Acta Mathematica Vietnamica
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Issue 4, Volume 46, December 2021
2021
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On Some Classes of Weighted Spaces of Weakly Holomorphic Functions Thai Thuan Quang PDF645-661
▸ Abstract
Let $v$ be a weight on a domain $D$ in a metrizable locally convex space $E$ and $F$ be a complete locally convex space. Denote by $H_v(D, F)$ the weighted space of $F$-valued holomorphic functions on $D$ satisfying that $v.f$ is bounded, $A_v(D)$ subspace of $H_v(D, \mathbb C)$ with the unit ball is compact for the open-compact topology. The aim of this paper is to study linearization theorems and approximation properties in several different topologies for weighted spaces $A_v(D, F)$ of functions $f\in H_v(D, F)$ such that $u∘f\in A_v(D)$ for every continuous linear functional $u$ on $F$.
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Multiplicity of the Saturated Special Fiber Ring of Height Three Gorenstein Ideals Yairon Cid-Ruiz , Vivek Mukundan PDF663-674
▸ Abstract
Let $R$ be a polynomial ring over a field and let $I\subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from Busé et al. (Proc. Lond. Math. Soc. 121(4):743–787, 2020) we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.
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Boundary Conditions for Constrained Hyperbolic Systems of Partial Differential Equations Nicolae Tarfulea PDF675-688
▸ Abstract
Important mathematical models in science and technology are based on first-order symmetric hyperbolic systems of differential equations whose solutions must satisfy certain constraints. When the models are restricted to bounded domains, the problem of well-posed, constraint-preserving boundary conditions arises naturally. However, for numerical solutions, finding such boundary conditions may represent just a step in the right direction. Including the constraints as dynamical variables of a larger, unconstrained system associated to the original one could provide better numerical results, as the constraints are kept under control during evolution. One of the main goals of this work is to investigate this idea in the case of constrained constant-coefficient first-order symmetric hyperbolic systems of differential equations subject to maximal nonnegative boundary conditions.
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Rational Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One Nguyen Tri Dat , Ngo Lam Xuan Chau PDF689-700
▸ Abstract
We give a necessary and sufficient condition for an autonomous first-order AODE to have a rational liouvillian solution. We also give an algorithm to compute a rational liouvillian general solution if it exists. The algorithm is based on the rational parametrization of the corresponding algebraic curve of the first-order autonomous AODE and the existence of a rational liouvillian element over $\mathbb C$. When the corresponding algebraic curve is rational, this method covers the known cases of rational solutions and radical solutions.
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On Some $\overrightarrow {p(x)}$ Anisotropic Elliptic Equations in Unbounded Domain Ahmed Aberqi , Benali Aharrouch , Jaouad Bennouna PDF701-718
▸ Abstract
We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain $\varOmega \subset \mathbb{R}^{N}$ $(N \geq 2)$. We prove the existence of entropy solutions avoiding sign condition and coercivity on the lower order terms.
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Variations of Weyl Type Theorems for Upper Triangular Operator Matrices Mohammad H. M. Rashid PDF719-735
▸ Abstract
Let $\mathcal X$ be a Banach space and let $T$ be a bounded linear operator on $\mathcal X$. We denote by $S(T)$ the set of all complex $\lambda\in\mathcal C$ such that $T$ does not have the single-valued extension property. In this paper it is shown that if $M_C$ is a $2\times 2$ upper triangular operator matrix acting on the Banach space $\mathcal X\oplus\mathcal Y$, then the passage from $\sigma_{LD}(A)\cup\sigma_{LD}(B)$ to $\sigma_{LD}(M_C)$ is accomplished by removing certain open subsets of $\sigma_d(A)\cap\sigma_{LD}(B)$ from the former, that is, there is the equality $\sigma_{LD}(A)\cup\sigma_{LD}(B)=\sigma_{LD}(M_C)\cup\aleph,$ where $\aleph$ is the union of certain of the holes in $\sigma_{LD}(M_C)$ which happen to be subsets of $\sigma_d(A)\cap\sigma_{LD}(B)$. Generalized Weyl's theorem and generalized Browder's theorem are liable to fail for $2\times 2$ operator matrices. In this paper, we also explore how generalized Weyl' theorem, generalized Browder's theorem, generalized a-Weyl's theorem and generalized a-Browder's theorem survive for $2\times 2$ upper triangular operator matrices on the Banach space. -
The Complex Monge-Ampère Type Equation for $\mathcal F$-plurisubharmonic Functions Nguyen Thi Lien , Hoang Van Can , Hoang Viet PDF737-746
▸ Abstract
The purpose of this paper is to establish the existence solutions of the complex Monge-Ampère type equation in plurifinely open subsets of $\mathbb C^n$.
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Hilbert-Kirby Polynomials in Generalized Local Cohomology Modules M. Shafiei , A. Khojali , A. Azari , N. Zamani PDF747-759
▸ Abstract
Let $R=\oplus_{n\in\mathbb{N}_0}R_n$ be a Noetherian homogeneous ring with irrelevant ideal $R_+=\oplus_{n\in\mathbb{N}}R_n$ and with local base ring $(R_0,\mathfrak{m}_0)$. Let $M$, $N$ be two finitely generated $\mathbb{Z}$-graded $R$-modules. We show that the lengths of the graded components of various graded submodules and quotients of the $i$-th generalized local cohomology $H^i_{R_+}(M, N)$ are anti-polynomial. Under some mild assumptions, the Artinianness of $H^i_{R_+}(M, N)$ and the asymptotic behaviour of the $R_0$-modules $H^i_{R_+}(M, N)_n$ for $n\rightarrow -\infty$ in the range $i\leq\inf\{i\in \mathbb{N}_{0} \vert \sharp\{n\vert\ell_{R_{0}}(H^i _{ R_+}(M , N)_n)$ $ = \infty\}=\infty\}$ will be studied. Moreover, it has been proved that, if $u$ is the least integer $i$ for which $H^i_{R_+}(M,N)$ is not Artinian and $\mathfrak q_0$ is an $\mathfrak m_0$-primary ideal of $R_0$, then $H^u_{R_+}(M,N)/\mathfrak q_0H^u_{R_+}(M,$ $N)$ is Artinian with Hilbert-Kirby polynomial of degree less than $u$. In particular, with $M=R$, we deduce the correspondence result for ordinary local cohomology module $H^i_{R_+}(N)$.
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Minimal Representations of a Face of a Convex Polyhedron and Some Applications Ta Van Tu PDF761-778
▸ Abstract
In this paper, we propose a method for determining all minimal representations of a face of a polyhedron defined by a system of linear inequalities. Main difficulties for determining prime and minimal representations of a face are that the deletion of one redundant constraint can change the redundancy of other constraints and the number of descriptor index pairs for the face can be huge. To reduce computational efforts in finding all minimal representations of a face, we prove and use properties that deleting strongly redundant constraints does not change the redundancy of other constraints and all minimal representations of a face can be found only in the set of all prime representations of the face corresponding to the maximal descriptor index set for it. The proposed method is based on a top-down search strategy, is easy to implement, and has many computational advantages. Based on minimal representations of a face, a reduction of degeneracy degrees of the face and ideas to improve some known methods for finding all maximal efficient faces in multiple objective linear programming are presented. Numerical examples are given to illustrate the method.
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Multiplicative Subgroups in Weakly Locally Finite Division Rings Bui Xuan Hai , Huynh Viet Khanh PDF779-794
▸ Abstract
Describing the subgroup structure of a non-commutative division ring is the subject of an intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In the present paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings.
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Uncertain Fractional Evolution Equations with Non-Lipschitz Conditions Using the Condensing Mapping Approach Nguyen Thi Kim Son PDF795-820
▸ Abstract
This paper studies the solvability of Cauchy problems for fractional evolution equations with uncertainty. By using a new approach based on the concept of non-compactness measure and the principle of condensing mappings in the spaces without linearity, we prove the existence of $C^0$−solutions without assuming the Lipschitz continuity of the function on the right-hand side. The present results extend previous results when external forces are always required to satisfy some kinds of generalized Lipschitz conditions. Moreover, the principle of condensing mappings for fuzzy-valued functions or set-valued functions found as an application of noncompactness measure is a useful result when studying dynamical systems containing uncertainties. -
Algebra of Polynomials Bounded on a Semi-algebraic Set $\left [ f\le r ight ]$ Du Thu Trang , Ho Minh Toan , Nguyen Thi Hong PDF821-838
▸ Abstract
The algebra of polynomials in $\mathbb{R}[x]$ which are bounded on a semi-algebraic set determined by a polynomial inequality $f(x) \le r$ with $f(0) =0$ is studied and the case when it is generated by a finite set of monomials is discussed. A large class of polynomials which are asymptotic to finitely many monomials (including nondegenerate polynomials) is introduced and the algebra of polynomials bounded on $[f\le r]$ can be determined by a cone and is independent on $r>0,$ where $f$ belongs to this class. Note that the set of all polynomials whose supports lie in a given closed convex cone in the first quadrant forms an algebra generated by a finite set of monomials. In other cases, we can give upper and lower bounds of the algebra via outer normal cones of the faces of the Newton polyhedron. As a consequence, some sufficient conditions which ensure that the algebra under consideration is generated by finitely many monomials is given.
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Graded Betti Numbers of Balanced Simplicial Complexes Martina Juhnke-Kubitzke , Lorenzo Venturello PDF839-871
▸ Abstract
We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings $S/I$, where $S$ is a polynomial ring and $I\subseteq S$ is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.