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).     

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).     

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- \({\Sigma^0_2}\) .     

Parametric Cutoffs for Interacting Fermi Liquids

This paper is a sequel to Disertori et al. (Annales Henri Poincaré 2, 733–806, 2001). We introduce a new multiscale decomposition of the Fermi propagator based on its parametric representation. We prove that the corresponding sliced propagator obeys the same direct space bounds than the decomposition used in Disertori et al. (Annales Henri Poincaré 2, 733–806, 2001). Therefore the non perturbative bounds on completely convergent contributions of Disertori et al. (Annales Henri Poincaré 2, 733–806, 2001) still hold. In addition the new slicing better preserves momenta, hence should become an important new technical tool for the rigorous analysis of condensed matter systems. In particular it should allow to complete the proof that a three dimensional interacting system of Fermions with spherical Fermi surface is a Fermi liquid in the sense of Salmhofer’s criterion.     

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.     

Efficient model for interval goal programming with arbitrary penalty function

Penalty function is a key factor in interval goal programming (IGP), especially for decision makers weighing resources vis-à-vis goals. Many approaches (Chang et al. J Oper Res Soc 57:469–473, 2006; Chang and Lin Eur J Oper Res 199, 9–20, 2009; Jones et al. Omega 23, 41–48, 1995; Romero Eur J Oper Res 153, 675–686, 2004; Vitoriano and Romero J Oper Res Soc 50, 1280–1283, 1999)have been proposed for treating several types of penalty functions in the past several decades. The recent approach of Chang and Lin (Eur J Oper Res 199, 9–20, 2009) considers the S-shaped penalty function. Although there are many approaches cited in literature, all are complicated and inefficient. The current paper proposes a novel and concise uniform model to treat any arbitrary penalty function in IGP. The efficiency and usefulness of the proposed model are demonstrated in several numeric examples.     

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.     

The numerical range of positive operators on Hilbert lattices

We show symmetry properties of the numerical range of positive operators on Hilbert lattices. These results generalise the respective properties for positive matrices shown in Li et al. (Linear Algebra Appl 350:1–23, 2002) and Maroulas et al. (Linear Algebra Appl 348:49–62, 2002). Similar assertions are also valid for the block numerical range of positive operators.     

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).     

The best papers from BRIMS 2011: models of users and teams interacting

This special issue is similar to our previous special issues (Kennedy et al. in Comput. Math. Organ. Theory 16(3):217–219, 2010; 17(3):225–228, 2011) in that it includes articles based on the award winning conference papers of the, here, 2011 BRiMS Annual Conference. These articles were reviewed by the editors, extended to journal article length, and then peer-reviewed and revised before being accepted. The articles include a new way to evaluate designs of interfaces for safety critical systems (Bolton in Comput. Math. Organ. Theory, 2012), an article that extends our understanding of how to model situation awareness (SA) in a cognitive architecture (Rodgers et al. in Comput. Math. Organ. Theory, 2012), an article that presents electroencephalography (EEG) data used to derive dynamic neurophysiologic models of engagement in teamwork (Stevens et al. in Comput. Math. Organ. Theory, 2012), and an article that demonstrates using machine learning to generate models and an example application of that tool (Best in Comput. Math. Organ. Theory, 2012). After presenting a brief summary of each paper we will see some recurrent themes of task analysis, team and individual models, spatial reasoning, usability issues, and particularly that they are models that interact with each other or systems.     

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.     

On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem

In this paper, under new assumptions, which do not contain any information about the solution set, the lower semicontinuity of solution mappings to a parametric generalized strong vector equilibrium problem are established by using a scalarization method. These results extend and generalize the corresponding ones in Gong and Yao (J Optim Theory Appl 138:197–205, 2008), Chen and Li (Pac J Optim 6:141–151, 2010) and Li et al. (2012, submitted). Some examples are given to illustrate our results.     

Two Equivalent Properties of \mathcal{Z}_3-Connectivity

The concept of group connectivity was introduced by Jaeger et al. (J Comb Theory Ser B 56:165–182, 1992) for the study of integer flows. The concept of all generalized Tutte-orientations was introduced by Barát and Thomassen (J Graph Theory 52:135–146, 2006) for the study of claw-decompositions of graphs. In this paper, we establish the equivalence of the following 3 properties: a graph is $\mathcal{Z}_3$ -connected, a graph admits all generalized Tutte-orientations and a graph is 3-flow contractible. We also give some applications of this result.     

Averaging Fluctuations in Resolvents of Random Band Matrices

We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of Erd?s et al. (Ann Probab, arXiv:1103.1919, 2013; Commun Math Phys, arXiv:1103.3869, 2013; J Combin 1(2):15–85, 2011) which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices and improve previous estimates from order 2 to order 4 in the cases relevant to applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper (Erd?s et al., arXiv:1205.5669, 2013).     

Layer Lengths, Torsion Theories and the Finitistic Dimension

Let $\mathcal{C}$ be a length-category. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for objects of $\mathcal{C}$ . As an application, we use the layer lengths and the Igusa–Todorov function to get a theorem (see Theorem 6.4) having as corollaries the main results of Huard et al. (Bull Lond Math Soc 41:367–376, 2009) and Wang (Commun Algebra 22(7):419–449, 1994).     

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.     

Greedy Algorithms for Reduced Bases in Banach Spaces

Given a Banach space X and one of its compact sets $\mathcal{F}$ , we consider the problem of finding a good n-dimensional space X _n ?X which can be used to approximate the elements of $\mathcal{F}$ . The best possible error we can achieve for such an approximation is given by the Kolmogorov width $d_{n}(\mathcal{F})_{X}$ . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) in the case $X=\mathcal{H}$ is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457–1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.     

An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of common fixed points of an infinite family of notself strict pseudocontractions and the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings in q-uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results improve and extend the recent results announced by Qin et al. (J Comput Appl Math 233:231–240, 2009), Yao et al. (Acta Appl Math 110:1211–1244, 2010) and many others.     

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).