Correction: Nearly Quartic Mappings in β-Homogeneous F-Spaces [Results Math. 63 (2013) 529–541]

In Chang et al. (Results Math. 63:529–541, 2013), Eshaghi Gordji et al. proved the Hyers-Ulam stability of a quartic functional equation in β-homogeneous F-spaces. In the main step of the proof of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2), there is a fatal error. We correct the statement of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2).     

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- \({\Sigma^0_2}\) .     

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).     

Continuity of the solution mapping to parametric generalized vector equilibrium problems

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010), Chen and Li (Pac J Optim 6:141–152, 2010), Chen et al. (J Glob Optim 45:309–318, 2009), Cheng and Zhu (J Glob Optim 32:543–550, 2005), Gong (J Optim Theory Appl 139:35–46, 2008), Li and Fang (J Optim Theory Appl 147:507–515, 2010), Li et al. (Bull Aust Math Soc 81:85–95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011).     

Cyclic group blocking polyhedra

Gomory (Linear Algebra Appl 2:451–558, 1969) gave a subadditive characterization of the facets of the group polyhedron. Although there are exponentially many facets (see Gomory and Johnson in Math Program 3:359–389, 1972, Example 4.6) and exponentially many vertices for the group polyhedron for the master cyclic group problem, Gomory’s characterization of the non-trivial facets has polynomially many subadditive inequalities, in fact of order |G|² for a finite Abelian group G. We reduce this subadditive inequality system to its minimal representation by a triple system of the same order and show the dimensionality of the polytope of non-trivial facets. The system of all triples corresponds to all solution vectors of length three into which every solution vector can be decomposed. Our minimal representation leads to a characterization of the vertices of length three. Gomory et al. (Math Program 96:321–339, 2003) introduced a shooting experiment involving solving the shooting linear program repeatedly to find important facets. We develop a topological network flow model of the dual problem of the shooting linear program in a reverse procedure from the decomposition of solution vectors into triples. Hunsaker (2003) gave a knapsack shooting experiment, which we use to find a simple pattern for the most hit knapsack facets.     

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.     

Blocks with Defect Group \boldsymbol{Q_{2^n}\times C_{2^m}} and \boldsymbol{SD_{2^n}\times C_{2^m}}

We determine the numerical invariants of blocks with defect group $Q_{2^n}\times C_{2^m}$ and $SD_{2^n}\times C_{2^m}$ , where $Q_{2^n}$ denotes a quaternion group of order 2 ⁿ , $C_{2^m}$ denotes a cyclic group of order 2 ^m , and $SD_{2^n}$ denotes a semidihedral group of order 2 ⁿ . This generalizes Olsson’s results for m?=?0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).     

L p -theory for second-order elliptic operators with unbounded coefficients in an endpoint class

The m-accretivity and m-sectoriality of the minimal and maximal realizations of second-order elliptic operators of the form ${Au=-{\rm div}(a \nabla u)+F\cdot \nabla u +Vu}$ in ${L^p(\mathbb{R}^N)}$ are shown, where the coefficients a, F and V are unbounded. The result may be regarded as an endpoint assertion of the previous result in Sobajima (J Evol Equ 12:957–971, 2012) and an improvement of that in Metafune et al. (Forum Math 22:583–601, 2010). Moreover, an L ^p -generalization of Kato’s self-adjoint problem in Kato (1981, Appendix 2) is discussed. The proof is based on Sobajima (J Evol Equ 12:957–971, 2012). As examples, the operators ${-\Delta \pm |x|^{\beta-1}x \cdot \nabla +c|x|^{\gamma}}$ are also dealt with, which are mentioned in Metafune et al. (Forum Math 22:583–601, 2010).     

Families of Finite-Dimensional Hopf Algebras with the Chevalley Property

We introduce and study new families of finite-dimensional Hopf algebras with the Chevalley property that are not pointed nor semisimple arising as twistings of quantum linear spaces. These Hopf algebras generalize the examples introduced in Andruskiewitsch et al. (Mich Math J 49(2):277–298, 2001), Etingof and Gelaki (Int Math Res Not 14:757–768, 2002, Math Res Lett 8:249–255, 2001).     

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).     

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).     

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.     

Inversion of Catalan matrix plus one

Motivated by a statistical problem, in (Aggarwala and Lamoureux in Am. Math. Mon., 109:371–377, 2002) it is shown how to invert a linear combination of the Pascal matrix with the identity matrix. Continuing this idea, we invert various linear combinations of the Catalan matrix, introduced in (Stanimirovi? et al. in Appl. Math. Comput. 215:796–805, 2009), with the identity matrix. In (Aggarwala and Lamoureux in Am. Math. Mon., 109:371–377, 2002) the occurrence of the polylogarithm function is observed. Inverses of linear combinations of the Catalan and the identity matrix are expressed in terms of Catalan numbers, the pochhammer function and the generalized hypergeometric function.     

Approximating common fixed points of non-self asymptotically nonexpansive mapping in Banach spaces

In this paper, we consider a composite iterative algorithm with errors for approximating a common fixed points of non-self asymptotically nonexpansive mappings in the framework of Hilbert spaces. Our results improve and extend Chidume et al. (J. Math. Anal. Appl. 280:364–374, [2003]), Shahzad (Nonlinear Anal. 61:1031–1039, [2005]), Su and Qin (J. Appl. Math. Comput. 24:437–448, [2007]) and many others.     

The von Kármán theory for incompressible elastic shells

We rigorously derive the von Kármán shell theory for incompressible materials, starting from the 3D nonlinear elasticity. In case of thin plates, the Euler-Lagrange equations of the limiting energy functional reduce to the incompressible version of the classical von Kármán equations, obtained formally in the limit of Poisson’s ratio ν → 1/2. More generally, the midsurface of the shell to which our analysis applies, is only assumed to have the following approximation property: ${\mathcal C^3}$ first order infinitesimal isometries are dense in the space of all W ^2,2 infinitesimal isometries. The class of surfaces with this property includes: subsets of ${\mathbb R^2}$ , convex surfaces, developable surfaces and rotationally invariant surfaces. Our analysis relies on the methods and extends the results of Conti and Dolzmann (Calc Var PDE 34:531–551, 2009, Lewicka et al. (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX:253–295, 2010, Friesecke et al. (Comm. Pure. Appl. Math. 55, no. 2, 1461–1506, 2002).     

A new approach to the description of one-parameter groups of formal power series in one indeterminate

The aim of the paper is to describe one-parameter groups of formal power series, that is to find a general form of all homomorphisms \({\Theta_G : G \to \Gamma}\) , \({\Theta_G(t) = \sum_{k=1}^{\infty} c_k(t)X^k}\) , \({c_1 : G \to \mathbb{K} \setminus\{0\}}\) , \({c_k : G \to \mathbb{K}}\) for k ≥ 2, from a commutative group (G, + ) into the group \({(\Gamma, \circ)}\) of invertible formal power series with coefficients in \({\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}}\) . Considering one-parameter groups of formal power series and one-parameter groups of truncated formal power series, we give explicit formulas for the coefficient functions c _k with more details in the case where either c ₁ = 1 or c ₁ takes infinitely many values. Here we give the results much more simply than they were presented in Jab?oński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008). Also the case im c ₁ = E _m (here E _m stands for the group of all complex roots of order m of 1), not considered in Jab?oński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008), will be discussed.     

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).     

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).     

The Local Langlands Correspondence for GL n over p-adic fields

We extend our methods from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL _n over p-adic fields as well as the existence of ?-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497–544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) describing the inertia-invariant nearby cycles in certain regular situations.