A note on “Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality” [Positivity. 18, 449–473(2014)]

The note points out that the sufficiency of proposition 2.1 in Anh (Positivity 18:449–473, 2014) is erroneous and we provide an example to illustrate it. Also the proof of proposition 2.2 in Anh (Positivity 18:449–473, 2014) is incorrect and we give a new proof.     

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).     

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.     

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     

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow-up scenario for the 3D Euler equation proposed by Hou and Luo (Multiscale Model Simul 12:1722–1776, 2014) based on extensive numerical simulations. These models generalize the 1D Hou–Luo model suggested in Hou and Luo Luo and Hou (2014), for which finite-time blowup has been established in Choi et al. (arXiv preprint. arXiv:1407.4776, 2014). The main new aspects of this work are twofold. First, we establish finite-time blowup for a model that is a closer approximation of the three-dimensional case than the original Hou–Luo model, in the sense that it contains relevant lower-order terms in the Biot–Savart law that have been discarded in Hou and Luo Choi et al. (2014). Secondly, we show that the blow-up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou–Luo model. Such blow-up stability result may be useful in further work on understanding the 3D hyperbolic blow-up scenario.     

Strong convergence of Browder’s and Halpern’s type iterations in Hilbert spaces

We use results from theory of nonexpansive mappings to unify and deduce the recent results of Lin and Takahashi (Positivity 16:429–453, 2012) and of Takahashi (J Optim Theory Appl 157:781–802, 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.     

On the generalization of the Salem test

The assertion that the Salem test [5] for the uniform convergence of a trigonometric Fourier series is improvable, is proved. In particular, an example of a continuous function, which does not fulfill the condition of the Salem test but satisfies the condition of the generalized Salem test [10], is constructed.Besides, the theorem which improves Golubov’s [3,4] result for continuous functions of two variables, is given.     

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).     

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).     

Fatou Components of Attracting Skew-Products

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188 [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.     

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).     

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.     

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).     

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 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.     

Convergence in Law for the Branching Random Walk Seen from Its Tip

Consider a critical branching random walk on the real line. In a recent paper, Aïdékon (2011) developed a powerful method to obtain the convergence in law of its minimum after a log-factor translation. By an adaptation of this method, we show that the point process formed by the branching random walk seen from the minimum converges in law to a decorated Poisson point process. This result, confirming a conjecture of Brunet and Derrida (J Stat Phys 143:420–446, 2011), can be viewed as a discrete analog of the corresponding results for the branching Brownian motion, previously established by Arguin et al. (2010, 2011) and Aïdékon et al. (2011).     

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.     

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.