On the Rationality of the Spectrum

Let \(\Omega \subset {\mathbb R}\) be a compact set with measure 1. If there exists a subset \(\Lambda \subset {\mathbb R}\) such that the set of exponential functions \(E_{\Lambda }:=\{e_\lambda (x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda \}\) is an orthonormal basis for \(L^2(\Omega )\), then \(\Lambda \) is called a spectrum for the set \(\Omega \). A set \(\Omega \) is said to tile \({\mathbb R}\) if there exists a set \(\mathcal T\) such that \(\Omega + \mathcal T = {\mathbb R}\), the set \(\mathcal T\) is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum.     

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an \(\alpha \)-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case \(\alpha \le 1\). In the functional setting, we complement the existing results on \(M_1\)-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the \(S\) topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).     

Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

An algorithmic upper bound on the domination number \(\gamma \) of graphs in terms of the order n and the minimum degree \(\delta \) is proved. It is demonstrated that the bound improves best previous bounds for any \(5\le \delta \le 50\). In particular, for \(\delta =5\), Xing et al. (Graphs Comb. 22:127–143, 2006) proved that \(\gamma \le 5n/14 < 0.3572 n\). This bound is improved to 0.3440 n. For \(\delta =6\), Clark et al. (Congr. Numer. 132:99–123, 1998) established \(\gamma <0.3377 n\), while Biró et al. (Bull. Inst. Comb. Appl. 64:73–83, 2012) recently improved it to \(\gamma <0.3340 n\). Here the bound is further improved to \(\gamma < 0.3159n\). For \(\delta =7\), the best earlier bound 0.3088n is improved to \(\gamma < 0.2927n\).     

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.     

New heuristics for the vertex coloring problem based on semidefinite programming

The Lovász theta number Lovász (IEEE Trans Inf Theory 25:1–7, 1979) is a well-known lower bound on the chromatic number of a graph \(G\), and \(\varPsi _K(G)\) is its impressive strengthening Gvozdenovi? and Laurent (SIAM J Optim 19(2):592–615, 2008). The bound \(\varPsi _K(G)\) was introduced in very specific and abstract setting which is tough to translate into usual mathematical programming framework. In the first part of this paper we unify the motivations and approaches to both bounds and rewrite them in a very similar settings which are easy to understand and straightforward to implement. In the second part of the paper we provide explanations how to solve efficiently the resulting semidefinite programs and how to use optimal solutions to get good coloring heuristics. We propose two vertex coloring heuristics based on \(\varPsi _K(G)\) and present numerical results on medium sized graphs.     

Absolute Cesàro Series Spaces and Matrix Operators

In this paper we derive a series space \(\vert C_{\lambda,\mu} \vert _{k}\) using the well known absolute Cesàro summability \(\vert C_{\lambda,\mu} \vert _{k}\) of Das (Proc. Camb. Philol. Soc. 67:321–326, 1970), compute its \(\beta\)-dual, give some algebraic and topological properties, and characterize some matrix operators defined on that space. So we generalize some results of Bosanquet (J. Lond. Math. Soc. 20:39–48, 1945), Flett (Proc. Lond. Math. Soc. 7:113–141, 1957), Mehdi (Proc. Lond. Math. Soc. (3)10:180–199, 1960), Mazhar (Tohoku Math. J. 23:433–451, 1971), Orhan and Sar?göl (Rocky Mt. J. Math. 23(3):1091–1097, 1993) and Sar?göl (Commun. Math. Appl. 7(1):11–22, 2016; Math. Comput. Model. 55:1763–1769, 2012).     

Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $$\mathbb {C}^2$$

Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).

     

On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonic Weyl curvature, i.e., \(\delta W=0\). Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product \( \mathbb {R}^2 \times N_{\lambda }\) of the Euclidean metric and a 2-d Riemannian manifold of constant curvature \({\lambda } \ne 0\), a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao–Chen’s works (in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013) and Derdziński’s study on Codazzi tensors (in Math Z 172:273–280, 1980). Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with \(\delta W=0\). For the shrinking case, it re-proves the rigidity result (Fernández-López and García-Río in Math Z 269:461–466, 2011; Munteanu and Sesum in J. Geom Anal 23:539–561, 2013) in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally flat ones with \(\delta W=0\). We also characterize locally 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.     

On a question of Gromov about the Wirtinger inequalities

We prove that Gromov’s \(\mathrm {Cycl}_4(0)\) condition implies \(\mathrm {Wir}_k\) inequalities for any \(k \ge 4\), which answers a question of Gromov (J Math Sci N Y 119(2):178–200, 2004).     

Phase transition in loop percolation

We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters \(\alpha \) and \(\kappa \). The non-negative parameter \(\alpha \) measures the amount of loops, and \(\kappa \) plays the role of killing on vertices penalizing (\(\kappa >0\)) or favoring (\(\kappa <0\)) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed \(\kappa \) and large enough \(\alpha \), there exists an infinite cluster in the loop percolation on \({\mathbb {Z}}^d\). In the present article, we show a non-trivial phase transition on the integer lattice \({\mathbb {Z}}^d\) (\(d\ge 3\)) for \(\kappa =0\). More precisely, we show that there is no loop percolation for \(\kappa =0\) and \(\alpha \) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of \(\alpha \), namely, for \(\kappa =0\) and any sub-critical value of \(\alpha \), the probability of one-arm event decays at most polynomially. For \(d\ge 5\), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For \(\alpha \) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For \(d=3\) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity \(\alpha \).     

Dirichlet series with functional equations and arithmetical identities

Exploiting the functional equation of Hecke-type associated with a function satisfying a modular relation with a residual function as developed in Bochner (J Indian Math Soc 16:99–102, 1952), Chandrasekharan and Narasimhan (Ann Math 74:1–23, 1961) derived the equivalence of the functional equation to two arithmetical identities. Hawkins and Knopp (Contemp Math 143:451–475, 1993) showed the equivalence of the functional equation to modular integrals with rational period functions of weight 2k, \(k \in \mathbb {Z}^+\) on the theta group \(\Gamma _\vartheta \). The aim of the current work is to show that results analogous to those of Chandrasekharan and Narasimhan can be developed in the Hawkins and Knopp context, but with respect to the full modular group \(\Gamma (1)\), rather than the theta group \(\Gamma _\vartheta \).     

A Theory of Super-Resolution from Short-Time Fourier Transform Measurements

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, \(\Delta \), between spikes is not too small. Specifically, for a measurement cutoff frequency of \(f_c\), Donoho (SIAM J Math Anal 23(5):1303–1331, 1992) showed that exact recovery is possible if the spikes (on \(\mathbb {R}\)) lie on a lattice and \(\Delta > 1/f_c\), but does not specify a corresponding recovery method. Candès and Fernandez-Granda (Commun Pure Appl Math 67(6):906–956, 2014; Inform Inference 5(3):251–303, 2016) provide a convex programming method for the recovery of periodic spike trains (i.e., spike trains on the torus \(\mathbb {T}\)), which succeeds provably if \(\Delta > 2/f_c\) and \(f_c \ge 128\) or if \(\Delta > 1.26/f_c\) and \(f_c \ge 10^3\), and does not need the spikes within the fundamental period to lie on a lattice. In this paper, we develop a theory of super-resolution from short-time Fourier transform (STFT) measurements. Specifically, we present a recovery method similar in spirit to the one in Candès and Fernandez-Granda (2014) for pure Fourier measurements. For a STFT Gaussian window function of width \(\sigma = 1/(4f_c)\) this method succeeds provably if \(\Delta > 1/f_c\), without restrictions on \(f_c\). Our theory is based on a measure-theoretic formulation of the recovery problem, which leads to considerable generality in the sense of the results being grid-free and applying to spike trains on both \(\mathbb {R}\) and \(\mathbb {T}\). The case of spike trains on \(\mathbb {R}\) comes with significant technical challenges. For recovery of spike trains on \(\mathbb {T}\) we prove that the correct solution can be approximated—in weak-* topology—by solving a sequence of finite-dimensional convex programming problems.     

Adaptive inexact fast augmented Lagrangian methods for constrained convex optimization

In this paper we study two inexact fast augmented Lagrangian algorithms for solving linearly constrained convex optimization problems. Our methods rely on a combination of the excessive-gap-like smoothing technique introduced in Nesterov (SIAM J Optim 16(1):235–249, 2005) and the general inexact oracle framework studied in Devolder (Math Program 146:37–75, 2014). We develop and analyze two augmented based algorithmic instances with constant and adaptive smoothness parameters, and derive a total computational complexity estimate in terms of projections on a simple primal feasible set for each algorithm. For the constant parameter algorithm we obtain the overall computational complexity of order \(\mathcal {O}(\frac{1}{\epsilon ^{5/4}})\), while for the adaptive one we obtain \(\mathcal {O}(\frac{1}{\epsilon })\) total number of projections onto the primal feasible set in order to achieve an \(\epsilon \)-optimal solution for the original problem.     

Weakly horospherically convex hypersurfaces in hyperbolic space

In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces \(\phi :M^n \rightarrow \mathbb {H}^{n+1}\) and a class of conformal metrics on domains of the round sphere \(\mathbb {S}^n\). Some of the key aspects of the correspondence and its consequences have dimensional restrictions \(n\ge 3\) due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of \(\mathbb {S}^n\). In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions \(n\ge 2\) in a unified way. In the case of a single point boundary \(\partial _{\infty }\phi (M)=\{x\} \subset \mathbb {S}^n\), we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in \(\mathbb {H}^{3}\).     

Locally piecewise affine functions and their order structure

Piecewise affine functions on subsets of \(\mathbb R^m\) were studied in Aliprantis et al. (Macroecon Dyn 10(1):77–99, 2006), Aliprantis et al. (J Econometrics 136(2):431–456, 2007), Aliprantis and Tourky (Cones and duality, 2007), Ovchinnikov (Beitr\(\ddot{\mathrm{a}}\)ge Algebra Geom 43:297–302, 2002). In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in \(C(\mathbb R^m)\), while piecewise affine functions are sequentially order dense in \(C(\mathbb R^m)\). This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014)     

Global Solutions to a Nonlocal Fisher-KPP Type Problem

We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term \(u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) \), where \(\varOmega\) is a bounded domain in \(\mathbb{R}^{n}(n \ge1)\). The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of \(\alpha\), \(\beta\). More precisely, for \(1 \le\alpha <1+ ( 1-2/p ) \beta\), where \(p\) is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of \(n \ge3\) and \(\beta=1\), \(\alpha<1+2/n\) is the known Fujita exponent (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966). Comparing to Fujita equation (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966), this paper will give an opposite result to our nonlocal problem.     

Taut contact circles and bi-contact metric structures on three-manifolds

Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), \({\widetilde{SL}}(2,{\mathbb {R}})\), \({\widetilde{E}}(2)\), E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure \((\eta _1,\eta _2)\) with the so-called Webster function \({\mathcal {W}}\) constant along the flow of \(\xi _1\) (equivalently \(\xi _2\)). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).     

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Gradient Ricci solitons and metrics with half harmonic Weyl curvature are two natural generalizations of Einstein metrics on four-manifolds. In this paper we prove that if a metric has structures of both gradient shrinking Ricci soliton and half harmonic Weyl curvature, then except for three examples, it has to be an Einstein metric with positive scalar curvature. Precisely, we prove that a four-dimensional gradient shrinking Ricci soliton with \(\delta W^{\pm }=0\) is either Einstein, or a finite quotient of \(S^3\times \mathbb {R}\), \(S^2\times \mathbb {R}^2\) or \(\mathbb {R}^4\). We also prove that a four-dimensional gradient Ricci soliton with constant scalar curvature is either Kähler–Einstein, or a finite quotient of \(M\times \mathbb {C}\), where M is a Riemann surface. The method of our proof is to construct a weighted subharmonic function using curvature decompositions and the Weitzenböck formula for half Weyl curvature, and the method was motivated by previous work (Gursky and LeBrun in Ann Glob Anal Geom 17:315–328, 1999; Wu in Einstein four-manifolds of three-nonnegative curvature operator 2013; Trans Am Math Soc 369:1079–1096, 2017; Yang in Invent Math 142:435–450, 2000) on the rigidity of Einstein four-manifolds with positive sectional curvature, and previous work (Cao and Chen in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013; Catino in Math Ann 35:629–635, 2013) on the rigidity of gradient Ricci solitons.     

On a class of maximality principles

We study various classes of maximality principles, \(\mathrm {MP}(\kappa ,\Gamma )\), introduced by Hamkins (J Symb Log 68(2):527–550, 2003), where \(\Gamma \) defines a class of forcing posets and \(\kappa \) is an infinite cardinal. We explore the consistency strength and the relationship of \(\textsf {MP}(\kappa ,\Gamma )\) with various forcing axioms when \(\kappa \in \{\omega ,\omega _1\}\). In particular, we give a characterization of bounded forcing axioms for a class of forcings \(\Gamma \) in terms of maximality principles MP\((\omega _1,\Gamma )\) for \(\Sigma _1\) formulas. A significant part of the paper is devoted to studying the principle MP\((\kappa ,\Gamma )\) where \(\kappa \in \{\omega ,\omega _1\}\) and \(\Gamma \) defines the class of stationary set preserving forcings. We show that MP\((\kappa ,\Gamma )\) has high consistency strength; on the other hand, if \(\Gamma \) defines the class of proper forcings or semi-proper forcings, then by Hamkins (2003), MP\((\kappa ,\Gamma )\) is consistent relative to \(V=L\).     

Rulesets for Beatty games

We describe a ruleset for a 2-pile subtraction game with P-positions \(\{(\lfloor \alpha n \rfloor ,\lfloor \beta n \rfloor ) : n \in \mathbb Z_{\ge 0} \}\) for any irrational \(1< \alpha < 2\), and \(\beta \) such that \(1/\alpha +1/\beta = 1\). We determine the \(\alpha \)’s for which the game can be represented as a finite modification of t-Wythoff (Holladay, Math Mag 41:7–13, 1968; Fraenkel, Am Math Mon 89(6):353–361, 1982) and describe this modification.