112 research outputs found
Two-divisibility of the coefficients of certain weakly holomorphic modular forms
We study a canonical basis for spaces of weakly holomorphic modular forms of
weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a
relation between the Fourier coefficients of modular forms in this canonical
basis and a generalized Ramanujan tau-function, and use this to prove that
these Fourier coefficients are often highly divisible by 2.Comment: Corrected typos. To appear in the Ramanujan Journa
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions
Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall ex-hibit congruences for Fourier coefficients of a slightly different type. Let f(z) =P∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coef-ficients. If ` is prime, then we shall be interested in congruences of the form a(`N) ≡ 0 mod ` where N is any quadratic residue (resp. non-residue) modulo `. For every prime `> 3 we exhibit a natural holomorphic weight ` 2 +1 modular form whose coefficients satisfy the congruence a(`N) ≡ 0 mod ` for every N satisfying `−
Protecting 30% of the planet for nature: costs, benefits, and economic implications:Working paper analysing the economic implications of the proposed 30% target for areal protection in the draft post-2020 Global Biodiversity Framework
Protecting 30% of the planet for nature: costs, benefits, and economic implications:Working paper analysing the economic implications of the proposed 30% target for areal protection in the draft post-2020 Global Biodiversity Framework
Short-Lived Trace Gases in the Surface Ocean and the Atmosphere
The two-way exchange of trace gases between the ocean and the atmosphere is important for both the chemistry and physics of the atmosphere and the biogeochemistry of the oceans, including the global cycling of elements. Here we review these exchanges and their importance for a range of gases whose lifetimes are generally short compared to the main greenhouse gases and which are, in most cases, more reactive than them. Gases considered include sulphur and related compounds, organohalogens, non-methane hydrocarbons, ozone, ammonia and related compounds, hydrogen and carbon monoxide. Finally, we stress the interactivity of the system, the importance of process understanding for modeling, the need for more extensive field measurements and their better seasonal coverage, the importance of inter-calibration exercises and finally the need to show the importance of air-sea exchanges for global cycling and how the field fits into the broader context of Earth System Science
The Radiation-Induced Chain Reaction between Nitrous Oxide and Hydrogen in Aqueous Solutions 1
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