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).     

Remarks on defective Fano manifolds

This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016).     

Improved local convergence analysis of the Gauss–Newton method under a majorant condition

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.     

Vanishing of Rabinowitz Floer homology on negative line bundles

Following Frauenfelder (Rabinowitz action functional on very negative line bundles, Habilitationsschrift, Munich/München, 2008), Albers and Frauenfelder (Bubbles and onis, 2014. arXiv:1412.4360) we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. Ritter (Adv Math 262:1035–1106, 2014) showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem \(\mathrm {SH}=0\Leftrightarrow \mathrm {RFH}=0\) (Ritter in J Topol 6(2):391–489, 2013), does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak–Frauenfelder–Oancea long exact sequence Cieliebak et al. (Ann Sci Éc Norm Supér (4) 43(6):957–1015, 2010).     

A volume decreasing theorem for $$\mathbf{}$$-harmonic maps and applications

We establish a volume decreasing result for V-harmonic maps between Riemannian manifolds. We apply this result to obtain corresponding results for Weyl harmonic maps from conformal Weyl manifolds to Riemannian manifolds. We also obtain corresponding results for holomorphic maps from almost Hermitian manifolds to quasi-Kähler manifolds, which generalize or improve the partial results in Goldberg and Har’El (Bull Soc Math Grèce 18(1):141–148, 1977, J Differ Geom 14(1):67–80, 1979).     

A brief note on the coarea formula

In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into \({\mathbb {R}}\)) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions.     

Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with \(L^{\infty }\)-variable coefficients in \(L^2\)-based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in \(L^2\)-based Sobolev spaces by using the remarkable Nec?as–Babus?ka–Brezzi technique (see Babus?ka in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Nec?as in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in \(L^2\)-based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos? and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.     

Data dependence for the amplitude equation of surface waves

We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves (Hamilton et al. in J Acoust Soc Am 97:891–897, 1995), surface waves on current-vortex sheets in incompressible MHD (Alì and Hunter in Q Appl Math 61(3):451–474, 2003; Alì et al. in Stud Appl Math 108(3):305–321, 2002) and on the incompressible plasma–vacuum interface (Secchi in Q Appl Math 73(4):711–737, 2015). The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in Hunter (J Hyperbolic Differ Equ 3(2):247–267, 2006), Secchi (Q Appl Math 73(4):711–737, 2015). In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.     

Extensions of the Noncommutative Integration

In this paper we will continue the analysis undertaken in Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), Bongiorno et al. (Rocky Mt J Math 40(6):1745–1777, 2010), Triolo (Rend Circ Mat Palermo (2) 60(3):409–416, 2011) on the general problem of extending the noncommutative integration in a *-algebra of measurable operators. As in Aiena et al. (Filomat 28(2):263–273, 2014), Bagarello (Stud Math 172(3):289–305, 2006) and Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), the main problem is to represent different types of partial *-algebras into a *-algebra of measurable operators in Segal’s sense, provided that these partial *-algebras posses a sufficient family of positive linear functionals (states) (Fragoulopoulou et al., J Math Anal Appl 388(2):1180–1193, 2012; Trapani and Triolo, Stud Math 184(2):133–148, 2008; Trapani and Triolo, Rend Circolo Mat Palermo 59:295–302, 2010; La Russa and Triolo, J Oper Theory, 69:2, 2013; Triolo, J Pure Appl Math, 43(6):601–617, 2012). In this paper, a new condition is given in an attempt to provide a extension of the non commutative integration.     

Efficient importance sampling in low dimensions using affine arithmetic

Despite the development of sophisticated techniques such as sequential Monte Carlo (Del Moral et al. in J R Stat Soc Ser B 68(3):411–436, 2006), importance sampling (IS) remains an important Monte Carlo method for low dimensional target distributions (Chopin and Ridgway in Leave Pima Indians alone: binary regression as a benchmark for Bayesian computation, 32:64–87, 2017). This paper describes a new technique for constructing proposal distributions for IS, using affine arithmetic (de Figueiredo and Stolfi in Numer Algorithms 37(1–4):147–158, 2004). This work builds on the Moore rejection sampler (Sainudiin in Machine interval experiments, Cornell University, Ithaca, 2005; Sainudiin and York in Algorithms Mol Biol 4(1):1, 2009) to which we provide a comparison.     

Performance of convex underestimators in a branch-and-bound framework

The efficient determination of tight lower bounds in a branch-and-bound algorithm is crucial for the global optimization of models spanning numerous applications and fields. The global optimization method \(\alpha \)-branch-and-bound (\(\alpha \)BB, Adjiman et al. in Comput Chem Eng 22(9):1159–1179, 1998b, Comput Chem Eng 22(9):1137–1158, 1998a; Adjiman and Floudas in J Global Optim 9(1):23–40, 1996; Androulakis et al. J Global Optim 7(4):337–363, 1995; Floudas in Deterministic Global Optimization: Theory, Methods and Applications, vol. 37. Springer, Berlin, 2000; Maranas and Floudas in J Chem Phys 97(10):7667–7678, 1992, J Chem Phys 100(2):1247–1261, 1994a, J Global Optim 4(2):135–170, 1994), guarantees a global optimum with \(\epsilon \)-convergence for any \(\mathcal {C}^2\)-continuous function within a finite number of iterations via fathoming nodes of a branch-and-bound tree. We explored the performance of the \(\alpha \)BB method and a number of competing methods designed to provide tight, convex underestimators, including the piecewise (Meyer and Floudas in J Global Optim 32(2):221–258, 2005), generalized (Akrotirianakis and Floudas in J Global Optim 30(4):367–390, 2004a, J Global Optim 29(3):249–264, 2004b), and nondiagonal (Skjäl et al. in J Optim Theory Appl 154(2):462–490, 2012) \(\alpha \)BB methods, the Brauer and Rohn+E (Skjäl et al. in J Global Optim 58(3):411–427, 2014) \(\alpha \)BB methods, and the moment method (Lasserre and Thanh in J Global Optim 56(1):1–25, 2013). Using a test suite of 40 multivariate, box-constrained, nonconvex functions, the methods were compared based on the tightness of generated underestimators and the efficiency of convergence of a branch-and-bound global optimization algorithm.     

Local convergence and the dynamics of a two-point four parameter Jarratt-like method under weak conditions

We present a local convergence analysis of a two-point four parameter Jarratt-like method of high convergence order in order to approximate a locally unique solution of a nonlinear equation. In contrast to earlier studies such us (Amat et al. Aequat. Math. 69(3), 212–223 2015; Amat et al. J. Math. Anal. Appl. 366(3), 24–32 2010; Behl, R. 2013; Bruns and Bailey Chem. Eng. Sci. 32, 257–264 1977; Candela and Marquina. Computing 44, 169–184 1990; Candela and Marquina. Computing 45(4), 355–367 1990; Chun. Appl. Math. Comput. 190(2), 1432–1437 2007; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2007; Deghan. Comput. Appl Math. 29(1), 19–30 2010; Deghan. Comput. Math. Math. Phys. 51(4), 513–519 2011; Deghan and Masoud. Eng. Comput. 29(4), 356–365 15; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2012; Deghan and Masoud. Eng. Comput. 29(4), 356–365 2012; Ezquerro and Hernández. Appl. Math. Optim. 41(2), 227–236 2000; Ezquerro and Hernández. BIT Numer. Math. 49, 325–342 2009; Ezquerro and Hernández. J. Math. Anal. Appl. 303, 591–601 2005; Gutiérrez and Hernández. Comput. Math. Appl. 36(7), 1–8 1998; Ganesh and Joshi. IMA J. Numer. Anal. 11, 21–31 1991; González-Crespo et al. Expert Syst. Appl. 40(18), 7381–7390 2013; Hernández. Comput. Math. Appl. 41(3-4), 433–455 2001; Hernández and Salanova. Southwest J. Pure Appl. Math. 1, 29–40 1999; Jarratt. Math. Comput. 20(95), 434–437 1966; Kou and Li. Appl. Math. Comput. 189, 1816–1821 2007; Kou and Wang. Numer. Algor. 60, 369–390 2012; Lorenzo et al. Int. J. Interact. Multimed. Artif. Intell. 1(3), 60–66 2010; Magreñán. Appl. Math. Comput. 233, 29–38 2014; Magreñán. Appl. Math. Comput. 248, 215–224 2014; Parhi and Gupta. J. Comput. Appl. Math. 206(2), 873–887 2007; Rall 1979; Ren et al. Numer. Algor. 52(4), 585–603 2009; Rheinboldt Pol. Acad. Sci. Banach Ctr. Publ. 3, 129–142 1978; Sicilia et al. J. Comput. Appl. Math. 291, 468–477 2016; Traub 1964; Wang et al. Numer. Algor. 57, 441–456 2011) using hypotheses up to the fifth derivative, our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. The dynamics of the family for choices of the parameters such that it is optimal is also shown. Numerical examples are also provided in this study     

Finite time blow up for a solution to system of the drift–diffusion equations in higher dimensions

We discuss the existence of a blow-up solution for a multi-component parabolic–elliptic drift–diffusion model in higher space dimensions. We show that the local existence, uniqueness and well-posedness of a solution in the weighted \(L^2\) spaces. Moreover we prove that if the initial data satisfies certain conditions, then the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift–diffusion equation proved by Nagai (J Inequal Appl 6:37–55, 2001) and Nagai et al. (Hiroshima J Math 30:463–497, 2000) and gravitational interaction of particles by Biler (Colloq Math 68:229–239, 1995), Biler and Nadzieja (Colloq Math 66:319–334, 1994, Adv Differ Equ 3:177–197, 1998). We generalize the result in Kurokiba and Ogawa (Differ Integral Equ 16:427–452, 2003, Differ Integral Equ 28:441–472, 2015) and Kurokiba (Differ Integral Equ 27(5–6):425–446, 2014) for the multi-component problem and give a sufficient condition for the finite time blow up of the solution. The condition is different from the one obtained by Corrias et al. (Milan J Math 72:1–28, 2004).     

On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of \({\mathbb C}\) and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [8], see also [7, 9].     

A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc

In this short note, we generalized an energy estimate due to Malchiodi–Martinazzi (J Eur Math Soc 16:893–908, 2014) and Mancini–Martinazzi (Calc Var 56:94, 2017). As an application, we used it to reprove existence of extremals for Trudinger–Moser inequalities of Adimurthi–Druet type on the unit disc. Such existence problems in general cases had been considered by Yang  (Trans Am Math Soc 359:5761–5776, 2007; J Differ Equ 258:3161–3193, 2015) and Lu–Yang (Discrete Contin Dyn Syst 25:963–979, 2009) by using another method.     

Optimal software-implemented Itoh–Tsujii inversion for $$\mathbb {F}_{2^{m}}$$

Field inversion in \(\mathbb {F}_{2^{m}}\) dominates the cost of modern software implementations of certain elliptic curve cryptographic operations, such as point encoding/hashing into elliptic curves (Brown et al. in: Submission to NIST, 2008; Brown in: IACR Cryptology ePrint Archive 2008:12, 2008; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014) Itoh–Tsujii inversion using a polynomial basis and precomputed table-based multi-squaring has been demonstrated to be highly effective for software implementations (Taverne et al. in: CHES 2011, 2011; Oliveira et al. in: J Cryptogr Eng 4(1):3–17, 2014; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014), but the performance and memory use depend critically on the choice of addition chain and multi-squaring tables, which in prior work have been determined only by suboptimal ad-hoc methods and manual selection. We thoroughly investigated the performance/memory tradeoff for table-based linear transforms used for efficient multi-squaring. Based upon the results of that investigation, we devised a comprehensive cost model for Itoh–Tsujii inversion and a corresponding optimization procedure that is empirically fast and provably finds globally-optimal solutions. We tested this method on eight binary fields commonly used for elliptic curve cryptography; our method found lower-cost solutions than the ad-hoc methods used previously, and for the first time enables a principled exploration of the time/memory tradeoff of inversion implementations.     

On spectral analysis in varieties containing the solutions of inhomogeneous linear functional equations

The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in the case of homogeneous linear functional equations. The foundations of the theory can be found in Kiss and Varga (Aequat Math 88(1):151–162, 2014) and Kiss and Laczkovich (Aequat Math 89(2):301–328, 2015). We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to Kocl?ga-Kulpa and Szostok (Ann Math Sylesianae 22:27–40, 2008), see also Kocl?ga-Kulpa and Szostok (Georgian Math J 16:725–736, 2009; Acta Math Hung 130(4):340–348, 2011). They are motivated by quadrature rules of approximate integration.     

New characterizations of the Takagi function via functional equations

We provide two new characterizations of the Takagi function as the unique bounded solution of some systems of two functional equations. The results are independent of those obtained by Kairies (Wy? Szko? Ped Krakow Rocznik Nauk Dydakt Prace Mat 196:73–82, 1998), Kairies (Aequ Math 53:207–241, 1997), Kairies (Aequ Math 58:183–191, 1999) and Kairies et al. (Rad Mat 4:361–374, 1989; Errata, Rad Mat 5:179–180, 1989).     

Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems

An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007; J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. \(\sqrt{\mu}\) and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.     

On a rigidity result of singularities

The aim of this note is to prove, in the spirit of a rigidity result for isolated singularities of Schlessinger see Schlessinger (Invent Math 14:17–26, 1971) or also Kleiman and Landolfi (Compositio Math 23:407–434, 1971), a variant of a rigidity criterion for arbitrary singularities (Theorem 2.1 below). The proof of this result does not use Schlessinger’s Deformation Theory [Schlessinger (Trans Am Math Soc 130:208–222, 1968) and Schlessinger (Invent Math 14:17–26, 1971)]. Instead it makes use of Local Grothendieck-Lefschetz Theory, see (Grothendieck 1968, Éxposé 9, Proposition 1.4, page 106) and a Lemma of Zariski, see (Zariski, Am J Math 87:507–536, 1965, Lemma 4, page 526). I hope that this proof, although works only in characteristic zero, might also have some interest in its own.