Tensor weight structures and t-structures on derived categories of Noetherian schemes

We give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called $\otimes ^c$-weight structures. More precisely, for a Noetherian separated scheme $X$, we give a bijection between the set of compactly generated $\otimes ^c$-weight structures on $\mathbf{D} (\mathrm{Qcoh\hspace{1mm}}X)$ and the set of Thomason filtrations of $X$. We achieve this classification in two steps. First, we show that the bijection of S\v{t}ov\'{\i}\v{c}ek and Posp\'{\i}\v{s}il restricts to give a bijection between the set of compactly generated $\otimes ^c$-weight structures and the set of compactly generated tensor t-structures. We then use our earlier classification of compactly generated tensor t-structures to obtain the desired result. We also study some immediate consequences of these classifications in the particular case of the projective line. We show that in contrast to the case of tensor t-structures, there are no non-trivial tensor weight structures on $\mathbf{D}^b (\mathrm{Coh \hspace{1mm}} \mathbb{P}^1_k)$.Comment: 11 pages, comments are welcome

Transfer of Vertical Graphene Nanosheets onto Flexible Substrates towards Supercapacitor Application

Vertical graphene nanosheets (VGNs) are the material of choice for next-generation electronic device applications. The growing demand for flexible devices in electronic industry brings in restriction on growth temperature of the material of interest. However, VGNs with better structural quality is usually achieved at high growth temperatures. The difficulty associated with the direct growth on flexible substrates can overcome by adopting an effective strategy of transferring the well grown VGNs onto arbitrary flexible substrates through soft chemistry route. Hence, we demonstrated a simple, inexpensive and scalable technique for the transfer of VGNs onto arbitrary substrates without disrupting its morphology and structural properties. After transfer, the morphology, chemical structure and electronic properties are analyzed by scanning electron microscopy, Raman spectroscopy and four probe resistive methods, respectively. Associated characterization investigation indicates the retention of morphological, structural and electrical properties of transferred VGNs compared to as-grown one. Furthermore the storage capacity of the VGNs transferred onto flexible substrates is also examined. A very lower sheet resistance of 0.67 kOhm/sq. and excellent supercapacitance of 158 micro-Farrad/cm2 with 91.4% retention after 2000 cycles confirms the great prospective of this damage-free transfer approach of VGNs for flexible nanoelectronic device application

Are Survival Outcomes Different for Young and Old Patients with Oral and Oropharyngeal Squamous Cell Carcinoma? A Systematic Review and Meta-Analysis

This systematic review and meta-analysis aims to address whether age can be a determinant of overall survival (OS), disease-free survival (DFS), recurrence, distant metastasis (DM) and second primary (SP) in surgically treated oral and oropharyngeal squamous cell carcinoma (OOPSCC). A total of 4981 cases and 44254 controls from 25 comparative observational studies were included in the analysis. A significantly better OS (matched subgroup analysis: OR 1.64; 95% CI 1.31–2.04, overall analysis: OR 1.48; 95% CI 1.09–2.01) was observed in young patients compared to older adults, with heterogeneity ranging from moderate to severe. Worse DFS (unmatched subgroup analysis OR 0.43; 95% CI 0.27–0.68) was observed in young patients compared to older adults with minimal to moderate heterogeneity. The frequency of recurrence (OR 1.49; 95% CI 1.10–2.02) and DM (OR 1.83; 95% CI 1.10–3.03) was significantly higher in the young patients, as found in unmatched and matched subgroup analysis, with the least heterogeneities. Young age can be considered as an independent prognostic factor for recurrence and distant metastases in OOP-SCC. Larger and methodologically robust observational studies with longer follow-up are needed to establish the definitive role of age as an independent prognostic factor on OS and DFS in OOPSCC

Results on Laplacian spectra of graphs with pockets

Let F , H v be simple connected graphs on n and m + 1 vertices, respectively. Let v be a specified vertex of H v and u 1 , … , u k ∈ F . Then the graph G = G [ F , u 1 , … , u k , H v ] obtained by taking one copy of F and k copies of H v , and then attaching the i th copy of H v to the vertex u i , i = 1 , … , k , at the vertex v of H v (identify u i with the vertex v of the i th copy) is called a graph with k pockets. In 2008, Barik raised the question that ‘how far can the Laplacian spectrum of G be described by using the Laplacian spectra of F and H v ?’ and discussed the case when deg ( v ) = m in H v . In this article, we study the problem for more general cases and describe the Laplacian spectrum. As an application, we construct new nonisomorphic Laplacian cospectral graphs from the known ones. Keywords: Laplacian matrix, Laplacian spectrum, Join, Pocket

Compactly generated tensor t-structures on the derived category of a Noetherian scheme

We introduce a tensor compatibility condition for t-structures. For any Noetherian scheme $X$, we prove that there is a one-to-one correspondence between the set of filtrations of Thomason subsets and the set of aisles of compactly generated tensor compatible t-structures on the derived category of $X$. This generalizes the earlier classification of compactly generated t-structures for commutative rings to schemes. Hrbek and Nakamura have reformulated the famous telescope conjecture for t-structures. As an application of our main theorem, we prove that a tensor version of the conjecture is true for separated Noetherian schemes

On singularity and properties of eigenvectors of complex Laplacian matrix of multidigraphs

AbstractIn this article, we associate a Hermitian matrix to a multidigraph G. We call it the complex Laplacian matrix of G and denote it by [Formula: see text]. It is shown that the complex Laplacian matrix is a generalization of the Laplacian matrix of a graph. But, unlike the Laplacian matrix of a graph, the complex Laplacian matrix of a multidigraph may not always be singular. We obtain a necessary and sufficient condition for the complex Laplacian matrix of a multidigraph to be singular. For a multidigraph G, if [Formula: see text] is singular, we say G is [Formula: see text]-singular. We generalize some properties of the Fiedler vectors of undirected graphs to the eigenvectors corresponding to the second smallest eigenvalue of [Formula: see text]-singular multidigraphs