Issue 4, Volume 38, Demcember 2013

2013

1. Wave-front tracking for the equations of non-isentropic gas dynamics—basic lemmas Fumioki Asakura PDF

   ▸ Abstract

   In the random choice and its alternative wave-front tracking methods, approximate solutions are constructed by solving exactly or approximately the Riemann problem in each neighborhood of a certain finite set of jump points depending on the time. In order to obtain global in time BV solutions, one has to get a priori estimates for the total variation in x at the time t of approximate solutions, which is, roughly speaking, the summation of amplitudes of waves at t constituting the solutions to the Riemann problems. Since amplitudes may increase through the interaction of neighboring waves, the crucial point is to estimate the amplitudes of outgoing waves by those of incoming waves in a single Riemann solution, which is called the local interaction estimates.

   The aim of this note is to provide a detailed description of the Riemann problem to the equations of polytropic gas dynamics and a complete proof of the basic lemmas on which the local interaction estimates are based. Although all of them, except for Lemmas 4.2 and 5.1, are presented in T.-P. Liu (Indiana Univ. Math. J. 26:147–177, 1977), that paper is difficult and not well understood even at the present day in spite of its importance. For the sake of completeness, this note includes proofs of the local interaction estimates.

   487-516
2. Cubic derivations on Banach algebras Abasalt Bodaghi PDF

   ▸ Abstract

   Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. A mapping $D: A\to X$ is a cubic derivation if $D$ is a cubic homogeneous mapping, that is, $D $ is cubic and $D(\lambda a)=\lambda^3 D(a)$ for any complex number $\lambds$ and all $a \in A$, and $D(ab)=D(a)\cdot b^3+a^3\cdot D(b)$ for all $a,b\in A$. In this paper, we prove the stability of a cubic derivation with direct method. We also employ a fixed point method to establish the stability and the superstability of cubic derivations.

   517-528
3. Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities Bui Van Dinh , Le Dung Muu PDF

   ▸ Abstract

   We propose algorithms for finding the projection of a given point onto the solution set of the pseudomonotone variational inequality problem. This problem arises in the Tikhonov regularization method for pseudomonotone variational inequality. Since the solution set of the variational inequality is not given explicitly, the available methods of mathematical programming and variational inequalities cannot be applied directly.

   529-540
4. Castelnuovo–Mumford regularity of associated graded modules in dimension one Le Xuan Dung PDF

   ▸ Abstract

   An upper bound for the Castelnuovo–Mumford regularity of the associated graded module of an one-dimensional module is given in term of its Hilbert coefficients. It is also investigated when the bound is attained.

   541-550
5. A linear recursive scheme associated with the love equation Le Thi Phuong Ngoc , Nguyen Tuan Duy , Nguyen Thanh Long PDF

   ▸ Abstract

   This paper shows the existence of a unique weak solution of the following Dirichlet problem for a nonlinear Love equation $$\left\{ \begin{aligned} &u_{tt}-u_{xx}-\varepsilon u_{xxtt}=f(x,t,u,u_{x},u_{t},u_{xt}), \quad 0 < x < L,~ 0 < t < T, \\ &u(0,t)=u(L,t)=0, \\ &u(x,0)=\tilde{u}_{0}(x),\qquad u_{t}(x,0)= \tilde{u}_{1}(x), \end{aligned} \right.$$ where $\varepsilon > 0$ is a constant and $\tilde{u}_{0}, \tilde{u}_{1}, f$ are given functions. This is done by combining the linearization method for a nonlinear term, the Faedo–Galerkin method and the weak compactness method.

   551-562
6. On the gonality sequence of smooth curves: normalizations of singular curves in a quadric surface Edoardo Ballico PDF

   ▸ Abstract

   Let $C$ be a smooth curve of genus $g$. For each positive integer $r$, the $r$-gonality $d_r(C)$ of $C$ is the minimal integer $t$ such that there is $L \in \mathrm{Pic}^t(C)$ with $h^0(C,L)=r+1$. In this paper, for all $g \ge 40805$ we construct several examples of smooth curves $C$ of genus $g$ with $d_3(C)/3 < d_4(C)/4$, i.e., for which a slope inequality fails.

   563-573
7. On compactness of the $\bar{\partial}$-Neumann operator on $r$-convex domains Le Mau Hai , Nguyen Xuan Hong PDF

   ▸ Abstract

   In this paper, we show that on a bounded $r$-convex domain $\Omega$ in $\mathbb C^n$ such that for every $z\in \partial\Omega$ there exists a complex affine subspace $L_z$ of dimension $\le  q−1$ through $z$ such that $L_z \cap\partial\Omega$ is a $q$-peak set with $r\le q\le n$ the $\bar{\partial}$-Neumann operator $N_{q,\Omega}$ is compact.

   575-586
8. Compact composition operators on spaces of exponential Cauchy kernels Yusuf Abu Muhanna , El-Bachir Yallaoui PDF

   ▸ Abstract

   We study the action of the composition operator on the analytic function spaces whose kernels are of exponential Cauchy type. These function spaces become Banach spaces when the kernels are integrated with respect to the complex Borel measures of the unit circle. Necessary and sufficient conditions for the composition operator to be compact are found.

   587-595
9. Uniqueness of meromorphic functions by their defects Amer H. H. Al-Khaladi PDF

   ▸ Abstract

   In this paper we obtain a uniqueness theorem of a meromorphic function and its kth derivative that share one small function. So we give some results which are related to a conjecture of R. Brück, and improve some results given by Zhang and Yang (Ann. Acad. Sci. Fenn., Ser. A 1 Math., 32:141–149, 2007) and also answer the open questions posed by Yu (JIPAM. J. Inequal. Pure Appl. Math., 4:1–7, 2003). Furthermore, examples are provided to show that our results are sharp.

   597-606
10. Smooth model of system with diode nonlinearity Nguyen Thi Hien PDF

    ▸ Abstract

    A smooth model of a system with diode nonlinearity following the ordinary differential equation with a large parameter $K$ is proposed. It offers a convenient tool for numerical analysis using the advanced packages of the applied programs. The main result of this paper is presented in a theorem that gives the upper bound of the modeling error and shows that the model solution uniformly converges to the exact solution of system with diode nonlinearity when $K$ tends to infinity. The usefulness of this model is illustrated via a concrete example.

    607-616
11. On some classes of $\omega$-plurisubharmonic functions on compact Kähler manifolds Nguyen Van Phu , Vu Viet Hung PDF

    ▸ Abstract

    In this paper we establish a relation between classes of $\omega$-plurisubharmonic functions $\mathcal{E}_{p}(X,\omega), p>0$ and $\mathrm{DMA}_{\mathrm{loc}}(X,\omega)$ introduced and investigated by Guedj and Zeriahi (J. Funct. Anal., 250:442–482, 2007) and Cegrell (Acta Math., 180:187–217, 1998; Ann. Inst. Fourier (Grenoble), 54:159–179, 2004), respectively, on a compact Kähler manifold $$. We show that $\mathcal{E}_{n-1}(X,\omega)\subset\mathrm{DMA}_{\mathrm {loc}}(X,\omega)$ but $\bigcap_{0 < p < n-1}\mathcal{E}_{p}(\mathbb{CP}^{n},\omega)\not \subset\mathrm{DMA}_{\mathrm{loc}}(\mathbb{CP}^{n},\omega)$. At the same time we investigate a relation between the classes $\mathrm{DMA} (\mathbb{C}^{n})\cap\mathcal{L}(\mathbb{C}^{n})$ and $\mathrm {DMA}(\mathbb{CP}^{n},\omega)$ as well as $\widehat{\mathrm{DMA}}(\mathbb{C}^{n})\cap\mathcal {L}(\mathbb{C}^{n})$ and $\widehat{\mathrm{DMA}}(\mathbb{CP}^{n},\omega)$.

    617-625
12. On $\lambda$-ideal convergent interval valued difference classes defined by Musielak–Orlicz function Bipan Hazarika PDF

    ▸ Abstract

    An ideal $I$ is a family of subsets of positive integers $\mathbb N$ which is closed under taking finite unions and subsets of its elements. In this paper, using $\lambda$-ideal convergence as a variant of the notion of ideal convergence, the difference operator $\Delta^n$ and Musielak–Orlicz functions, we introduce and examine some generalized difference sequences of interval numbers, where $\lambda=(\lambda_m )$ is a nondecreasing sequence of positive real numbers such that $λ_{m+1}\le \lambda_m +1,\lambda_1=1,\lambda_m \to \infty~(m\to \infty)$. We prove completeness properties of these spaces. Further, we investigate some inclusion relations related to these spaces.

    627-639
13. Chains of idempotents in endomorphism monoids B. A. F. Wehrfritz PDF

    ▸ Abstract

    If $G$ is a group with finite Hirsch number and with its maximal locally finite normal subgroup satisfying the minimal condition on subgroups, e.g. if $G$ is a finite extension of a torsion-free soluble group of finite rank, then there exists an integer $k=k(G)$ such that for every subgroup $H$ of $G$ any chain of idempotents in the endomorphism monoid $\mathrm{End}(H)$ of $H$ has length at most $k$.

    641-646