New error bounds for linear complementarity problems of Nekrasov matrices and <Emphasis Type="Italic">B</Emphasis>-Nekrasov matrices

New error bounds for the linear complementarity problems are given respectively when the involved matrices are Nekrasov matrices and B-Nekrasov matrices. Numerical examples are given to show that the new bounds are better respectively than those provided by García-Esnaola and Peña (Numer. Algor. 67(3), 655–667, 2014 and Numer. Algor. 72(2), 435–445, 2016) in some cases.     

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.     

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 \(n\ge 3\).     

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.     

The Relaxed Edge-Coloring Game and <Emphasis Type="Italic">k</Emphasis>-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ?j and d≥2j+2 for 0≤j≤Δ?1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k?1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k?j and d≥h(k,j).     

Best <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces

In this article, we study the problem of best L ₁ approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem (Proc. Am. Math. Soc., 16, 665–670, 1965) into an appropriate framework and prove the unicity of best L ₁ approximation of Heaviside-type functions from an even-dimensional Chebyshev space under some assumptions on the dimension of the subspaces composed of the odd and even functions. We also apply the results to compute best L ₁ approximations of Heaviside-type functions by polynomials and Hermite polynomial splines with fixed knots.     

Generalized growth and best approximation of entire functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-norm in several complex variables

The paper deals with the characterization of generalized order and generalized type of entire functions in several complex variables in terms of the coefficients of the development with respect to the sequence of extremal polynomials and the best L ^p -approximation and interpolation errors, 0 < p ≤ ∞, on a compact set K with respect to the set

$K_r = \left\{z \in \mathbb{C}^n, {\rm exp} (V_K (z)) \leq r\right\}$

where V _K is the Siciak extremal function of a L-regular compact set K or V _K is the pluricomplex Green function with a pole at infinity. It has been noticed that in the study of growth of entire functions, the set K _r has not been used so extensively in comparison to disk. Our results apply satisfactorily for slow growth in \({\mathbb{C}^n}\) , replacing the circle \({\{z \in \mathbb{C}; |z| = r\}}\) by the set K _r and improve and extend various results of Harfaoui (Int J Maths Math Sci 2010:1–15, 2010), Seremeta (Am Math Soc Transl 88(2):291–301, 1970), Shah (J Approx Theory 19:315–324, 1977) and Vakarchuk and Zhir (Ukr Math J 54(9):1393–1401, 2002).

     

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.     

A note on diagonal dominance,Schur complements and some classes of <Emphasis Type="Italic">H</Emphasis>-matrices and <Emphasis Type="Italic">P</Emphasis>-matrices

In this note, an error on the proof of an important Theorem in a recent article by Peña (Adv. Comput. Math. 35, 357–373, 2011) is pointed out by a numerical counter example. Then a new proof for the corresponding result is presented.     

On <Emphasis Type="Italic">m</Emphasis>-ovoids of regular near polygons

We generalise the work of Segre (Ann Mat Pura Appl 4(70):1–201, 1965), Cameron et al. (J Algebra 55(2):257–280, 1978), and Vanhove (J Algebr Comb 34(3):357–373, 2011) by showing that nontrivial m-ovoids of the dual polar spaces \(\mathsf {DQ}(2d, q)\), \(\mathsf {DW}(2d-1,q)\) and \(\mathsf {DH}(2d-1,q^2)\) (\(d\geqslant 3\)) are hemisystems. We also provide a more general result that holds for regular near polygons.     

A new characterization for some extensions of PSL(2, <Emphasis Type="Italic">q)</Emphasis> for some <Emphasis Type="Italic">q</Emphasis> by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p²) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p²) of order 2|PSL(2,p²)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p³. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.     

Eigenvalue estimates for submanifolds with bounded <Emphasis Type="BoldItalic">f</Emphasis>-mean curvature

In this paper, we obtain an extrinsic low bound to the first non-zero eigenvalue of the f-Laplacian on complete noncompact submanifolds of the weighted Riemannian manifold (H ^m (?1),e^?f dv) with respect to the f-mean curvature. In particular, our results generalize those of Cheung and Leung in Math. Z. 236 (2001) 525–530.     

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.     

Generalizing the ENO-DB2<Emphasis Type="Italic">p</Emphasis> transform using the inverse wavelet transform

The essentially non-oscillatory (ENO)-wavelet transform developed by Chan and Zhou (SIAM J. Numer. Anal. 40(4), 1369–1404, 2002) is based on a combination of the Daubechies-2p wavelet transform and the ENO technique. It uses extrapolation methods to compute the scaling coefficients without differencing function values across jumps and obtains a multiresolution framework (essentially) free of edge artifacts. In this work, we present a different way to compute the ENO-DB2p wavelet transform of Chan and Zhou which allows us to simplify the process and easily generalize it to other families of orthonormal wavelets.     

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 .     

Additive properties of sequences of pseudo <Emphasis Type="Italic">s</Emphasis>-th powers

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erd?s and Rényi (Acta Arith 6:83–110, 1960). Goguel (J Reine Angew Math 278/279:63–77, 1975) proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order \(s+\epsilon \) for any \(\epsilon >0\). We then study the s-fold sumset \(sA=A+\cdots +A\) (s times) and in particular the minimal size of an additive complement, that is a set B such that \(sA+B\) contains all large enough integers. With respect to this problem, we prove quite precise theorems which are tantamount to asserting that a threshold phenomenon occurs.     

<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.     

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).     

On the homotopy type of (<Emphasis Type="Italic">n</Emphasis>-1)-connected (3<Emphasis Type="Italic">n</Emphasis>+1)-dimensional free chain Lie algebra

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ?Q, then p=∞. Denote by DGL _n ^np , n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL _n ^np . In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGL _n ³ⁿ⁺² and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGL _n ³ⁿ⁺² . How many homotopy equivalence classes of objects (L(W), δ) in DGL _n ³ⁿ⁺² such that H _* (W, d′)?H _* (V, d) are there? Note that DGL _n ³ⁿ⁺² is a subcategory of DGL _n ^np when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.