a:81033) 3. Avron, J. E., Seiler, R., and Simon, B., ”Charge deficiency, charge transport and comparison of