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In this article we study complement ideals, and the dual concept of supplement ideals, in nearrings, both of which are generalizations of the concept of complement in a bounded modular lattice. We prove fundamental properties of complements and supplements in arbitrary nearrings. We then establish Galois connections between the ideal lattices of a nearring and of its matrix nearrings, yielding one-to-one correspondences between their respective complement and supplement ideals. We also define graphs associated with complement and supplement ideals of nearrings and study some of their combinatorial properties such as girth and clique number.

Training large foundation models for agentic tasks is increasingly impractical due to the high computational costs, long iteration cycles, and rapid obsolescence as new models are continuously released. Instead of post-training massive models for every new task or domain, we propose Supplement Generation Training (SGT), a more efficient and sustainable strategy. SGT trains a smaller LLM to generate useful supplemental text that, when appended to the original input, helps the larger LLM solve the task more effectively. These lightweight models can dynamically adapt supplements to task requirements, improving performance without modifying the underlying large models. This approach decouples task-specific optimization from large foundation models and enables more flexible, cost-effective deployment of LLM-powered agents in real-world applications.

Let R be an associative ring with unity and let M be an R-module. We call M (ample) Rad-supplementing if M has a (ample) Rad-supplement in every extension. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal and reduced, and the direct sum of countably many copies of R is Rad-supplementing if and only if R is reduced and the direct sum of countably many copies of R is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is a left perfect ring, where P(R) is the sum of all left ideals of R such that Rad I=I.

Coding agents can fix a failing example without preserving the domain rule that made it fail, so later generations can repeat the same plausible mistake. We present agentic synthesis against counterexample-supplemented sketches, a repository-native method for systems whose governing policy is discovered during implementation. A human starts with a partial, code-shaped sketch, and a coding agent generates the first implementation. When a concrete failure exposes missing or mistaken policy, an operator explicitly approves the corrected behavior and rule. The agent then revises the sketch and repairs or regenerates code and prompt surfaces for that one counterexample. The full archive preserves provenance; a selected regression set gates each revision before the next candidate is revealed; and periodic clean regeneration tests whether the evolved sketch, rather than prompt history or accumulated examples, carries the learned policy. We demonstrate the method with CatSynth, a synthetic browser application and captured coding-agent experiment. In one open-world run with GPT-5.4-mini, 8 of 14 frozen candidate cases became counterexamples. The rebuild controls inherited that promotion sch

Large Language Models (LLMs) have demonstrated remarkable capabilities in various tasks, yet still produce errors in domain-specific tasks. To further improve their performance, we propose KSOD (Knowledge Supplement for LLMs On Demand), a novel framework that empowers LLMs to improve their capabilities with knowledge-based supervised fine-tuning (SFT). KSOD analyzes the causes of errors from the perspective of knowledge deficiency by identifying potential missing knowledge in LLM that may lead to the errors. Subsequently, KSOD tunes a knowledge module on knowledge dataset and verifies whether the LLM lacks the identified knowledge based on it. If the knowledge is verified, KSOD supplements the LLM with the identified knowledge using the knowledge module. Tuning LLMs on specific knowledge instead of specific task decouples task and knowledge and our experiments on two domain-specific benchmarks and four general benchmarks empirically demonstrate that KSOD enhances the performance of LLMs on tasks requiring the supplemented knowledge while preserving their performance on other tasks. Our findings shed light on the potential of improving the capabilities of LLMs with knowledge-based S

Summary Osteoporosis is a skeletal disorder, characterized by a decrease in bone strength and puts the individual at risk for fracture. On the other hand, rheumatoid arthritis is a systemic disease of unknown etiology that causes inflammation of the joints of the organs. Purpose Due to the destructive effects of these diseases and its increasing prevalence and lack of appropriate medication for treatment, the present study aimed to evaluate the therapeutic effect of a new type of healthy and live food supplement on rheumatoid arthritis and induced osteoporosis in rats. Methods In this research, healthy and live food powder were synthesized by a new and green route. This organic biomaterial was named NBS. The NBS food supplement had various vitamins, macro and micro molecules, and ingredients. The new healthy and nutritious diet showed that the use of this supplement led to the return of the parameters to normal levels. Results The concentration of 12.5 mg/ kg showed the least therapeutic effect and 50 mg/ kg had the highest therapeutic effect for osteoporosis. The results of blood parameters involved in inflammation in both healthy and patient groups showed that the use of complete

Anemia is a decrease in hemoglobin and red blood cells and due to a decrease in hemoglobin, oxygen carrying capacity reduce. In this disease, the red blood cell the amount and volume decrease. In this research, healthy and live food powder were synthesized by a green route. This organic biomaterial was named NBS. The NBS healthy and live food powder has various vitamins, macro and micro molecules, and ingredients. Twenty Wistar rats were randomly divided into 4 equal groups, including control and treatment groups 1, 2 and 3. Nutritional supplements for healthy living were administered orally via gavage to rats in groups 1, 2, and 3 at 12.5, 25, and 50 mg/ kg, respectively, and within a period of 20 days, one day in between. There was no intervention in the control group in order to reach baseline blood factors. At the end of the study, blood samples were taken from the heart, including blood-red blood cells, hemoglobin, hematocrit and platelets using a fully automated blood cell counting machine. The results showed that the new dietary supplement reduced the level of hematocrit and platelets in the studied rats. The healthy and live food supplement at a concentration of 50 mg / kg

Dietary supplements are widely used but not always safe. With the rapid development of the Internet, consumers usually seek health information including dietary supplement information online. To help consumers access quality online dietary supplement information, we have identified trustworthy dietary supplement information sources and built an evidence-based knowledge base of dietary supplement information-the integrated DIetary Supplement Knowledge base (iDISK) that integrates and standardizes dietary supplement related information across these different sources. However, as information in iDISK was collected from scientific sources, the complex medical jargon is a barrier for consumers' comprehension. To assess how different approaches to simplify and represent dietary supplement information from iDISK will affect lay consumers' comprehension, using a crowdsourcing platform, we recruited participants to read dietary supplement information in four different representations from iDISK: original text, syntactic and lexical text simplification, manual text simplification, and a graph-based visualization. We then assessed how the different simplification and representation strategies

Pollinators are critical to the world's ecosystems and food supply, yet recent studies have found pollination shortfalls in several crops, including strawberry. This is troubling because wild and managed pollinators are currently experiencing declines. One possibility is to try and provide supplemental pollination solutions. These solutions should be affordable and simple for farmers to implement if their use is to be widespread; quadcopters are a great example, already used for monitoring on many farms. This paper investigates a new method for artificial pollination based on wind pollination that bears further investigation. After determining the height where the lateral flow is maximized, we performed field experiments with a quadcopter assisting natural pollinators. Although our results in the field were inconclusive, lab studies show that the idea shows promise and could be adapted for better field results.

This narrative review evaluates the effect of chromium supplementation on glycemia and serum lipids, with an emphasis on patients with type 2 diabetes mellitus (T2DM). Additionally, this narrative review evaluates the essentiality of the trace mineral chromium to human nutrition. Meta-analyses and reviews were included, while certain clinical trials were specifically included to discuss flaws or impact. Overall, this narrative review concludes that chromium supplementation likely has no beneficial effect on glycemia or serum lipids (in subjects with or without T2DM). This narrative review also concludes the essentiality of chromium to human nutrition has become increasingly challenged over time, with some investigators postulating that chromium is pharmacologically active rather than an essential trace mineral. However, further randomized controlled trials (RCTs) are necessary to come to solid conclusions about the effect of chromium supplementation and the essentiality of chromium to human nutrition. This investigation is necessary as manufacturers continue to market chromium supplements as helpful aids to T2DM patients based on flawed studies.

Previous methods usually only extract the image modality's information to recognize group activity. However, mining image information is approaching saturation, making it difficult to extract richer information. Therefore, extracting complementary information from other modalities to supplement image information has become increasingly important. In fact, action labels provide clear text information to express the action's semantics, which existing methods often overlook. Thus, we propose ActivityCLIP, a plug-and-play method for mining the text information contained in the action labels to supplement the image information for enhancing group activity recognition. ActivityCLIP consists of text and image branches, where the text branch is plugged into the image branch (The off-the-shelf image-based method). The text branch includes Image2Text and relation modeling modules. Specifically, we propose the knowledge transfer module, Image2Text, which adapts image information into text information extracted by CLIP via knowledge distillation. Further, to keep our method convenient, we add fewer trainable parameters based on the relation module of the image branch to model interaction relat

Video-based scene graph generation (VidSGG) is an approach that aims to represent video content in a dynamic graph by identifying visual entities and their relationships. Due to the inherently biased distribution and missing annotations in the training data, current VidSGG methods have been found to perform poorly on less-represented predicates. In this paper, we propose an explicit solution to address this under-explored issue by supplementing missing predicates that should be appear in the ground-truth annotations. Dubbed Trico, our method seeks to supplement the missing predicates by exploring three complementary spatio-temporal correlations. Guided by these correlations, the missing labels can be effectively supplemented thus achieving an unbiased predicate predictions. We validate the effectiveness of Trico on the most widely used VidSGG datasets, i.e., VidVRD and VidOR. Extensive experiments demonstrate the state-of-the-art performance achieved by Trico, particularly on those tail predicates.

Variational Autoencoders are widespread in Machine Learning, but are typically explained with dense math notation or static code examples. This paper presents VAE Explainer, an interactive Variational Autoencoder running in the browser to supplement existing static documentation (e.g., Keras Code Examples). VAE Explainer adds interactions to the VAE summary with interactive model inputs, latent space, and output. VAE Explainer connects the high-level understanding with the implementation: annotated code and a live computational graph. The VAE Explainer interactive visualization is live at https://xnought.github.io/vae-explainer and the code is open source at https://github.com/xnought/vae-explainer.

Humanity is close to characterizing the atmospheres of rocky exoplanets due to the advent of JWST. These astronomical observations motivate us to understand exoplanetary atmospheres to constrain habitability. We study the influence greenhouse gas supplement has on the atmosphere of TRAPPIST-1e, an Earth-like exoplanet, and Earth itself by analyzing ExoCAM and CMIP6 model simulations. We find an analogous relationship between CO2 supplement and amplified warming at non-irradiated regions (night side and polar) - such spatial heterogeneity results in significant global circulation changes. A dynamical systems framework provides additional insight into the vertical dynamics of the atmospheres. Indeed, we demonstrate that adding CO2 increases temporal stability near the surface and decreases stability at low pressures. Although Earth and TRAPPIST-1e take entirely different climate states, they share the relative response between climate dynamics and greenhouse gas supplements.

For a Lie algebra $L$ and a subalgebra $M$ of $L$ we say that a subalgebra $U$ of $L$ is a {\em supplement} to $M$ in $L$ if $L = M + U$. We investigate those Lie algebras all of whose maximal subalgebras have abelian supplements, those that have nilpotent supplements, those that have nil supplements, and those that have supplements with the property that their derived algebra is inside the maximal subalgebra being supplemented. For the algebras over an algebraically closed field of characteristic zero in the last three of these classes we find complete descriptions; for those in the first class partial results are obtained.

This paper tackles the goal of conclusion-supplement answer generation for non-factoid questions, which is a critical issue in the field of Natural Language Processing (NLP) and Artificial Intelligence (AI), as users often require supplementary information before accepting a conclusion. The current encoder-decoder framework, however, has difficulty generating such answers, since it may become confused when it tries to learn several different long answers to the same non-factoid question. Our solution, called an ensemble network, goes beyond single short sentences and fuses logically connected conclusion statements and supplementary statements. It extracts the context from the conclusion decoder's output sequence and uses it to create supplementary decoder states on the basis of an attention mechanism. It also assesses the closeness of the question encoder's output sequence and the separate outputs of the conclusion and supplement decoders as well as their combination. As a result, it generates answers that match the questions and have natural-sounding supplementary sequences in line with the context expressed by the conclusion sequence. Evaluations conducted on datasets including "

This document is an internet supplement to my book "Partially Observed Markov Decision Processes - From Filtering to Controlled Sensing" published by Cambridge University Press in 2016. This internet supplement contains exercises, examples and case studies. The material appears in this internet supplement (instead of the book) so that it can be updated. This document will evolve over time and further discussion and examples will be added. This internet supplement document is work in progress and will be updated periodically. I welcome constructive comments from readers of the book and this internet supplement.

Dietary supplements are used by a large portion of the population, but information on their pharmacologic interactions is incomplete. To address this challenge, we present SUPP.AI, an application for browsing evidence of supplement-drug interactions (SDIs) extracted from the biomedical literature. We train a model to automatically extract supplement information and identify such interactions from the scientific literature. To address the lack of labeled data for SDI identification, we use labels of the closely related task of identifying drug-drug interactions (DDIs) for supervision. We fine-tune the contextualized word representations of the RoBERTa language model using labeled DDI data, and apply the fine-tuned model to identify supplement interactions. We extract 195k evidence sentences from 22M articles (P=0.82, R=0.58, F1=0.68) for 60k interactions. We create the SUPP.AI application for users to search evidence sentences extracted by our model. SUPP.AI is an attempt to close the information gap on dietary supplements by making up-to-date evidence on SDIs more discoverable for researchers, clinicians, and consumers.

A subalgebra $B$ of a Lie algebra $L$ is {\em c-supplemented} in $L$ if there is a subalgebra $C$ of $L$ with $L = B + C$ and $B \cap C \leq B_L$, where $B_L$ is the core of $B$ in $L$. This is analogous to the corresponding concept of a c-supplemented subgroup in a finite group. We say that $L$ is {\em c-supplemented} if every subalgebra of $L$ is c-supplemented in $L$. We give here a complete characterisation of c-supplemented Lie algebras over a general field.