We calculate the correction to the electronic density of states in a disordered ferromagnetic metal induced by spin-wave mediated interaction between the electrons. Our calculation is valid for the case that the exchange splitting in the ferromagnet is much smaller than the Fermi energy, but we make no assumption on the relative magnitude of and the elastic electronic scattering time τ el . In the 'clean limit' τ el /h 1 we find a correction with a T d/2 temperature dependence, where d is the effective dimensionality of the ferromagnet. In the 'dirty limit' τ el /h 1, the density-of-states correction is a non-monotonous function of energy and temperature.

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We calculate the correction to the conductivity of a disordered ferromagnetic metal due to spinwave-mediated electron-electron interactions. This correction is the generalization of the AltshulerAronov correction to spin-wave-mediated interactions. We derive a general expression for the conductivity correction to lowest order in the spin-wave-mediated interaction and for the limit that the exchange splitting ∆ is much smaller than the Fermi energy. For a "clean" ferromagnet with ∆τ el /h 1, with τ el the mean time for impurity scattering, we find a correction δσ ∝ −T 5/2 at temperatures T above the spin wave gap. In the opposite, "dirty" limit, ∆τ el /h 1, the correction is a non-monotonous function of temperature.

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We propose DIPS (Difficulty-based Incentives for Problem Solving), a simple modification of the Bitcoin proof-of-work algorithm that rewards blockchain miners for solving optimization problems of scientific interest. The result is a blockchain which redirects some of the computational resources invested in hash-based mining towards scientific computation, effectively reducing the amount of energy ‘wasted’ on mining. DIPS builds the solving incentive directly in the proof-of-work by providing a reduction in block hashing difficulty when optimization improvements are found. A key advantage of this scheme is that decentralization is not greatly compromised while maintaining a simple blockchain design. We study two incentivization schemes and provide simulation results showing that DIPS is able to reduce the amount of hash-power used in the network while generating solutions to optimization problems.