Where L1 γ and L2 γ are respective automorphisms of IRn and IRp, L3 : IRn → IRp is a linear map, and l1 γ and l2 γ are respective elements of IRn and IRp .
Non abelian cohomology: the point of view of gerbed tower
Let TF be the translation group of (F, ∇F ), that is the group of aﬃne automorphisms of (F, ∇F ) whose elements lift to translations of IRp .
Non abelian cohomology: the point of view of gerbed tower
The ob jects of CF (N ) are classiﬁed by H 1 (N, p′ F ), the 1-cohomology group of the sheaf of aﬃne F ) is deﬁned by an aﬃne C3 : IRn → IRp which is a 1-cocycle sections of p′ F .
Non abelian cohomology: the point of view of gerbed tower
We thus obtain L[k=1 [n∈Np :n≤N irp (k ; n) ∩ brp (L − k ; n, N ), where the union is of disjoint sets.
Coincidence of Lyapunov exponents for random walks in weak random potentials
IRn such that IE[Y ] = X β and cov(Y ) = σ2 In where X ∈ Mn,p(IR) is a known matrix (rows of X are the explanatory variables) and where β ∈ IRp and σ2 ∈ IR+ are the unknown parameters (to be estimated).
A new graphical tool of outliers detection in regression models based on recursive estimation
For δ > 0 and a ∈ IRp we denote B(a, δ) = {x ∈ IRp | d(x, a) ≤ δ}.
Componentwise condition numbers of random sparse matrices
The answer depends on how massive the set A(r) = {γ : kγ k◦ ≤ r} is in terms of the standard Gaussian measure on IRp .
Sharp deviation bounds for quadratic forms
Recall that the quadratic norm kεk2 of a standard Gaussian vector ε in IRp concentrates around p at least for p large.
Sharp deviation bounds for quadratic forms
Let ε be a standard normal vector in IRp and u ∈ IRp .
Sharp deviation bounds for quadratic forms
IRp and IPξ means the conditional probability given ξ .
Sharp deviation bounds for quadratic forms
Let ε be a standard normal vector in IRp and u ∈ IRp .
Sharp deviation bounds for quadratic forms
We are interested in testing H0 : m ∈ MΘ,G = {γ (·, θ, g) : θ ∈ Θ, g ∈ G } for some known function γ, some compact set Θ ⊂ IRp and some function set G of real valued functions.
Empirical likelihood based testing for regression
IRp and some function set G of real valued functions.
Empirical likelihood based testing for regression
I is an indicator matrix and entry Irp is 1 if it was observed and 0 otherwise.
A Framework for Optimizing Paper Matching
For each pair of IRP 3 ’s, one can slide one through the other, with the square of this slide being trivial (since π1 (IRP 3 ) = Z2 ), making a total of N (N − 1) independent order 2 generators.
Geon Statistics and UIR's of the Mapping Class Group
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