Let H1 and H2 be Hilbert spaces and let B(H1,H2) be the space of bounded linear operators A : H1 → H2. An operator A ∈ B(H1,H2) is said to be Fredholm if first, the kernel ofA is finite-dimensional, and second the image ofA is closed and has finite codimension. An application of the open mapping theorem shows that the closedness requirement on the image is redundant. A well-known example of Fre...

Let (Gi | i ∈ I) be a family of groups, let F be a free group, and let G = F ∗ ∗ i∈I Gi, the free product of F and all the Gi. Let F denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, H∩Ggi = {1}. By the Kurosh Subgroup Theorem, every element of F is a free group. For each free group H, the reduced rank of H, denoted r̄(H), is d...

Let F/Q be a number field closed under complex conjugation. Denote by L0(F (t)) the Witt group of hermitian forms over F (t). We find full invariants for detecting non–zero elements in L0(F (t))⊗Q, this group plays an important role in topology in the work done by Casson and Gordon. 1. L-groups and signatures Let R be a ring with (possibly trivial) involution. An –hermitian ( = ±1) form is a se...

Let X1,…,Xn be i.i.d. random variables sampled from a normal distribution N(μ,Σ) in Rd with unknown parameter θ=(μ,Σ)∈Θ:=Rd×C+d, where C+d is the cone of positively definite covariance operators Rd. Given smooth functional f:Θ↦R1, goal to estimate f(θ) based on X1,…,Xn. Θ(a;d):=Rd×{Σ∈C+d:σ(Σ)⊂[1/a,a]},a≥1, σ(Σ) spectrum Σ. θˆ:=(μˆ,Σˆ), μˆ sample mean and Σˆ covariance, observations For an arbit...

Let H1 be an n × n invertible Hermitian matrix, and let U(H1) be the group of n × n H1-unitary matrices, i.e., matrices A satisfying A H1A = H1. Suppose H2 is an m × m invertible Hermitian matrix. We show that a linear transformation φ : Mn → Mm satisfies φ(U(H1)) ⊆ U(H2) if and only if there exist invertible matrices S ∈ Mm, U, V ∈ U(H2) such that SH2S = [(Ia ⊕−Ib)⊗H1]⊕ [(Ic ⊕−Id)⊗ (H−1 1 )], ...

Simplicity is analyzed topologically. Ockham’s razor is shown, under very general conditions, to be the unique, convergent strategy that jointly minimizes errors, retractions, and retraction times prior to convergence to the true theory. 1 Empirical Questions and Solutions 1.1 Input Streams Let I be a set of potential inputs. An input stream is an element of Iω. If w is an input stream let w|i ...

Given a family of graphs F , a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G, some element of F is a subgraph of G + e. Let sat(n,F) denote the minimum number of edges in an F-saturated graph of order n, which we refer to as the saturation number or saturation function of F . If F = {F}, then we instead say that G is F -saturated and write sat(n, F ). For...

With an optimal design one wants to determine the ideal allocation of observations for the estimation of the unknown parameter vector θ of a given model ( ) ( ) θ ; i i x f y E = , with [ ] p T θ θ θ L 1 = , ( ) n T x x X ,..., 1 = , the xi from a given experimental region ER.. Let ( ) X V ; θ be the asymptotical variance-covariance matrix of the least squares estimatorθ̂ of θ . We consider the ...

Let Gn : C → C denote the PGD updates at the n-sample level—that is, as applied to the objective function Ln(θ) = 1 n ∑n i=1 `i(θ;Zi). Similarly, let G denote the projected gradient operator as applied at the population level (to the function L̄ = ELn(θ)). Our goal is to bound the error ‖θ − θ‖2. Last lecture, we showed that if the population loss L̄ : C → R is (γ`, γμ) {strongly convex, smooth},...

On the p-adic Leopoldt transform of a power series BrunoAngì es 1 Let p be an odd prime number. Let X be the projective limit for the norm maps of the p-Sylow subgroups of the ideal class groups of Q(ζ p n+1), n ≥ 0. Let ∆ = Gal(Q(ζ p)/Q) and let θ be an even and non-trivial character of ∆. Then X is a Z p [[T ]]-module and the characteristic ideal of the isotypic component X(ωθ −1) is generate...