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If a graph G is chordal, Ha and Van Tuyl proved that $${{\,\mathrm{reg}\,}}(G) = \nu _0(G)$$ , where $$\nu _0(G)$$ is the induced matching number of G. In this paper, we produce a lower bound for $${{\,\mathrm{reg}\,}}(G)$$ which coincides with the result of Ha and Van Tuyl in the case of chordal graphs. We also consider the case G is vertex-decomposable.

We classify all Gorenstein claw-free graphs. Moreover, we provide a new way to construct a Gorenstein graph from another one.

Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

To meet the increasing demand for mass-produced microalgae, production processes must be optimised and monitored. The herein described optical sensors provide an instantaneous and direct opportunity to monitor biological and biochemical information for microalgae cultivation. Combination of expertise in natural science and engineering, as well as the application of novel data acquisition and analysis methods, are required to develop an appropriate system for industrial purpose. Prior to this, the correlation between biological and physical parameters must be determined. In this work, we combined colour science with the experimentally determined dataset from the cultivation of Microchloropsis salina. Our results indicate a resilient correlation between algal biomass and its specific pigment concentration with the determined reflection spectrum. Data evaluation allowed us to establish identification models for contactless quantification of the respective biological parameter in large-scale automated bioreactors.

New high-k gate dielectrics are highly necessary in facilitating the continuous down-scaling of metal–oxide–semiconductor devices to the sub-10 nm range. This study presents ultrathin organic hydrocarbon (HC) films as a novel high-k gate insulator for metal–insulator–semiconductor (MIS) devices. During inductively-coupled plasma chemical vapor deposition with CH4 and H2 gases, the growth temperature greatly affects the structure of the carbon layers and consequently their dielectric characteristics. Specifically, sp2-rich dielectric HC layers are formed below 600 °C, whereas highly-ordered sp2-hybridized graphene is formed at 950 °C. The k value of the resulting HC films increases up to a maximum value of 90 at 350 °C. Moreover, the MIS devices exhibit excellent gate-insulating properties, including almost no hysteresis in the capacitance–voltage curve, low leakage current, and high dielectric strength, which surpass those of existing high-k gate oxides. These results reveal that the organic HC films are a promising next-generation high-k gate dielectric material for sub-10 nm node Si and organic semiconductor technologies.

Let G be a simple graph and let ?(?) be the matching number of G. It is well-known that reg?(?)⩽?(?)+1. In this paper we show that reg?(?)=?(?)+1 if and only if every connected component of G is either a pentagon or a Cameron–Walker graph.

Vertically-standing, few-layer MoS2 flakes were uniformly and conformally grown on closely-packed TiO2 nanowire substrates using metal-organic chemical vapor deposition. The TiO2/MoS2 heterostructures exhibited remarkably improved photoelectrochemical (PEC) performance and long-term stability owing to enhanced photogenerated electron–hole separation and transfer properties across the heterojunction. Due to the built-in heterojunction potential, the onset potential of TiO2/MoS2 heterostructures was cathodic-shifted to −0.3 V, thereby resulting in maximum PEC reactivity at approximately zero potential. In contrast, MoS2 flakes on fluorine-doped tin oxide substrates exhibited an onset potential of approximately 0.3 V. Furthermore, PEC performance was critically affected by the size of the vertically-standing MoS2 flakes, indicating that controlled conformal growth of MoS2 flakes on appropriate nanostructures is essential to exploit these nanostructures as efficient, non-toxic, inexpensive, and earth-abundant PEC electrodes.

Let Δ be a matroid complex. In this paper, we explicitly compute the regularity of all the symbolic powers of its Stanley–Reisner ideal in terms of combinatorial data of Δ. In order to do that, we provide a sharp bound between the arboricity of Δ and the circumference of its dual Δ * {\Delta^{*}} .

One-dimensional (1D) WO3/two-dimensional (2D) WS2 heterojunction nanostructures are of great interest in various photoelectrochemical (PEC) and electrochemical applications. In this study, we introduced a simple and effective route to synthesize 2D WS2 shell layer on WO3 nanorods by utilizing WO3∙0.33H2O phase. The formation of WO3∙0.33H2O nanorods was critically affected by hydrothermal reaction temperature. The WO3∙0.33H2O phase was preferentially transformed into hexagonal WO3 and 2D WS2 through oxidation and sulfurization at 450 °C, respectively. The 2D WS2 shell layer was more favorably formed on WO3 nanorods, which were synthesized at 180 °C and possessed a significant amount of WO3∙0.33H2O phase (referred to as W-180), than on WO3 nanorods with predominant monoclinic WO3 phase, which were synthesized at 200 °C (referred to as W-200). Thus, the WO3/WS2 core@shell nanorods from W-180 exhibited significantly enhanced PEC performance because of the improved charge transfer properties attributed to the advantageous heterojunction effect of 1D WO3/2D WS2. The new synthesis route from WO3∙0.33H2O phase to 2D WS2 can be applied to synthesize various WO3/2D WS2 heterojunction nanostructures, such as thin films, nanoparticles, and nanorods.

Let I be a monomial ideal in a polynomial ring $R = k[x_{1},\dots ,x_{r}]$ . In this paper, we give an upper bound on $\overline {\text {dstab}} (I)$ in terms of r and the maximal generating degree d(I) of I such that $\text {depth} R/\overline {I^{n}}$ is constant for all $n\geqslant \overline {\text {dstab}}(I)$ . As an application, we classify the class of monomial ideals I such that $\overline {I^{n}}$ is Cohen-Macaulay for some integer n ≫ 0.

Ultrathin Ag nanowires (AgNWs) with a high aspect ratio of over 1300 were synthesized through NaCl and KBr-mediated polyol reaction. AgNWs became thinner and longer with the increase in NaCl concentration, whereas Ag nanoparticles were formed without NaCl. When 0.4 mM of KBr was added, the diameter of the AgNWs decreased further to 20–50 nm and its length increased to 30–60 µm. The appropriate mixture of metal salts was crucial in synthesizing ultrathin and long AgNWs. Transparent conductive films of AgNWs yielded excellent properties with a sheet electrical resistance of 25 Ω/sq and a visible-light transmittance of over 90%.

We give a bound for n 0 ( I ) from which the depth function depth R / I ( n ) of the quotient ring by symbolic powers of a squarefree monomial ideal I stabilizes. In the unmixed codimension two case we show that depth R / I ( n ) is a non-increasing function of n and use this property to provide a sharp bound for n 0 ( I ) .

Ketocarotenoids represent a special group of carotenoids characterized by presence of one or several carbonyl (C=O) groups in their ionone rings. Due to their excellent anti-oxidative characteristics, several ketocarotenoids such as canthaxanthin and astaxanthin are valuable pigments highly sought by feed, cosmetical and nutraceutical industries. In this work I aimed to introduce ketocarotenoid biosynthetic pathway into the model organism Chlamydomonas reinhardtii via overexpression of its native key enzyme ß-carotene ketolase (CrBKT). High transgene expression and transgene stability were attained with help of the bicistronic Ble2A system in which, the selection marker - zeocin-resistance conferring ble gene - is directly linked to gene-of-interest via the so-called self-cleavage foot-and-mounth-disease-virus (FMDV) 2A sequence. Functionality of Ble2A system was proven by successful production of the fluorescence protein mCherry in Chlamydomonas. CrBKT’s enzymatic activity was also successfully confirmed by heterologous production in carotenoid-producing E. coli, indicated by the almost complete conversion of ß-carotene to canthaxanthin and of zeaxanthin to astaxanthin. Two strains of Chlamydomonas were chosen for metabolic engineering: strain UVM-4 which boasted improved transgene expression and strain CC-4102 in which only ß-carotene and zeaxanthin are accumulated due to mutations in beta-carotene biosynthetic pathway and xanthophyll cycle. Transformation of both strains with CrBKT overepxression vectors yielded hundreds of zeocin-resistant colonies but only in ~10% of them could the integration of CrBKT be confirmed via PCR. Ketocarotenoid production was not detected in any PCR-positive lines either. Success was only achieved when transformation conditions were changed, namely algal cells were transformed and recovered in dark on growth medium supplemented with yeast extract and tryptone. Under these conditions, I was able to isolate four CrBKT-overproducing transformants. All four lines were characterized by their pale green color as well as their drastically reduced chlorophyll contents. Canthaxanthin production was also detected in two lines, whose concentration stood at about 10% of total cellular carotenoids (~0.1 pg/cell). An intriguing phenomenon was also observed with these pale green canthaxanthin-producing transformants. When cultivated in light, they promptly reverted back to the normal dark green color. Canthaxanthin could no longer be detected and zeocin resistance was seeminly impaired as well. Though the cause of this phenomenon could not be pinpointed with certainty, it did help explain the failure of my earlier experiments. Taken together, these results formed the foundation for future projects of ketocarotenoid metabolic engineering in microalgae. Several potential pitfalls that might be encountered were identified and strategies to overcome them were also suggested.

We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing open question of Ratliff on the associated primes of powers of ideals., 9 pages. This paper is split from the first version of the paper "Symbolic powers of sums of ideals", arXiv:1702.01766, due to a recommendation of its referee

We prove that a planar graph is Gorenstein if and only if its independence complex is Eulerian.

We report the direct and self-selective synthesis of Ag nanowires (NWs) on graphene patterns through a modified citrate reduction reaction approach. High-density NWs were selectively formed only on the graphene areas, whereas NWs were not synthesized on the SiO2/Si and glass substrates. The size and density of the Ag NWs gradually increased with increasing reaction time and AgNO3 concentration. When the Ag NWs were synthesized with 60 mM AgNO3 at 90 °C for 30 min, the average width and length of the NWs were approximately 150 nm and 3 μm, respectively. Graphene–Ag NW hybrids showed excellent light transmittance (91.5% at 550 nm) with low sheet resistance (102 Ω sq−1). These properties can be further optimized by manipulating the size and density of the Ag NWs. This simple and cost-effective process for the fabrication of graphene–Ag NW hybrid electrode patterns is useful for a variety of applications.

We prove that the cover ideals of all unimodular hypergraphs have the non-increasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables.

Let $G$ be a graph and let $I := I (G)$ be its edge ideal. In this paper, we provide an upper bound of $n$ from which $\depth R/ I(G)^n$ is stationary, and compute this limit explicitly. This bound is always achieved if $G$ has no cycles of length $4$ and every its connected component is either a tree or a unicyclic graph., Comment: To appear in Journal of Algebra

Two-dimensional (2D) SnS2 nanosheets vertically grown on graphene electrodes have still not been demonstrated despite their considerable research interest and potential for photoelectrochemical (PEC) applications. Herein, the controllable growth and enhanced PEC water-splitting performance of vertically aligned 2D SnS2 nanosheets on graphene are reported. The heterojunction quality of the graphene/2D SnS2 nanosheets was ensured by low-temperature growth at 230 °C using metalorganic chemical vapor deposition, resulting in significantly improved charge transfer properties. The PEC photocurrent density and photoconversion efficiency significantly increased upon insertion of a graphene layer. In addition, the 2D SnS2 nanosheet microscopic structure dependency of the PEC performance was systematically studied. The best PEC reactivity of the 2D SnS2 nanosheets was achieved at a height of ∼0.85 µm, corresponding to nearly the limit of the carrier diffusion length, and a thickness of ∼70 nm to balance the low interfacial contact resistance with an effective photogenerated carrier dynamics. These results, including the low-temperature vertical growth of 2D SnS2 on graphene, can effectively be utilized to exploit the full potential of 2D SnS2 for various PEC applications.

To control the growth of layered two-dimensional transition metal dichalcogenide (TMD) materials or complex structures such as heterostructures, it is important to understand the growth mechanism. In this study, we demonstrated the chemical vapor deposition (CVD) growth of vertically aligned MoO2/MoS2 flakes by using sulfur and molybdenum trioxide (MoO3) powders as precursors. Moreover, based on various experimental measurements, a detailed growth mechanism is proposed for elucidating the conversion of MoO2 to MoS2 in the CVD process to achieve vertically aligned MoO2/MoS2 flakes. Our results reveal that two competing pathways for sulfurization reactions occur during the CVD growth process: the dominance of the reactions in the vapor-phase pathway facilitates the formation of parallel-aligned MoS2 flakes, while the dominance of the reactions in the solid-phase pathway determines the formation of vertically aligned MoO2/MoS2 flakes. In addition, the photoelectrochemical (PEC) activity of the vertically aligned MoO2/MoS2 flakes was evaluated for hydrogen production applications based on PEC water splitting. Our results open the door to promote the process for designing the growth of other TMD materials or complex structures, which can be beneficial for use in various applications.

Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I^s for all s > 0. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I^s) as s > 0, and the initial value of s starting from which reg(I^s) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph., Changed title, 16 pages, 3 figures

We present a simple, ultrasonic vibration-assisted lift-off-based patterning approach for graphene and graphene–Ag nanowire (NW) hybrid films. A 20 μm width pattern with uniform and smooth pattern edges was neatly defined on various rigid and flexible substrates. The patterned graphene–Ag NW electrodes showed a low sheet resistance of 19 Ω/sq with a high transmittance of 93% at 550 nm, a robust stability against oxidation, and a high reliability under a bending test. The electrodes also exhibited markedly higher performance than that of commercial fluorine-doped tin oxide electrodes for dye-sensitized solar cells. Given its low-cost, high throughput, and nondamaging effect, this simple and reliable patterning approach stimulates the practical applications of graphene-based flexible transparent electrodes in soft electronic and optoelectronic devices.

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}.$$ If $A$ and $B$ are polynomial rings and if chara$(k) = 0$ or if $I$ and $J$ are monomial ideals, we give exact formulas for the depth and the Castelnuovo-Mumford regularity of $(I+J)^{(n)}$, which depend on the interplay between the symbolic powers of $I$ and $J$. The proof involves a result of independent interest which states that under the above assumption, the induced map Tor$_i^A(k,I^{(n)}) \to$ Tor$_i^A(k,I^{(n-1)})$ is zero for all $i \ge 0$, $n \ge 0$. We also investigate other properties and invariants of $(I+J)^{(n)}$ such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen-Macaulayness., 22 pages, to appear in Math. Z.; This version does not contain the result on the depth function of powers of a homogeneous ideal, due to a recommendation of the referee

Let $I$ be a square-free monomial ideal of a polynomial ring $R$ such that $\dim (R/I) = 2$. We give explicit formulas for computing the $a_i$-invariants $a_i(R/I^{(n)})$, $i=1,2$, and the Castelnuovo-Mumford regularity $\reg (R/I^{(n)})$ for all $n$. The values of these functions depend on the structure of an associated graph. It turns out that these functions are linear functions of $n$ for all $n \ge 2$.

ZnO was prepared by hydrothermal method. The result of scanning electron microscopy showed that the materials had nano rod structures. Ag-doped ZnO was prepared by UV-photoreduction. Crystalline phases and optical absorption of the prepared Ag-doped ZnO samples were determined by X-ray diffraction, Raman spectrum, UV-visible, and UV- photoreduction spectrophotometer. X-ray analyses revealed that Ag was doped ZnO crystallizes in hexagonal wurtzite structure. The incorporation of Ag+ in the site of Zn2+ provoked an increase in the size of nanocrystals as compared to pure ZnO. The photocatalytic and photoluminescence properties of materials were considered.

We graph-theoretically characterize the class of graphs $G$ such that $I(G)^2$ are Buchsbaum., 20 pages, to appear in JAA