The purpose of this paper is to prove various independence of ` results for correspondences acting on the étale cohomology of algebraic varieties in positive characteristic. The most basic result in this direction is the following: Theorem 1.1 (special case of 7.12). Let k be an algebraically closed field, and let X/k be a separated scheme of finite type. Let u : X → X be a quasi-finite morphis...

We describe a general method for modelling γ -ray burst (GRB) prompt emission, and determine the range of magnetic field strength, electron energy, Lorentz factor of the source and the distance of the source from the central explosion that is needed to account for the prompt γ -ray emission of a typical long-duration burst. We find that for the burst to be produced via the synchrotron process u...

Db(Coh(Aθ λ )) ∼ −→ D(Coh(Aλ)) for two different generic θ, θ′. These derived equivalences will play a crucial role in proving that ImCCv ⊂ Lω[ν]. They were introduced in [BPW, Section 6] under the name of twisting functors. Let us recall from Lecture 3 that abelian localization holds for (λ + nθ, θ) for all n sufficiently large, see Proposition 2.2 there. Also recall that Coh(Aλ) ∼ −→ Coh(Aλ+n...

This result is strikingly similar to the following hoped-for picture of affine Lie algebra representation theory, explained to us by A.Beilinson. Let M1,M2 be two modules over an affine Lie algebra ĝ on the critical level. One hopes that there is a localization functor which associates to these modules perverse sheaves ∆(M1),∆(M2) over the semiinfinite flag space Ĝ/B̂. Suppose that ∆(M1) and ∆(M...

Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of a lattice Γ with a rotated copy RΓ of itself. They are useful for classifying grain boundaries and have been studied extensively since the mid sixties. Recently the interests turned to so-called multiple CSLs, i.e. intersections of n rotated copies of a given lattice Γ, in particular in connection with lattice quantize...

To make this meaningful to machine learning, we need to replace Ef with some form of risk. Today will discuss three choices. 1. R` where ` is Lipschitz. We covered this last time but will recap a little. 2. Rz(f) := Pr[f(X) 6= Y ]; for this we’ll use finite classes and discuss shatter coefficients and VC dimension. 3. Rγ(f) = R`γ where `γ(z) := max{0,min{z/γ+ 1, 1}} will lead to nice bounds for...

Let [Formula: see text] be a Galois extension of text]-adic number fields and let de Rham representation the absolute group text]. In case text], equivariant local text]-constant conjecture describes compatibility Tamagawa with functional equation Artin text]-functions it can formulated as vanishing certain element in text]; similar approach followed also unramified twists (see [W. Bley A. Cobb...

In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. The heat flux ~ Q is given by the Fourier law T−ν ~ Q proportional to ∇T , where ν > 1 is the thermal...

We consider in details the effects of the 13-mixing (sin θ13) and of the type of mass hierarchy/ordering (sign[∆m13]) on neutrino signals from the gravitational collapses of stars. The observables (characteristics of the energy spectra of νe and ν̄e events) sensitive to sin θ13 and sign[∆m 2 13] have been calculated. They include the ratio of average energies of the spectra, rE ≡ 〈E〉/〈Ē〉, the ra...

The goal of this short paper is to give a slightly different perspective on the comparison between crystalline cohomology and de Rham cohomology. Most notably, we reprove Berthelot’s comparison result without using pd-stratifications, linearisations, and pd-differential operators. Crystalline cohomology is a p-adic cohomology theory for varieties in characteristic p created by Berthelot [Ber74]...