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).     

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.     

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- \({\Sigma^0_2}\) .     

F-harmonic maps as global maxima

In this note, we show that some $F$ -harmonic maps into spheres are global maxima of the variations of their energy functional on the conformal group of the sphere. Our result extends partially those obtained in El Soufi and Lejune [C.R.A.S. 315(Serie I):1189–1192, 1992] and El Soufi [Compositio Math 95:343–362,1995] for harmonic and $p$ -harmonic maps.     

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).     

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.     

Counterexamples to a triality theorem for quadratic-exponential minimization problems

The main goal of this note is to give a counterexample to the Triality Theorem in Gao and Ruan (Math Methods Oper Res 67:479–491, 2008). This is done first by considering a more general optimization problem with the aim to encompass several examples from Gao and Ruan (Math Methods Oper Res 67:479–491, 2008) and other papers by Gao and his collaborators (see f.i. Gao Duality principles in nonconvex systems. Theory, methods and applications. Kluwer, Dordrecht, 2000; Gao and Sherali Advances in applied mathematics and global optimization. Springer, Berlin, 2009). We perform a thorough analysis of the general optimization problem in terms of local extrema while presenting several counterexamples.     

On weak D-solutions to the non-stationary Navier–Stokes equations in a three-dimensional exterior domain

In this note we study the Navier–Stokes initial boundary value problem in exterior domains. We assume that the initial data has just finite Dirichlet norm. We call the solution \(D\) -solution. It is well known that the analogous steady problem is solved in Galdi (An Introduction to the Mathematical Theory of the Navier–Stokes Equations II. Springer, Berlin, 1994), as well as the existence of time periodic solutions in Maremonti et al. (J Math Sci 93(5):719–746, 1999, Zap. Nauchn. Semin. POMI 233:142–182, 1996). So it is natural to inquire about the case of the nonstationary problem.     

Tilting Theory and Functor Categories I. Classical Tilting

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel (1988) and Cline et al. (J Algebra 304:397–409 1986) proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard (J Lond Math Soc 39:436–456, 1989) to develop a general Morita theory of derived categories. On the other hand, functor categories were introduced in representation theory by Auslander (I Commun Algebra 1(3):177–268, 1974), Auslander (1971) and used in his proof of the first Brauer–Thrall conjecture (Auslander 1978) and later on, used systematically in his joint work with I. Reiten on stable equivalence (Auslander and Reiten, Adv Math 12(3):306–366, 1974), Auslander and Reiten (1973) and many other applications. Recently, functor categories were used in Martínez-Villa and Solberg (J Algebra 323(5):1369–1407, 2010) to study the Auslander–Reiten components of finite dimensional algebras. The aim of this paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod (mod_Λ), with Λ a finite dimensional algebra.     

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 strong convergence of a three-step algorithm for the split feasibility problem

Based on the very recent work by Dang and Gao (Invers Probl 27:1–9, 2011) and Wang and Xu (J Inequal Appl, doi:10.1155/2010/102085, 2010), and inspired by Yao (Appl Math Comput 186:1551–1558, 2007), Noor (J Math Anal Appl 251:217–229, 2000), and Xu (Invers Probl 22:2021–2034, 2006), we suggest a three-step KM-CQ-like method for solving the split common fixed-point problems in Hilbert spaces. Our results improve and develop previously discussed feasibility problem and related algorithms.     

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).     

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).     

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.     

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61–97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy–Boussinesq equations, which is a degenerate version of the Navier–Stokes–Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge–Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375–417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141–1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131–151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20–31, 1991, Arch. Ration. Mech. Anal. 185:341–363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450–1461, 1998; Loeper in SIAM J. Math. Anal. 38:795–823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov–Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29–59, 2001) and of the Keller–Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225–234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819–824, 1992; Chalub et al. in Mon. Math. 142:123–141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier–Stokes–Boussinesq equations. Finally, we show how a “stringy” generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology (see Arnold and Khesin in Topological methods in hydrodynamics. Applied mathematical sciences, vol. 125, Springer, Berlin, 1998; Moffatt in J. Fluid Mech. 159:359–378, 1985, Topological aspects of the dynamics of fluids and plasmas. NATO adv. sci. inst. ser. E, appl. sci., vol. 218, Kluwer, Dordrecht, 1992; Schonbek in Theory of the Navier–Stokes equations, Ser. adv. math. appl. sci., vol. 47, pp. 179–184, World Sci., Singapore, 1998; Vladimirov et al. in J. Fluid Mech. 390:127–150, 1999; Nishiyama in Bull. Inst. Math. Acad. Sin. (N.S.) 2:139–154, 2007).     

When a vector quasimonotone mapping is a vector monotone mapping

In this paper we provide a sufficient conditions that under them a vector quasimonotone set-valued mapping transfer to a vector monotone set-valued mapping. In fact this note is a vector version of the papers (Farajzadeh, J Ineq Appl 2012:192, 2012) and (Hadjisavvas, Appl Math Lett 19:913–915, 2006).     

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.     

Right-convergence of sparse random graphs

The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs \(\mathbb{G }_N, N\ge 1\) to some target graph \(W\) . The theory of dense graph convergence, including random dense graphs, is now well understood (Borgs et al. in Ann Math 176:151–219, 2012; Borgs et al. in Adv Math 219:1801–1851, 2008; Chatterjee and Varadhan in Eur J Comb 32:1000–1017, 2011; Lovász and Szegedy in J Comb Theory Ser B 96:933–957, 2006), but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the limits of appropriately normalized log-partition functions, also known as free energy limits, for the Gibbs distribution associated with \(W\) . In this paper we prove that the sequence of sparse Erdös-Rényi graphs is right-converging when the tensor product associated with the target graph \(W\) satisfies a certain convexity property. We treat the case of discrete and continuous target graphs \(W\) . The latter case allows us to prove a special case of Talagrand’s recent conjecture [more accurately stated as level III Research Problem 6.7.2 in his recent book (Talagrand in Mean Field Models for Spin Glasses: Volume I: Basic examples. Springer, Berlin, 2010)], concerning the existence of the limit of the measure of a set obtained from \(\mathbb{R }^N\) by intersecting it with linearly in \(N\) many subsets, generated according to some common probability law. Our proof is based on the interpolation technique, introduced first by Guerra and Toninelli (Commun Math Phys 230:71–79, 2002) and developed further in (Abbe and Montanari in On the concentration of the number of solutions of random satisfiability formulas, 2013; Bayati et al. in Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010; Contucci et al. in Antiferromagnetic Potts model on the Erdös-Rényi random graph, 2011; Franz and Leone in J Stat Phys 111(3/4):535–564, 2003; Franz et al. in J Phys A Math Gen 36:10967–10985, 2003; Montanari in IEEE Trans Inf Theory 51(9):3221–3246, 2005; Panchenko and Talagrand in Probab Theory Relat Fields 130:312–336, 2004). Specifically, Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) establishes the right-convergence property for Erdös-Rényi graphs for some special cases of \(W\) . In this paper most of the results in Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) follow as a special case of our main theorem.     

The Shapley value for shortest path games: a non-graph-based approach

The shortest path games are considered in this paper. The transportation of a good in a network has costs and benefits. The problem is to divide the profit of the transportation among the players. Fragnelli et al. (Math Methods Oper Res 52: 251–264, 2000) introduce the class of shortest path games and show it coincides with the class of monotone games. They also give a characterization of the Shapley value on this class of games. In this paper we consider further five characterizations of the Shapley value (Hart and Mas-Colell’s in Econometrica 57:589–614, 1989; Shapley’s in Contributions to the theory of games II, annals of mathematics studies, vol 28. Princeton University Press, Princeton, pp 307–317, 1953; Young’s in Int J Game Theory 14:65–72, 1985, Chun’s in Games Econ Behav 45:119–130, 1989; van den Brink’s in Int J Game Theory 30:309–319, 2001 axiomatizations), and conclude that all the mentioned axiomatizations are valid for the shortest path games. Fragnelli et al. (Math Methods Oper Res 52:251–264, 2000)’s axioms are based on the graph behind the problem, in this paper we do not consider graph specific axioms, we take $TU$ axioms only, that is we consider all shortest path problems and we take the viewpoint of an abstract decision maker who focuses rather on the abstract problem than on the concrete situations.     

On the scalarization method in cone metric spaces

Recently, Du (J Nonlinear Anal 72:2259–2261, 2010) by using a nonlinear scalarization function, in the setting of locally convex topological vector spaces, could transfer a cone metric space to a usual metric space. Simultaneously, Amini-Harandi and Fakhar (Com Math Appl 59:3529–3534, 2010) by using a notion of base for the cone , in the setting of Banach spaces, could do the same. In this note we will see that two methods coincide and moreover they are valid for topological vector spaces and it is not necessary that we only consider the cones which have a compact base. Finally, it is worth noting that the nature of this note is similar to Caglar and Ercan (Order-unit-metric spaces, arXiv:1305.6070 [math.FA], 2013).