Exponential polynomials on commutative hypergroups

Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2–4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.     

Burgers type equations, Gelfand’s problem and Schumpeterian dynamics

Burgers?? equations have been introduced to study different models of fluids (Bateman, 1915, Burgers, 1939, Hopf, 1950, Cole, 1951, Lighthill andWhitham, 1955, etc.). The difference-differential analogues of these equations have been proposed for Schumpeterian models of economic development (Iwai, 1984, Polterovich and Henkin, 1988, Belenky, 1990, Henkin and Polterovich, 1999, Tashlitskaya and Shananin, 2000, etc.). This paper gives a short survey of the results and conjectures on Burgers type equations, motivated both by fluid mechanics and by Schumpeterian dynamics. Proofs of some new results are given. This paper is an extension and an improvement of (Henkin, 2007, 2011).     

Computing Final Polynomials and Final Syzygies Using Buchberger’s Gröbner Bases Method

Final polynomials and final syzygies provide an explicit representation of polynomial identities promised by Hilbert’s Nullstellensatz. Such representations have been studied independently by Bokowski [2,3,4] and Whiteley [23,24] to derive invariant algebraic proofs for statements in geometry. In the present paper we relate these methods to some recent developments in computational algebraic geometry. As the main new result we give an algorithm based on B. Buchberger’s Gröbner bases method for computing final polynomials and final syzygies over the complex numbers. Degree upper bound for final polynomials are derived from theorems of Lazard and Brownawell, and a topological criterion is proved for the existence of final syzygies. The second part of this paper is expository and discusses applications of our algorithm to real projective geometry, invariant theory and matrix theory.     

Finite simple 3′-groups are cyclic or Suzuki groups

In this note we prove that all finite simple 3′-groups are cyclic of prime order or Suzuki groups. This is well known in the sense that it is mentioned frequently in the literature, often referring to unpublished work of Thompson. Recently an explicit proof was given by Aschbacher [3], as a corollary of the classification of ${\mathcal{S}_3}$ -free fusion systems. We argue differently, following Glauberman’s comment in the preface to the second printing of his booklet [8]. We use a result by Stellmacher (see [12]), and instead of quoting Goldschmidt’s result in its full strength, we give explicit arguments along his ideas in [10] for our special case of 3′-groups.     

Estimating upper bounds on the limit points of majorizing sequences for Newton’s method

We present new sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of an equation in a Banach space setting. Upper bounds on the limit points of majorizing sequences are also given. Numerical examples are provided, where our new results compare favorably to earlier ones such as Argyros (J Math Anal Appl 298:374–397, 2004), Argyros and Hilout (J Comput Appl Math 234:2993-3006, 2010, 2011), Ortega and Rheinboldt (1970) and Potra and Pták (1984).     

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).     

Quasi-stationary stefan problem with values on the front depending on its geometry

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5–8, 10, 12, 13].     

Error analysis of a hybrid method for computing Lyapunov exponents

In a previous paper (Beyn and Lust in Numer Math 113:357–375, 2009) we suggested a numerical method for computing all Lyapunov exponents of a dynamical system by spatial integration with respect to an ergodic measure. The method extended an earlier approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) for the largest Lyapunov exponent by integrating the diagonal entries from the $QR$ -decomposition of the Jacobian for an iterated map. In this paper we provide an asymptotic error analysis of the method for the case in which all Lyapunov exponents are simple. We employ Oseledec multiplicative ergodic theorem and impose certain hyperbolicity conditions on the invariant subspaces that belong to neighboring exponents. The resulting error expansion shows that one step of extrapolation is enough to obtain exponential decay of errors.     

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.     

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.     

Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping

We consider the nonlinear viscoelastic equation $$u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(\tau)\,d\tau +a(x)|u_{t}|^{m}u_{t}+b|u|^{\gamma }u=0$$ in a bounded domain and establish exponential or polynomial decay result which depend on the rate of the decay of the relaxation function g. This result improves an earlier one given by Berrimi and Messaoudi (Electron. J. Differ. Equ. (88):1–10, 2004).     

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.     

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).     

Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\) . For \(d\le 7\) , all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\) ; see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\) , see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\) ; see Doi et al. (Osaka J. Math. 37:667–687, 2000). In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\) , up to projectivity. For \(d=11,13,17,19\) , this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).     

Second-order properties of risk concentrations without the condition of asymptotic smoothness

For the purpose of risk management, the quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. Consider a portfolio of n independent and identically distributed loss random variables with a common survival function $\overline {F}$ possessing the property of second-order regular variation. Under the additional assumption that $\overline {F}$ is asymptotically smooth, Degen et al. (Insur Math Econ 46:541–546, 2010) and Mao et al. (Insur Math Econ 51:449–456, 2012) derived second-order approximations of the risk concentrations based on the risk measures of Value-at-Risk and conditional tail expectation, respectively. In this paper, we remove the assumption of the asymptotic smoothness, and reestablish the second-order approximations of these two risk concentrations.     

A Note on the Volume Growth Criterion for Stochastic Completeness of Weighted Graphs

This paper gives complementary results of Folz (Trans Am Math Soc, 2013). We first generalize the weak Omori–Yau maximum principle to the setting of strongly local Dirichlet forms. As an application, we obtain an analytic approach to compare the stochastic completeness of a weighted graph with that of an associated metric graph. This comparison result played an essential role in the volume growth criterion of Folz (Trans Am Math Soc, 2013), who first proved it via a probabilistic approach. We also give an alternative analytic proof based on a criterion in Fukushima et al. (1994).     

Period- and mirror-maps for the quartic K3

We study in detail mirror symmetry for the quartic K3 surface in ${\mathbb{P}^3}$ and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison (Mirror symmetry II, 1997), mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with the one defined by Morrison (Essays on mirror manifolds, 1992) near the point of maximal unipotent monodromy. Our results rely on earlier work by Narumiyah–Shiga (Proceedings on Moonshine and related topics, 2001), Dolgachev (J. Math. Sci., 1996) and Nagura–Sugiyama (Int. J. Mod. Phys. A, 1995).     

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).     

Classroom-based interventions in mathematics education: relevance, significance, and applicability

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth 2013); (2) Development of Ways of Thinking (Jahnke and Wambach 2013; Lehrer, Kobiela and Weinberg 2013; Mariotti 2013; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides 2013; Tabach, Hershkowitz and Dreyfus 2013); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart 2013); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann 2013).     

A note on Schanuel’s conjectures for exponential logarithmic power series fields

In Ax (Ann. Math. 93(2):252–268, 1971), J. Ax proved a transcendency theorem for certain differential fields of characteristic zero : the differential counterpart of the still open Schanuel conjecture about the exponential function over ${\mathbb{C}}$ (Lang, Introduction to transcendental numbers, 1966). In this article, we derive from Ax’s theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields, and Exponential-Logarithmic power series fields.