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Rahul Saha

Publications and source records attributed to Rahul Saha.

10 recordsLinked to original sources

New Lower and Upper Bounds for the Grothendieck Constant

We establish new bounds on the Grothendieck constant $K_G$: \[ \frac{6\pi}{11} \le K_G \le \frac{\pi}{2\log(1+\sqrt2)} - 10^{-4}. \] Methodologically, our lower bound approach differs from previous works by establishing limitations on the asymptotically optimal Krivine schemes, rather than giving explicit constructions of gap instances. Our upper bound is obtained by proposing and analyzing the first asymptotic construction of rounding schemes, whereas previous works only consider low-dimensional schemes. Together, these bounds determine the previously unknown tenths digit of $K_G$ to be $7$. The bounds were discovered by a long-running collaborative effort of humans and a long-horizon AI research system that we engineered.

cs.CC

Long-Horizon AI Research for Grothendieck Constant: A Case Study in Human-AI Mathematical Collaboration

AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant $K_G$, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of $K_G$ is not known, we recently tightened the best known bounds to \[ \frac{6\pi}{11} \;\le\; K_G \;\le\; \frac{\pi}{2\log(1+\sqrt2)} - 10^{-4}. \] Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.

cs.AI

GeoAI-based post-segmentation quality validation of building footprints via spatial feature engineering

Deep learning-based building footprint extraction from high-resolution imagery often produces topologically inconsistent vectors unfit for direct GIS database ingestion. To address this, we present a multidomain GeoAI quality control framework that automates error detection to systematically purify vector footprint databases. Candidate footprints were generated across five UAV survey sites in Bangladesh using U-Net (ResNet-34) and SAM-LoRA (ViT-B). The extracted raster masks were vectorized, geometrically regularized, and consolidated under a spatial-exclusivity constraint to eliminate duplicate representations. We used twenty-four predictors capturing geometric, spatial-contextual, and raster-derived spectral and texture properties. Machine Learning (ML) classifiers were trained on a development partition (Sites B-D) and rigorously validated on a spatially independent test set (Site E) excluded from hyperparameter tuning and class balancing. The experimental results demonstrate that geometric and spatial-contextual predictors using Decision Tree (DT) provide the most effective discriminatory evidence for identifying object-level boundary deformations. DT achieved an accuracy of 95.31%, an F1-score of 91.06%, and a Matthews correlation coefficient (MCC) of 0.880 on the unseen testing site. At the database level, this framework successfully identified 87.34% of erroneous footprints while maintaining 98.31% of acceptable structures, reducing the residual error proportion from 27.32% to 4.62% and improving final database purity to 95.38%. This translates into a relative error reduction of 83.09%. The findings indicate that post-segmentation object-level ML provides a highly transferable, robust mechanism for automated quality assurance in production-ready geographic information system (GIS) workflows.

cs.CV

Learning Interestingness in Automated Mathematical Theory Formation

We take two key steps in automating the open-ended discovery of new mathematical theories, a grand challenge in artificial intelligence. First, we introduce $\emph{FERMAT}$, a reinforcement learning (RL) environment that models concept discovery and theorem-proving using a set of symbolic actions, opening up a range of RL problems relevant to theory discovery. Second, we explore a specific problem through $\emph{FERMAT}$: automatically scoring the $\emph{interestingness}$ of mathematical objects. We investigate evolutionary algorithms for synthesizing nontrivial interestingness measures. In particular, we introduce an LLM-based evolutionary algorithm that features function abstraction, leading to notable improvements in discovering elementary number theory and finite fields over hard-coded baselines. We open-source the $\emph{FERMAT}$ environment at this URL(https://github.com/trishullab/Fermat).

cs.AI

Swarm UAVs Communication

The advancement in cyber-physical systems has opened a new way in disaster management and rescue operations. The usage of UAVs is very promising in this context. UAVs, mainly quadcopters, are small in size and their payload capacity is limited. A single UAV can not traverse the whole area. Hence multiple UAVs or swarms of UAVs come into the picture managing the entire payload in a modular and equiproportional manner. In this work we have explored a vast topic related to UAVs. Among the UAVs quadcopter is the main focus. We explored the types of quadcopters, their flying strategy,their communication protocols, architecture and controlling techniques, followed by the swarm behaviour in nature and UAVs. Swarm behaviour and a few swarm optimization algorithms has been explored here. Swarm architecture and communication in between swarm UAV networks also got a special attention in our work. In disaster management the UAV swarm network must have to search a large area. And for this proper path planning algorithm is required. We have discussed the existing path planning algorithm, their advantages and disadvantages in great detail. Formation maintenance of the swarm network is an important issue which has been explored through leader-follower technique. The wireless path loss model has been modelled using friis and ground ray reflection model. Using this path loss models we have managed to create the link budget and simulate the variation of communication link performance with the variation of distance.

cs.DC

LLMSTEP: LLM proofstep suggestions in Lean

We present LLMSTEP, a tool for integrating a language model into the Lean proof assistant. LLMSTEP is a Lean 4 tactic that sends a user's proof state to a server hosting a language model. The language model generates suggestions, which are checked in Lean and displayed to a user in their development environment. We provide a baseline language model, along with code for fine-tuning and evaluation to support further development. We provide server implementations that run on CPU, a CUDA GPU, or a Google Colab notebook, as a step towards fast, effective language model suggestions for any user.

cs.AI

A New Approach Towards Autoformalization

Verifying mathematical proofs is difficult, but can be automated with the assistance of a computer. Autoformalization is the task of automatically translating natural language mathematics into a formal language that can be verified by a program. This is a challenging task, and especially for higher-level mathematics found in research papers. Research paper mathematics requires large amounts of background and context. In this paper, we propose an avenue towards tackling autoformalization for research-level mathematics, by breaking the task into easier and more approachable subtasks: unlinked formalization (formalization with unlinked definitions and theorems), entity linking (linking to the proper theorems and definitions), and finally adjusting types so it passes the type checker. In addition, we present arXiv2Formal, a benchmark dataset for unlinked formalization consisting of 50 theorems formalized for the Lean theorem prover sampled from papers on arXiv.org. We welcome any contributions from the community to future versions of this dataset.

cs.CL

On Three-Term Linear Relations for Theta Series of Positive-Definite Binary Quadratic Forms

In this paper, we investigate three-term linear relations among theta series of positive-definite integral binary quadratic forms. We extend Schiemann's methods to characterize all possible three-term linear relations among theta series of such forms, providing necessary and sufficient conditions for such relations to exist. To accomplish this, we develop, implement, and execute a novel extended refinement algorithm on polyhedral cones. We show that there is exactly one non-trivial three-term linear relation: it involves quadratic forms with discriminants $-3, -12, -48$, all in the same rational squareclass $-3(\mathbb{Q}^\times)^2$.

math.NT

Identifying Reference Objects by Hierarchical Clustering in Java Environment

Recently Java programming environment has become so popular. Java programming language is a language that is designed to be portable enough to be executed in wide range of computers ranging from cell phones to supercomputers. Computer programs written in Java are compiled into Java Byte code instructions that are suitable for execution by a Java Virtual Machine implementation. Java virtual Machine is commonly implemented in software by means of an interpreter for the Java Virtual Machine instruction set. As an object oriented language, Java utilizes the concept of objects. Our idea is to identify the candidate objects' references in a Java environment through hierarchical cluster analysis using reference stack and execution stack.

cs.SE