Program extraction for 2-random reals

Let ${2-\textsf{RAN}}$ be the statement that for each real X a real 2-random relative to X exists. We apply program extraction techniques we developed in Kreuzer and Kohlenbach (J. Symb. Log. 77(3):853–895, 2012. doi:10.2178/jsl/1344862165), Kreuzer (Notre Dame J. Formal Log. 53(2):245–265, 2012. doi:10.1215/00294527-1715716) to this principle. Let ${{\textsf{WKL}_0^\omega}}$ be the finite type extension of ${\textsf{WKL}_0}$ . We obtain that one can extract primitive recursive realizers from proofs in ${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN}}$ , i.e., if ${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN} \, {\vdash} \, \forall{f}\, {\exists}{x} A_{qf}(f,x)}$ then one can extract from the proof a primitive recursive term t(f) such that ${A_{qf}(f,t(f))}$ . As a consequence, we obtain that ${{\textsf{WKL}_0}+ \Pi^0_1 - {\textsf{CP}} + 2-\textsf{RAN}}$ is ${\Pi^0_3}$ -conservative over ${\textsf{RCA}_0}$ .     

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.     

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).     

The General Solution of Abel-Type Functional Equations

The general measurable solution of (A) was found by Stamate [8]. Aczél [3] and Lajkô [6] proved that the general solution of (A) for unknown functions ψ, g, h: ? → ? are (1), (2) and (3), respectively. Filipescu [5] found the general measurable solution of (B). We establish an elementary prof for the general solution of equation (A) (Theorem 1.). Our method is suitable for finding the general solution of (B) (Theorem 2.).     

Jensen’s inequality on convex spaces

We prove a Jensen’s inequality on $p$ -uniformly convex space in terms of $p$ -barycenters of probability measures with $(p-1)$ -th moment with $p\in ]1,\infty [$ under a geometric condition, which extends the results in Kuwae (Jensen’s inequality over CAT $(\kappa )$ -space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, 2009) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, 2001) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, 2003). As an application, we give a Liouville’s theorem for harmonic maps described by Markov chains into $2$ -uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen’s inequality over Banach spaces is also presented.     

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).     

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- \({\Sigma^0_2}\) .     

A note on the global convergence theorem of the scaled conjugate gradient algorithms proposed by Andrei

In (Andrei, Comput. Optim. Appl. 38:402?C416, 2007), the efficient scaled conjugate gradient algorithm SCALCG is proposed for solving unconstrained optimization problems. However, due to a wrong inequality used in (Andrei, Comput. Optim. Appl. 38:402?C416, 2007) to show the sufficient descent property for the search directions of SCALCG, the proof of Theorem?2, the global convergence theorem of SCALCG, is incorrect. Here, in order to complete the proof of Theorem?2 in (Andrei, Comput. Optim. Appl. 38:402?C416, 2007), we show that the search directions of SCALCG satisfy the sufficient descent condition. It is remarkable that the convergence analyses in (Andrei, Optim. Methods Softw. 22:561?C571, 2007; Eur. J. Oper. Res. 204:410?C420, 2010) should be revised similarly.     

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).     

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).     

On elementary equivalence in fuzzy predicate logics

Our work is a contribution to the model theory of fuzzy predicate logics. In this paper we characterize elementary equivalence between models of fuzzy predicate logic using elementary mappings. Refining the method of diagrams we give a solution to an open problem of Hájek and Cintula (J Symb Log 71(3):863–880, 2006, Conjectures 1 and 2). We investigate also the properties of elementary extensions in witnessed and quasi-witnessed theories, generalizing some results of Section 7 of Hájek and Cintula (J Symb Log 71(3):863–880, 2006) and of Section 4 of Cerami and Esteva (Arch Math Log 50(5/6):625–641, 2011) to non-exhaustive models.     

Lyapunov stability for semigroup actions

In this paper we introduce a theory of Lyapunov stability of sets for semigroup actions on Tychonoff spaces. We also present the main properties and the main results relating these new concepts. We generalize several concepts and results of Lyapunov stable sets from Bhatia and Hajek (Local Semi-Dynamical Systems. Lecture Notes in Mathematics, vol. 90. Springer, Berlin, 1969), Bhatia and Szegö (Dynamical Systems: Stability Theory and Applications. Lecture Notes in Mathematics, vol. 35. Springer, Berlin, 1967; and Stability Theory of Dynamical Systems. Springer, Berlin, 1970).     

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].     

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).     

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.     

A Weighted Generalization of the Classical Kantorovich Operator. II: Saturation

In [1], we have introduced a new weighted type of modification of the classical Kantorovich operator. The advantage of this operator is that there is no restriction on the parameters of the weight, and the class of functions is wider than in the earlier version of the weighted operator (cf. the monograph of Ditzian and Totik [3]). Direct and converse theorems and a Voronovskaya-type relation were proved. Here we solve the saturation problem of the operator (Theorem 2.1). We follow the method developed in [3], but the details are much more involved. A surprising fact emerges in determining the trivial class of saturation (Theorem 3.1).     

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 average values of convex functions

The aim of this paper is to give an extension of an inequality proved by Wulbert (Math Comput Model 37:1383–1391, 2003, Lemma 2.5) and to define Stolarsky type means as an application of this inequality. Further, we discuss some properties of averages of a continuous convex function, some consequences of a double inequality given by Wulbert (Math Comput Model 37:1383–1391, 2003, Theorem 3.3) and obtain improvement results of Wulbert (Math Comput Model 37:1383–1391, 2003, Corollary 4.3).