On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a \({{\mathrm{{{\mathrm{Spin}}}^c}}}\)-Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).     

An Easton like theorem in the presence of Shelah cardinals

We show that Shelah cardinals are preserved under the canonical \({{\mathrm{GCH}}}\) forcing notion. We also show that if \({{\mathrm{GCH}}}\) holds and \(F:{{\mathrm{REG}}}\rightarrow {{\mathrm{CARD}}}\) is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies \(\forall \kappa \in {{\mathrm{REG}}},~ 2^{\kappa }=F(\kappa )\). This gives a partial answer to a question asked by Cody (Arch Math Logic 52(5–6):569–591, 2013) and independently by Honzik (Acta Univ Carol 1:55–72, 2015). We also prove an indestructibility result for Shelah cardinals.     

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.     

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.     

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.     

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)     

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     

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.     

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).     

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).     

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)).     

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 the Semicircular Law of Large-Dimensional Random Quaternion Matrices

It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erd?s in Russ Math Surv 66(3):507–626, 2011; Erd?s in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erd?s et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.     

On the solution of systems of equations with constant rank derivatives

The famous for its simplicity and clarity Newton–Kantorovich hypothesis of Newton’s method has been used for a long time as the sufficient convergence condition for solving nonlinear equations. Recently, in the elegant study by Hu et al. (J Comput Appl Math 219:110–122, 2008), a Kantorovich-type convergence analysis for the Gauss–Newton method (GNM) was given improving earlier results by Häubler (Numer Math 48:119–125, 1986), and extending some results by Argyros (Adv Nonlinear Var Inequal 8:93–99, 2005, 2007) to hold for systems of equations with constant rank derivatives. In this study, we use our new idea of recurrent functions to extend the applicability of (GNM) by replacing existing conditions by weaker ones. Finally, we provide numerical examples to solve equations in cases not covered before (Häubler, Numer Math 48:119–125, 1986; Hu et al., J Comput Appl Math 219:110–122, 2008; Kontorovich and Akilov 2004).     

Spectral Properties of Unbounded Jacobi Matrices with Almost Monotonic Weights

We present a unified framework to identify spectra of Jacobi matrices. We give applications of the long-standing problem of Chihara (Mt J Math 21(1):121–137, 1991, J Comput Appl Math 153(1–2):535–536, 2003) concerning one-quarter class of orthogonal polynomials, to the conjecture posed by Roehner and Valent (SIAM J Appl Math 42(5):1020–1046, 1982) concerning continuous spectra of generators of birth and death processes, and to spectral properties of operators studied by Janas and Moszyńki (Integral Equ Oper Theory 43(4):397–416, 2002) and Pedersen (Proc Am Math Soc 130(8):2369–2376, 2002).     

Optimal Rearrangement Invariant Sobolev Embeddings in Mixed Norm Spaces

We improve the Sobolev-type embeddings due to Gagliardo (Ric Mat 7:102–137, 1958) and Nirenberg (Ann Sc Norm Sup Pisa 13:115–162, 1959) in the setting of rearrangement invariant (r.i.) spaces. In particular, we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space \(W^{1}L^{p}\) by Poornima (Bull Sci Math 107(3):253–259,  1983), O’Neil (Duke Math J 30:129–142,  1963) and Peetre (Ann Inst Fourier 16(1):279–317,  1966) (\(1 \le p < n\)), and by Hansson (Math Scand 45(1):77–102,  1979, Brezis and Wainger (Commun Partial Differ Equ 5(7):773–789,  1980) and Maz’ya (Sobolev spaces,  1985) (\(p=n\)) can be further strengthened by considering mixed norms on the target spaces.     

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).     

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).     

Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Let V be a locally bounded measurable function on \({\mathbb {R}}^d\) such that \(\mu _V(\mathrm{d}x)=C_V \mathrm{e}^{-V(x)}\,\mathrm{d}x\) is a probability measure. Explicit criteria are presented for weighted Poincaré inequalities of the following non-local Dirichlet form

$$\begin{aligned} \hat{D}_{\rho ,V}(f,f)=\iint _{\{|x-y|>1\}}(f(y)-f(x))^2\rho (|y-x|)\,\mathrm{d}y\, \mu _V(\mathrm{d}x). \end{aligned}$$

Taking \(\rho (r)={\mathrm{e}^{-\delta r}}{r^{-(d+\alpha )}}\) with \(0<\alpha <2\) and \(\delta \geqslant 0\), we get new conclusions for (exponentially) tempered fractional Dirichlet forms, which not only complete our recent work (Chen and Wang in Stoch Process Their Appl 124:123–153, 2014; Wang and Wang in J Theor Probab 28:423–448, 2015), but also improve the main result in Mouhot et al. (J Math Pures Appl 95:72–84, 2011).