(ii) Denote this map by φ, which is induced by the compositions with εi : Xi ↪→ ⊔ j Xj , i ∈ I. If φ(fi) = φ(gj) for some fi ∈ Hom(Y,Xi) and gj ∈ Hom(Y,Xj), then we must have i = j because φ(fi) and φ(gj) have the same image. Since φ(fi) = εi ◦ fi, φ(gi) = εi ◦ gi and εi is an injective map, we deduce that fi(y) = gi(y), ∀y ∈ Y , i.e. fi = gi. (iii) This map is induced by the composition with t...

We discuss semiparametric regression when only the ranks of responses are observed. The model is Yi = F (x ′ iβ0 + εi), where Yi is the unobserved response, F is a monotone increasing function, xi is a known p−vector of covariates, β0 is an unknown p-vector of interest, and εi is an error term independent of xi. We observe {(xi, Rn(Yi)) : i = 1, . . . , n}, where Rn is the ordinal rank function...

*Correspondence: [email protected] 1School of Information and Statistics, Guangxi University of Finance and Economics, Nanning, 530003, P.R. China Full list of author information is available at the end of the article Abstract In this paper, considering the nonparametric regression model Yni = g(ti) + εi (1≤ i≤ n), where εi =∑∞j=–∞ ajei–j and ei–j are identically distributed and ρ-mixing s...

In case of an anisotropic superconductor non-magnetic disorder leads eventually to pair breaking effect via Abrikosov–Gorkov formula [1, 2, 3, 4, 5], like for magnetic impurities, for any value of a coherence length ξ. By analogy to an isotropic s–wave pairing, for superconductors with an enough large coherence length ξ, the amplitude of paring potential |∆(ij)| tends be the same for all bonds ...

Let F : X → X be a C 2 loc map in a Banach space X, and A be its Frèchet derivative at the element w := w ε , which solves the problem (*) ˙ w = −A −1 ε (F (w) + εw), w(0) = w 0 , where A ε := A + εI. Assume that A −1 ε ≤ cε −k , 0 < k ≤ 1, 0 < ε > ε 0. Then (*) has a unique global solution, w(t), there exists w(∞), and (* *) F (w(∞)) + εw(∞) = 0. Thus the DSM (Dynamical Systems Method) is just...

• y(u) aims at reproducing some real phenomenon which we denote by y 1.2 Validation of computer models •Question of interest: Does the computer model adequately represent reality? •Obtain field data yi that results from observing reality in a physical experiment yFi = y +εi , εi ∼ Np(0,Σ F ), ,Σ = Λ R Λ, Λ = diag(σ i ) •Combine model and field data with a statistical model y = y(u) + b (Kennedy...

* Correspondence: lym1019@163. com Department of Mathematics, Shangrao Normal University, Shangrao 334001, China Full list of author information is available at the end of the article Abstract Consider the semiparametric regression model Yi = xi b + g (ti) + εi, i = 1, . . . , n, where the linear process errors εi = ∑∞ j=−∞ ajei−j with ∑∞ j=−∞ ∣∣aj∣∣ < ∞ , and {ei} are identically distributed a...

and Applied Analysis 3 If any solution x of 1.1 is either oscillatory, or satisfies the condition 1.7 , or admits the asymptotic representation x i c 1 sin t − α i εi t , i 0, 1, 2, 3 , 1.8 where c / 0 and α are constants, the continuous functions εi i 0, 1, 2, 3 vanish at infinity and ε0 satisfies the inequality cε0 t > 0 for large t, then we say that 1.1 has weak property A. For n 3, the resu...

Suppose we observe data of the form Yi = Di(Si + εi) ∈ R or Yi = DiSi + εi ∈ R, i = 1, . . . , n, where Di ∈ Rp×p are known diagonal matrices, εi are noise, and we wish to perform principal component analysis (PCA) on the unobserved signals Si ∈ R. The first model arises in missing data problems, where the Di are binary. The second model captures noisy deconvolution problems, where the Di are t...

In the common time series model Xi,n=μ(i∕n)+εi,n with non-stationary errors we consider problem of detecting a significant deviation mean function μ from benchmark g(μ) (such as initial value μ(0) or average trend ∫01μ(t)dt). The is motivated by more realistic modelling change point analysis, where one interested in identifying relevant deviations smoothly varying sequence means (μ(i∕n))i=1,…,n...