Mathematics

Time series forecasting performance is strongly influenced by the structural properties of the underlying data, yet learning-based models are often applied without sufficient validation of this dependency. This study evaluates a uniformly configured Long Short-Term Memory (LSTM) model on five real-world weekly time series with different levels of periodicity, noise, and volatility. Forecasting is performed in a single-step setting using a fixed sliding window of 12 weeks under a consistent training, validation, and testing framework. Model performance is assessed using mean squared error (MSE) and the coefficient of determination R². The results show that for well-structured series, both the LSTM model and Holt’s exponential smoothing achieve very low MSE values with R² scores close to one, indicating excellent predictive accuracy. For other items, performance varies across methods, with either the LSTM or Holt model providing the best results depending on the data structure. These findings confirm that high forecasting accuracy can be achieved with both advanced and classical methods, and that data characteristics play a more decisive role than model complexity.

Topology identification and signal inference are cornerstone tasks in graph signal processing (GSP). Structural Equation Modeling (SEM) is particularly effective for network inference as it explicitly captures causal dependencies. However, a major bottleneck in existing SEM-based approaches is the reliance on fully observable exogenous inputs. In many practical applications, systems are driven by latent stimuli, rendering traditional estimation methods ineffective. To overcome this, we propose a novel SEM framework for the joint inference of graph topology and unknown exogenous inputs. The core innovation lies in the spatio-temporal modeling of these latent inputs: each stimulus is decomposed into a rank-one component characterized by nodal sparsity (spatial localization) and temporal piecewise smoothness (temporal persistence). This structured formulation transforms an otherwise ill-posed blind identification problem into a tractable regularized optimization task. We develop an efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve the resulting convex problem. Numerical experiments on synthetic and real-world datasets demonstrate that the proposed method effectively disentangles endogenous network interactions from latent exogenous influences, outperforming baseline approaches in both topology and signal recovery.

Reconstructing a genome from collections of short DNA fragments is a fundamental problem in modern sequencing. Although genome assembly algorithms are widely used in practice, the mathematical conditions that allow exact reconstruction are not always clear. This study develops a graph-theoretic framework for genome reconstruction using De Bruijn graphs and Eulerian paths in an idealized, error-free setting. Each k-mer is represented as a directed edge connecting its -length prefix and suffix. The resulting overlap graph is constructed using a balanced search tree and traversed with a stack-based Eulerian algorithm. Numerical experiments over a broad range of genome lengths and fragment lengths reveal a sharp transition in reconstruction accuracy. This transition is explained by a probabilistic model for prefix collisions in the directed graph. The theoretical predictions agree with simulation results and provide conditions on the fragment length required for reliable reconstruction. These results show that the difficulty of genome assembly is governed primarily by the combinatorial structure of the underlying graph rather than by algorithmic heuristics.