Text-overlap comparison
This pair was found by matching on title alone (find_title_bucket_dupes.py) — the usual paragraph-embedding similarity search never generated a candidate for it at any threshold. The exhibits below are near-exact word-shingle matches instead — a multi-word verbatim run is the more convincing evidence of shared text than embedding similarity, with the matched span highlighted.
PAPER A
PAPER B
EARLIER¶1-5
Qin Guo1* and Peixin Ye1 *Correspondence: guoqin_1985@163.com 1School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China Abstract We consider the moving least-square (MLS) method by the coefficient-based regression framework with lq-regularizer (1 ≤ q ≤ 2) and the sample dependent hypothesis spaces. The data dependent characteristic of the new algorithm provides flexibility and adaptivity for MLS. We carry out a rigorous error analysis by using the stepping stone technique in the error decomposition. The concentration technique with the l2-empirical covering number is also employed in our study to improve the sample error. We derive the satisfactory learning rate that can be arbitrarily close to the best rate O(m–1) under more natural and much simpler conditions. 1 Introduction The least-square (LS) method is an important global approximate method based on the regular or concentrated data sample points. However, there are still some irregular or scattered samples which are obtained in many practical applications such as engineering and machine learning [1–4]. They also need to be analyzed to achieve their special usefulness. For example, in geographical contour drawing, it is important to derive a set of contours but the height is available only for some scattered data sample points. Therefore, it is vital to seek a suitable local approximation method to deal with scattered data. The moving least-square (MLS) method was introduced by McLain in [4] to draw a set of contours based on a cluster of scattered data sample points. The central idea of the MLS method consists of two steps: first, one takes an arbitrary fixed point and forms a local approximation formula; second, since the fixed point is arbitrary, therefore, one can let it move over the whole domain. It turns out that MLS method is a useful local approximation tool in various mathematics fields such as approximation theory, data smoothing [5], statistics [6] and numerical analysis [7]. In computer graphics, the MLS method is useful for reconstructing a surface from a set of points. Often it is used to create a 3D surface from a point cloud. Recently, a research effort has been made to study the regression learning algorithm by the MLS method; see [8–12]. It has advantages over classical learning algorithms in the sense that its involved hypothesis space can be very simple such as the space of linear functions or a polynomial space. © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Guo and Ye Journal of Inequalities and Applications ( 2018) 2018:262 Page 2 of 15 We recall the regression learning problem by the MLS method briefly. Functions for learning are defined on a compact subset X (input space) of Rn and take values in Y = R (output space). The sampling process is controlled by a unknown Borel probability measure on Z = X × Y. The regression function is given by fρ(x) = � Y ydρ(y|x), where ρ(·|x) is the conditional probability measure induced by ρ on Y given x ∈ X. The goal of regression learning is to find a good approximation of the regression function fρ based on a set of random samples z = {zi}m i=1 = {(xi,yi)}m i=1 ∈ Zm drawn according to the measure ρ independently and identically. In [11], Tong and Wu considered the following regularized MLS regression algorithm. The hypothesis space is a reproducing kernel Hilbert space (RKHS) HK induced by a Mercer kernel K, which is a continuous, symmetric, and positive semi-definite function on X × X. The RKHS HK is the completion of the linear span of the set of functions {Kx := K(x,·) : x ∈ X} with respect to the inner product ⟨�n i=1 αiKxi,�m j=1 βjKyj⟩K := �n i=1 �m j=1 αiβjK(xi,yj). The reproducing property in HK is
LATER¶2-6
Abstract The application of the new criteria for minimally thin sets with respect to the Schrödinger operator to an approximate solution of singular Schrödinger-type boundary value problems are discussed in this study. The method is based on approximating functions and their derivatives by using the natural and weakened total energies. This study shows that the new criteria are very effective and powerful tools in solving such problems. At the end of the paper, we are also concerned with the boundary behaviors of solutions for a kind of quasilinear Schrödinger equation. izt = –�z + W(x)z – a(x)h � |z|2� z – k�l � |z|2� l′� |z|2� z, (1) where x ∈ Rn, z : R × Rn → C, a,W : Rn → R is a given potential, k is real constant, and l and h are real functions. The above quasilinear equations have been accepted as models of several physical phenomena corresponding to various types of l; we refer to [2] and the references given therein for physical applications of these problems. Specifically, we would like to mention that the superfluid film equation in plasma physics has this structure for l(s) = s (see e.g. [3, 4]), while in the case l(s) = (1 + s)1/2, (1) models the self-channeling of a high-power ultrashort laser in matter (see e.g. [5, 6]). The standing waves solutions of (1); that is, solutions of the type z(t,x) = exp(–iEt)u(x) where E ∈ R and u > 0 is a real function. Inserting z into (1), with l(s) = s and l(s) = (1 + s2)1/2, turns, respectively, the following equations (see e.g. [7]): –�u + V∞u – k� � u2� )u = a(x)h(u), –�u + V∞u – k� �� 1 + u2�1/2� u (1 + u2)1/2 = a(x)h(u), where x ∈ Rn and V∞ = W – E. © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Meng Boundary Value Problems ( 2019) 2019:91 Page 2 of 13 It is well known that an unknown Borel probability measure on W = S × T controls the sampling process, where T = R and S is a compact metric space in Rn. As in [8], the exact weak solutions of (1) can be defined by gρ(s) = � T ydρ(t|s), where ρ(·|s) is the conditional probability measure induced by ρ on T given s ∈ S. To our knowledge, the criteria for minimally thin sets with respect to the Schrödinger operator (1) was introduced for the first time in the context of the stationary Schrödinger equations in [9, 10]. In 2018, Jiang, Zhang and Li (see [11]) further improved this complex method and applied to study meromorphic solutions for the linear differential equations with analytic coefficients and obtain some applications. Recently, Zhang (see [12, 13]) defined a new type of minimal thinness with respect to the stationary Schrödinger operator, established new criteria for it and applied the result to study growth properties at infinity of the maximum modulus with respect to the Schrödinger operator. In this paper, we will continue to apply new criteria for solutions for a kind of quasilinear Schrödinger equations. Although we are motivated here by [9–13], there were substantial difficulties to adapt the above approach to the present situation. Let HE be the completion of the linear span of the set of functions {Es := E(s,·) : s ∈ S} equipped with (see [8, 14])
EARLIER¶3-7
© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Guo and Ye Journal of Inequalities and Applications ( 2018) 2018:262 Page 2 of 15 We recall the regression learning problem by the MLS method briefly. Functions for learning are defined on a compact subset X (input space) of Rn and take values in Y = R (output space). The sampling process is controlled by a unknown Borel probability measure on Z = X × Y. The regression function is given by fρ(x) = � Y ydρ(y|x), where ρ(·|x) is the conditional probability measure induced by ρ on Y given x ∈ X. The goal of regression learning is to find a good approximation of the regression function fρ based on a set of random samples z = {zi}m i=1 = {(xi,yi)}m i=1 ∈ Zm drawn according to the measure ρ independently and identically. In [11], Tong and Wu considered the following regularized MLS regression algorithm. The hypothesis space is a reproducing kernel Hilbert space (RKHS) HK induced by a Mercer kernel K, which is a continuous, symmetric, and positive semi-definite function on X × X. The RKHS HK is the completion of the linear span of the set of functions {Kx := K(x,·) : x ∈ X} with respect to the inner product ⟨�n i=1 αiKxi,�m j=1 βjKyj⟩K := �n i=1 �m j=1 αiβjK(xi,yj). The reproducing property in HK is f (x) = ⟨f ,Kx⟩K, for all f ∈ HK,x ∈ X. (1.1) Denote C(X) as the space of continuous functions on X with the norm ∥ · ∥∞. Since K is continuous in X, HK ⊆ C(X). Let κ := supt,x∈X |K(x,t)| < ∞. Then, by (1.1), we have ∥f ∥∞ ≤ κ∥f ∥K, ∀f ∈ HK, (1.2) We define the approximation fz,λ of fρ pointwise: fz,λ(x) = fz,σ,λ,x(x) = fz,σ,λ,x(u)|u=x, (1.3) fz,σ,λ,x := arg min f ∈HK
LATER¶4-8
–�u + V∞u – k� � u2� )u = a(x)h(u), –�u + V∞u – k� �� 1 + u2�1/2� u (1 + u2)1/2 = a(x)h(u), where x ∈ Rn and V∞ = W – E. © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Meng Boundary Value Problems ( 2019) 2019:91 Page 2 of 13 It is well known that an unknown Borel probability measure on W = S × T controls the sampling process, where T = R and S is a compact metric space in Rn. As in [8], the exact weak solutions of (1) can be defined by gρ(s) = � T ydρ(t|s), where ρ(·|s) is the conditional probability measure induced by ρ on T given s ∈ S. To our knowledge, the criteria for minimally thin sets with respect to the Schrödinger operator (1) was introduced for the first time in the context of the stationary Schrödinger equations in [9, 10]. In 2018, Jiang, Zhang and Li (see [11]) further improved this complex method and applied to study meromorphic solutions for the linear differential equations with analytic coefficients and obtain some applications. Recently, Zhang (see [12, 13]) defined a new type of minimal thinness with respect to the stationary Schrödinger operator, established new criteria for it and applied the result to study growth properties at infinity of the maximum modulus with respect to the Schrödinger operator. In this paper, we will continue to apply new criteria for solutions for a kind of quasilinear Schrödinger equations. Although we are motivated here by [9–13], there were substantial difficulties to adapt the above approach to the present situation. Let HE be the completion of the linear span of the set of functions {Es := E(s,·) : s ∈ S} equipped with (see [8, 14]) � n � i=1 ξiEsi, m � l=1 øjEtj � E := n � i=1 m � l=1 ξiøjE(si,tj). Let s ∈ S and g ∈ HE. Define (see [15, Remark 2.3]) g(s) = ⟨g, Es⟩E. (2) It follows from (2) that (see [16]) ∥g∥∞ ≤ κ∥g∥E, (3) where κ := sup t,s∈S ��E(s,t) �� < ∞. Define (see [17]) gw,χ(s) = gw,ζ,χ,s(s) = gw,ζ,χ,s(u)|u=s, gw,ζ,χ,s := arg min f ∈HE � 1 m m �
EARLIER¶11-15
fz,η(x) = fz,σ,η,x(x) = fz,σ,η,x(u)|u=x, (1.8) fz,σ,η,x = arg min f ∈HK,z � 1 m m � i=1 � � x σ , xi σ �� f (xi) – yi �2 + η�z(f ) � , η = η(m) > 0, (1.9) where HK,z = � f (x) = m � i=1 αiK(x,xi) : α = (α1,...,αm) ∈ Rm,m ∈ N � , �z(f ) = inf � m � i=1 |αi|q � , 1 ≤ q ≤ 2. The data dependence nature of the kernel-based hypothesis space provides flexibility for the learning algorithm such as choosing the lq-norm regularizer of a function expansion involving samples. Compared with the scheme (1.3)–(1.4) in a reproducing kernel Hilbert space, the first advantage of the algorithm (1.9) is the effectivity of computations without any optimization processes. Another advantage is that we can choose the suitable parameter q according to the research interest such as smoothness and sparsity. To study the approximation quality of fz,η, we derive the upper bound of the error ∥fz,η – fρ∥ρX with ∥f (·)∥ρX := ( � X |f (·)|2dρX) 1 2 and its convergence rates as m → ∞; see [8–11, 13, 14]. The remainder of this paper is organized as follows. In Sect. 2, we will provide the main result. The error decomposition analysis and the upper bounds of the hypothesis error, the approximation error and the sample error will be given in Sects. 3. In Sect. 4, we will prove the main result. Finally, Sect. 5 concludes the paper with future research lines. 2 Main result We firstly formulate some basic notations and assumptions. Let ρX be the marginal distribution of ρ on X and L2 ρX(X) be the Hilbert space of functions from X to Y square-integrable with respect to ρX with the norm denoted by ∥ · ∥ρX. The integral operator LK : L2 ρX(X) → L2 ρX(X) is defined by (LKf )(x) = � X K(x,t)f (t)dρX(t), x ∈ X. Since X is compact and K is continuous, LK is a compact operator. Its fractional power operator Lr K : L2 ρX(X) → L2 ρX(X),r > 0 is defined by
LATER¶9-13
i=1 Φ � s ζ , si ζ �� ti – g(si) �2 + χ∥g∥2 E � , (4) where Φ(s,t) ≤ 1, ∀s,t ∈ Rn, (5) Φ(s,t) ≥ cq, ∀|s – t| ≤ 1. (6) Scheme (4) yields (see [18, 19]) ζ �� g(si) – ti �2 + ς m � i=1 |ξi|q � , and HE,w = � g(s) = m � i=1 ξiE(s,si) : ξ = (ξ1,...,ξm) ∈ Rm,m ∈ N � . In order to study the boundary behaviors of gw,ς, we derive ∥gw,ς – gρ∥ρS with (see [20–23] for more details) ��g(·) �� ρS := �� S ��g(·) ��2 dρS � 1 2 . The remainder of this paper is organized as follows. In Sect. 2, we will provide the main results. In Sect. 3, some basic but important estimates and properties are summarized. The proofs of main results will be given in Sect. 4. Section 5 contains the conclusions of the paper. 2 Main results The integral operator LE : L2 ρS(S) → L2 ρS(S) is defined by (LEg)(s) = � S E(s,t)g(t)dρS(t). Let {μi} be the eigenvalues of LE and {ei} be the corresponding eigenfunctions. Then we define Lr E(g) = ∞ � i=1 μr i⟨g,ei⟩L2ρS ei for g ∈ L2 ρS(S). We assume that gρ satisfies L–r E gρ ∈ L2 ρS, where r is a positive constant depending on the size of the initial data in a suitable norm. Let cp (0 < p < 2) be a positive constant. Define (see [24])
EARLIER¶56-61
� mλI + QxK[x] � az = Qxy, Guo and Ye Journal of Inequalities and Applications ( 2018) 2018:262 Page 10 of 15 where I is the identity matrix, y = (y1,y2,y3,...,ym)T, K[x] = (K(xi,xj))m i,j=1 and Qx = diag(�( x σ , xi σ ) : i = 1,2,...,m). This implies λmaz = Qx � y – K[x]az� . Therefore, for i = 1,2,...,m, we get az i = 1 λm� � x σ , xi σ �� yi – m � j=1 az j K(xi,xj) � = 1 λm� � x σ , xi σ �� yi – fz,σ,λ,x(xi) � . By the Hölder inequality, we have m � i=1 ��az i ��q = 1 (λm)q m � i=1 ����� � x σ , xi σ �� yi – fz,σ,λ,x(xi) ����� q ≤ 1 (λm)q � m � i=1 � � x σ , xi σ � q 2–q �1– q 2 × � m � i=1 � x σ , xi σ �� yi – fz,σ,λ,x(xi) �2 � q 2 . By (1.5), we have m � i=1 ��az i ��q ≤ m (λm)q �Ez,x(fz,σ,λ,x) � q 2 . Thus, Ez,x � πM(fz,σ,η,x) � + η�z(fz,σ,η,x) ≤ Ez,x(fz,σ,η,x) + η�z(fz,σ,η,x) ≤ Ez,x(fz,σ,λ,x) + η�z(fz,σ,λ,x) ≤ Ez,x(fz,σ,λ,x) + mη (λm)q �Ez,x(fz,σ,λ,x) � q 2 ≤ Ez,x(fz,σ,λ,x) + λ∥fz,σ,λ,x∥2 K + mη (λm)q �Ez,x(fz,σ,λ,x) + λ∥fz,σ,λ,x∥2 K � q 2 . Since Ez,x(fz,σ,λ,x) + λ∥fz,σ,λ,x∥2 K ≤ Ez,x(0) + λ∥0∥2 K,
LATER¶37-42
� S �Eg(gχ;t)dt ≤ C δ(√δ0 – δ) � γM(gρ;0) + �Eg(gχ;0) � + C (√δ0 – δ)2 � S γM(gρ;t)dt, which proves the inequality (20). Meng Boundary Value Problems ( 2019) 2019:91 Page 8 of 13 δ0 ,1 �� S γM(gρ;t)dt. □ 4 Proofs of main results Now we derive the learning rates. Proof of Theorem 1 Let y = (t1,t2,t3,...,tm)τ, K[s] = (E(si,sj))m i,j=1 and aw = (aw 1 ,...,aw m) be the coefficient of gw,ς. It follows from the representation theorem (see [27, 28]) that aw i = 1 χmΨ � s ζ , si ζ �� ti – gw,ζ,χ,s(si) � for i = 1,2,...,m. By the Hölder inequality, we have m � i=1 ��aw i ��q = 1 (χm)q m � i=1 ����Ψ � s ζ , si ζ �� ti – gw,ζ,χ,s(si) ����� q ≤ 1 (χm)q � m � i=1 Ψ � s ζ , si ζ � 1 2–q �1– q 2 × � m � i=1 � s ζ , si ζ �� ti – gw,ζ,χ,s(si) �2 � q 2 . It follows that m � i=1 ��aw i ��q ≤ m (χm)q � Ew,s(gw,ζ,χ,s) � q 2 from (5). Thus Ew,s � γM(gw,ζ,ς,s) � + ςΩw(gw,ζ,ς,s) ≤ Ew,s(gw,ζ,ς,s) + ςΩw(gw,ζ,ς,s) ≤ Ew,s(gw,ζ,χ,s) + ςΩw(gw,ζ,χ,s) ≤ Ew,s(gw,ζ,χ,s) + mς (χm)q � Ew,s(gw,ζ,χ,s) � q 2 ≤ Ew,s(gw,ζ,χ,s) + χ∥gw,ζ,χ,s∥2 E + mς (χm)q � Ew,s(gw,ζ,χ,s) + χ∥gw,ζ,χ,s∥2 E � q 2 . Since Ew,s(gw,ζ,χ,s) + χ∥gw,ζ,χ,s∥2 E ≤ Ew,s(0) + χ∥0∥2 E,
EARLIER¶75-79
X � � x σ , u σ � dρX(x). Hence, E � g2� ≤ 16M2 � X �� X � � x σ , u σ �� πM(f )(u) – fρ(u) �2 dρX(u) � dρX(x). It has been proved in [8] that � X � � x ��� πM(f )(u) – y �2 – � fρ(u) – y �2� dρ(u,y) � dρX(x) = 16M2 � Z �� X � � x σ , u σ ��� πM(f )(u) – y �2 – � fρ(u) – y �2� dρX(x) � dρ(u,y) = 16M2E(g). � BR, ε 4M � = N2 � B1, ε 4MR � . Thus from the capacity condition (2.1), we have logN2(GR,ε) ≤ cp(4M)pRpε–p. Now we can apply Proposition C.1 to G with Q = 8M2, α = 1, τ = 16M2 and a = cp(4M)pRp. Thus for any 0 < δ < 1, with confidence 1 – δ/2, we have Eg – 1 m m � i=1 g(zi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p , where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = fz,σ,η,x and derive the following bound of fz,σ,η,x by using the same method as in Lemma 3 of [18] and (1.5): ∥fz,η∥K ≤ κm1– 1 q �M2 η � 1 q . Finally, we complete the proof of Proposition 3.6 by taking R = Rη = κm1– 1 q ( M2 η ) 1 q . □ Funding This work was supported by the Natural Science Foundation of China (grant nos. 11271199, 11671213).
LATER¶50-54
ζ �� γM(g)(u) – gρ(u) �2 dρS(s) � S Ψ � s ζ , u ζ � dρS(s). So E � h2� ≤ 16M2 � S �� S Ψ � s ζ , u ζ �� γM(g)(u) – gρ(u) �2 dρS(u) � dρS(s). It has been proved in [13, 29] that ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρ(u,t) � dρS(s) = 16M2 � Z �� S Ψ � s ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρS(s) � dρ(u,t) = 16M2E(h). N2(GR,ε) ≤ N2 � BR, ε 4M � = N2 � B1, ε 4MR � . It follows from the capacity condition (7) that logN2(GR,ε) ≤ cp(4M)pRpε–p. By applying Lemma 2 to G with Q = 8M2 we have Eg – 1 m m � i=1 h(wi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p for any 0 < δ < 1, where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = gw,ζ,ς,s and derive the following bound of gw,ζ,ς,s by using the same method in [9, Lemma 3] and (5): ∥gw,ς∥E ≤ κm1– 1 q �M2 ς � 1 q . If we take R = Rς = κm1– 1 q �M2 ς � 1 q , then we can complete the proof of Theorem 3. □ 5 Conclusion The application of the new criteria for minimally thin sets with respect to the Schrödinger operator to an approximate solution of singular Schrödinger-type boundary value problems were discussed in this study. The method was based on approximating functions and
EARLIER¶75-80
X � � x σ , u σ � dρX(x). Hence, E � g2� ≤ 16M2 � X �� X � � x σ , u σ �� πM(f )(u) – fρ(u) �2 dρX(u) � dρX(x). It has been proved in [8] that � X � � x ��� πM(f )(u) – y �2 – � fρ(u) – y �2� dρ(u,y) � dρX(x) = 16M2 � Z �� X � � x σ , u σ ��� πM(f )(u) – y �2 – � fρ(u) – y �2� dρX(x) � dρ(u,y) = 16M2E(g). � BR, ε 4M � = N2 � B1, ε 4MR � . Thus from the capacity condition (2.1), we have logN2(GR,ε) ≤ cp(4M)pRpε–p. Now we can apply Proposition C.1 to G with Q = 8M2, α = 1, τ = 16M2 and a = cp(4M)pRp. Thus for any 0 < δ < 1, with confidence 1 – δ/2, we have Eg – 1 m m � i=1 g(zi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p , where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = fz,σ,η,x and derive the following bound of fz,σ,η,x by using the same method as in Lemma 3 of [18] and (1.5): ∥fz,η∥K ≤ κm1– 1 q �M2 η � 1 q . Finally, we complete the proof of Proposition 3.6 by taking R = Rη = κm1– 1 q ( M2 η ) 1 q . □ Funding This work was supported by the Natural Science Foundation of China (grant nos. 11271199, 11671213). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
LATER¶50-55
ζ �� γM(g)(u) – gρ(u) �2 dρS(s) � S Ψ � s ζ , u ζ � dρS(s). So E � h2� ≤ 16M2 � S �� S Ψ � s ζ , u ζ �� γM(g)(u) – gρ(u) �2 dρS(u) � dρS(s). It has been proved in [13, 29] that ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρ(u,t) � dρS(s) = 16M2 � Z �� S Ψ � s ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρS(s) � dρ(u,t) = 16M2E(h). N2(GR,ε) ≤ N2 � BR, ε 4M � = N2 � B1, ε 4MR � . It follows from the capacity condition (7) that logN2(GR,ε) ≤ cp(4M)pRpε–p. By applying Lemma 2 to G with Q = 8M2 we have Eg – 1 m m � i=1 h(wi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p for any 0 < δ < 1, where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = gw,ζ,ς,s and derive the following bound of gw,ζ,ς,s by using the same method in [9, Lemma 3] and (5): ∥gw,ς∥E ≤ κm1– 1 q �M2 ς � 1 q . If we take R = Rς = κm1– 1 q �M2 ς � 1 q , then we can complete the proof of Theorem 3. □ 5 Conclusion The application of the new criteria for minimally thin sets with respect to the Schrödinger operator to an approximate solution of singular Schrödinger-type boundary value problems were discussed in this study. The method was based on approximating functions and Meng Boundary Value Problems ( 2019) 2019:91 Page 12 of 13 their derivatives by using the natural and weakened total energies. This study showed that the new criteria were very effective and powerful tools in solving such problems. At the end of the paper, we were also concerned with the boundary behaviors of solutions for a kind of quasilinear Schrödinger equation.
EARLIER¶38-42
Next the estimation for S1(z,η) is more difficult in the sense that it involves the complexity of the function space HK,z. Hence we need the uniform concentration inequality from [22]. Proposition 3.6 Under the assumptions of Theorem 2.1, for any 0 < δ < 1, with confidence 1 – δ/2, S1(z,η) ≤ 1 2 � X �Ex � πM(fz,σ,η,x) � – Ex(fρ) � dρX(x) + 176M2 m log �2 δ � + Cp,MR 2p 2+p η m– 2 2+p , (3.8) where Rη = κm1– 1 q ( M2 η ) 1 q . 4 Proof of the main result Now we derive the learning rates. Guo and Ye Journal of Inequalities and Applications ( 2018) 2018:262 Page 7 of 15 Proof of Theorem 2.1 Combining the four bounds of Proposition 3.3, 3.4, 3.5 and 3.6 with (3.5), with confidence 1 – δ, we have � X �Ex � πM(fz,σ,η,x) � – Ex(fρ) � dρX(x) ≤ D1 log �2 δ �� λmin{2r,1} + m–1λmin{2r–1,0} + m1–qηλ–q + m –2q–2p+2pq (2+p)q η– 2p q(2+p) � . (4.1) By substituting (4.1) into (3.1), we have
LATER¶51-55
ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρ(u,t) � dρS(s) = 16M2 � Z �� S Ψ � s ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρS(s) � dρ(u,t) = 16M2E(h). N2(GR,ε) ≤ N2 � BR, ε 4M � = N2 � B1, ε 4MR � . It follows from the capacity condition (7) that logN2(GR,ε) ≤ cp(4M)pRpε–p. By applying Lemma 2 to G with Q = 8M2 we have Eg – 1 m m � i=1 h(wi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p for any 0 < δ < 1, where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = gw,ζ,ς,s and derive the following bound of gw,ζ,ς,s by using the same method in [9, Lemma 3] and (5): ∥gw,ς∥E ≤ κm1– 1 q �M2 ς � 1 q . If we take R = Rς = κm1– 1 q �M2 ς � 1 q , then we can complete the proof of Theorem 3. □ 5 Conclusion The application of the new criteria for minimally thin sets with respect to the Schrödinger operator to an approximate solution of singular Schrödinger-type boundary value problems were discussed in this study. The method was based on approximating functions and Meng Boundary Value Problems ( 2019) 2019:91 Page 12 of 13 their derivatives by using the natural and weakened total energies. This study showed that the new criteria were very effective and powerful tools in solving such problems. At the end of the paper, we were also concerned with the boundary behaviors of solutions for a kind of quasilinear Schrödinger equation.
EARLIER¶76-80
��� πM(f )(u) – y �2 – � fρ(u) – y �2� dρ(u,y) � dρX(x) = 16M2 � Z �� X � � x σ , u σ ��� πM(f )(u) – y �2 – � fρ(u) – y �2� dρX(x) � dρ(u,y) = 16M2E(g). � BR, ε 4M � = N2 � B1, ε 4MR � . Thus from the capacity condition (2.1), we have logN2(GR,ε) ≤ cp(4M)pRpε–p. Now we can apply Proposition C.1 to G with Q = 8M2, α = 1, τ = 16M2 and a = cp(4M)pRp. Thus for any 0 < δ < 1, with confidence 1 – δ/2, we have Eg – 1 m m � i=1 g(zi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p , where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = fz,σ,η,x and derive the following bound of fz,σ,η,x by using the same method as in Lemma 3 of [18] and (1.5): ∥fz,η∥K ≤ κm1– 1 q �M2 η � 1 q . Finally, we complete the proof of Proposition 3.6 by taking R = Rη = κm1– 1 q ( M2 η ) 1 q . □ Funding This work was supported by the Natural Science Foundation of China (grant nos. 11271199, 11671213). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
LATER¶51-55
ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρ(u,t) � dρS(s) = 16M2 � Z �� S Ψ � s ζ , u ζ ��� γM(g)(u) – t �2 – � gρ(u) – t �2� dρS(s) � dρ(u,t) = 16M2E(h). N2(GR,ε) ≤ N2 � BR, ε 4M � = N2 � B1, ε 4MR � . It follows from the capacity condition (7) that logN2(GR,ε) ≤ cp(4M)pRpε–p. By applying Lemma 2 to G with Q = 8M2 we have Eg – 1 m m � i=1 h(wi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p for any 0 < δ < 1, where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = gw,ζ,ς,s and derive the following bound of gw,ζ,ς,s by using the same method in [9, Lemma 3] and (5): ∥gw,ς∥E ≤ κm1– 1 q �M2 ς � 1 q . If we take R = Rς = κm1– 1 q �M2 ς � 1 q , then we can complete the proof of Theorem 3. □ 5 Conclusion The application of the new criteria for minimally thin sets with respect to the Schrödinger operator to an approximate solution of singular Schrödinger-type boundary value problems were discussed in this study. The method was based on approximating functions and Meng Boundary Value Problems ( 2019) 2019:91 Page 12 of 13 their derivatives by using the natural and weakened total energies. This study showed that the new criteria were very effective and powerful tools in solving such problems. At the end of the paper, we were also concerned with the boundary behaviors of solutions for a kind of quasilinear Schrödinger equation.
EARLIER¶78-81
Eg – 1 m m � i=1 g(zi) ≤ Eg 2 + 176M2 m log �2 δ � + Cp,MR 2p 2+p m– 2 2+p , where Cp,M = c′ p(4M) 4 2+p c 2 2+p p . Moreover, we take f = fz,σ,η,x and derive the following bound of fz,σ,η,x by using the same method as in Lemma 3 of [18] and (1.5): ∥fz,η∥K ≤ κm1– 1 q �M2 η � 1 q . Finally, we complete the proof of Proposition 3.6 by taking R = Rη = κm1– 1 q ( M2 η ) 1 q . □ Funding This work was supported by the Natural Science Foundation of China (grant nos. 11271199, 11671213). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 20 June 2018 Accepted: 18 September 2018
LATER¶56-58
Acknowledgements Not applicable. Funding This work was supported by the Post-Doctoral Applied Research Projects of Qingdao (no. 2015122) and the Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (no. 2014RCJJ032). Abbreviations Not applicable. Availability of data and materials Not applicable. Ethics approval and consent to participate Not applicable. Competing interests The author declares that he has no competing interests. Consent for publication Not applicable. Authors’ contributions The author read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.