An improved local convergence analysis for a two-step Steffensen-type method

We study a class of Steffensen-type algorithm for solving nonsmooth variational inclusions in Banach spaces. We provide a local convergence analysis under ω-conditioned divided difference, and the Aubin continuity property. This work on the one hand extends the results on local convergence of Steffensen’s method related to the resolution of nonlinear equations (see Amat and Busquier in Comput. Math. Appl. 49:13–22, 2005; J. Math. Anal. Appl. 324:1084–1092, 2006; Argyros in Southwest J. Pure Appl. Math. 1:23–29, 1997; Nonlinear Anal. 62:179–194, 2005; J. Math. Anal. Appl. 322:146–157, 2006; Rev. Colomb. Math. 40:65–73, 2006; Computational Theory of Iterative Methods, 2007). On the other hand our approach improves the ratio of convergence and enlarges the convergence ball under weaker hypotheses than one given in Hilout (Commun. Appl. Nonlinear Anal. 14:27–34, 2007).     

More about λ-support iterations of (<λ)-complete forcing notions

This article continues Ros?anowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ ⁺ (for a strongly inaccessible cardinal λ).     

Small tight sets of hyperbolic quadrics

An x-tight set of a hyperbolic quadric Q ⁺(2n + 1, q) can be described as a set M of points with the property that the number of points of M in the tangent hyperplanes of points of M is as big as possible. We show that such a set is necessarily the union of x mutually disjoint generators provided that x ≤ q and n ≤ 3, or that x < q, n ≥ 4 and q ≥ 71. This unifies and generalizes many results on x-tight sets that are presently known, see (J Comb Theory Ser A 114(7):1293–1314 [1], J Comb Des 16(4):342–349 [5], Des Codes Cryptogr 50:187–201 [4], Adv Geom 4(3):279–286 [8], Bull Lond Math Soc 42(6):991–996 [11]).     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.     

The Number of Generations Entirely Visited for Recurrent Random Walks in a Random Environment

In this paper we deal with a random walk in a random environment on a super-critical Galton–Watson tree. We focus on the recurrent cases already studied by Hu and Shi (Ann. Probab. 35:1978–1997, 2007; Probab. Theory Relat. Fields 138:521–549, 2007), Faraud et al. (Probab. Theory Relat. Fields, 2011, in press), and Faraud (Electron. J. Probab. 16(6):174–215, 2011). We prove that the largest generation entirely visited by these walks behaves like logn, and that the constant of normalization, which differs from one case to another, is a function of the inverse of the constant of Biggins’ law of large numbers for branching random walks (Biggins in Adv. Appl. Probab. 8:446–459, 1976).     

L p -theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form ${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$ in ${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$ are considered, which are the same as in recent papers Metafune et?al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et?al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et?al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in ${L^{p}(\mathbb{R}^{N})}$ is presented via a certain identity that behaves like a sesquilinear form over L ^p ×?L ^p'. It partially improves the results in (Metafune et?al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et?al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠?2. The simplicity is illustrated with the typical example ${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$ in ${L^p(\mathbb{R})}$ which is dealt with in (Arendt et?al. in J Operator Theory 55:185–211, 2006; Arendt et?al. in J Math Anal Appl 338: 505–517, 2008).     

Directional Lipschitzness of Minimal Time Functions in Hausdorff Topological Vector Spaces

In a general Hausdorff topological vector space E, we associate to a given nonempty closed set S???E and a bounded closed set Ω???E, the minimal time function T _S,Ω defined by $T_{S,\Omega}(x):= \inf \{ t> 0: S\cap (x+t\Omega)\not = \emptyset\}$ . The study of this function has been the subject of various recent works (see Bounkhel (2012, submitted, 2013, accepted); Colombo and Wolenski (J Global Optim 28:269–282, 2004, J Convex Anal 11:335–361, 2004); He and Ng (J Math Anal Appl 321:896–910, 2006); Jiang and He (J Math Anal Appl 358:410–418, 2009); Mordukhovich and Nam (J Global Optim 46(4):615–633, 2010) and the references therein). The main objective of this work is in this vein. We characterize, for a given Ω, the class of all closed sets S in E for which T _S,Ω is directionally Lipschitz in the sense of Rockafellar (Proc Lond Math Soc 39:331–355, 1979). Those sets S are called Ω-epi-Lipschitz. This class of sets covers three important classes of sets: epi-Lipschitz sets introduced in Rockafellar (Proc Lond Math Soc 39:331–355, 1979), compactly epi-Lipschitz sets introduced in Borwein and Strojwas (Part I: Theory, Canad J Math No. 2:431–452, 1986), and K-directional Lipschitz sets introduced recently in Correa et al. (SIAM J Optim 20(4):1766–1785, 2010). Various characterizations of this class have been established. In particular, we characterize the Ω-epi-Lipschitz sets by the nonemptiness of a new tangent cone, called Ω-hypertangent cone. As for epi-Lipschitz sets in Rockafellar (Canad J Math 39:257–280, 1980) we characterize the new class of Ω-epi-Lipschitz sets with the help of other cones. The spacial case of closed convex sets is also studied. Our main results extend various existing results proved in Borwein et al. (J Convex Anal 7:375–393, 2000), Correa et al. (SIAM J Optim 20(4):1766–1785, 2010) from Banach spaces and normed spaces to Hausdorff topological vector spaces.     

Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980; Knightly in SIAM J. Math. Anal. 3:506–511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633–652, 2012, Quad. Mat. 4:27–68, 1999, Nonlinear Anal. 47:4151–4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159–175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719–732, 2008; Maremonti in Mat. Ves. 61:81–91, 2009, Appl. Anal. 90:125–139, 2011).     

On Ulm’s method for Fréchet differentiable operators

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Izv. Akad. Nauk Est. SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) do not. We also show that under the same hypotheses and computational cost as (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) finer error sequences can be obtained. Numerical examples are also provided further validating the results.     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

Index sets and Scott sentences

For a computable structure \({\mathcal{A}}\) , there may not be a computable infinitary Scott sentence. When there is a computable infinitary Scott sentence \({\varphi}\) , then the complexity of the index set \({I(\mathcal{A})}\) is bounded by that of \({\varphi}\) . There are results (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print) giving “optimal” Scott sentences for structures of various familiar kinds. These results have been driven by the thesis that the complexity of the index set should match that of an optimal Scott sentence (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012). In this note, it is shown that the thesis does not always hold. For a certain subgroup of \({\mathbb{Q}}\) , there is no computable d- \({\Sigma_2}\) Scott sentence, even though (as shown in Ash and Knight in Scott sentences for certain groups, pre-print) the index set is d- \({\Sigma^0_2}\) .     

Rigidity and the Lower Bound Theorem for Doubly Cohen–Macaulay Complexes

We prove that for d≥3, the 1-skeleton of any (d?1)-dimensional doubly Cohen–Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121–125, 1971; Barnette, D. Pac. J. Math. 46:349–354, 1973) hold for every 2-CM complex of dimension ≥2 (see Kalai, G. Invent. Math. 88:125–151, 1987). Moreover, the initial part (g ₀,g ₁,g ₂) of the g-vector of a 2-CM complex (of dimension ≥3) is an M-sequence. It was conjectured by Björner and Swartz (J. Comb. Theory Ser. A 113:1305–1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Constructive finite free resolutions

Northcott’s book Finite Free Resolutions (1976), as well as the paper (J. Reine Angew. Math. 262/263:205–219, 1973), present some key results of Buchsbaum and Eisenbud (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974) both in a simplified way and without Noetherian hypotheses, using the notion of latent nonzero divisor introduced by Hochster. The goal of this paper is to simplify further the proofs of these results, which become now elementary in a logical sense (no use of prime ideals, or minimal prime ideals) and, we hope, more perspicuous. Some formulations are new and more general than in the references (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974; Finite Free Resolutions 1976) (Theorem 7.2, Lemma 8.2 and Corollary 8.5).     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Completion by Derived Double Centralizer

Let A be a commutative ring, and let $\mathfrak{a}$ be a weakly proregular ideal in A. (If A is noetherian then any ideal in it is weakly proregular.) Suppose M is a compact generator of the category of cohomologically $\mathfrak{a}$ -torsion complexes. We prove that the derived double centralizer of M is isomorphic to the $\mathfrak{a}$ -adic completion of A. The proof relies on the MGM equivalence from Porta et al. (Algebr Represent Theor, 2013) and on derived Morita equivalence. Our result extends earlier work of Dwyer et al. (Adv Math 200:357–402, 2006) and Efimov (2010).     

A remark on Feuerbach hyperbolas

Recently many authors have studied properties of triangles and the theory of perspective triangles in the Euclidean plane (see Kimberling et al. J Geom Graph 14:1–14, 2010; Kimberling et al. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html, 2012; Moses and Kimberling J Geom Graph 13:15–24, 2009; Moses and Kimberling Forum Geom 11:83–93, 2011; Odehnal Elem Math 61:74–80, 2006; Odehnal Forum Geom 10:35–40, 2010; Odehnal J Geom Graph 15: 45–67, 2011). The aim of this paper is to present a new approach to the construction of points on the Feuerbach hyperbola. Surprisingly, these points can be obtained as centers of perspectivity of a triangle ABC and a certain one-parametric set of triangles A′B′C′. The presented construction is based on partitions of the triangle’s sides and—in a way—dual to the construction of points on the Kiepert hyperbola. It can also be generalized to spherical triangles. The proofs are based on an affine property of triangles, which amazingly can also be used for the proof of the spherical theorem.     

Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.     

The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups

The general surface group conjecture asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in Fine et al. (Sci Math A 1:1–15, 2008). First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if G is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then G is free, a surface group or a solvable Baumslag–Solitar Group. Further combining results in Fine et al. (Sci Math A 1:1–15, 2008) on Property IF with a theorem of Wilton (Geom Topol, 2012) and results of Stallings (Ann Math 2(88):312–334, 1968) and Kharlampovich and Myasnikov (Trans Am Math Soc 350(2):571–613, 1998) we show that Surface Group Conjecture C proposed in Fine et al. (Sci Math A 1:1–15, 2008) is true, namely: If G is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then G is a surface group.