Issue 3, Volume 46, September 2021

2021

1. Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes Martin Kreuzer , Tran N. K. Linh , Le Ngoc Long PDF

   ▸ Abstract

   Given a fat point scheme $\mathbb {W}=m_{1}P_{1}+\cdots +m_{s}P_{s} $ in the projective $n$-space $\mathbb{P}^n$ over a field $K$ of characteristic zero, the modules of Kähler differential $k$-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of $\mathbb{W}$ when $k\in \{1,\dots , n+1\}$. In this paper, we determine the value of its Hilbert polynomial explicitly for the case $k=n+1$, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme $\mathbb {Y} = (m_{1}-1)P_{1} + {\cdots } + (m_{s}-1)P_{s}$. For $n=2$, this allows us to determine the Hilbert polynomials of the modules of Kähler differential $k$-forms for $k=1, 2, 3$, and to produce a sharp bound for the regularity index for $k=2$.

   441-455
2. On Minimax Modules and Generalized Local Cohomology with Respect to a Pair of Ideals Nguyen Thanh Nam , Tran Tuan Nam , Nguyen Minh Tri PDF

   ▸ Abstract

   We study the minimax properties and the artinianness of the generalized local cohomology modules $H^{i}_{I,J}(M,N)$ with respect to a pair of ideals $(I, J)$. We also show some results on top generalized local cohomology modules.

   457-470
3. Resurgence and Castelnuovo-Mumford Regularity of Certain Monomial Curves in $\mathbb{A}^3$ Clare D’Cruz PDF

   ▸ Abstract

   Let \mathfrak{p} be the defining ideal of the monomial curve ${\mathcal {C}}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space $\mathbb {A}_{k}^{3}$ parameterised by $(x^{2q+1}, x^{2q+1+m}, x^{2q+1+2m})$, where $\mathbb {A}_{k}^{3}$. In this paper we compute the resurgence of \mathfrak{p}, the Waldschmidt constant of $\mathfrak {p}$ and the Castelnuovo-Mumford regularity of the symbolic powers of $\mathfrak {p} $.

   471-481
4. Algebraic Cycles and Special Horikawa Surfaces Robert Laterveer PDF

   ▸ Abstract

   This note is about a certain 16-dimensional family of surfaces of general type with $p_g=2$ and $q=0$ and $k^2=1$, called “special Horikawa surfaces”. These surfaces, studied by Pearlstein–Zhang and by Garbagnati, are related to $K_3$ surfaces. We show that special Horikawa surfaces have a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of special Horikawa surfaces displays $K_3$-like behavior.

   483-497
5. $fL_k$-Harmonic Maps and $fL_k$-Harmonic Morphisms Mehran Aminian , Mehran Namjoo PDF

   ▸ Abstract

   In this paper, we introduce $fL_k$-energy functionals; and by deriving variations of these functionals, we define $fL_k$-harmonic maps between Riemannian manifolds. Hereafter, by using these definitions, we introduce $fL_k$-harmonic morphisms, and then we find a relation between $fL_k$-harmonic maps and $fL_k$-harmonic morphisms.

   499-507
6. The Bergman and Szegő Kernels: a Direct Relationship Steven G. Krantz PDF

   ▸ Abstract

   In this paper, we prove, for a fairly general class of domains, that the Bergman kernel of a domain is closely related to the normal derivative of the Szegő kernel. Such a result is useful in passing back and forth between estimates for the Bergman projection and estimates for the Szegő projection. We also make some remarks about comparability of the singularity of the Bergman kernel and the singularity of the Szegő kernel.

   509-514
7. An Algorithm for a Class of Bilevel Variational Inequalities with Split Variational Inequality and Fixed Point Problem Constraints Nguyen Minh Hai , Le Huynh My Van , Tran Viet Anh PDF

   ▸ Abstract

   In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of a split variational inequality and fixed point problem. Strong convergence of the iterative process is proved. In particular, the problem of finding a common solution to a variational inequality with pseudomonotone mapping and a fixed point problem involving demicontractive mapping is also studied. Besides, we get a strongly convergent algorithm for finding the minimum-norm solution to the split feasibility problem, which requires only two projections at each step. A simple numerical example is given to illustrate the proposed algorithm.

   515-530
8. Analytic Study on Boundary Value Problem of Implicit Differential Equations via Composite Fractional Derivative D. Vivek , E. M. Elsayed , K. Kanagarajan PDF

   ▸ Abstract

   This paper is devoted to investigate implicit differential equations with boundary condition, which involves the composite fractional derivative in weighted space. The existence and uniqueness of the solution are obtained using the classic fixed point theorems. As an application, an example is presented to illustrate the main results.

   531-543
9. Generalized Wavelet Transform Associated with the Heckman-Opdam-Jacobi Theory on $\mathbb {R}^{d+1}$ Fida Bahba PDF

   ▸ Abstract

   We introduce, in this paper, the harmonic analysis associated with the Heckman-Opdam-Jacobi operator on $\mathbb {R}^{d+1}$ in the W -invariant case. Next, we study the generalized wavelets and the generalized wavelet transform associated with this operator and we establish their properties. In particular, we prove for the generalized wavelet transform Plancherel and inversion formulas.

   545-566
10. Weyl-Heisenberg Frame Operators and Kohn-Nirenberg Symbols T. C. Easwaran Nambudiri , K. Parthasarathy PDF

    ▸ Abstract

    An explicit expression for the Kohn-Nirenberg symbol of a Weyl-Heisenberg frame operator on $L^{2}(\mathbb {R})$ is obtained directly from the Gabor atom coming from new classes of window functions. This new approach, using only elementary Fourier analysis, is independent of the theory of distributions and works strictly inside $L^{2}(\mathbb {R}) $. Kohn-Nirenberg operators are introduced and are shown to be Weyl-Heisenberg frame operators in suitable cases.

    567-578
11. Dynamics of Non-autonomous Quasilinear Degenerate Parabolic Equations: the Non-compact Case Tran Thi Quynh Chi , Le Thi Thuy , Nguyen Xuan Tu PDF

    ▸ Abstract

    We prove the existence of pullback attractors in various spaces for the following non-autonomous quasilinear degenerate parabolic equations involving weighted p-Laplacian operators on $\mathbb {R}^{N}$

    $$\frac{\partial u}{\partial t}-\text{div}(\sigma(x)|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u+f(u)=g(x,t),$$

    under a new condition concerning a variable non-negative diffusivity $\sigma(x)$, an arbitrary polynomial growth order of the non-linearity f, and an exponential growth of the external force. To overcome the essential difficulty arising due to the unboundedness of the domain, the results are proved by combining the tail estimates method and the asymptotic a priori estimate method.

     

    579-598
12. Symbolic Powers of Certain Cover Ideals of Graphs Arvind Kumar , Rajiv Kumar , Rajib Sarkar , S. Selvaraja PDF

    ▸ Abstract

    In this paper, we compute the regularity and Hilbert series of symbolic powers of cover ideal of a graph G when G is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover ideals in terms of the number of edges.

    599-611
13. Existence and Asymptotic Behavior of Solutions for Degenerate Nonlinear Kirchhoff Strings with Variable-Exponent Nonlinearities Rahmoune Abita PDF

    ▸ Abstract

    In this paper, we investigate the existence of a local solution in time and discuss the exponential asymptotic behavior to a weakly damped wave equation involving the variable-exponents

    \begin{array}{@{}rcl@{}} &&u_{tt}-M\left( \left\vert \nabla u\left( t\right) \right\vert^{2}\right) {\Delta} u+{{\int}_{0}^{t}}g\left( t-s\right) {\Delta} u\left( s\right) ds+\gamma_{1}u_{t}+\left\vert u_{t}\right\vert^{k\left( x\right) -1}u_{t}\\ &=&\left\vert u\right\vert^{p\left( x\right) -1}u \text{ in }{\Omega} \times \mathbb{R}^{+} \end{array}

    with simply supported boundary condition, where $\Omega$ is a bounded domain of $\mathbb{R}^n, g>0$ is a memory kernel that decays exponentially, and $M(s)$ is a locally Lipschitz function. This kind of problem without the memory term when $k(.)$ and $p(.)$ are constants models viscoelastic Kirchhoff equation.

    613-643