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## Positive periodic solutions of discrete Lotka–Volterra cooperative systems with delays

### Acta Mathematica Vietnamica (2013-09-01) 38: 461-470 , September 01, 2013

The existence of positive periodic solutions of discrete nonautonomous Lotka–Volterra cooperative systems with delays is studied by applying the continuation theorem of coincidence degree theory.

## Global attractivity of almost periodic solutions for competitive Lotka–Volterra diffusion system

### Acta Mathematica Vietnamica (2014-06-01) 39: 151-161 , June 01, 2014

In this paper, two competitive Lotka–Volterra populations in the two-patch-system with diffusion are considered. Each of the two species can diffuse independently and discretely between its intrapatch and interpatch. By means of a Lyapunov function, under a moderate condition, the system has a unique almost periodic solution, which is asymptotically stable and globally attractive.

## Syzygies of Hibi Rings

### Acta Mathematica Vietnamica (2015-09-01) 40: 403-446 , September 01, 2015

We survey recent results on resolutions of Hibi rings.

## Wave-front tracking for the equations of non-isentropic gas dynamics—basic lemmas

### Acta Mathematica Vietnamica (2013-12-01) 38: 487-516 , December 01, 2013

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.

## Some inequalities for continuous functions of selfadjoint operators in hilbert spaces

### Acta Mathematica Vietnamica (2014-09-01) 39: 287-303 , September 01, 2014

If
$\{ E_{\lambda} \} _{\lambda\in\mathbb{R}}$
is the spectral family of a bounded selfadjoint operator *A* on a Hilbert space *H* and *m*=min*Sp*(*A*) and *M*=max*Sp*(*A*), we show that for any continuous function *φ*:
$[ m,M ] \rightarrow \mathbb{C}$
, we have the inequality
$$\begin{aligned} \bigl\vert \bigl\langle \varphi ( A ) x,y \bigr\rangle \bigr\vert ^{2} \leq& \Biggl( \int_{m-0}^{M}\bigl\vert \varphi ( t ) \bigr\vert \,d \Biggl( \bigvee_{m-0}^{t} \bigl( \langle E_{ ( \cdot ) }x,y \rangle \bigr) \Biggr) \Biggr) ^{2} \\ \leq& \bigl\langle \bigl\vert \varphi ( A ) \bigr\vert x,x \bigr\rangle \bigl\langle \bigl\vert \varphi ( A ) \bigr\vert y,y \bigr\rangle \end{aligned}$$
for any vectors *x* and *y* from *H*. Some related results and applications are also given.

## A Remark on Vanishing of Chain Complexes

### Acta Mathematica Vietnamica (2015-03-01) 40: 173-177 , March 01, 2015

For chain complexes
$W \in D^{\pm }_{fg} (R)$
and *X* ∈ *D*(*R*) in the derived category over a commutative noetherian ring *R*, we prove that RHom_{R}(*W*, *X*) = 0 if and only if *W*^{L}⊗_{R}*X* = 0.

## On a Minimal Set of Generators for the Polynomial Algebra of Five Variables as a Module over the Steenrod Algebra

### Acta Mathematica Vietnamica (2017-03-01) 42: 149-162 , March 01, 2017

Let *P*_{k} be the graded polynomial algebra
$\mathbb {F}_{2}[x_{1},x_{2},{\ldots } ,x_{k}]$
over the prime field of two elements,
$\mathbb {F}_{2}$
, with the degree of each *x*_{i} being 1. We study the *hit problem*, set up by Frank Peterson, of finding a minimal set of generators for *P*_{k} as a module over the mod-2 Steenrod algebra,
$\mathcal {A}$
. In this paper, we explicitly determine a minimal set of
$\mathcal {A}$
-generators for *P*_{k} in the case *k* = 5 and the degree 4(2^{d}−1) with *d* an arbitrary positive integer.

## A linear recursive scheme associated with the love equation

### Acta Mathematica Vietnamica (2013-12-01) 38: 551-562 , December 01, 2013

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 *ε*>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.

## The Stratification by Rank for Homogeneous Polynomials with Border Rank 5 which Essentially Depend on Five Variables

### Acta Mathematica Vietnamica (2017-09-01) 42: 509-531 , September 01, 2017

We give the stratification by the symmetric tensor rank of all degree *d* ≥ 9 homogeneous polynomials with border rank 5 and which depend essentially on at least five variables, extending previous works (A. Bernardi, A. Gimigliano, M. Idà, E. Ballico) on lower border ranks. For the polynomials which depend on at least five variables, only five ranks are possible: 5, *d* + 3, 2*d* + 1, 3*d* − 1, 4*d* − 3, but each of the ranks 3*d* − 1 and 2*d* + 1 is achieved in two geometrically different situations. These ranks are uniquely determined by a certain degree 5 zero-dimensional scheme *A* associated with the polynomial. The polynomial *f* depends essentially on at least five variables if and only if *A* is linearly independent (in all cases, *f* essentially depends on exactly five variables). The polynomial has rank 4*d* − 3 (resp. 3*d* − 1, resp. 2*d* + 1, resp. *d* + 3, resp. 5) if *A* has 1 (resp. 2, resp. 3, resp. 4, resp. 5) connected component. The assumption *d* ≥ 9 guarantees that each polynomial has a uniquely determined associated scheme *A*. In each case, we describe the dimension of the families of the polynomials with prescribed rank, each irreducible family being determined by the degrees of the connected components of the associated scheme *A*.

## THE STABILITY OF PROPERTY (gw) UNDER COMPACT PERTURBATION

### Acta Mathematica Vietnamica (2014-09-01) 39: 325-336 , September 01, 2014

Let
$\mathcal {H}$
be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator *T* on
$\mathcal {H}$
to satisfy that *f*(*T*) obeys property (*g**w*) for each function *f* analytic on some neighborhood of *σ*(*T*). Also, we investigate the stability of property (*g**w*) under (small) compact perturbations.