Algebraische Unabhängigkeit der Werte gewisser Lückenreihen in nicht-archimedisch bewerteten Körpern

In 1844 Liouville proved the transcendence of α = ∑_h≥1 10^{?h h!} over Q. The number α can be considered as the value of the gap power series ∧(x) =∑_h≥1 at tne point 1/10 Since then, the above result has been generalized in this direction by different authors by applying improved “Liouville-estimates”. For instance, in 1973 Cijsouw and Tijdeman [2] showed that a gap series with algebraic coefficients takes on transcendental values (over Q) at non-zero algebraic points under some conditions on the growth of the coefficients and the gaps. In 1988 Bundschuh [1] resp. Zhu [9] proved the algebraic independence (over Q) of the values of several gap series at different algebraic points. In particular this result includes the algebraic independence of A(α₁),…, α(α_s) for non-zero algebraic numbers α₁,…, α_s of distinct absolute values less than 1. Moreover in [1] a set of continuum-many algebraically independent numbers was constructed. In 1978 Geijsel [4] obtained a result analogous to that of Cijsouw and Tijdeman underlying a non-archimedian valued function field over a finite field, and in 1983 Sieburg [7] was concerned with the algebraic independence of “Liouville-series” in non-archimedian valued fields of characteristic zero. In this paper some of the results of [1] resp. [9] will be transfered to the situation of some non-archimedian valued fields. If the characteristic of the field is prime, we have to require stronger conditions as in the “classical case”. An example shows that in this case the numbers A(c*i),..., A(aa) need not to be algebraically independent. But a set of continuum-many algebraically independent numbers still exists. In characteristic zero, results of the same kind will be obtained like in the “classical case”.     

Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations

In this paper we present new concepts of efficiency for uncertain multi-objective optimization problems. We analyze the connection between the concept of minmax robust efficiency presented by Ehrgott et al. (Eur J Oper Res, 2014, doi:10.1016/j.ejor.2014.03.013) and the upper set less order relation \(\preceq _s^u\) introduced by Kuroiwa (1998, 1999). From this connection we derive new concepts of efficiency for uncertain multi-objective optimization problems by replacing the set ordering with other set orderings. Those are namely the lower set less ordering (see Kuroiwa 1998, 1999), the set less ordering (see Nishnianidze in Soobshch Akad Nauk Gruzin SSR 114(3):489–491, 1984; Young in Math Ann 104(1):260–290, 1931, doi:10.1007/BF01457934; Eichfelder and Jahn in Vector Optimization. Springer, Berlin, 2012), the certainly less ordering (see Eichfelder and Jahn in Vector Optimization. Springer, Berlin, 2012), and the alternative set less ordering (see Ide et al. in Fixed Point Theory Appl, 2014, doi:10.1186/1687-1812-2014-83; Köbis 2014). We analyze the resulting concepts of efficiency and present numerical results on the occurrence of the various concepts. We conclude the paper with a short comparison between the concepts, and an outlook to further work.     

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence C_N . The mixed middle curvature and mixed curvature on C_N are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].     

Explicit Polynomial Generators for the Ring of Quasisymmetric Functions over the Integers

In (Hazewinkel in Adv. Math. 164:283–300, 2001, and CWI preprint, 2001) it has been proved that the ring of quasisymmetric functions over the integers is free polynomial. This is a matter that has been of great interest since 1972; for instance because of the role this statement plays in a classification theory for noncommutative formal groups that has been in development since then, see (Ditters in Invent. Math. 17:1–20, 1972; in Scholtens’ Thesis, Free Univ. of Amsterdam, 1996) and the references in the latter. Meanwhile quasisymmetric functions have found many more applications (see Gel’fand et al. in Adv. Math. 112:218–348, 1995). However, the proofs of the author in the aforementioned papers do not give explicit polynomial generators for QSymm over the integers. In this note I give a (really quite simple) set of polynomial generators for QSymm over the integers.     

On complex lie supergroups and split homogeneous supermanifolds

It is well known that the category of real Lie supergroups is equivalent to the category of the so-called (real) Harish-Chandra pairs, see [DM], [Kost], [Kosz]. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra. In this paper we give a proof of this result in the complex-analytic case. Furthermore, if (G, $ \mathcal{O} $ _G ) is a complex Lie supergroup and H ? G is a closed Lie subgroup, i.e., it is a Lie subsupergroup of (G, $ \mathcal{O} $ _G ) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H, $ \mathcal{O} $ _G/H ) is split. In particular, any complex Lie supergroup is a split supermanifold. It is well known that a complex homogeneous supermanifold may be nonsplit (see, e.g., [OS1]). We find here necessary and sufficient conditions for a complex homogeneous supermanifold to be split.     

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- \({\Sigma^0_2}\) .     

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].     

Generalizations Of A “folk-Theorem” On Simple Functional Equations In A Single Variable

It is known (“mathematical folklore”) that, to every function defined on [1,2], there exists a solution of f(2x) = 2f(x) on ]0,∞[ of which the given function is a restriction to [1,2]. With a little care in the definition on [1,2], with still a lot of arbitrariness left, the resulting solution will be continuous, even C^∞ on ]0,∞[ (a behaviour markedly different from that of the Cauchy equation f(x + y) = f(x) + f(y), which has f(x) = cx as only continuous solution on ]0,∞[, even though, with y = x, it degenerates into the above equation). If 0 is added to the domain and we choose the “arbitrary function” bounded on [1,2[, then the solution will even be continuous (from the right) at 0. However, if f is supposed to be differentiable at 0 (from the right), then f(x) = cx is the only solution on [0,∞[. p In this paper we present similar and further results concerning general, Cⁿ (n ≤ ∞), analytic, locally monotonie or γ-th order convex solutions of the somewhat more general equation f(kx) = k^γf(x) (k ≠ 1 a positive, γ a real constant), which seems to be of importance in meterology. Some of the results are not quite what one expects.     

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].     

Further results on factorization of meromorphic solutions of partial differential equations

This paper is a continuation of Hu-Yang [2]. Here we extend Malmquist type theorem ofalgebraic differential equations of Steinmetz [3] and Tu [4] to higher order partial differential equations. The results also generalize Theorems 4.2 and 4.3 in [2].     

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.     

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.     

Combinatorial geometry of belt bodies

In this paper we consider a new class of convex bodies which was introduced in [11]. This is the class of belt bodies, and it is a natural generalization of the class of zonoids (see the surveys [18, 28, 24]). While the class of zonoids is not dense in the family of all centrally symmetric, convex bodies, the class of belt bodies is dense in the set of all convex bodies. Nevertheless, we shall extend solutions of combinatorial problems for zonoids (cf. [2, 12]) to the class of belt bodies. Therefore, we first introduce the set of belt bodies by using zonoids as starting point. (To make the paper self-contained, a few parts of the approach from [11] are given repeatedly.) Second, complete solutions of three well-known (and generally unsolved) problems from the combinatorial geometry of convex bodies are given for the class of belt bodies. The first of these, connected with the names of I. Gohberg and H. Hadwiger, is the problem of covering a convex body with smaller homothetic copies, or the equivalent illumination problem. The second is the Szökefalvi-Nagy problem, which asks for the determination of the convex bodies whose families of translates have a given Helly dimension. The third problem concerns special fixing systems, a notion which is due to L. Fejes Tóth. These solutions consist of improved and more general approaches to recently solved problems (as in the case of the Helly-dimensional classification of belt bodies) or new results (as those concerning minimal fixing systems, providing also an answer to a problem of B. Grünbaum which is not only restricted to belt bodies).     

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).     

Transfinite Extension to Q m-normality Theory

The theory of Q _m-normal families, m ∈ ?, was developed by P. Montel for the cases m = 0 (normal families) [5] and m = 1 (quasinormal families) [4] and later generalized by C.T. Chuang [2] for any m ≥ 0. In this paper, we extend the definition to an arbitrary ordinal number α as follows. Given E ? D, define the α-th derived set $E^{(\alpha)}_D$ of E with respect to D by $(E^{(\alpha-1)}_D)^{(1)}_D$ if α has an immediate predecessor and by ${\mathop \bigcap\limits_{\beta<\alpha}} E^{(\beta)}_D$ if α is a limit ordinal. Then a family ${\cal F}$ of meromorphic functions on a plane domain D is Q_α-normal if each sequence S of functions in ${\cal F}$ has a subsequence which converges locally χ-uniformaly on the domain DE, where E = E(S) ? D satisfies $E^{(\alpha)}_{D}=\emptyset$ . Inparticular, a Q ₀ -normal family is a normal family, and a Q ₁ -normal family is a quasi- normal family. We also give analogues to some basic results in Q_m-normality theory and extend Zalcman’s Lemma to Q _α -normal families where α is an infinite countable (enumerable) ordinal number.     

Ideal Divisibility Monoids

Inspired by the monograph of Larsen/McCarthy, [26], in [10] and [11] the author started a series of articles concerning abstract multiplicative ideal theory along the problem lines of [26]. In this paper we turn to multiplicative lattices having the left Priifer property, that is to m-lattices satisfying the implication a¹ + … + a_n ? B ? a₁ +… + a_n ¦? B or even the multiplication property A ? B ? A ¦B, respectively. Clearly, studying such structures includes studying substructures of d-semigroups.     

Comparison Theorems For Nonlinear Iterative Functional Inequality

We present two comparison theorems for inequality (1). These theorems are generalizations of similar comparison theorems proved in [1] for linear homogeneous iterative functional inequalities (see also [3] pp. 482–483).     

A Characterization of the Finite Veronesean by Intersection Properties

A combinatorial characterization of the Veronese variety of all quadrics in PG(n, q) by means of its intersection properties with respect to subspaces is obtained. The result relies on a similar combinatorial result on the Veronesean of all conics in the plane PG(2, q) by Ferri [Atti Accad. Naz. Lincei Rend. 61(6), 603?C610 (1976)], Hirschfeld and Thas [General Galois Geometries. Oxford University Press, New York (1991)], and Thas and Van Maldeghem [European J. Combin. 25(2), 275?C285 (2004)], and a structural characterization of the quadric Veronesean by Thas and Van Maldeghem [Q. J. Math. 55(1), 99?C113 (2004)].     

The Totally Nonnegative Part of G/P is a CW Complex

The totally nonnegative part of a partial ag variety G/P has been shown in [18], [17] to be a union of semialgebraic cells. Moreover, the closure of a cell was shown in [19] to be a union of smaller cells. In this paper we provide glueing maps for each of the cells to prove that (G/P)_?0 is a CW complex. This generalizes a result of Postnikov, Speyer and the second author [15] for Grassmannians.