Bhaskara Reddy Sannapureddy’s Post

The Geometry of Concepts in LLMs Studies the geometry of categorical concepts and how the hierarchical relations between them are encoded in LLMs. Finding from the paper: "Simple categorical concepts are represented as simplices, hierarchically related concepts are orthogonal in a sense we make precise, and (in consequence) complex concepts are represented as polytopes constructed from direct sums of simplices, reflecting the hierarchical structure." (link in the comments)

📃Scientific paper: A First Order Theory of Diagram Chasing;Une théorie du premier ordre pour la chasse au diagramme Abstract: International audience; This paper discusses the formalization of proofs "by diagram chasing", a standard technique for proving properties in abelian categories. We discuss how the essence of diagram chases can be captured by a simple many-sorted first-order theory, and we study the models and decidability of this theory. The longer-term motivation of this work is the design of a computer-aided instrument for writing reliable proofs in homological algebra, based on interactive theorem provers. Continued on ES/IODE ➡️ https://etcse.fr/5N9vx ------- If you find this interesting, feel free to follow, comment and share. We need your help to enhance our visibility, so that our platform continues to serve you.

Check out my third paper 'n-dependent continuous theories and hyperdefinable sets' on arXiv! ¡Mi tercer artículo "n-dependent continuous theories and hyperdefinable sets" ya está en arXiv!

My first solo authored paper has just officially been published in Communications in Algebra! It is an explicit characterization of a universal valuation structure introduced by Jeffrey and Noah Giansiracusa, along with an extension and applications to noncommutative geometry using it. Check it out here: https://lnkd.in/e2A8FJKE

Excited to share this valuable insight on the composition and merging of assume-guarantee contracts, now available in the new blog post titled "Composition and Merging of Assume-Guarantee Contracts Are Tensor Products" on arXiv (2405.06052v1). The post demonstrates how these operations are integrated into the tensor product structure of the contract algebra. Check it out here: https://bit.ly/4dyXACv.

We just finished today's event in the Si Learning Series on Complexity Theory with lots of big ideas. Here's one of them: network theory. You can find all our resources on complexity theory, nonlinear systems, and network theory here: https://t.ly/9Jho0 You can find the next event on complex adaptive systems next week here: https://t.ly/rMmiS

I would like to share my recent arXiv paper with you, about Navier-Stokes equations. Something I have never intended to write. https://lnkd.in/dWCNSHse

Very proud to share my recent work in collaboration with Prof. D. L. Karabalis. In this work, we study the linear #SDOF and #MDOF problems for #StructuralDynamics in full mathematical rigor, and, as a result, we obtain a family of numerical methods that vastly outperform the classical Newmark-type time-step methods. In part I, we obtain the Weak Form of the equations of motion for both the SDOF and MDOF problems for #StructuralDynamics, and show how to obtain particular numerical algorithms and their bounds. In part II we construct a particular case of an algorithm for the SDOF problem, using Bernstein polynomials for interpolation. This algorithm has two free parameters contributing to convergence: the degree of approximation (unlike traditional timestep methods where interpolation is restricted to second-degree polynomials) and the timestep. The estimates provided in the article show very fast convergence both when the polynomial degree increases and/or the timestep goes to 0. The power of the algorithm can already be seen in the graph of the error in eigenperiod estimation. We consider an undamped SDOF system, for which initial displacement is set to 1, and initial velocity and external force are both set to 0. The time of the first full oscillation is then calculated for different degrees of polynomial approximation p and timestep h. We plot the error rate in the estimation of the eigenperiod. The error rate of Newmark-type methods is somewhere on the top left of the chart, which is in log-scale. We also calculate the amplitude at the estimated eigenperiod and plot the corresponding error rate. In part III, to appear soon, we will present an exhaustive numerical study using a codebase that will be made available on Gitlab. A first version of the codebase is already available on Github. The papers are on the arXiv: https://lnkd.in/drWzZv75 https://lnkd.in/d3fqmEGj The preliminary version of the codebase is on Github: https://lnkd.in/dd6iS8ex #CivilEngineering #MechanicalEngineering #research #opensource #numericalmethods #numericalmethod #numericalanalysis

Recently I started working on my second paper for publishing... In this paper, I embark on a geometric journey to uncover the essence of π through a novel examination of internal distances within equilateral polygons inscribed in a circle. Revisiting the classical methods pioneered by Archimedes, I introduce a fresh perspective by focusing on the internal structure of both even and odd-sided polygons. For even-sided polygons, we derive the distance between opposite vertices, revealing a direct relationship to the circle's circumference. Similarly, for odd-sided polygons, we examine the distance from a vertex to the midpoint of the opposite side, uncovering a unique geometric path to π. These derivations show how the perimeters of these polygons converge to the circle's circumference, highlighting the universal emergence of π. My approach not only aligns with historical methods but also offers new educational insights, making π more intuitive and accessible. This perspective enriches the understanding of π, bridging ancient geometric techniques with modern mathematical concepts. Join me in exploring how the simple shapes of polygons reveal the profound constant π, deepening our appreciation of its geometric beauty and mathematical significance.

Shards https://ift.tt/iHtmxYl Shards. Continuing to explore this system. This type swapping out the typeface for a typeface created with the Fontastic library in Processing creating abstract shards.