Issue 2, Volume 42, June 2017

2017

1. The Zero Loci of $F$-triangles Kyoji Saito PDF

   ▸ Abstract

   We are interested in the zero locus of a Chapoton’s $F$-triangle as a polynomial in two real variables $x$ and $y$. An expectation is that (1) the $F$-triangle of rank $l$ as a polynomial in $x$ for each fixed $y\in [0,1]$ has exactly $l$ distinct real roots in $[0,1]$, and (2) ith root $x_i(y)$ $(1\geq i\geq l)$ as a function on $y\in [0,1]$ is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of $F$-triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above expectation is true. We formulate inductive conjectures and questions for further rank cases. This study gives a new insight on the zero loci of $f^+$- and $f$-polynomials.

   209-236
2. On Innermost Circles of the Sets of Singular Values for Generic Deformations of Isolated Singularities Kazumasa Inaba , Masaharu Ishikawa , Masayuki Kawashima , Nguyen Tat Thang PDF

   ▸ Abstract

   We will show that for each $k\not =1$, there exists an isolated singularity of a real analytic map from $\mathbb R^4$ to $\mathbb R^2$ which admits a real analytic deformation such that the set of singular values of the deformed map has a simple, innermost component with $k$ outward cusps and no inward cusps. Conversely, such a singularity does not exist if $k=1$.

   237-247
3. The Higher Topological Complexity of Complement of Fiber Type Arrangement Nguyen Viet Dung , Nguyen Van Ninh PDF

   ▸ Abstract

   Let $\mathcal A$ be a fiber type arrangement of hyperplanes in $\mathbb C_n$ with complement $M(\mathcal A)$ (see Orlik, O., Terao, H. 1992). In this paper, we will give an explicit formula for the higher topological complexity $T C_n$ for the complement $M(\mathcal A)$ in terms of exponents of the arrangement $\mathcal A$.

   249-256
4. On Relations for Zeros of $f$-Polynomials and $f^+$-Polynomials Tadashi Ishibe PDF

   ▸ Abstract

   Let $\Phi$ be an irreducible (possibly noncrystallographic) root system of rank $l$ of type P. For the corresponding cluster complex $\Delta(P)$, which is known as pure $(l − 1)$-dimensional simplicial complex, we define the generating function of the number of faces of $\Delta(P)$ with dimension $i − 1$, which is called $f$-polynomial. We show that the $f$-polynomial has exactly $l$ simple real zeros on the interval $(0, 1)$ and the smallest root for the infinite series of type $A_l , B_l$, and $D_l$ monotone decreasingly converges to zero as the rank $l$ tends to infinity. We also consider the generating function (called the $f^+$-polynomial) of the number of faces of the positive part $\Delta_+(P)$ of the complex $\Delta(P)$ with dimension $i − 1$, whose zeros are real and simple and are located in the interval $(0, 1)$, including a simple root at $t = 1$. We show that the roots in decreasing order of $f$-polynomial alternate with the roots in decreasing order of $f^+$-polynomial.

   257-277
5. Computing $\mu^*$-Sequences of Hypersurface Isolated Singularities via Parametric Local Cohomology Systems Katsusuke Nabeshima , Shinichi Tajima PDF

   ▸ Abstract

   Complex analytic invariants of hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of $\mu^*$-sequences that introduced by B. Teissier to study the Whitney equisingularity of deformations of complex hypersurfaces. A new algorithm that utilizes parametric local cohomology systems is proposed to compute $\mu^*$-sequences. Lists of $\mu^*$-sequences of some typical cases are also given.

   279-288
6. A Short Survey on the Integral Identity Conjecture and Theories of Motivic Integration Lê Quy Thuong PDF

   ▸ Abstract

   In Kontsevich-Soibelman’s theory of motivic Donaldson-Thomas invariants for 3-dimensional noncommutative Calabi-Yau varieties, the integral identity conjecture plays a crucial role as it involves the existence of these invariants. A purpose of this note is to show how the conjecture arises. Because of the integral identity’s nature, we shall give a quick tour on theories of motivic integration, which lead to a proof of the conjecture for algebraically closed ground fields of characteristic zero.

   289-310
7. Continuity and Directional Differentiability of the Value Function in Parametric Quadratically Constrained Nonconvex Quadratic Programs Tran Van Nghi , Nguyen Nang Tam PDF

   ▸ Abstract

   The aim of this paper is to investigate the continuity and the directional differentiability of the value function in quadratically constrained nonconvex quadratic programming problem. Our result can be used in some cases where the existing results on differential stability in nonlinear programming (applied to quadratic programming) cannot be used.

   311-336
8. Conjugate Duality and Optimization over Weakly Efficient Set Tran Van Thang PDF

   ▸ Abstract

   In this article, we present a conjugate duality for nonconvex optimization problems. This duality scheme is symmetric and has zero gap. As applied to a vector-maximization problem, it transforms the latter into an optimization problem over a weakly efficient set which can be solved by monotonic optimization methods.

   337-355
9. On the Stationary Solutions to 2D $g$-Navier-Stokes Equations Dao Trong Quyet , Nguyen Viet Tuan PDF

   ▸ Abstract

   We consider $g$-Navier-Stokes equations in a two-dimensional smooth bounded domain $\Omega$. First, we study the existence and exponential stability of a stationary solution under some certain conditions. Second, we prove that any unstable steady state can be stabilized by proportional controller with support in an open subset $\omega \subset {\Omega }$ such that $\Omega∖ \omega$ is sufficiently “small.”

   357-367
10. Some Inequalities for Partial Derivatives on Time Scales Tran Dinh Phung PDF

    ▸ Abstract

    We prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial differential dynamic equations are also considered.

    369-394
11. On a Mixed Boundary Value Problem for the Biharmonic Equation in a Strip Nguyen Van Ngoc PDF

    ▸ Abstract

    The aim of the present work is to consider a mixed boundary value problem for the biharmonic equation in a strip. The problem may be interpreted as a deflection surface of a strip plate with the edges $y=0$, $y = h$ having clamped conditions on intervals $|x|\geq a$ and hinged support conditions for $|x| < a$. Using the Fourier transform, the problem is reduced to studying a system of dual integral equations on the edges of the strip. The uniqueness and existence theorems of solution of system of dual integral equations are established in appropriate Sobolev spaces. A method for reducing the dual integral equation to infinite system of linear algebraic equations is also proposed.

    395-411