These profiles are fed back to the LLM, which then revises the code to reduce overhead.

Code generation models have increasingly become integral to aiding software development.

Using a magnet to connect the transmitting and receiving coils, electrons behave more consistently and the phone is less likely to overheat. Breakthroughs, discoveries, and DIY tips sent six days a week. In-car wireless chargers are notoriously finicky. Your phone can slide off the slippery charging pad at a sudden stop, or overheat and stop charging; the case can also prevent your phone from connecting. Often, it's a pain in the neck, not to mention an added distraction while you're behind the wheel.

The U.S. State Department reissued its Russia travel warning as the Ukraine conflict intensifies, telling U.S. citizens to leave immediately due to detention risks and limited embassy help.

We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~$\mathbb{Q}$, combining explicit resolvent methods with machine learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants~$J_0, \dots, J_4$ derived from binary transvections. For each polynomial, we compute resolvent factorizations to determine its Galois group among the seven transitive subgroups of~$S_7$ identified by Foulkes. Using this dataset, we train a neurosymbolic classifier that integrates invariant-theoretic features with supervised learning, yielding improved accuracy in detecting rare solvable groups compared to coefficient-based models. The resulting database provides a reproducible resource for constructive Galois theory and supports empirical investigations into group distribution under height constraints. The methodology extends to higher-degree cases and illustrates the utility of hybrid symbolic-numeric techniques in computational algebra.

Organizers say the "No Kings" protests drew more than 7 million people across 2,700 cities. The crowds included high-profile politicians, A-list celebrities, and more than a few creative inflatables. On Saturday, crowds gathered in cities across the United States to protest President Donald Trump and his administration. Organizers of the No Kings rallies claim that more than 7 million people attended in all, across 2,700 cities in the Unites States and beyond. The gatherings provided a clear picture not only of how widespread the resistance to the Trump administration has become, but also the diversity of the coalition driving it.

These profiles are fed back to the LLM, which then revises the code to reduce overhead.

We introduce the Graded Transformer framework, a novel class of sequence models that embeds algebraic inductive biases through grading transformations on vector spaces. Extending the theory of Graded Neural Networks (GNNs), we propose two architectures: the Linearly Graded Transformer (LGT) and the Exponentially Graded Transformer (EGT). These models apply parameterized scaling operators-governed by fixed or learnable grading tuples and, for EGT, exponential factors to infuse hierarchical structure into attention and representation layers, enhancing efficiency for structured data. We derive rigorous theoretical guarantees, including universal approximation theorems for continuous and Sobolev functions, reduced sample complexity via effective VC dimension bounds, Lipschitz continuity of graded operations, and robustness to adversarial perturbations. A graded loss function ensures gradient stability and alignment with domain priors during optimization. By treating grades as differentiable parameters, the framework enables adaptive feature prioritization, overcoming limitations of fixed grades in prior work. The Graded Transformer holds transformative potential for hierarchical learning and neurosymbolic reasoning, with applications spanning algebraic geometry (e.g., moduli spaces and zeta functions), physics (e.g., multiscale simulations), natural language processing (e.g., syntactic parsing), biological sequence analysis (e.g., variant prediction), and emerging areas like graph neural networks and financial modeling. This work advances structured deep learning by fusing geometric and algebraic principles with attention mechanisms, offering a mathematically grounded alternative to data-driven models and paving the way for interpretable, efficient systems in complex domains.