AAdditional Experiments620 Task 1 - Grouping In addition to grouping clue words using token embeddings (discussed in621 the main paper 4), we also ran grouping the words by clustering on'contextual' embeddings. We622 experimentally induce'context' by joining the sixteen (16) word tokens (in a random order) into a623 single pseudo-sentence. The embeddings for each token were different based on the ordering of the624 tokens. We repeat the random ordering sixteen times and report the mean and variance of the results625 obtained in Table 6.626 Mean standard deviation over 16 random seeds is shown. Task 2 - Connections In addition to prompting based results on GPT-4 (discussed in 4), we ran627 experiments on additional LLMs like LLaMa [67] (7B, 13B) using pre-trained configuration weights628 obtained by permission from Meta AI. However, without additional fine-tuning on the specific task,629 these LLMs were unable to solve the task in a meaningful manner.

Theorem proving in natural mathematical language - the mixture of symbolic and natural language used by humans - plays a central role in mathematical advances and education, and tests aspects of reasoning that are core to intelligence. Yet it has remained underexplored with modern generative models. We study largescale language models on two new generation tasks: suggesting the next step in a mathematical proof, and full proof generation. We develop NATURALPROVER,a language model that generates proofs by conditioning on background references (e.g.

In this supplementary material, we present additional details of the proposed Flare7K dataset and experimental settings and show more results. Figure 1: Illustration of a simplified lens system. In the lens and aperture plane, the light passes through the dirty aperture and lens system, leaving a scattering flare on the image plane. In this section, we use a simplified Fourier optics model to illustrate how different kinds of scattering flares occur. A basic lens system can be viewed as a combination of one convex lens, one aperture, and an image plane as shown in Figure 1. We set the optical center as the origin of a coordinate system. Then, the light source's position is (x0,y0, z0). It is a combination of aperture function eAλ(x,y) and a lens function eTL(x,y). Supposing the focus of the lens is f and the lens is ideal. After adjusting the origin of x1 and x2, Equation (11) can be viewed as a standard Fourier transformation. Thus, the point spread function (PSF) which is the square of the amplitude of the image plane's optical field can be written as: PSFλ = |F{eAλ(x,y)}|2. Since stains with depth may bring phase shift for the aperture function, the PSFλ may vary with the wavelength λof the light source.

His Supporters Are Already Looking for the Next Target. "Grok has thoughts on who to look at next." Acting Attorney General Todd Blanche and FBI Director Kash Patel hold a press conference on their prosecution of the Southern Poverty Law Center on April 21, 2026. Get your news from a source that's not owned and controlled by oligarchs. The Justice Department this week announced criminal charges against the Southern Poverty Law Center, alleging that the longtime civil rights watchdog had defrauded its own donors by secretly paying large sums of money to informants within various hate groups.

A.1 Access instructions OpenProteinSet is hosted by the Registry of Open Data on AWS (RODA) and can be accessed at the following link: registry.opendata.aws/openfold/. A.2 Documentation and intended uses We include a datasheet [1] in Section B. Detailed documentation on the precise structure and content of the dataset is provided on the dataset's landing page. A.3 Data format All OpenProteinSet files are in standard plaintext formats (A3M for MSAs, HHSearch format for template hits, and PDB for structure files) that can be read by a wide variety of bioinformatics software. A.5 License OpenProteinSet is made available under the CCBY 4.0 license. A copy of the license is provided with the dataset.

The Department of Justice has announced that it's intervening on the behalf of xAI in the company's recent lawsuit against the state of Colorado. The law is set to go into effect in June, and the DOJ is now asking a Colorado District Court to declare it unconstitutional. In xAI's original argument, Colorado Bill SB24-205 violated the company's First Amendment rights by forcing its developers to change how they create AI products and compelling them to align their products with Colorado's views on diversity and discrimination. The DOJ acknowledges those concerns in its complaint, but specifically focuses its argument on the idea that the law violates the Equal Protection Clause of the Fourteenth Amendment. According to the DOJ, because the law relies on demographics and statistical disparities as evidence of discrimination, it will essentially require developers to distort an AI system's outputs and discriminate based on race, sex, religion and other protected characteristics, a violation of the Fourteenth Amendment.