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.     

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

Let \({\alpha}\) be a bounded linear operator in a Banach space \({\mathbb{X}}\), and let A be a closed operator in this space. Suppose that for \({\Phi_1, \Phi_2}\) mapping D(A) to another Banach space \({\mathbb{Y}}\), \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) are generators of strongly continuous semigroups in \({\mathbb{X}}\). Assume finally that \({A_{|{\rm ker}\, \Phi_\text{a}}}\), where \({\Phi_\text{a} = \Phi_1 \alpha + \Phi_2 \beta}\) and \({\beta = I_\mathbb{X} - \alpha}\), is a generator also. In the case where \({\mathbb{X}}\) is an L ¹-type space, and \({\alpha}\) is an operator of multiplication by a function \({0 \le \alpha \le 1}\), it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of \({A_{|{\rm ker}\, \Phi_1}}\) with probability \({\alpha (x)}\) and is subject to the rules of \({A_{|{\rm ker}\, \Phi_2}}\) with probability \({\beta (x)= 1 - \alpha (x)}\). We provide an approximation (a singular perturbation) of the semigroup generated by \({A_{|{\rm ker}\, \Phi_\text{a}}}\) by semigroups built from those generated by \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM J Appl Math 60(2):408–436, 1999–2000; Banasiak and Goswami in Discrete Continuous Dyn Syst Ser A 35(2):617–635, 2015; Banasiak et al. in J Evol Equ 11:121–154, 2011, Mediterr J Math 11(2):533–559, 2014; Banasiak and Lachowicz in Methods of small parameter in mathematical biology, Birkhäuser, 2014; Sanchez et al. in J Math Anal Appl 323:680–699, 2006) and is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555–565, 2007), Bobrowski and Bogucki (Stud Math 189:287–300, 2008), where semigroups generated by convex combinations of Feller’s generators were studied.     

A general pinching principle for mean curvature flow and applications

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken (J Differ Geom 20(1):237–266, 1984), the convexity estimates of Huisken–Sinestrari (Acta Math 183(1):45–70, 1999) and the cylindrical estimate of Huisken–Sinestrari (Invent Math 175(1):137–221, 2009; see also Andrews and Langford in Anal PDE 7(5):1091–1107, 2014; Huisken and Sinestrari in J Differ Geom 101(2):267–287, 2015). Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is \((m+1)\)-convex, then the curvature of the solution pinches onto the convex hull of the curvatures of the shrinking cylinders \(\mathbb {R}^m\times S^{n-m}_{\sqrt{2(n-m)(1-t)}}\), \(t<1\). In particular, this yields a sharp estimate for the largest principal curvature, which we use to obtain a new proof of a sharp estimate for the inscribed curvature for embedded solutions (Brendle in Invent Math 202(1):217–237, 2015; Haslhofer and Kleiner in Int Math Res Not 15:6558–6561, 2015; Langford in Proc Am Math Soc 143(12):5395–5398, 2015). Making use of a recent idea of Huisken–Sinestrari (2015), we then obtain a series of sharp estimates for ancient solutions. In particular, we obtain a convexity estimate for ancient solutions which allows us to strengthen recent characterizations of the shrinking sphere due to Huisken–Sinestrari (2015) and Haslhofer–Hershkovits (Commun Anal Geom 24(3):593–604, 2016).     

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.     

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 \(\mathbb {H}^{3}\).     

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 \(\Gamma _\vartheta \).     

On manifolds of tensors of fixed TT-rank

Recently, the format of TT tensors (Hackbusch and Kühn in J Fourier Anal Appl 15:706–722, 2009; Oseledets in SIAM J Sci Comput 2009, submitted; Oseledets and Tyrtyshnikov in SIAM J Sci Comput 31:5, 2009; Oseledets and Tyrtyshnikov in Linear Algebra Appl 2009, submitted) has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor \({U \in \mathbb{R}^{n_1\times \cdots\times n_d}}\) and for the manifold of tensors of TT-rank \({\underline{r}}\) . As a first result, we prove that the TT (or compression) ranks r _i of a tensor U are unique and equal to the respective separation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that the set \({\mathbb{T}}\) of TT tensors of fixed rank \({\underline{r}}\) locally forms an embedded manifold in \({\mathbb{R}^{n_1\times\cdots\times n_d}}\) , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices (Conte and Lubich in M2AN 44:759, 2010), we introduce certain gauge conditions to obtain a unique representation of the tangent space \({\mathcal{T}_U\mathbb{T}}\) of \({\mathbb{T}}\) and deduce a local parametrization of the TT manifold. The parametrisation of \({\mathcal{T}_{U}\mathbb{T}}\) is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems (Lubich in From quantum to classical molecular dynamics: reduced methods and numerical analysis, 2008). We conclude with remarks on those applications and present some numerical examples.     

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.     

The Reich–Zaslavski property and fixed points of non-self multivalued mappings

Let (X, d) be a metric space, Y be a nonempty subset of X, and let \(T:Y \rightarrow P(X)\) be a non-self multivalued mapping. In this paper, by a new technique we study the fixed point theory of multivalued mappings under the assumption of the existence of a bounded sequence \((x_n)_n\) in Y such that \(T^nx_n\subseteq Y,\) for each \(n \in \mathbb {N}\). Our main result generalizes fixed point theorems due to Matkowski (Diss. Math. 127, 1975), W?grzyk (Diss. Math. (Rozprawy Mat.) 201, 1982), Reich and Zaslavski (Fixed Point Theory 8:303–307, 2007), Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and provides a solution to the problems posed in Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and Rus and ?erban (Miskolc Math. Notes 17:1021–1031, 2016).     

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).     

On Weyl’s Asymptotics and Remainder Term for the Orthogonal and Unitary Groups

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by Avakumovic (Math Z 65:327–344, [1]) and Hörmander (Acta Math 121:193–218, [15]) and show that for the scale of orthogonal and unitary groups \(\mathbf{SO}(N)\), \(\mathbf{SU}(N)\), \(\mathbf{U}(N)\) and \(\mathbf{Spin}(N)\) it is not sharp. While for negative sectional curvature improvements are possible and known, cf. e.g., Duistermaat and Guillemin (Invent Math 29:39–79, [7]), here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for \(\mathbf{U}(N)\)]. Furthermore here the improvements are sharp and quantitative relating to the dimension and rank of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum.     

Rigidity theorem for integral pinched shrinking Ricci solitons

We prove that an n-dimensional, \(n\ge 4\), compact gradient shrinking Ricci soliton satisfying a \(L^{\frac{n}{2}}\)-pinching condition is isometric to a quotient of the round \(\mathbb {S}^n\), which improves the rigidity theorem given by Catino (Integral pinched shrinking Ricci solitons, 2016), in dimension \(4\le n\le 6\).     

Nonexistence of two classes of generalized bent functions

We obtain some new nonexistence results of generalized bent functions from \({\mathbb {Z}}^n_q\) to \({\mathbb {Z}}_q\) (called type [n, q]) in the case that there exist cyclotomic integers in \( {\mathbb {Z}}[\zeta _{q}]\) with absolute value \(q^{\frac{n}{2}}\). This result generalizes two previous nonexistence results \([n,q]=[1,2\times 7]\) of Pei (Lect Notes Pure Appl Math 141:165–172, 1993) and \([3,2\times 23^e]\) of Jiang and Deng (Des Codes Cryptogr 75:375–385, 2015). We also remark that by using a same method one can get similar nonexistence results of GBFs from \({\mathbb {Z}}^n_2\) to \({\mathbb {Z}}_m\).     

New constructions of permutation polynomials of the form $$x^rh\left( x^{q-1}\right) $$ over $${\mathbb F}_{q^2}$$Fq2

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic.     

Second Order Riesz Transforms on Multiply–Connected Lie Groups and Processes with Jumps

We study a class of combinations of second order Riesz transforms on Lie groups \(\mathbb {G}=\mathbb {G}_{x} \times \mathbb {G}_{y}\) that are multiply connected, composed of a discrete abelian component \(\mathbb {G}_{x}\) and a compact connected component \(\mathbb {G}_{y}\). We prove sharp L^p estimates for these operators, therefore generalizing previous results (Volberg, A., Nazarov, F.: St Petersburg Math J. 15 (14), 563–573 2004; Domelevo, K., Petermichl, S.: Adv. Math. 262, 932–952 2014; Bañuelos, R., Baudoin, F.: Potential Anal. 38(4), 1071–1089 2013). We construct stochastic integrals with jump components adapted to functions defined on the semi-discrete set \(\mathbb {G}_{x} \times \mathbb {G}_{y}\). We show that these second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The analysis shows that Itô integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local. Previous representations of Riesz transforms through stochastic integrals in this direction do not consider discrete components and jump processes.     

ACCR ring of the forms $$\mathcal {A}[X]$$ and $$\mathcal {A}[[X]]$$A[[X]]

Let \(\mathcal {A}=(A_n)_{n\in \mathbb {N}}\) be an ascending chain of commutative rings with identity and let \(\mathcal {A}[X]\) (respectively, \(\mathcal {A}[[X]]\)) be the ring of polynomials (respectively, power series) with coefficient of degree n in \(A_n\) for each \(n\in \mathbb {N}\) (Hamed and Hizem in Commun Algebra 43:3848–3856, 2015; Haouat in Thèse de doctorat. Faculté des Sciences de Tunis, 1988). An A-module M is said to satisfy ACCR if the ascending chain of residuals of the form \(N:B\subseteq N:B^2\subseteq N:B^3\subseteq \cdots \) terminates for every submodule N of M and for every finitely generated ideal B of A (Lu in Proc Am Math Soc 117:5–10, 1993). We give necessary and sufficient condition for the ring \(\mathcal {A}[X]\) (respectively, \(\mathcal {A}[[X]]\)) to satisfy ACCR.     

Remarks on Kähler Ricci Flow

We show the convergence of Kähler Ricci flow directly if the α-invariant of the canonical class is greater than \(\frac{n}{n+1}\). Applying these convergence theorems, we can give a Kähler Ricci flow proof of Calabi conjecture on such Fano manifolds. In particular, the existence of KE metrics on a lot of Fano surfaces can be proved by flow method. Note that this geometric conclusion (based on the same assumption) was established earlier via elliptic method by Tian (Invent. Math. 89(2):225–246, 1987; Invent. Math. 101(1):101–172, 1990; Invent. Math. 130:1–39, 1997). However, a new proof based on Kähler Ricci flow should be still interesting in its own right.     

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 \(\Omega \).

     

O $$\varvec{}$$-orm preservers ad the Aleksadrov coservative $$\varvec{}$$-distace problem

The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for \(n \ge 3\) any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine (\(n \ge 2\)) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and ?emrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions (\(n\ge 2\)) is automatically an n-isometry.     

New Characterizations of the Clifford Torus as a Lagrangian Self-Shrinker

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus \({\mathbb {S}}^1(1)\times {\mathbb {S}}^1(1)\) is the unique compact orientable Lagrangian self-shrinker in \({\mathbb {C}}^2\) with \(|A|^2\le 2\), which gives an affirmative answer to Castro–Lerma’s conjecture in Castro and Lerma (Int Math Res Not 6:1515–1527; 2014). We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in \({\mathbb {C}}^2\).