Moderate deviation principle for <Emphasis Type="Italic">m</Emphasis>-dependent random variables<Superscript>*</Superscript>

Let {X _n }_n?≥?1 be a sequence of strictly stationary m-dependent random variableswith EX₁ = 0 and \( \mathrm{E}{X}_1^2<\infty \), and let (b _n ) be an increasing sequence of positive numbers such that b _n ?↑?∞ and \( {b}_n/\sqrt{n}\downarrow 0\kern0.5em \mathrm{as}\kern0.5em n\to \infty \). We establish a moderate deviation principle of \( {\left({b}_n\sqrt{n}\right)}^{-1}{\sum}_{i=1}^n{X}_i \) under the condition

$$ \underset{n\to \infty }{\lim \sup}\frac{1}{b_n^2}\log \left[n\mathbf{P}\left(\left|{X}_1\right|>{b}_n\sqrt{n}\right)\right]=-\infty, $$

which is weaker than the classical exponential integrability condition. The results in the present paper weaken the assumptions of Chen [5] and extend partially the results of Eichelsbacher and Löwe [10].     

Finite time blowup of the <Emphasis Type="Italic">n</Emphasis>-harmonic flow on <Emphasis Type="Italic">n</Emphasis>-manifolds

In this paper, we generalize the no-neck result of Qing and Tian (in Commun Pure Appl Math 50:295–310, 1997) to show that there is no neck during blowing up for the n-harmonic flow as \(t\rightarrow \infty \). As an application of the no-neck result, we settle a conjecture of Hungerbühler (in Ann Scuola Norm Sup Pisa Cl Sci 4:593–631, 1997) by constructing an example to show that the n-harmonic map flow on an n-dimensional Riemannian manifold blows up in finite time for \(n\ge 3\).     

Small Representations for Affine <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

When the parameter \(q\in \mathbb {C}^{*}\) is not a root of unity, simple modules of affine q-Schur algebras have been classified in terms of Frenkel–Mukhin’s dominant Drinfeld polynomials (Deng et al. 2012). We compute these Drinfeld polynomials associated with the simple modules of an affine q-Schur algebra which come from the simple modules of the corresponding q-Schur algebra via the evaluation maps.     

Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

Congruences of two-variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions of congruent modular forms of different weights

Vatsal (Duke Math J 98(2):397–419, 1999) proved that there are congruences between the p-adic L-functions (constructed by Mazur and Swinnerton-Dyer in Invent Math 25:1–61, 1974) of congruent modular forms of the same weight under some conditions. On the other hand, Kim (J Number Theory 144: 188–218, 2014), the second author, constructed two-variable p-adic L-functions of modular forms attached to imaginary quadratic fields generalizing Hida’s work (Invent Math 79:159–195, 1985), and the novelty of his construction was that it works whether p is an ordinary prime or not. In this paper, we prove congruences between the two-variable p-adic L-functions (of the second author) of congruent modular forms of different but congruent weights under some conditions when p is a nonordinary prime for the modular forms. This result generalizes the work of Emerton et al. (Invent Math 163(3): 523–580, 2006), who proved similar congruences between the p-adic L-functions of congruent modular forms of congruent weights when p is an ordinary prime.     

Constructing New Braided <Emphasis Type="Italic">T</Emphasis>-Categories over Weak Hopf Algebras

Let Aut _{weak Hopf} (H) denote the set of all automorphisms of a weak Hopf algebra H with bijective antipode in the sense of Böhm et al. (J Algebra 221:385–438, 1999) and let G be a certain crossed product group Aut _{weak Hopf} (H)×Aut _{weak Hopf} (H). The main purpose of this paper is to provide further examples of braided T-categories in the sense of Turaev (1994, 2008). For this, we first introduce a class of new categories \( _{H}{\mathcal {WYD}}^{H}(\alpha, \beta)\) of weak (α, β)-Yetter-Drinfeld modules with α, β?∈?Aut _{weak Hopf} (H) and we show that the category \({\mathcal WYD}(H) =\{{}_{H}\mathcal {WYD}^{H}(\alpha, \beta)\}_{(\alpha , \beta )\in G}\) becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic (Isr J Math 158:349–365, 2007). Finally, when H is finite-dimensional we construct a quasitriangular weak T-coalgebra WD(H)?=?{WD(H)_{(α, β)}}_{(α, β)?∈?G} in the sense of Van Daele and Wang (Comm Algebra, 2008) over a family of weak smash product algebras \(\{\overline{H^{*cop}\# H_{(\alpha,\beta)}}\}_{(\alpha , \beta)\in G}\), and we obtain that \({\mathcal {WYD}}(H)\) is isomorphic to the representation category of the quasitriangular weak T-coalgebra WD(H).     

Ordinal indices of Small Subspaces of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We calculate the ordinal L _p index defined in [3] for Rosenthal’s space X _p , \({\ell_p}\) and \({\ell_2}\). We show that an infinite-dimensional subspace of L _p \({(2 < p < \infty)}\) non-isomorphic to \({\ell_2}\) embeds in \({\ell_p}\) if and only if its ordinal index is the minimal possible. We also give a sufficient condition for a \({\mathcal{L}_p}\) subspace of \({\ell_p \oplus \ell_2}\) to be isomorphic to X _p .     

The <Emphasis Type="Italic">p</Emphasis>-Affine Capacity Redux

Continuing from Xiao (Adv Math 268:906–914, 2015; J Geom Anal 26:947–966, 2016), this note is devoted to the discovery of new geometric properties of the so-called \([1,n)\ni p\)-affine capacity in the Euclidean n-space.     

<Emphasis Type="Italic">um</Emphasis>-Topology in multi-normed vector lattices

Let \({\mathcal {M}}=\{m_\lambda \}_{\lambda \in \Lambda }\) be a separating family of lattice seminorms on a vector lattice X, then \((X,{\mathcal {M}})\) is called a multi-normed vector lattice (or MNVL). We write \(x_\alpha \xrightarrow {\mathrm {m}} x\) if \(m_\lambda (x_\alpha -x)\rightarrow 0\) for all \(\lambda \in \Lambda \). A net \(x_\alpha \) in an MNVL \(X=(X,{\mathcal {M}})\) is said to be unbounded m-convergent (or um-convergent) to x if \(|x_\alpha -x |\wedge u \xrightarrow {\mathrm {m}} 0\) for all \(u\in X_+\). um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963–974, 2017; Kandi? et al. in J Math Anal Appl 451:259–279, 2017) and uaw-convergence (Zabeti in Positivity, 2017. doi: 10.1007/s11117-017-0524-7), and specializes up-convergence (Ayd?n et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and \(u\tau \)-convergence (Dabboorasad et al. in \(u\tau \)-Convergence in locally solid vector lattices. arXiv:1706.02006v3). um-Convergence is always topological, whose corresponding topology is called unbounded m-topology (or um-topology). We show that, for an m-complete metrizable MNVL \((X,{\mathcal {M}})\), the um-topology is metrizable iff X has a countable topological orthogonal system. In terms of um-completeness, we present a characterization of MNVLs possessing both Lebesgue’s and Levi’s properties. Then, we characterize MNVLs possessing simultaneously the \(\sigma \)-Lebesgue and \(\sigma \)-Levi properties in terms of sequential um-completeness. Finally, we prove that every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue’s and Levi’s properties.     

Sharp <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Bound for Holomorphic Functions on the Unit Disc

For any 1 < p < ∞ and any \({X, Y\in \mathbb{R}}\) satisfying \({|X|\leq Y}\) , we determine the optimal constant C _p (X,Y) such that the following holds. If F is a holomorphic function on the unit disc satisfying ReF(0) = X and \({||{\rm Re}F||_{L^{p}(\mathbb{T})}=Y}\) , then

$$||F||_{L^p(\mathbb{T})}\geq C_p(X,Y).$$

This can be regarded as a reverse version of the classical estimates of Riesz and Essén. The proof rests on the exploitation of certain families of special subharmonic functions on the plane.

     

A High-Order <Emphasis Type="Italic">B</Emphasis>-Spline Collocation Method for Solving Nonlinear Singular Boundary Value Problems Arising in Engineering and Applied Science

In this paper, a high-order B-spline collocation method on a uniform mesh is presented for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions:

$$\begin{aligned} (p(x)y')'= & {} p(x)f(x,y), \quad 0<x\le 1, \\ y'(0)= & {} 0,\quad ay(1)+by'(1)=c, \end{aligned}$$

where \(p(x)=x^{\alpha }g(x),\alpha \ge 0\) is a general class of non-negative function. The error analysis for the quartic B-spline interpolation is discussed. To demonstrate the applicability and efficiency of our method we consider eight numerical examples, seven of which arise in various branches of applied science and engineering: (1) equilibrium of isothermal gas sphere; (2) thermal explosion; (3) thermal distribution in the human head; (4) oxygen diffusion in a spherical cell; (5) stress distribution on shallow membrane cap; (6) reaction diffusion process in a spherical permeable catalyst; (7) heat and mass transfer in a spherical catalyst. It is shown that our method has fourth-order convergence and is more accurate than finite difference methods (Chawla et al., in BIT 28:88–97, 1988; Pandey et al. in J Comput Appl Math 224:734–742, 2009) and B-spline collocation methods (Abukhaled et al. in Int J Numer Anal Model 8:353–363, 2011; Khuri and Sayfy in Int J Comput Methods 11(1):1350052, 2014).

     

Weak Type (1,1) Behavior for the Maximal Operator with <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dini Kernel

Let M _Ω be the maximal operator with homogeneous kernel Ω. In the present paper, we show that if Ω satisfies the L ¹-Dini condition on ?? ^n?1, then the following weak type (1,1) behaviors

$$\lim\limits _{\lambda \rightarrow 0_{+}}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})=\frac {1}{n} \|\Omega \|_{1} \|f\|_{1},$$

$$\sup\limits_{\lambda >0}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})\lesssim {\bigg ((\log n)\|\Omega \|_{1}+{\int }_{0}^{1/n}\frac {\tilde {\omega }_{1}(\delta )}{\delta }d\delta \bigg )}\|f\|_{1}$$

hold for the maximal operator M _Ω and \(f\in L^{1}(\mathbb {R}^{n})\), here \(\tilde {\omega }_{1}\) denotes the L ¹ integral modulus of continuity of Ω defined by translation in \(\mathbb {R}^{n}\).     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

Probably certifiably correct <Emphasis Type="Italic">k</Emphasis>-means clustering

Recently, Bandeira (C R Math, 2015) introduced a new type of algorithm (the so-called probably certifiably correct algorithm) that combines fast solvers with the optimality certificates provided by convex relaxations. In this paper, we devise such an algorithm for the problem of k-means clustering. First, we prove that Peng and Wei’s semidefinite relaxation of k-means Peng and Wei (SIAM J Optim 18(1):186–205, 2007) is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next, we show how to test the optimality of a proposed k-means solution using this dual certificate in quasilinear time. Finally, we analyze a version of spectral clustering from Peng and Wei (SIAM J Optim 18(1):186–205, 2007) that is designed to solve k-means in the case of two clusters. In particular, we show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model.     

The localization sequence for the algebraic <Emphasis Type="Italic">K</Emphasis>-theory of topological <Emphasis Type="Italic">K</Emphasis>-theory

We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra \(K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})\) for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the K-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective \(A_\infty\) ring spectrum R with the Quillen K-theory of the abelian category of finitely generated \(\pi_{0}R\)-modules.     

Two operator representations for the trivariate <Emphasis Type="Italic">q</Emphasis>-polynomials and Hahn polynomials

In this paper, we introduce a trivariate q-polynomials \(F_n(x,y,z;q)\) as a general form of Hahn polynomials \(\psi _n^{(a)}(x|q)\) and \(\psi _n^{(a)}(x,y|q)\). We represent \(F_n(x,y,z;q)\) by two operators: the homogeneous q-shift operator \(L(b\theta _{xy})\) given by Saad and Sukhi (Appl Math Comput 215:4332–4339, 2010), and the Cauchy companion operator \(E(a,b;\theta )\) given by Chen (q-Difference Operator and Basic Hypergeometric Series, 2009) to derive the generating function, symmetric property, Mehler’s formula, Rogers formula, another Roger-type formula, linearization formula, and an extended Rogers formula for the trivariate q-polynomials. Then, we give the corresponding formulas for our new definitions of Hahn polynomials \(\psi _n^{(a)}(x|q)\) and \(\psi _n^{(a)}(x,y|q)\) by representing Hahn polynomials by the operators \(L(b\theta _{xy})\) and \(E(a,b;\theta )\), and by a special substitution in the trivariate q-polynomials \(F_n(x,y,z;q)\).     

Symmetries on the Lattice of <Emphasis Type="Italic">k</Emphasis>-Bounded Partitions

In 2002, Suter [25] identified a dihedral symmetry on certain order ideals in Young’s lattice and gave a combinatorial action on the partitions in these order ideals. Viewing this result geometrically, the order ideals can be seen to be in bijection with the alcoves in a 2- fold dilation in the geometric realization of the affine symmetric group. By considering the m-fold dilation we observe a larger set of order ideals in the k-bounded partition lattice that was considered by Lapointe, Lascoux, and Morse [14] in the study of k-Schur functions. We identify the order ideal and the cyclic action on it explicitly in a geometric and combinatorial form.     

<Emphasis Type="Italic">l</Emphasis><Subscript>1</Subscript>-<Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript> regularization of split feasibility problems

Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable, possibly composed with linear operators and that in turn can be transformed to split feasibility problems (SFP); see for example Censor and Elfving (Numer. Algorithms 8, 221–239 1994). Each function is typically either a data fidelity term or a regularization term enforcing some properties on the solution; see for example Chaux et al. (SIAM J. Imag. Sci. 2, 730–762 2009) and references therein. In this paper, we are interested in split feasibility problems which can be seen as a general form of Q-Lasso introduced in Alghamdi et al. (2013) that extended the well-known Lasso of Tibshirani (J. R. Stat. Soc. Ser. B 58, 267–288 1996). Q is a closed convex subset of a Euclidean m-space, for some integer m ≥ 1, that can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the Lasso. Inspired by recent works by Lou and Yan (2016), Xu (IEEE Trans. Neural Netw. Learn. Syst. 23, 1013–1027 2012), we are interested in a nonconvex regularization of SFP and propose three split algorithms for solving this general case. The first one is based on the DC (difference of convex) algorithm (DCA) introduced by Pham Dinh Tao, the second one is nothing else than the celebrate forward-backward algorithm, and the third one uses a method introduced by Mine and Fukushima. It is worth mentioning that the SFP model a number of applied problems arising from signal/image processing and specially optimization problems for intensity-modulated radiation therapy (IMRT) treatment planning; see for example Censor et al. (Phys. Med. Biol. 51, 2353–2365, 2006).     

Classification and nondegeneracy of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>+1) Toda system with singular sources

We consider the following Toda system

Open image in new window

where γ _i >?1, δ ₀ is Dirac measure at 0, and the coefficients a _ij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:

$\sum_{j=1}^n a_{ij}\int_{\mathbb{R}^2}e^{u_j} dx = 4\pi(2+\gamma _i+\gamma_{n+1-i}), \quad\forall\;1\leq i \leq n.$

This generalizes the classification result by Jost and Wang for γ _i =0, \(\forall\;1\leq i\leq n\). (ii) We prove that if γ _i +γ _i+1+?+γ _j ?? for all 1≤i≤j≤n, then any solution u _i is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.