Higher-order cone-pseudoconvex, quasiconvex and other related functions in vector optimization

Recently Bhatia (Optim. Lett. doi:10.1007/s11590-010-0248-0, 2010) introduced higher-order cone-convex functions and used them to obtain higher-order sufficient optimality conditions and duality results for a vector optimization problem over cones. The concepts of higher-order (strongly) cone-pseudoconvex and cone-quasiconvex functions were also defined by Bhatia (Optim. Lett. doi:10.1007/s11590-010-0248-0, 2010). In this paper we introduce the notions of higher-order naturally cone-pseudoconvex, strictly cone-pseudoconvex and weakly cone-quasiconvex functions and study various interrelations between the above mentioned functions. Higher-order sufficient optimality conditions have been established by using these functions. Generalized Mond–Weir type higher-order dual is formulated and various duality results have been established under the conditions of higher-order strongly cone-pseudoconvexity and higher-order cone quasiconvexity.     

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.     

Remark on Itô’s Diffusion in Multidimensional Scattering with Sign-Indefinite Potentials

This paper extends some results of Denisov and Kupin (Int Math Res Not, doi:10.1093/imrn/rnr131, 2011) to the case of sign–indefinite potentials by applying methods developed in Denisov (J Funct Anal 254:2186–2226, 2008). This enables us to prove the presence of a.c. spectrum for the generic coupling constant.     

A note “On some primary subgroups and the structure of finite groups”

We point out an error in the paper (Monatsh Math. doi:10.1007/s00605-013-0492-3, 2013) by Kang, and give new brief proofs of the main results.     

Privacy-preserving horizontally partitioned linear programs with inequality constraints

In this paper we solve the open problem, finding the solutions for privacy-preserving horizontally partitioned linear programs with inequality constraints, proposed recently by Mangasarian (Optim Lett 2011, doi:10.1007/s11590-010-0268-9).     

Minimization of vectors of curvilinear functionals on the second order jet bundle: sufficient efficiency conditions

Strongly motivated by its possible applications in Mechanics, in our previous work (Pitea and Postolache (Optim. Lett. doi:10.1007/s11590-010-0272-0, 2011)), we initiated an optimization theory for the second order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multi-time multi-objective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary conditions. As natural continuation of our results in Pitea and Postolache (Optim. Lett. doi:10.1007/s11590-010-0272-0, 2011), the present work introduces a study of sufficient efficiency conditions. While the background in Sect. 2 is introductory, the theory in Sect. 3 is new as a whole, containing our results.     

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.     

Implicit and explicit algorithms for solving the split feasibility problem

Very recently, Dang and Gao (Inverse Probl 27:015007, 2011) introduced a KM-CQ algorithm with strong convergence for the split feasibility problem. In this paper, we will continue to consider the split feasibility problem. We present two algorithms. First, we introduce an implicit algorithm. Consequently, by discretizing the continuous implicit algorithm, we obtain an explicit algorithm. Under some weaker conditions, we show the strong convergence of presented algorithms to some solution of the split feasibility problem which solves some special variational inequality. As special cases, we obtain two algorithms which converge strongly to the minimum norm solution of the split feasibility problem. Results obtained in this paper include the corresponding results of Dang and Gao (2011) and extend a recent result of Wang and Xu (J Inequalities Appl 2010, doi:10.1155/2010/102085).     

Directional Lipschitzness of Minimal Time Functions in Hausdorff Topological Vector Spaces

In a general Hausdorff topological vector space E, we associate to a given nonempty closed set S???E and a bounded closed set Ω???E, the minimal time function T _S,Ω defined by $T_{S,\Omega}(x):= \inf \{ t> 0: S\cap (x+t\Omega)\not = \emptyset\}$ . The study of this function has been the subject of various recent works (see Bounkhel (2012, submitted, 2013, accepted); Colombo and Wolenski (J Global Optim 28:269–282, 2004, J Convex Anal 11:335–361, 2004); He and Ng (J Math Anal Appl 321:896–910, 2006); Jiang and He (J Math Anal Appl 358:410–418, 2009); Mordukhovich and Nam (J Global Optim 46(4):615–633, 2010) and the references therein). The main objective of this work is in this vein. We characterize, for a given Ω, the class of all closed sets S in E for which T _S,Ω is directionally Lipschitz in the sense of Rockafellar (Proc Lond Math Soc 39:331–355, 1979). Those sets S are called Ω-epi-Lipschitz. This class of sets covers three important classes of sets: epi-Lipschitz sets introduced in Rockafellar (Proc Lond Math Soc 39:331–355, 1979), compactly epi-Lipschitz sets introduced in Borwein and Strojwas (Part I: Theory, Canad J Math No. 2:431–452, 1986), and K-directional Lipschitz sets introduced recently in Correa et al. (SIAM J Optim 20(4):1766–1785, 2010). Various characterizations of this class have been established. In particular, we characterize the Ω-epi-Lipschitz sets by the nonemptiness of a new tangent cone, called Ω-hypertangent cone. As for epi-Lipschitz sets in Rockafellar (Canad J Math 39:257–280, 1980) we characterize the new class of Ω-epi-Lipschitz sets with the help of other cones. The spacial case of closed convex sets is also studied. Our main results extend various existing results proved in Borwein et al. (J Convex Anal 7:375–393, 2000), Correa et al. (SIAM J Optim 20(4):1766–1785, 2010) from Banach spaces and normed spaces to Hausdorff topological vector spaces.     

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

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).     

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).     

Weyl modules and q-Whittaker functions

Let $G$ be a semi-simple simply connected group over $\mathbb {C}$ . Following Gerasimov et al. (Comm Math Phys 294:97–119, 2010) we use the $q$ -Toda integrable system obtained by quantum group version of the Kostant–Whittaker reduction (cf. Etingof in Am Math Soc Trans Ser 2:9–25, 1999, Sevostyanov in Commun Math Phys 204:1–16, 1999) to define the notion of $q$ -Whittaker functions $\varPsi _{\check{\lambda }}(q,z)$ . This is a family of invariant polynomials on the maximal torus $T\subset G$ (here $z\in T$ ) depending on a dominant weight $\check{\lambda }$ of $G$ whose coefficients are rational functions in a variable $q\in \mathbb {C}^*$ . For a conjecturally the same (but a priori different) definition of the $q$ -Toda system these functions were studied by Ion (Duke Math J 116:1–16, 2003) and by Cherednik (Int Math Res Notices 20:3793–3842, 2009) [we shall denote the $q$ -Whittaker functions from Cherednik (Int Math Res Notices 20:3793–3842, 2009) by $\varPsi '_{\check{\lambda }}(q,z)$ ]. For $G=SL(N)$ these functions were extensively studied in Gerasimov et al. (Comm Math Phys 294:97–119, 2010; Comm Math Phys 294:121–143, 2010; Lett Math Phys 97:1–24, 2011). We show that when $G$ is simply laced, the function $\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}$ (here $I$ denotes the set of vertices of the Dynkin diagram of $G$ ) is equal to the character of a certain finite-dimensional $G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*$ -module $D(\check{\lambda })$ (the Demazure module). When $G$ is not simply laced a twisted version of the above statement holds. This result is known for $\varPsi _{\check{\lambda }}$ replaced by $\varPsi '_{\check{\lambda }}$ (cf. Sanderson in J Algebraic Combin 11:269–275, 2000 and Ion in Duke Math J 116:1–16, 2003); however our proofs are algebro-geometric [and rely on our previous work (Braverman, Finkelberg in Semi-infinite Schubert varieties and quantum $K$ -theory of flag manifolds, arXiv/1111.2266, 2011)] and thus they are completely different from Sanderson (J Algebraic Combin 11:269–275, 2000) and Ion (Duke Math J 116:1–16, 2003) [in particular, we give an apparently new algebro-geometric interpretation of the modules $D(\check{\lambda })]$ .     

Syntactic Rings

If the state set and the input set of an automaton are Ω-groups then near-rings are useful in the study of automata (see [5]). These near-rings, called syntactic near-rings, consist of mappings from the state set Q of the automaton into itself. If, as is often the case, Q bears the structure of a module, then the zerosymmetric part N₀(A) of syntactic near-rings is a commutative ring with identity. If N₀(A) is a syntactic ring then its ideals are useful for determining reachability in automata (see [1] or [2]). In this paper we investigate syntactic rings.     

Convergence of Pham Dinh–Le Thi’s algorithm for the trust-region subproblem

It is proved that any DCA sequence constructed by Pham Dinh–Le Thi’s algorithm for the trust-region subproblem (Pham Dinh and Le Thi, in SIAM J. Optim. 8:476–505, 1998) converges to a Karush–Kuhn–Tucker point of the problem. This result provides a complete solution for one open question raised by Le Thi et al. (J. Global Optim., Online First, doi:10.1007/s10898-011-9696-z, 2010).     

Growth,entropy and commutativity of algebras satisfying prescribed relations

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfies some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod–Shafarevich algebras. This paper provides bounds for the growth function on images of Golod–Shafarevich algebras based upon the number of defining relations. This extends results from Smoktunowicz and Bartholdi (Q J Math. doi:10.1093/qmath/hat005 2013) and Smoktunowicz (J Algebra 381:116–130, 2013). Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky (A private communication, 2013) by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov (2013). Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss (Noncommutative deformations and flops, ArXiv:1309.0698v2 [math.AG]).     

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).     

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).     

Relaxation methods for solving linear inequality systems: converging results

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. In González-Gutiérrez and Todorov (Optim. Lett. doi:10.1007/s11590-010-0244-4, 2011), an algorithm, called extended relaxation method, for solving the feasibility problem has been proposed by the authors. Convergence of the algorithm has been proven. In this paper, we consider a class of extended relaxation methods depending on a parameter and prove their convergence. Numerical experiments have been provided, as well.