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].     

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].     

To the question on the almost everywhere convergence of the Riesz means of double orthogonal series

Sufficient conditions of the classical type ensuring the almost everywhere (a.e.) convergence of the nonnegative-order Riesz means of double orthogonal series are indicated. Analogies of the onedimensional results of Kolmogoroff [7] and Kaczmarz?CZygmund [5, 12] have been obtained for the Cesaro means and those of Zygmund [13] for the Riesz means. These analogies establish the a.e. equiconvergence of the lacunary subsequences of rectangular partial sums and of the entire sequence of Riesz means, generalize the corresponding results of Moricz [9] for the Cesaro a.e. summability by (C, 1, 1), (C, 1, 0), and (C, 0, 1) methods of double orthogonal series, and were announced earlier without proofs in the author??s work [3].     

A probabilistic position value

In this article, we generalize the position value, defined by Meessen (Master??s thesis, 1988) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations (Gómez et al. in European Journal of Operational Research 190:539?C556, 2008). We provide two characterizations of this new allocation rule. Following in Slikker??s (International Journal of Game Theory 33:505?C514, 2005a) footsteps, we characterize the probabilistic position value using probabilistic versions of component efficiency and balanced link contributions. Then we generalize the notion of link potential, defined by Slikker (International Game Theory Review 7:473?C489, 2005b) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations, and use it to characterize our allocation rule.     

Product of Almost-Hermitian Manifolds

This paper continues the work about the nonexistence of some complete metrics on the product of two manifolds studied by Tam and Yu (Asian J. Math., 14(2):235–242, 2010), and is motivated by the result of Tosatti (Commun. Anal. Geom., 15(5):1063–1086, 2007). We generalize the corresponding results of Tam and Yu (Asian J. Math., 14(2):235–242, 2010) to the almost-Hermitian case.     

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).     

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].     

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.     

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.     

The composition of solutions of the multidimensional translation equation

Under certain conditions, the general solution of the multidimensional translation equation was constructed locally in L. Berg [4] and globally in J. Aczél, L. Berg and Z. Moszner [2]. By composition of two of these solutions there arise new functional equations, which are solved here locally using generalized inverses, cf. A. Ben-Israel and T.N.E. Greville [5]. The results are illustrated by the linear case.     

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].     

Differentiated Generalized Voronovskaja’s Theorem in Compact Disks

In this paper we obtain the differentiated generalized Voronovskaja’s theorem in complex setting with upper and exact quantitative estimates. The results extend that obtained in the real case on [0, 1] in Gonska and Ra?a (Mat Vesnik 61:53–60, 2009) and generalize those obtained in the complex case in Gal (Results Math 53:257–268, 2009).     

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.     

Monotone hulls for {\mathcal {N}}\cap {\mathcal {M}}

Using the method of decisive creatures [see Kellner and Shelah (J Symb Log 74:73–104, 2009)] we show the consistency of “there is no increasing \(\omega _2\) –chain of Borel sets and \(\mathrm{non}({\mathcal N})= \mathrm{non}({\mathcal M})=\mathrm{non}({\mathcal N}\cap {\mathcal M})=\omega _2=2^\omega \) ”. Hence, consistently, there are no monotone Borel hulls for the ideal \({\mathcal M}\cap {\mathcal N}\) . This answers Balcerzak and Filipczak (Math Log Q 57:186–193, 2011 [Questions 23, 24]). Next we use finite support iteration of ccc forcing notions to show that there may be monotone Borel hulls for the ideals \({\mathcal M},{\mathcal N}\) even if they are not generated by towers.     

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).     

A symmetry property for q-weighted Robinson–Schensted and other branching insertion algorithms

In [19], a \(q\) -weighted version of the Robinson–Schensted algorithm was introduced. In this paper, we show that this algorithm has a symmetry property analogous to the well-known symmetry property of the usual Robinson–Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin [5–8]. This approach, which uses ‘growth graphs’, can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other \(q\) -weighted versions of the Robinson–Schensted algorithm which have recently been introduced by Borodin–Petrov [2].     

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.).     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Characterizations for General Besov-Type Space in Clifford Analysis

In this paper, we obtain some characterizations for general Besov-type spaces by employing certain power series satisfying certain growth conditions in lieu of the weight function in Clifford analysis. The obtained results extend and generalize the corresponding results which are given in [6, 23].