Heterochromatic Higher Order Transversals for Convex Sets

In this short paper, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families compact $(r, R)$-fat convex sets in $\mathbb{R}^{d}$ and if every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $k+2$ convex sets that can be pierced by a $k$-flat then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by finitely many $k$-flats. Additionally, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families of compact convex sets in $\mathbb{R}^{d}$ where each $\mathcal{F}_{n}$ is a family of closed balls (axis parallel boxes) in $\mathbb{R}^{d}$ and every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $2$ intersecting closed balls (boxes) then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by a finite number of points from $\mathbb{R}^{d}$. To complement the above results, we also establish some impossibility of proving similar results for other more general families of convex sets. Our results are a generalization of $(\aleph_0,k+2)$-Theorem for $k$-transversals of convex sets by Keller and Perles (Symposium on Computational Geometry 2022), and can also be seen as a colorful infinite variant of $(p,q)$-Theorems of Alon and Klietman (Advances in Mathematics 1992), and Alon and Kalai (Discrete & Computational Geometry 1995).Comment: 16 pages and 5 figures. Section 3 rewritte

Spectral Properties of Disordered Interacting Non-Hermitian Systems

Non-hermitian systems have gained a lot of interest in recent years. However, notions of chaos and localization in such systems have not reached the same level of maturity as in the Hermitian systems. Here, we consider non-hermitian interacting disordered Hamiltonians and attempt to analyze their chaotic behavior or lack of it through the lens of the recently introduced non-hermitian analog of the spectral form factor and the complex spacing ratio. We consider three widely relevant non-hermitian models which are unique in their ways and serve as excellent platforms for such investigations. Two of the models considered are short-ranged and have different symmetries. The third model is long-ranged, whose hermitian counterpart has itself become a subject of growing interest. All these models exhibit a deep connection with the non-hermitian Random Matrix Theory of corresponding symmetry classes at relatively weak disorder. At relatively strong disorder, the models show the absence of complex eigenvalue correlation, thereby, corresponding to Poisson statistics. Our thorough analysis is expected to play a crucial role in understanding disordered open quantum systems in general.Comment: 12 pages, 15 figures, 3 table

Probing the Mechanism of Viral Inhibition by the Radical S-adenosyl-L-methionine (SAM) Dependent Enzyme- Viperin

Viperin (Virus Inhibitory Protein; Endoplasmic Reticulum associated, INterferon inducible) is an endoplasmic reticulum (ER)-associated antiviral responsive protein that is highly up-regulated in eukaryotic cells upon viral infection. Viperin is a radical S-adenosyl-L-methionine (SAM) enzyme, that catalyses the synthesis of antiviral nucleotide 3’-deoxy-3’, 4’-didehydro-CTP (ddhCTP) exploiting radical SAM chemistry. However, the modulation of its catalytic activity by other intracellular proteins is not well understood and needs further investigation. In this dissertation, I use enzymology-based approaches to investigate how viperin’s enzymatic activity is regulated through its interaction with various cellular and viral proteins that are involved in cellular metabolic and signalling pathways and viral replication. I showed that viperin can reduce the intracellular expression level of the cholesterol biosynthetic enzyme, farnesyl pyrophosphate synthase (FPPS). This, in turn perturbing the intracellular cholesterol synthesis, thereby retarding budding of enveloped viruses from cholesterol-rich lipid rafts of host cell membranes. I also undertook a proteomics study that revealed that viperin interacts with several other endogenous cholesterol biosynthetic enzymes. I also demonstrated that viperin promotes the degradation of viral non-structural protein A (NS5A) from hepatitis C virus through proteasome-mediated degradation in the presence of sterol-regulatory protein VAP-33. In turn, co-expression of viperin with VAP-33 and NS5A reduced the specific activity of viperin by ~ 3-fold. Lastly, this study showed that viperin is activated by innate immune signalling proteins kinase IRAK1 and ubiquitin ligase TRAF6, as it facilitates the ubiquitination of IRAK1 by TRAF6. The results provide valuable insights into the mechanism of action of viperin in regulating these target proteins and its significance as a SAM-dependent enzyme.PHDChemistryUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155178/1/soumigho_1.pd

Stabbing boxes with finitely many axis-parallel lines and flats

We give necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p,q)$-problem.Comment: 13 page

Dimension Independent Helly Theorem for Lines and Flats

We give a generalization of dimension independent Helly Theorem of Adiprasito, B\'{a}r\'{a}ny, Mustafa, and Terpai (Discrete & Computational Geometry 2022) to higher dimensional transversal. We also prove some impossibility results that establish the tightness of our extension.Comment: 10 page

Almost covering all the layers of hypercube with multiplicities

Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of $\mathcal{Q}^{n}_{k}$. In this short note, we show that $$deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_{1}, \dots, H_{m}$ in $\mathbb{R}^{n}$, where $m = \max\left\{ k, n-k\right\}+2t-2$, such that every point of $\mathcal{Q}^{n}_{k}$ will be covered exactly $t-1$ times, and every other point of $\mathcal{Q}^{n}$ will be covered at least $t$ times. Note that putting $k = 0$ and $t=1$, we recover the much celebrated covering result of Alon and F\"uredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube $\mathcal{Q}^{n}$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.Comment: 16 pages, substantial changes from previous version, title and abstract changed to better reflect the content of the pape

Statistical optimalization of α-Amylase production from Penicillium notatum NCIM 923 and kinetics study of the purified enzyme

In this study, response surface methodology (RSM) was employed to optimize the production of α-amylase by Penicillium notatum NCIM 923 through solid-state fermentation. The individual and combinational effects of the factors, i.e. substrate amount, initial moisture, fermentation time, temperature and size of inoculum were found to have significant effects on α-amylase production: the optimum values of the tested variables were 5 g, 70%, 94 h, 28 °C and 20%, respectively. The predicted amylase production (2819.24 U/g) was in good agreement with the value measured under optimized surrounding (2810.33 U/g). The molecular mass of purified α-amylase was about 52 kDa. The enzyme activity exhibited its pH optimum between pH 4.6 and 6.6, and it had maximal activity at 50 °C. The apparent Km and Vmax of α-amylase for starch were 4.1 mg/ml and 247.6 μmol/min, respectively. The activation energy (Ea) for starch hydrolysis was found to be 14.133 kJ/mol. The enzyme was thermostable with half-life (t1/2) of 110 min at 80 °C and temperature coefficient (Q10) value of 1.0. Purified enzyme was activated by Ca2+ and inhibited by Hg2+ ions. EDTA also inhibited the enzyme activity, indicating that the purified enzyme is a metalloenzyme

On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube

Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin. Their proof is among the early instances of the polynomial method, which considers a natural polynomial (a product of linear factors) associated to the hyperplane arrangement, and gives a lower bound on its degree, whilst being oblivious to the (product) structure of the polynomial. Thus, their proof gives a lower bound for a weaker polynomial covering problem, and it turns out that this bound is tight for the stronger hyperplane covering problem. In a similar vein, solutions to some other hyperplane covering problems were obtained, via solutions of corresponding weaker polynomial covering problems, in some special cases in the works of the fourth author (Electron. J. Combin. 2022), and the first three authors (Discrete Math. 2023). In this work, we build on these and solve a hyperplane covering problem for general symmetric sets of the hypercube, where we consider hyperplane covers with higher multiplicities. We see that even in this generality, it is enough to solve the corresponding polynomial covering problem. Further, this seems to be the limit of this approach as far as covering symmetry preserving subsets of the hypercube is concerned. We gather evidence for this by considering the class of blockwise symmetric sets of the hypercube (which is a strictly larger class than symmetric sets), and note that the same proof technique seems to only solve the polynomial covering problem

Combination therapy in the treatment of Stevens-Johnson syndrome/toxic epidermal necrolysis: a case series and review of literature

Stevens-Johnson syndrome (SJS) and toxic epidermal necrolysis (TEN) are life-threatening disease of skin and mucous membrane that are mostly caused by drugs. Many studies have focussed on treatment that modify immunologic responses like corticosteroid, IVIG, cyclosporine, biologics like TNF-α inhibitors etanercept, infliximab etc. But there are few studies available on using two immunomodifier drugs simultaneously. However, no standardized treatment protocol has been established for SJS/TEN patients. We present a case-series of 10 SJS-TEN patients treated with both systemic corticosteroid and cyclosporine. We provide a review of literature on individual systemic corticosteroid, cyclosporine and also simultaneous use of both agents for SJS/TEN, including various outcome measures-stabilization, mortality rate, hospital length of stay and comparison to other systemic agents