Black holes, as the most mysterious celestial bodies in the universe, have received limited research from humans. This article reviews the history of observing black holes and present the dynamical method for measuring their mass. Drawing on reported dynamical estimates, I built a dataset of stellar-mass black holes and obtained an average mass of 8.016 M⊙, which is consistent with the theoretical lower limit of 5M⊙. Although I verified that most black holes apply to the theoretical lower limit of 5M⊙ proposed in previous studies, it is possible for Swift J1727.8-1613, XTE J1650-500, GRS 1716-249, GRS 1009-45, GRO J0422+32, H 1705-250, 3A 1524-617, 1H 1659-487 with a minimum mass below 5M⊙.

This paper uses X-ray reflection light spectrum method as the key. It studies how to measure the spin number (a*) of star-mass black holes. The spin number (a*) is linked to the radius of the Innermost Stable Circular Orbit (ISCO). The big change of the Fe Kα line in the reflection light spectrum (caused by Doppler effect and gravity red-shift) is the key to find out a*. We use three telescopes together to get data: NICER, NuSTAR, Insight-HXMT. We use the relxill model series to fit the data. We get results for 3 typical black holes: Cygnus X-1: Its spin is extremely fast (a*>0.9985). 4U 1543-47: Its spin is medium. GX 339-4: Its spin is very fast, and its iron amount is 5 times more than the Sun’s. We compare this method with the continuous spectrum fitting method. The reflection light spectrum method is more accurate for checking spin, tilt angle and iron amount. But it depends on the model (for example, the result of GX 339-4 is conflicting). We need to check it many times to reduce mistakes. The conclusion says: Long-time material absorption makes the black hole’s spin reach the maximum. Jet feedback stops the spin from getting faster. In the future, with new telescopes (eXTP, Athena) and studies on medium-mass black holes, we can understand the spin’s change rules better.

Predicting traffic volumes is crucial for reducing congestion on roads while avoiding unnecessary over-spending on road infrastructure. This study uses AutoRegressive (AR), AutoRegressive Integrated Moving Average (ARIMA), and Seasonal ARIMA models with eXogenous regressors (SARIMAX) models to make hourly traffic volume predictions at seven locations in the vicinity of the Adelaide Central Market; the AR and ARIMA models were trained on daily data, and the SARIMAX model was trained on hourly data. For improved performance, a logarithm return function was combined with a z-score test and an IQR test to remove data outliers before using time-weighted linear interpolation or exponential smoothing to fill in missing data. Statistical tests such as the Augmented Dickey-Fuller (ADF) and Partial Autocorrelation Function (PACF) were applied to the models to determine their optimal coefficients. The accuracy of each model was measured using R-square (R​2), root mean square error (RMSE), mean absolute percentage error (MAPE), and symmetric MAPE (sMAPE). All three models had lower MAPE and sMAPE values, all less than 20%. The SARIMAX produced the best R​2results with all values at least 0.90, while the other two models performed poorly in this test. Overall, the SARIMAX model yielded the most productive results; urban planners can use this model to forecast future traffic volumes and make adjustments to road networks to mitigate congestion.

As the economy grows non-stop, stocks have become a part of people's daily lives. So, guessing stock prices correctly is a very important topic. If we can guess stock prices right, it can help people who invest money get more money back. It also helps the country's economy grow. To solve this problem, many people who study this stuff have used "neural networks" (a kind of computer tool) and got good results. In this study, we used a "fully connected neural network" as the tool to guess stock prices. The data we used is Coca-Cola's daily stock prices from October 23, 2012 to October 23, 2022. We got this data from Kaggle (a website with data). We compared many test results using two ways: MAE and MSE (two simple ways to check if the guess is good). Then we found the right settings for the tool. The study says: this tool is good for guessing stock prices when the data has some time-related links, some useful information, and is not a simple straight-line relationship.

This essay traces the historical development of optimization, focusing on one-variable problems and their applications. We discuss the role of non-negative polynomials in optimization and carefully examine a classical result by Motzkin. Building on this result, we construct a new polynomial that is non-negative but not a sum-of-squares polynomial, propose a method to check whether a polynomial is sum-of-squares, and explore the relationship between the set of sum-of-squares polynomials and the set of non-negative polynomials.

The connections between the Cube puzzle, group theory, and Cayley graphs is explored. It provides a comprehensive introduction to these concepts and their relation. The Cube puzzle is a popular puzzle, and the collection of all its possible states constitutes a group under face rotation operations. Focusing on the Cube puzzle group, we delve into the mathematical structure underlying this puzzle's structure. A key object of this exploration is the Cayley graph, which visually represents the Cube puzzle group. This graph connects the Cube puzzle with many other topics and lead to many applications. By modeling the Cube puzzle as a random walk on its corresponding Cayley graph, we gain insights into scrambling and solving processes. Another surprising application is its connection to musical patterns through modulation graphs. This paper aims to bridge the gap between abstract algebraic concepts and their tangible applications in the Cube puzzle.

The recurrent nova T CrB (T Coronae Borealis) is on the verge of its next dramatic eruption. Using decades of observations from the American Association of Variable Star Observers (AAVSO) Database, we set out to build an "eruption alarm" based on machine learning to predict the eruption. We focused primarily on the photometric behavior preceding the 1946 eruption, and emphasized the "pre-eruption dip" in 1945. The time series models, including Gaussian process regression (GPR) and long short-term memory (LSTM) networks, are fed with both current data and historical records to forecast the next eruption. It is likely to occur near JD 2461231.8 (July 10, 2026), with an estimated uncertainty window of ≈30 days. A timely prediction of the eruption epoch can enable coordinated follow-up campaigns across multiple messengers and wavelengths, thus providing a continuous spectral energy distribution (SED). Such coverage will offer new insights into the details of the mass accretion and evolution of this intriguing binary.

Music genre classification remains a fundamental challenge in music information retrieval, with applications spanning automated music recommendation and content organization on streaming platforms. This study compares two prevalent machine learning approaches: feature-based analysis using Random Forest (RF) classifiers and visual analysis using Convolutional Neural Networks (CNNs). Using the GTZAN dataset containing 1,000 audio samples across 10 genres, we evaluate both methodologies through comprehensive performance metrics and cross-validation. The feature-based approach employs manually extracted audio features including Mel-Frequency Cepstral Coefficients (MFCCs), spectral centroid, chroma, and temporal characteristics. The visual approach processes mel-spectrograms through a CNN architecture optimized for small datasets via Global Average Pooling, reducing parameters from 5.3 million to 392,000. Results demonstrate that the optimized CNN achieves superior performance with 67.67% mean accuracy and 5.46% standard deviation in 5-fold cross-validation, compared to the RF model's 54.50% accuracy with 20% variance. While both approaches struggle with genres like disco and rock, the CNN approach shows more consistent classification across all genres. These findings suggest that visual analysis through properly configured CNNs outperforms feature-based methods for music genre classification, particularly when architectural adjustments account for limited training data.

NASA's Kepler mission has discovered thousands of Transit Candidate Events (TCEs), though the accurate discrimination between genuine exoplanets and astrophysical false positives remains a challenge. Deep learning methods have attained very high accuracy but at the cost of poor interpretability and significant computational time. Presenting herein an interpretable machine learning framework using random forest classification with physically motivated feature engineering, this study extracts 31 diagnostic features from Kepler light curves and optimizes model parameters through systematic hyperparameter tuning. Testing on 1523 TCEs yields competitive performance (AUC = 0.967), ranking true planets above false positives 96.7% of the time. Two new planets are identified in our study: Kepler-80 g, adding to a five-planet resonant chain, and Kepler-80 i, equalizing Kepler-90 and the Sun as the star hosting the greatest number of known planets. This provides astronomers with a clear interpretation and a computation-time-friendly alternative to deep learning in exoplanet validation.