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     

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.     

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.     

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.     

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).     

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).     

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).     

Condition Length and Complexity for the Solution of Polynomial Systems

Smale’s 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale in Mathematical problems for the next century, American Mathematical Society, Providence, 2000). The main progress on Smale’s problem is Beltrán and Pardo (Found Comput Math 11(1):95–129, 2011) and Bürgisser and Cucker (Ann Math 174(3):1785–1836, 2011). In this paper, we will improve on both approaches and prove an interesting intermediate result on the average value of the condition number. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltrán and Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in Bürgisser and Cucker (2011), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last theorem is similar to the main result of Bürgisser and Cucker (2011) but relies only on homotopy methods, thus removing the need for the elimination theory methods used in Bürgisser and Cucker (2011). We build on methods developed in Armentano et al. (2014).     

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).     

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.     

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 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.     

Amalgamating players,symmetry, and the Banzhaf value

We suggest new characterizations of the Banzhaf value without the symmetry axiom, which reveal that the characterizations by Lehrer (Int J Game Theory 17:89–99, 1988) and Nowak (Int J Game Theory 26:137–141, 1997) as well as most of the characterizations by Casajus (Theory Decis 71:365–372, 2011b) are redundant. Further, we explore symmetry implications of Lehrer’s 2-efficiency axiom.     

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).     

Truthmaker Internalism and the Mind-Dependence of Propositions

It is generally thought that truthmaking has to be an internal relation because if it weren’t, then, as David Armstrong argues, “everything may be a truthmaker for any truth” (1997: 198). Depending on whether we take an internal relation to be one that is necessitated by the mere existence of its terms (Armstrong 1997: 87 and 2004: 9) or one that supervenes on the intrinsic properties of its relata (Lewis 1986: 62), the truthbearers involved in the truthmaking relation must either have their contents essentially or intrinsically. In this paper, I examine Armstrong’s account (1973; 1997 and 2004), according to which what is made true at the fundamental level are mental state tokens. The conclusion is reached that such tokens have their contents neither essentially nor intrinsically, and so, are simply the wrong kind of entities to be made true internally.     

Filtration de monodromie et cycles évanescents formels

Berkovich (Invent. Math. 125(2):367–390, 1996), Fujiwara (Duke Math. J. 80(1):15–57, 1995) and Huber (J. Algebraic Geom. 7(2):359–403, 1998) have proved that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by Boyer (Invent. Math. doi: 10.1007/s00222-009-0183-9, 2009).     

Hedging with temporary price impact

We consider the problem of hedging a European contingent claim in a Bachelier model with temporary price impact as proposed by Almgren and Chriss (J Risk 3:5–39, 2001). Following the approach of Rogers and Singh (Math Financ 20:597–615, 2010) and Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b) from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets as, e.g., Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b), Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), Rogers and Singh (Math Financ 20:597–615, 2010), Almgren and Li (Option hedging with smooth market impact. Preprint, 2015), Moreau et al. (Math Financ. doi: 10.1111/mafi.12098, 2015), Kallsen and Muhle-Karbe (High-resilience limits of block-shaped order books. Preprint, 2014), Guasoni and Weber (Mathematical Financ. doi: 10.1111/mafi.12099, 2015a; Nonlinear price impact and portfolio choice. Preprint, 2015b), where the frictionless hedging strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods.     

Twisting the Alive Particle Filter

This work focuses on sampling from hidden Markov models (Cappe et al. 2005) whose observations have intractable density functions. We develop a new sequential Monte Carlo (e.g. Doucet, 2011) algorithm and a new particle marginal Metropolis-Hastings (Andrieu et al J R Statist Soc Ser B 72:269-342, 2010) algorithm for these purposes. We build from Jasra et al (2013) and Whiteley and Lee (Ann Statist 42:115-141, 2014) to construct the sequential Monte Carlo (SMC) algorithm, which we call the alive twisted particle filter. Like the alive particle filter (Amrein and Künsch, 2011, Jasra et al, 2013), our new SMC algorithm adopts an approximate Bayesian computation (Tavare et al. Genetics 145:505-518, 1997) estimate of the HMM. Our alive twisted particle filter also uses a twisted proposal as in Whiteley and Lee (Ann Statist 42:115-141, 2014) to obtain a low-variance estimate of the HMM normalising constant. We demonstrate via numerical examples that, in some scenarios, this estimate has a much lower variance than that of the estimate obtained via the alive particle filter. The low variance of this normalising constant estimate encourages the implementation of our SMC algorithm within a particle marginal Metropolis-Hastings (PMMH) scheme, and we call the resulting methodology “alive twisted PMMH”. We numerically demonstrate, on a stochastic volatility model, how our alive twisted PMMH can converge faster than the standard alive PMMH of Jasra et al (2013).     

Symmetry axioms and perceived ambiguity

Since at least de Finetti (Annales de l’Institut Henri Poincare 7:1–68, 1937), preference symmetry assumptions have played an important role in models of decision making under uncertainty. In the current paper, we explore (1) the relationship between the symmetry assumption of Klibanoff et al. (KMS) (Econometrica 82:1945–1978, 2014) and alternative symmetry assumptions in the literature, and (2) assuming symmetry, the relationship between the set of relevant measures, shown by KMS (2014) to reflect only perceived ambiguity, and the set of measures (which we will refer to as the Bewley set) developed by Ghirardato et al. (J Econ Theory 118:133–173, 2004), Nehring (Ambiguity in the context of probabilistic beliefs, working paper, 2001, Bernoulli without Bayes: a theory of utility-sophisticated preference, working paper, 2007) and Ghirardato and Siniscalchi (A more robust definition of multiple priors, working paper, 2007, Econometrica 80:2827–2847, 2012). This Bewley set is the main alternative offered in the literature as possibly representing perceived ambiguity. Regarding symmetry assumptions, we show that, under relatively mild conditions, a variety of preference symmetry conditions from the literature [including that in KMS (2014)] are equivalent. In KMS (2014), we showed that, under symmetry, the Bewley set and the set of relevant measures are not always the same. Here, we establish a preference condition, No Half Measures, that is necessary and sufficient for the two to be the same under symmetry. This condition is rather stringent. Only when it is satisfied may the Bewley set be interpreted as reflecting only perceived ambiguity and not also taste aspects such as ambiguity aversion.     

Finite-Codimensional Compressions of Symmetric and Self-Adjoint Linear Relations in Krein Spaces

Theorems due to Stenger (Bull Am Math Soc 74:369–372, 1968) and Nudelman (Int Equ Oper Theory 70:301–305, 2011) in Hilbert spaces and their generalizations to Krein spaces in Azizov and Dijksma (Int Equ Oper Theory 74(2):259–269, 2012) and Azizov et al. (Linear Algebra Appl 439:771–792, 2013) generate additional questions about properties a finite-codimensional compression \({T_0}\) of a symmetric or self-adjoint linear relation \({T}\) may or may not inherit from \({T}\). These questions concern existence of invariant maximal nonnegative subspaces, definitizability, singular critical points and defect indices.