title: Linear law for the logarithms of the Riemann periods at simple critical zeta zeros

authors: Broughan, Kevin A.

subjects: mathematics

abstract: Each simple zero 1/2 + iγn of the Riemann zeta function on the critical line with γn > 0 is a center for the flow s˙ = ξ(s) of the Riemann xi function with an associated period Tn. It is shown that, as γn →∞, log Tn ≥ π/4 γn + O(log γn).
Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γn+1 − γn≥ γn-θ for some exponent θ > 0, we obtain the upper bound log Tn ≤ γn2 + θ Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log Tn = π/ 4 γn +O(log γn). Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.

text: Linear law for the logarithms of the Riemann periods at simple critical zeta zeros Each simple zero 1/2 + iγn of the Riemann zeta function on the critical line with γn > 0 is a center for the flow s˙ = ξ(s) of the Riemann xi function with an associated period Tn. It is shown that, as γn →∞, log Tn ≥ π/4 γn + O(log γn).
Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γn+1 − γn≥ γn-θ for some exponent θ > 0, we obtain the upper bound log Tn ≤ γn2 + θ Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log Tn = π/ 4 γn +O(log γn). Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.

year: 2006

title: Portimine: a bioactive metabolite from the benthic dinoflagellate Vulcanodinium rugosum

authors: Selwood, Andrew I.

subjects: pinnatoxin

abstract: Portimine, a new polycyclic ether toxin containing a cyclic imine moiety, was isolated from the marine benthic dinoflagellate Vulcanodinium rugosum collected from Northland, New Zealand. The structure of portimine, including the relative configurations, was elucidated by spectroscopic analyses. The cyclic imine moiety consists of an unprecedented five-membered ring with a spiro-link to a cyclohexene ring. This is the only structural similarity to the pinnatoxin group of polycyclic ethers also produced by V. rugosum, which all contain a six-membered cyclic imine ring. The LD50 of portimine to mice by intraperitoneal injection was 1570 μg/kg, indicating a much lower toxicity than many other cyclic imine shellfish toxins. In contrast, portimine was highly toxic to mammalian cells in vitro with an LC50 to P388 cells of 2.7 nM, and activation of caspases indicating apoptotic activity.

text: Portimine: a bioactive metabolite from the benthic dinoflagellate Vulcanodinium rugosum Portimine, a new polycyclic ether toxin containing a cyclic imine moiety, was isolated from the marine benthic dinoflagellate Vulcanodinium rugosum collected from Northland, New Zealand. The structure of portimine, including the relative configurations, was elucidated by spectroscopic analyses. The cyclic imine moiety consists of an unprecedented five-membered ring with a spiro-link to a cyclohexene ring. This is the only structural similarity to the pinnatoxin group of polycyclic ethers also produced by V. rugosum, which all contain a six-membered cyclic imine ring. The LD50 of portimine to mice by intraperitoneal injection was 1570 μg/kg, indicating a much lower toxicity than many other cyclic imine shellfish toxins. In contrast, portimine was highly toxic to mammalian cells in vitro with an LC50 to P388 cells of 2.7 nM, and activation of caspases indicating apoptotic activity.

year: 2013

title: Introduction to special section on Māori culture and education.

authors: Bishop, Russell

subjects: language policy

abstract: An introduction is presented in which the editor discusses various reports within the issue on topics including the need for coordinated language policy, the Treaty of Waitangi, and monoculturalism.

text: Introduction to special section on Māori culture and education. An introduction is presented in which the editor discusses various reports within the issue on topics including the need for coordinated language policy, the Treaty of Waitangi, and monoculturalism.

year: 2002