Government College of Engineering and Ceramic Technology

Albertson conjectured that every graph of chromatic number r has crossing number at least that of K_r. We prove the conjecture for every r. After the known case r <= 18, an r-critical counterexample is reduced to two order ranges. Near r, we use Gallai's decomposition, completion, and a reserved weak-immersion routing argument. In the remaining middle range, we compress repeated independent-triple reductions into an exact terminal edge bound and combine it with sampled crossing-number inequalities. The remaining finite and interval inequalities are verified by exact certificates.

Short-lived communities refer to groups of nodes in a network that form suddenly, persist briefly, and dissolve as the network evolves. These communities in real-world networks play a crucial role in capturing transient consumer interests that emerge around events, trends, or campaigns, making them essential for time-sensitive decision-making. Such interaction and transactional networks evolve over time and can be naturally modeled as temporal networks. Detecting such temporal communities in these evolving networks is critical for understanding their underlying temporal structures. However, most existing temporal community detection methods integrate all snapshots into a single community structure, thereby overlooking short-lived communities that form and dissolve around specific events or transient interests. These temporally localized groups are particularly important in applications where timing and contextual relevance are essential. To address these challenges, a novel approach called graph regularized incremental learning nonnegative matrix factorization (GRIL-NMF) is proposed. This method selects core-periphery subgraphs that exhibit the most significant topological changes at each time step. These selected subgraphs are then combined with adjacent snapshots to create an incremental learning framework. To effectively detect short-lived temporal communities and estimate their optimal duration, a cluster potential spectral bound is introduced that leverages the Courant-Fischer min-max principle. Extensive experiments on both real and synthetic datasets show that GRIL-NMF achieves an average runtime improvement of 8%-10%, with higher gains in later timesteps due to effective pruning of inactive subnetworks, compared to most state-of-the-art methods.

We propose a new quasiparticle – the phoniton – arising from strong hybridization of topological moiré phonons and correlated electrons in twisted WSe₂/MoSe₂ heterostructures. At magic twist angle 𝜃 = 1.82∘, the moiré superlattice simultaneously generates a flat electronic band with bandwidth 𝑊 ≈ 5 meV and a topological flat phonon band with frequency ℏ𝜔ph ≈ 42 meV and Chern number 𝐶 = 1. The electron-phonon coupling constant is 𝜆 = 2.73, leading to a superconducting transition temperature 𝑇𝑐 ≈ 287 K from the McMillan-Allen-Dynes equation. The pairing symmetry is 𝑠 + 𝑖𝑑, fully gapped but time-reversal breaking. We predict a quantized thermal Hall effect as a definitive experimental signature.

Quantifying disorder in composite microstructures is essential for predicting me chanical, thermal, and electrical properties, yet traditional order parameters require prior knowledge of reference structures and fail in high-noise regimes. We present a computational framework based on delay-coordinates dynamic mode decompo sition (DC-DMD) and Shannon entropy that provides a universal, parameter-free disorder metric. The method transforms a two-dimensional microstructure into a one-dimensional scan via a serpentine space-filling curve, embeds the scan into de lay coordinates, computes the DMD eigenvalue spectrum, and evaluates the Shan non entropy of normalized eigenvalue powers. On synthetic composites (N = 500 microstructures, disorder σ ∈ [0.02,0.8], volume fractions ϕ ∈ [0.15,0.35]), geo metric entropy increases monotonically with positional disorder (Pearson r = 0.98, p < 10−6), correlates with configurational entropy (R2 = 0.96), and maintains 95% classification accuracy at signal-to-noise ratio 5 dB, outperforming bond orientational order (60%) and pair distribution function peak height (45%). The method requires no parameter tuning, computes in 2.8±0.3 seconds per 256×256 image on standard hardware, and successfully distinguishes well-dispersed, aggre gated, and percolated dispersion states in polymer nanocomposites, quantifies crys tallinity in glass-ceramics, and detects early-stage demixing in polymer blends 17 minutes before conventional peak intensity methods. This framework provides a reproducible, computationally efficient alternative to traditional order parameters for microstructure characterization, quality control, and inverse materials design.

Parallel Processing is extremely important and ubiquitous nowadays. Synchronization Problem is a very important problem in parallel processing. Currently locks are used to solve synchronization problem in parallel processing. However locks suffer from various drawbacks. Software Transactional Memory (STM) is an alternative approach to solve synchronization problem which does not suffer from the drawbacks of locks. In this paper we have solved synchronization problems in a well-known classical problem- Sleeping Barber Problem and a real-life problem- Airline Reservation Problem using locks and STM and have compared their performances. As the performance of STM is as good as that of locks and it does not suffer from the drawbacks of locks hence STM is better than locks. Locks use pessimistic approach. But STM uses optimistic approach and is hence better. Also codes with locks can be converted to codes with STM just by using STM calls in place of lock calls. So STM can be used with legacy code which is a huge advantage.