Dense Subgraphs in Random Graphs

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then $\omega_{\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\alpha(\gamma,p)=\gamma\log\frac{\gamma}{p}+(1-\gamma)\log\frac{1-\gamma}{1-p}$, we show that $\omega_{\gamma}(G_{n,p})$ is one of the two integers closest to $\frac{2}{\alpha(\gamma,p)}\big(\log n-\log\log n+\log\frac{e\alpha(\gamma,p)}{2}\big)+\frac{1}{2}$, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to handle the more complicated ways in which these "quasi-cliques" may overlap

Auto-Grading for 3D Modeling Assignments in MOOCs

Bottlenecks such as the latency in correcting assignments and providing a grade for Massive Open Online Courses (MOOCs) could impact the levels of interest among learners. In this proposal for an auto-grading system, we present a method to simplify grading for an online course that focuses on 3D Modeling, thus addressing a critical component of the MOOC ecosystem that affects. Our approach involves a live auto-grader that is capable of attaching descriptive labels to assignments which will be deployed for evaluating submissions. This paper presents a brief overview of this auto-grading system and the reasoning behind its inception. Preliminary internal tests show that our system presents results comparable to human graders

Enzyme Assisted Biodegradation of Direct Red 81 By Micrococcus Glutamicus NCIM 2168

Azo dyes have been extensively used in textile, pharmaceutical, paper, paint industries. The industries manufacturing dyes generate a large volume of water. Wastewater containing dyes in most of the cases is discharged into water bodies without any treatment or impartial treatment. This hampers not only flora and fauna of the aquatic ecosystem but showed adverse effects on human beings. Existing physical and chemical methods have their advantages and disadvantages. Biodegradation of dyes finds an eco- friendly process. In the present study, Micrococcus glutamicus NCIM 2168 was used for decolourization of the dye Direct Red 81. The isolate decolourized 98.54% of the dye at pH 6 and 28°C in 9 hours. Degradation of the dye was confirmed by the change in λmax of the decolorized sample. Confirmation of the degradation was done by HPLC and GCMS studies. Degradation was brought about by Oxidoreductases. Toxicity studies revealed nontoxic nature of the product. The culture was found to decolourise mixture of five dyes. Hence, the selected bacterial culture can be successfully used for the treatment of dye containing wastewate

Raman Scattering on alpha-RuCl3 under high magnetic fields

The Kitaev Quantum Spin Liquid (QSL) is a theoretical phase of matter proposed by physicist Alexei Kitaev in 2006. It is characterized by the absence of conventional long-range magnetic order in the magnetic moments of electrons, even at absolute zero temperature, distinguishing it from traditional magnetic systems. This QSL phase arises from anisotropic, direction-dependent spin interactions among spin-1/2 electrons on a two-dimensional honeycomb lattice. alpha-RuCl3 has emerged as a promising candidate for the observation of the QSL phase due to its direction-dependent exchange interactions. However, the presence of isotropic Heisenberg exchange interactions results in a zigzag antiferromagnetic order below 7 K, with additional complexity introduced by anisotropic off-diagonal interactions. Despite these challenges, numerous studies suggest that a QSL phase might be stabilized in the field-induced phase of the material. In this study, we employ Raman scattering to investigate  alpha-RuCl3. We begin with angle-resolved Raman scattering to verify the crystal symmetry. Subsequently, we explore the field-induced phase, observing a transition in the Raman response from a featureless, gapless broad continuum and weak antiferromagnetic magnons in the magnetically ordered phase to sharp magnetic excitations and a gapped continuum in the disordered phase. Additionally, helicity-resolved Raman scattering with in-plane magnetic fields reveals the chiral nature of magnetic and phononic excitations. This technique identifies a distinct field regime between 7.5 and 10.5 T, correlating with the proposed stabilization of the QSL phase. Our study underscores the significance of the field-induced phase, providing insights into complex interactions such as chirality-selective hybridization between the continuum and phonons, and a particular non-reciprocal hybridized feature arising from phonon-magnon interaction

Covariance Realization Problem for Spin Systems

Let (Ω, Α) be a measurable space, F be a family of measurable functions f from Ω to R, and c: F→R be a given function. A generalized moment problem consists of finding all probabilities P on (Ω, Α) such that ∫ f dP = c(f) = cf for all f є F, and in providing conditions on ( c ) f є F for the existence of at least one such probability. Generalized moment problems of this kind have been widely studied, mainly in the theoretical engineering community, for continuous random variables. In this thesis, we consider the special case of the covariance realization problem for spin systems and discuss the necessary and sufficient conditions for the realizability of a correlation matrix of order n ≥ 2. Let Ωn = { -1, 1}ⁿ be the space of length n sequences which are denoted by σ = (σ1, σ 2, …, σn), where σi є { -1, 1 }. Define the spin random variables ξi : Ω →{ -1, 1} for 1 ≤ i ≤ n as ξi (σ ) = σ i . For a probability P on Ωn , we denote by EP the expectation with respect to P . Given a symmetric matrix C = (( c ij)), we ask the following question in this thesis: under what condition does there exist a probability P such that EP (ξi) = 0 and c ij = EP (ξi ξj) for 1 ≤ i ≤ j ≤ n ? In this case, we say that C is a spin correlation matrix. The necessary and sufficient conditions for a symmetric matrix of order n ≤ 4 to be a spin correlation matrix are already known. In this thesis, we obtain a general set of inequalities that are necessary and sufficient for any n . We also give a minimal set of necessary and sufficient conditions for n=5,6. Finally, we discuss methods to explicitly find the measure that realizes the given spin correlations (if they are feasible). We give a deterministic algorithm as well as a stochastic version of the same algorithm to find the measure explicitly. The efficiency of different algorithms is compared and some examples are worked out to illustrate the point