If we do gradient descent with η * (t), then almost surely, the algorithm converges (for the quadratic model). To prove that, we follow classical techniques based on Lyapunov stability theory (Bucy, 1965). Notice that the expected loss follows E J θ (t+1) | θ (t) = 1 2 h · E (1 − η * h)(θ (t) − θ *) + η * hσξ 2 + σ 2 = 1 2 h (1 − η * h) 2 (θ (t) − θ *) 2 + (η *) 2 h 2 σ 2 + σ 2 = 1 2 h σ 2 (θ (...

The symplectic group Sp(4, F ) has a particular interesting unitary representation for F a finite or local field. When F is finite this representation was denoted Θ10 [30], and somehow the name has stuck. We discuss this representation in some detail in the case of R. This material is mostly well known, at least to the experts, and this paper is intended as a reference for non–specialists. It a...

We consider the directed first passage percolation model on Z2. In this model, we assign independently to each edge e a passage time t(e) with a common distribution F . We denote by ~ T (0, (r, θ)) the passage time from the origin to (r, θ) by a northeast path for (r, θ) ∈ R+ × [0, π/2]. It is known that ~ T (0, (r, θ))/r converges to a time constant ~μF (θ). Let ~ pc denote the critical probab...

In Harding (1996; see also Harding, 1998, 1999), a method was given to construct an orthomodular poset Fact X from the direct product decompositions of a set. As this forms the basis of our study we briefly review the pertinent facts. For a set X let Eq X denote the set of equivalence relations on X, use ◦ for relational product, 1 for the least equivalence relation on X, and ∇ for the largest....

In this paper we study a class of infinite words on a finite alphabet A whose factors are closed under the image of an involutory antimorphism θ of the free monoid A∗. We show that given a recurrent infinite word ω ∈ AN, if there exists a positive integer K such that for each n ≥ 1 the word ω has 1) cardA + (n − 1)K distinct factors of length n, and 2) a unique right and a unique left special f...

Taking into account the terms ∼ m 2 in the effective chiral lagrangian , we show that, at θ = π, the theory with 2 light quarks of equal mass involves two degenerate vacuum states separated by a barrier. For N f = 3, the energy barrier between two vacua appears already in the leading order in mass. This corresponds to the first order phase transition at θ = π. The surface energy density of the ...

Let M be a separable compact Hausdorff space with dim M ≤ 2 and θ : M → M be a homeomorphism with prime period p (p ≥ 2). Set Mθ = {x ∈ M | θ(x) = x} 6= ∅ and M0 = M\Mθ. Suppose that M0 is dense in M and H(M0/θ,Z) ∼= 0, H(χ(M0/θ),Z) ∼= 0. Let M ′ be another separable compact Hausdorff space with dim M ′ ≤ 2 and θ′ be the self–homeomorphism of M ′ with prime period p. Suppose that M ′ 0 = M ′\M ...

We present improved upper bounds on the spanning ratio of a large family of θ-graphs. A θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m. We show that for any integer k ≥ 1, θ-graphs with 4k + 4 cones have spanning ratio at most 1 + 2 sin(θ/2)/(cos(θ/2) − sin(θ/2)). We also show that θ-graphs with 4k + 3 and 4k + 5 cones have spanning ratio at...