O $$\varvec{}$$-orm preservers ad the Aleksadrov coservative $$\varvec{}$$-distace problem

The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for \(n \ge 3\) any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine (\(n \ge 2\)) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and ?emrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions (\(n\ge 2\)) is automatically an n-isometry.     

The Reich–Zaslavski property and fixed points of non-self multivalued mappings

Let (X, d) be a metric space, Y be a nonempty subset of X, and let \(T:Y \rightarrow P(X)\) be a non-self multivalued mapping. In this paper, by a new technique we study the fixed point theory of multivalued mappings under the assumption of the existence of a bounded sequence \((x_n)_n\) in Y such that \(T^nx_n\subseteq Y,\) for each \(n \in \mathbb {N}\). Our main result generalizes fixed point theorems due to Matkowski (Diss. Math. 127, 1975), W?grzyk (Diss. Math. (Rozprawy Mat.) 201, 1982), Reich and Zaslavski (Fixed Point Theory 8:303–307, 2007), Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and provides a solution to the problems posed in Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and Rus and ?erban (Miskolc Math. Notes 17:1021–1031, 2016).     

Congruences of two-variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions of congruent modular forms of different weights

Vatsal (Duke Math J 98(2):397–419, 1999) proved that there are congruences between the p-adic L-functions (constructed by Mazur and Swinnerton-Dyer in Invent Math 25:1–61, 1974) of congruent modular forms of the same weight under some conditions. On the other hand, Kim (J Number Theory 144: 188–218, 2014), the second author, constructed two-variable p-adic L-functions of modular forms attached to imaginary quadratic fields generalizing Hida’s work (Invent Math 79:159–195, 1985), and the novelty of his construction was that it works whether p is an ordinary prime or not. In this paper, we prove congruences between the two-variable p-adic L-functions (of the second author) of congruent modular forms of different but congruent weights under some conditions when p is a nonordinary prime for the modular forms. This result generalizes the work of Emerton et al. (Invent Math 163(3): 523–580, 2006), who proved similar congruences between the p-adic L-functions of congruent modular forms of congruent weights when p is an ordinary prime.     

Fourier transforms and bent functions on faithful actions of finite abelian groups

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.     

Robust return risk measures

In this paper we provide an axiomatic foundation to Orlicz risk measures in terms of properties of their acceptance sets, by exploiting their natural correspondence with shortfall risk Föllmer and Schied (Stochastic finance. De Gruyter, Berlin, 2011), thus paralleling the characterization in Weber (Math Financ 16:419–442, 2006). From a financial point of view, Orlicz risk measures assess the stochastic nature of returns, in contrast to the common use of risk measures to assess the stochastic nature of a position’s monetary value. The correspondence with shortfall risk leads to several robustified versions of Orlicz risk measures, and of their optimized translation invariant extensions (Rockafellar and Uryasev in J Risk 2:21–42, 2000, Goovaerts et al. in Insur Math Econ 34:505–516, 2004), arising from an ambiguity averse approach as in Gilboa and Schmeidler (J Math Econ 18:141–153, 1989), Maccheroni et al. (Econometrica 74:1447–1498, 2006), Chateauneuf and Faro (J Math Econ 45:535–558, 2010), or from a multiplicity of Young functions. We study the properties of these robust Orlicz risk measures, derive their dual representations, and provide some examples and applications.     

Uniform Sublevel Radon-like Inequalities

This paper is concerned with establishing uniform weighted L ^p –L ^q estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a sublevel-type inequality is satisfied by certain associated measures (the inequality is of the sort studied by Oberlin (Math. Proc. Camb. Philos. Soc. 129(3):517–526, 2000), relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new L ^p -improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger (J. Reine Angew. Math. 455:35–56, 1994) for FIOs (building on the ideas of Sogge (Invent. Math. 104(2):349–376, 1991) and Mockenhaupt et al. (J. Am. Math. Soc. 6(1):65–130, 1993)); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.     

A brief note on the coarea formula

In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into \({\mathbb {R}}\)) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions.     

Reformulation of Hensel’s Lemma and extension of a theorem of Ore

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.     

A Nonlinear Age-Structured Model of Population Dynamics with Inherited Properties

In this paper, we present some results regarding existence and uniqueness of solution on L ^p -spaces, 1 < p < + ∞, to a nonlinear initial boundary value problem originally proposed by Lebowitz and Rubinow (J Math Biol 1:17–36, 1974) to model an age-structured cell population with inherited properties. Our results complete those obtained by Garcia-Falset (Math Meth Appl Sci 34:1658–1666, 2011).     

On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonic Weyl curvature, i.e., \(\delta W=0\). Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product \( \mathbb {R}^2 \times N_{\lambda }\) of the Euclidean metric and a 2-d Riemannian manifold of constant curvature \({\lambda } \ne 0\), a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao–Chen’s works (in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013) and Derdziński’s study on Codazzi tensors (in Math Z 172:273–280, 1980). Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with \(\delta W=0\). For the shrinking case, it re-proves the rigidity result (Fernández-López and García-Río in Math Z 269:461–466, 2011; Munteanu and Sesum in J. Geom Anal 23:539–561, 2013) in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally flat ones with \(\delta W=0\). We also characterize locally 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.     

On asymptotically equivalent difference sequences with respect to a modulus function

This paper presents new definitions which are a natural combination of the definition for asymptotically equivalence and Δ ^m -lacunary strongly summable with respect to a modulus f. Using this definitions we have proved the (f, Δ ^m )-asymptotically equivalence and Δ ^m -lacunary statistical asymptotically equivalence analogues of theorems of Tripathy and Et (Stud Univ Babe?-Bolyai Math (1):119–130, 2005) and Çolak’s theorems (Filomat 17:9–14, 2003).     

Taut contact circles and bi-contact metric structures on three-manifolds

Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), \({\widetilde{SL}}(2,{\mathbb {R}})\), \({\widetilde{E}}(2)\), E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure \((\eta _1,\eta _2)\) with the so-called Webster function \({\mathcal {W}}\) constant along the flow of \(\xi _1\) (equivalently \(\xi _2\)). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).     

Spectral transfer morphisms for unipotent affine Hecke algebras

We classify the spectral transfer morphisms (cf. Opdam in Adv Math 286:912–957, 2016) between affine Hecke algebras associated to the unipotent types of the various inner forms of an unramified absolutely simple algebraic group G defined over a non-archimedean local field k. This turns out to characterize Lusztig’s classification (Lusztig in Int Math Res Not 11:517–589, 1995; in Represent Theory 6:243–289, 2002) of unipotent characters of G in terms of the Plancherel measure, up to diagram automorphisms. As an application of these results, the spectral correspondences associated with such morphisms (Opdam 2016), and some results of Ciubotaru, Kato and Kato [CKK] (also see Ciubotaru and Opdam in A uniform classification of the discrete series representations of affine Hecke algebras. arXiv:1510.07274) we prove a conjecture of Hiraga, Ichino and Ikeda [HII] on formal degrees and adjoint gamma factors in the special case of unipotent discrete series characters of inner forms of unramified simple groups of adjoint type defined over k.     

Snarks,Hypohamiltonian Graphs and Non-Supereulerian Graphs

A graph G is hypohamiltonian if it is not Hamiltonian but for each \(v\in V(G)\), the graph \(G-v\) is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset \(R\subseteq V(G)\), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173–181, 1990] and in [J. Combin. Theory, Ser B 66:123–139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291–304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Université Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.     

Truncation error estimates for generalized Hermite sampling

The generalized Hermite sampling uses samples from the function itself and its derivatives up to order r. In this paper, we investigate truncation error estimates for the generalized Hermite sampling series on a complex domain for functions from Bernstein space. We will extend some known techniques to derive those estimates and the bounds of Jagerman (SIAM J. Appl. Math. 14, 714–723 1966), Li (J. Approx. Theory 93, 100–113 1998), Annaby-Asharabi (J. Korean Math. Soc. 47, 1299–1316 2010), and Ye and Song (Appl. Math. J. Chinese Univ. 27, 412–418 2012) will be special cases for our results. Some examples with tables and figures are given at the end of the paper.     

Improved local convergence analysis of the Gauss–Newton method under a majorant condition

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.     

Extensions of the Noncommutative Integration

In this paper we will continue the analysis undertaken in Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), Bongiorno et al. (Rocky Mt J Math 40(6):1745–1777, 2010), Triolo (Rend Circ Mat Palermo (2) 60(3):409–416, 2011) on the general problem of extending the noncommutative integration in a *-algebra of measurable operators. As in Aiena et al. (Filomat 28(2):263–273, 2014), Bagarello (Stud Math 172(3):289–305, 2006) and Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), the main problem is to represent different types of partial *-algebras into a *-algebra of measurable operators in Segal’s sense, provided that these partial *-algebras posses a sufficient family of positive linear functionals (states) (Fragoulopoulou et al., J Math Anal Appl 388(2):1180–1193, 2012; Trapani and Triolo, Stud Math 184(2):133–148, 2008; Trapani and Triolo, Rend Circolo Mat Palermo 59:295–302, 2010; La Russa and Triolo, J Oper Theory, 69:2, 2013; Triolo, J Pure Appl Math, 43(6):601–617, 2012). In this paper, a new condition is given in an attempt to provide a extension of the non commutative integration.     

Toeplitz Operators with Uniformly Continuous Symbols

Let T _f be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain \({\Omega \subset {\bf C}^n}\) with possibly unbounded symbol f. Combining recent results in Bauer et al. (J. Funct. Anal. 259:57–78, 2010), Bauer et al. (J. reine angew. Math. doi: 10.1515/crelle-2015-0016), Issa (Integr. Equ. Oper. Theory 70:569–582, 2011) we show that in the case of uniformly continuous symbols f with respect to the Euclidean metric on C ⁿ and the Bergman metric on \({\Omega}\), respectively, the operator T _f is bounded if and only if f is bounded. Moreover, T _f is compact if and only if f vanishes at the boundary of \({\Omega.}\) This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).     

Arc-Transitive Pentavalent Graphs of Square-Free Order

Hua et al. (Discrete Math 311, 2259–2267, 2011) and Yang et al. (Discrete Math. 339, 522–532, 2016) classify arc-transitive pentavalent graphs of order 2pq and of order 2pqr (with p, q, r distinct odd primes), respectively. In this paper, we extend their results by giving a classification of arc-transitive pentavalent graphs of any square-free order.