On certain equidimensional polymatroidal ideals

The class of equidimensional polymatroidal ideals are studied. In particular, we show that an unmixed polymatroidal ideal is connected in codimension one if and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in codimension one precisely when it is a squarefree Veronese ideal. As a consequence we indicate that for polymatroidal ideals, the Serre's condition $(S_n)$ for some $n\geq 2$ is equivalent to Cohen-Macaulay property. We also give a classification of generalized Cohen-Macaulay polymatroidal ideals.Comment: To appear in Manuscripta Mathematic

Pretty cleanness and filter-regular sequences

Let $K$ be a field and $S=K[x_1,\ldots, x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\ldots, u_r$ be monomials in $S$ which form a filter-regular sequence on $S/I$. We show that $S/I$ is pretty clean if and only if $S/(I,u_1,\ldots, u_r)$ is pretty clean.Comment: It will be published in Czechoslovak Mathematical Journa

A Novel Large-Scale, Multilayer, and Facilely Aligned Micropatterning Technique Based on Flexible and Reusable SU-8 Shadow Masks

A simple method to fabricate flexible, mechanically robust, and reusable SU-8 shadow masks is demonstrated. This shadow mask technology has high pattern flexibility as various shapes with different dimensions can be created. The fabricated shadow masks are characterized in terms of the resolution, reusability, and capability of multilayer surface micropatterning. Fabrication of a new plastic photomask for the exposure process simplifies the shadow mask fabrication process and results in higher resolution in the shadow mask structures compared to the commercial chromium photomasks. For the multilayer surface micropatterning technology, a simple and fast alignment technique based on SU-8 pillars and without usage of any microscopic tools is reported. This unique method leads to a less complicated alignment process with the alignment accuracy of ≈2 µm. The proposed shadow mask technology can be easily employed for wafer-scale micropatterning process. The capability of fabricated SU-8 shadow masks in micropatterning on polymer thin films is evaluated by fabricating metallic contacts on poly(3,4-ethylenedioxythiophene) samples and electrical characterization. © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei

Highly Symmetric and Extremely Compact Multiple Winding Microtubes by a Dry Rolling Mechanism

Rolled-up nanotechnology has received significant attention to self-assemble planar nanomembranes into 3D micro and nanotubular architectures. These tubular structures have been well recognized as novel building blocks in a variety of applications ranging from microelectronics and nanophotonics to microbatteries and microrobotics. However, fabrication of multiwinding microtubes with precise control over the winding interfaces, which is crucial for many complex applications, is not easy to achieve by existing materials and technologies. Here, a dry rolling approach is introduced to tackle this challenge and create tight windings in compact and highly symmetric cylindrical microstructures. This technique exploits hydrophobicity of fluorocarbon polymers and the thermal expansion mismatch of polymers and inorganic films upon thermal treatment. Quality parameters for rolled-up microtubes, against which different fabrication technologies can be benchmarked are defined. The technique offers to fabricate long freestanding multiwinding microtubes as well as hierarchical architectures incorporating rolled-up wrinkled nanomembranes. This work presents an important step forward toward the fabrication of more complex but well-controlled microtubes for advanced high-quality device architectures. © 2020 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei

Monomial ideals whose depth function has any given number of strict local maxima

We construct monomial ideals with the property that their depth function has any given number of strict local maxima

The cleanness of (symbolic) powers of Stanley-Reisner ideals

summary:Let $\Delta$ be a pure simplicial complex on the vertex set $[n]=\{1,\ldots ,n\}$ and $I_\Delta$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,\ldots ,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$) is clean for all $m\in \mathbb {N}$ and this is equivalent to saying that $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb {N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{(m)}$ is not (pretty) clean for all integer $m\geq 3$. If $\dim (\Delta )=1$, we also prove that $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$) is clean if and only if $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$, respectively) is Cohen-Macaulay