An explicit formula for the Skorokhod map Γ0,a on [0, a] for a > 0 is provided and related to similar formulas in the literature. Specifically, it is shown that on the space D[0,∞) of right-continuous functions with left limits taking values in R, Γ0,a(ψ)(t) = ψ(t) − [( ψ(0) − a )+ ∧ inf u∈[0,t] ψ(u) ] ∨ sup s∈[0,t] [ (ψ(s) − a) ∧ inf u∈[s,t] ψ(u) ] . is the unique function taking values in [0,...

As a new margin-based classifier, ψ-learning shows great potential for high accuracy. However, the optimization of ψ-learning involves non-convex minimization and is very challenging to implement. In this article, we convert the optimization of ψ-learning into a mixed integer programming (MIP) problem. This enables us to utilize the state-of-art algorithm of MIP to solve ψ-learning. Moreover, t...

A random variable Y1 is said to be smaller than Y2 in the increasing concave stochastic order if E[ϕ(Y1)]≤E[ϕ(Y2)] for all functions ϕ which expected values exist, and convex E[ψ(Y1)]≤E[ψ(Y2)] ψ. This article develops nonparametric estimators conditional cumulative distribution Fx(y)=ℙ(Y≤y|X=x) of a response Y given covariate X, solely under assumption that distributions are x or order. Uniform...

Several inequalities are presented which, in part, generalize inequalities by Weinstein and Weiss, giving rise to new lower bounds for the Bayes risk under squared error loss. Consider a Bayesian setting in which y denotes the observed data and θ denotes an unknown scalar parameter. Suppose that E [ (θ − E(θ|y)) |y ] and E [ (θ − E(θ|y)) ] exist. For any estimator δ(y) of θ, E [ (θ − δ(y))|y ] ...

ing Meta-Variables • Propositional reflection of meta-variables ∆, u:A[Ψ]; Γ ⊢ M : B ∆;Γ ⊢ λU.M : [Ψ]A → B →I ∆;Γ ⊢ M : [Ψ]A → B ∆;Ψ ⊢ N : A ∆;Γ ⊢ M (Ψ. N) : B →E • New reduction (λU.M) (Ψ. N) → [(Ψ. N)/U ]M RDP’07, Paris, June 07 – p.34 Incomplete Proofs, Rerevisited • Now open leaves have form ∆;Γ ⊢ ? : A • Need meta-variables U • New meta-hypothesis rule U : A[Σ;Ψ] ∈ ∆ ∆; ∆; Γ ⊢ (σ;σ) : (Σ;Ψ...

A set theoretical assertion ψ is forceable or possible, written ♦ψ, if ψ holds in some forcing extension, and necessary, written ψ, if ψ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2.

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's states that for any non-increasing approximation function ψ:N→(0,1/2) with ∑qψ(q)=∞ and number γ, the following setW(ψ,γ)={x∈[0,1]:|qx−p−γ|<ψ(q) infinitely many q,p∈N} has full Lebesgue measure. Since then, there are very few results in relaxing monotonicity condition. this paper, we sho...

While theorists expect that the glueballs exist as the flavorless hadrons made of gluons alone, the predicted masses vary from lattice QCD [1] to phenomenological models such as the bag-type model [2] and a naive constituent model [3]. Experiment provided a few candidates for the ground-state glueball of J = 0 in the past, but no consensus has been reached. The glueball O of 1−− is made of thre...

We establish conditions on ψ under which the small-scale affine system {ψ(ajx− k) : j ≥ J, k ∈ Zd} spans the Lebesgue space L(R) and the Sobolev space W(R), for 1 ≤ p < ∞ and J ∈ Z. The dilation matrices aj are expanding (meaning limj→∞ ‖a−1 j ‖ = 0) but they need not be diagonal. For spanning L our result assumes ∫ Rd ψ dx 6= 0 and, when p > 1, that the periodization of |ψ| or of 1{ψ 6=0} is b...