Tammy Kolda’s Post

[𝗢𝗽𝗲𝗻 𝗔𝗰𝗰𝗲𝘀𝘀 𝗔𝗿𝘁𝗶𝗰𝗹𝗲 𝗛𝗶𝗴𝗵𝗹𝗶𝗴𝗵𝘁 🌟] 𝐍𝐨𝐧𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 𝐋𝐨𝐰 𝐌𝐮𝐥𝐭𝐢-𝐫𝐚𝐧𝐤 𝐓𝐡𝐢𝐫𝐝-𝐨𝐫𝐝𝐞𝐫 𝐓𝐞𝐧𝐬𝐨𝐫 𝐀𝐩𝐩𝐫𝐨𝐱𝐢𝐦𝐚𝐭𝐢𝐨𝐧 𝐯𝐢𝐚 𝐓𝐫𝐚𝐧𝐬𝐟𝐨𝐫𝐦𝐚𝐭𝐢𝐨𝐧 (Numerical Linear Algebra with Applications, 2024) Michael K. Ng Department of Mathematics, Hong Kong Baptist University 𝗔𝗯𝘀𝘁𝗿𝗮𝗰𝘁: The main aim of this paper is to develop a new algorithm for computing a nonnegative low multi-rank tensor approximation for a nonnegative tensor. In the literature, there are several nonnegative tensor factorizations or decompositions, and their approaches are to enforce the nonnegativity constraints in the factors of tensor factorizations or decompositions. In this paper, we study nonnegativity constraints in tensor entries directly, and a low rank approximation for the transformed tensor by using discrete Fourier transformation matrix, discrete cosine transformation matrix, or unitary transformation matrix. This strategy is particularly useful in imaging science as nonnegative pixels appear in tensor entries and a low rank structure can be obtained in the transformation domain. We propose an alternating projections algorithm for computing such a nonnegative low multi-rank tensor approximation. The convergence of the proposed projection method is established. Numerical examples for multidimensional images are presented to demonstrate that the performance of the proposed method is better than that of nonnegative low Tucker rank tensor approximation and the other nonnegative tensor factorizations and decompositions. https://lnkd.in/gDwWjQVA ⭐ 𝗠𝗮𝗱𝗲 𝗢𝗽𝗲𝗻 𝗔𝗰𝗰𝗲𝘀𝘀 𝘁𝗵𝗿𝗼𝘂𝗴𝗵 𝘁𝗵𝗲 𝗟𝗶𝗯𝗿𝗮𝗿𝘆’𝘀 𝗢𝗔 𝗣𝘂𝗯𝗹𝗶𝘀𝗵𝗶𝗻𝗴 𝗔𝗴𝗿𝗲𝗲𝗺𝗲𝗻𝘁 Learn about our OA Agreements: https://bit.ly/hkbuOA #Tensor #Tensorflow #Tensorlearning #Nonnegative #LowRankApproximation #TensorDecomposition #machinelearning #MathematicsResearch #ImageProcessing #NumericalLinearAlgebra #FourierTransformation #AlgorithmDevelopment #PeerReviewed #MathematicalModeling #DataScience #DataScienceApplications #MachineLearning #TechInnovation #DataAnalysis #EdTech #Mathematics #Research #OpenAccess #Article #University #Education #HKBU HKBU Faculty of Science #HKBUScience #HKBUSCI #MATH #Library

"The last 200 years have seen the influence of mathematics [via the 'computing paradigm'] deepen across almost all domains of human activity, amply supported by torrents of data and dramatic increases in computing power. ... Sometimes, it seems the paradigm has reached its limits; that every field that can benefit from math has been introduced to it. But we may now be nearing the computational paradigm’s greatest success of all: modeling intelligence through math using large language models. In that sense, the computational paradigm may be reaching its logical conclusion: turning us all into math." – Bo Malmberg & Hannes Malmberg via Works In Progress https://lnkd.in/dfTQuYsT

My latest AMS review is available. Paper: "DEEP SIGNATURE ALGORITHM FOR MULTIDIMENSIONAL PATH-DEPENDENT OPTIONS" Author: Erhan Bayraktar, Qi Feng and Zhaoyu Zhang Journal: SIAM Journal of Financial Mathematics (2024) Vol. 15, Issue 1 Field: Applied Mathematics and Computational Finance. Technical report: https://lnkd.in/deYSyFCG Courtesy of American Mathematical Society - Mathematical Reviews. #AmericanMathematicalSociety #AMS #MathSciNet #MathematicalReviews

📚 Exciting New Research Alert! 🌟 I'm thrilled to share a recent publication in the Journal of Advances in Applied & Computational Mathematics by Prof. Youssef Raffoul on "Analysis of Functional and Neutral Differential Equations via Lyapunov Functionals." This insightful study explores the use of Lyapunov functionals to analyze stability and decay properties in nonlinear delay and neutral differential systems. 🔗 Read the full article here: https://lnkd.in/d9SPzQYk Prof. Raffoul's work provides valuable insights into: Boundedness and exponential decay of solutions Stability and exponential stability of zero solutions Illustrative examples demonstrating theoretical applications Congratulations to Prof. Youssef N. Raffoul on this significant contribution to applied mathematics! 🌐 Let's continue advancing mathematical knowledge together! 📊 #AppliedMathematics #ComputationalMathematics #LyapunovFunctionals #ResearchPublication

Check out our latest paper titled: New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties published in Communications on Applied Mathematics and Computation https://lnkd.in/eNt4FCAF

Dear colleagues and experts in mathematics, I am excited to share my latest research contribution to the field of number theory and chaos theory, now published on arXiv. The paper delves into the intriguing connections between chaotic dynamics and the Riemann hypothesis. Abstract: In this paper, we explore novel chaotic dynamics derived from the Riemann and von Mangoldt function formula regarding the distribution of nontrivial zeros of the Riemann zeta function. By computing Lyapunov exponents, we demonstrate that the derived dynamics exhibit chaotic behavior when the gaps between zeros are within a certain bound, specifically up to 2.4. Beyond this threshold, the dynamics do not display chaotic behavior. Furthermore, we derive a chaotic operator for the Riemann zeta function within the critical strip, utilizing the correction term from the Riemann-von Mangoldt formula. We establish the chaotic nature and Hermiticity of this operator, and discuss its diagonalization properties. Moreover, our study reveals a remarkable compatibility between our derived chaotic operator and the quantum hydrogen model, as evidenced by the analysis of its eigenvalues resembling the energy levels of hydrogen. Numerical evidence, including Lyapunov exponents, bifurcation analysis, and entropy computation, underscores the unpredictability of the system. Additionally, we establish a connection between our chaotic operator and the prime number theorem regarding the density of primes. Furthermore, our investigation suggests that this chaotic operator strongly supports the validity of the Riemann hypothesis, as proposed by Hilbert and Polya. These findings shed light on the intricate relationship between chaotic dynamics, number theory, and quantum mechanics, offering new perspectives on the behavior of the Riemann zeta function and its zeros. Finally, we demonstrate the Hermiticity and diagonalization properties of our operator using the spectral theorem, further elucidating its mathematical properties and unboundedness. 🔗https://lnkd.in/ev3mzQKe I kindly invite experts in the field to review my work and provide their valuable comments and insights. Your feedback will be greatly appreciated and will contribute to further refining this research. Thank you in advance for your time and expertise. Best regards, Zeraoulia Rafik University of Batna 2

"Strange new universes: Proof assistants and synthetic foundations" An interesting paper by Michael Shulman in the Bulletin of the American Mathematical Society about the integration of LLM and proofs checkers "Existing computer programs called proof assistants can verify the correctness of mathematical proofs but their specialized proof languages present a barrier to entry for many mathematicians. Large language models have the potential to lower this barrier, enabling mathematicians to interact with proof assistants in a more familiar vernacular. Among other advantages, this may allow mathematicians to explore radically new kinds of mathematics using an LLM-powered proof assistant to train their intuitions as well as ensure their arguments are correct." https://lnkd.in/dhcXvjX2

Did you know that most systems in nature are inherently nonlinear? Nonlinear systems are those in which the output doesn't change proportionally to the input. This makes them of interest to mathematicians, physicists, biologists, engineers, and many other scientists. Nonlinear dynamical systems can appear chaotic, unpredictable, or counterintuitive, making them difficult to solve. To approximate them, they are commonly approximated by linear equations, but this can hide interesting phenomena such as solitons, chaos, and singularities. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term "nonlinear science" for the study of nonlinear systems, but this term is disputed by others.

HEMANTH LINGAMGUNTA Unlocking the Power of Fractals: The Barnsley Fern Example The Barnsley fern is a fascinating example of a fractal image created by repeated mathematical transformations. Here’s how this concept can be integrated to define fractal structures: - Iterative Transformations: The Barnsley fern consists of a series of transformations applied repeatedly to points in a coordinate system, demonstrating how complex and lifelike structures can be created through simple, iterative processes alone. - Fractal Geometry: The Barnsley fern is an example of fractal geometry, which studies complex patterns that repeat at every scale, found in nature and various applications. Example Applications: - Nature: Fractals are found in nature, where they can be observed in plants, coastlines, and many other forms. - Computer Science: Fractals have applications in computer science, such as image compression and resolution. Citations: The Barnsley Fern: Mathematical Art https://lnkd.in/gkQYceej Top 5 applications of fractals | Mathematics - University of Waterloo https://lnkd.in/gh3eREKi Fractals in nature and applications https://lnkd.in/gydDQBDH Barnsley fern - Wikipedia https://lnkd.in/gB_-pC8X Exploring Fractal Geometry: Nature's Hidden Patterns https://lnkd.in/gfn9_iqQ **#FractalGeometry #BarnsleyFern #Mathematics #Nature #ComputerScience** Citations: [1] The Applications Of Fractal Geometry In Science And Technology https://lnkd.in/gs9hTkqE [2] The Fascinating World of Fractals: Patterns in Nature and Beyond https://lnkd.in/gZq_PaHs [3] The Barnsley Fern: Mathematical Art https://lnkd.in/gkQYceej [4] Exploring Fractal Geometry: Nature's Hidden Patterns https://lnkd.in/gfn9_iqQ