M.A.Lerma's Index of Preprints


  1. Putnam Training Problems (pdf) - Used for Putnam Training.
  2. Notes on Discrete Mathematics (pdf) - Used for teaching in a CS program.
  3. A Gradient Theorem for Lipschitz Continuous Functions (pdf) - A generalization of the Gradient Theorem for Line Integrals, used in another paper.
  4. Principal Components Analysis in 2D (pdf) - An overview. I had to use it in a programming project and wanted to leave a record of it for future reference.
  5. How inefficient can a sort algorithm be? (pdf) - https://arxiv.org/abs/1406.1077 - Tongue in cheek essay about how to build a pessimal "inefficient" sorting algorithm.
  6. The Bernoulli Periodic functions (pdf) - Overview.
  7. An Extremal Majorant for the Logarithm and its Aplications (pdf) - Dissertation.
  8. Some Applications of Extremal Functions in Fourier Analysis (pdf) - Related to my doctoral dissertation, advised by Prof. Vaaler.
  9. Construction of a Number Greater Than One Whose Powers Are Uniformly Distributed Modulo One (pdf) - Proof of concept (and rather complicated, may need revision).
  10. Distribution of Powers Modulo 1 and Related Topics (pdf) - https://arxiv.org/abs/1312.5996 - From my times as a graduate student.
  11. Some Results in Analysis (pdf) - A list of mathematical results placed here to have them handy.


  1. A Mirror Based Event Cloaking Device (pdf) - https://arxiv.org/abs/1308.2606 - Proof of concept, very simple.
  2. A Simple Derivation of the Equation for the Brachistochrone Curve (pdf) - Just for fun.

Computer Science.

With M.Lucas et al. (on Artificial Intelligence and Deep Learning).
  1. Visual Explanations from Deep Networks via Riemann-Stieltjes Integrated Gradient-based Localization (2022) - https://arxiv.org/abs/2205.10900 - Gradient based attribution method for deep neural networks that is not affected by the vanishing gradients problems and can be applied to hidden layers.
  2. Grad-CAM++ is Equivalent to Grad-CAM with Positive Gradients (2022) (pdf) - https://arxiv.org/abs/2205.10838 - We show that the Grad-CAM++ algorithm is equivalent to a very simple variation of Grad-CAM in which gradients are replaced with positive gradients.
  3. Symmetry-Preserving Paths in Integrated Gradients (2021) (pdf) - https://arxiv.org/abs/2103.13533 - Filling some "holes" in a mathematical argument found in a paper.
  4. Pre or Post-Softmax scores in gradient-based attribution methods, what is best? (2023) (pdf) - Advantages and disadvantages of using pre-softmax versus post-softmax network scores in gradient-based attribution methods.
  5. On Evaluating Explanation Methods for Deep Networks (2021) (pdf) - Slides. Challenges of getting empirical evaluations of explanation methods for deep networks.

Unfinished preprints.

  1. Physics: A Brief Summary (pdf) - Mostly based on Math E11 Spring 1999 taught by Prof. Zaslow. Section on Quantum Field Theory added and still unfinished.


  1. Lucas M; Lerma M; Furst JD; Raicu DS (2022): Explainable Model for Localization of Spiculation in Lung Nodules, in Workshop on AI-enabled Medical Image Analysis (AIMIA): Digital Pathology & Radiology/COVID19, held jointly with the European Conference on Computer Vision (ECCV) 2022, Tel-Aviv, Israel, October 23 to 27, 2022 (link)
  2. Lucas, M., Lerma M., Furst, J., Raicu, D. (2022). RSI-Grad-CAM: Visual Explanations from Deep Networks via Riemann-Stieltjes Integrated Gradient-Based Localization. In: Bebis, G. et al (Ed.), Advances in Visual Computing. ISVC 2022. Lecture Notes in Computer Science, vol 13598. Springer, Cham. https://doi.org/10.1007/978-3-031-20713-6_20 (link)
  3. Lerma M and Lucas M: Grad-CAM++ is Equivalent to Grad-CAM with Positive Gradients (2022). In proceedings of the 24th Irish Machine Vision and Image Processing Conference, pages 113–120 (link to proceedings) (pics).
  4. Lucas M; Lerma M; Furst JD; Raicu DS (2021): Robust Heatmap Template Generation for COVID-19 Biomarker Detection, EAI Endorsed Transactions on Bioengineering and Bioinformatics, issue 2, 2021, bebi 21:e2, doi:10.4108/eai.24-2-2021.168729
  5. Miguel A. Lerma (2021), Truck Versus Human 2.0: Mathematical Follow-Up Under Increasing Pressure, and How Kepler’s Laws Come to the Rescue, The College Mathematics Journal, 52:1, 22–30, DOI: 10.1080/07468342.2021.1847590
  6. Lucas M; Lerma M; Furst JD; Raicu DS (2020): Heatmap Template Generation for COVID-19 Biomarker Detection in Chest X-rays, The 20th IEEE International Conference on BioInformatics And BioEngineering (BIBE 2020), USA, October 26-28, 2020.
  7. Introducción a las redes de Hopfield, in Olmeda, I., and Barba-Romero, S. (Ed.): Redes Neuronales Artificiales, Fundamentos y Aplicaciones, Servicio de Publicaciones de la Universidad de Alcalá de Henares, Alcalá de Henares, Spain, 1993.
  8. with Castellanos, J.; Ríos, J.; Segovia, J. (1992): Approximation of Functions by Neural Networks: an Experimental Test, International Journal of Neural Networks, Vol.3, No.4, December 1992, pp.149–153.
  9. Ampliación al espacio de una aplicación de la integración en el campo complejo para la solución de una cuestión de informática gráfica, English summary: Extension to the Space of an Application of Complex Integration for Solving a Topic on Graphic Computation, Questiiò, Vol.16, N.1, 1992, pp.59–75, Barcelona, Spain.
  10. with Barrios, D.; Ríos, J.; Segovia, J. (1992): Fast Training of Feedforward Multilayer Neural Networks by Means of Linear Methods: Experimental Tests, Proceedings of the Fifth International Symposium on Knowledge Engineering, Sevilla, Spain, October 5-9, 1992, pp.170–178.
  11. with Castellanos, J.; Ríos, J.; Segovia, J. (1992)): Fast Training of Feedforward Multilayer Neural Networks By Means of a Weighted Least Mean Square Algorithm, Neural Network World 5/92, 423-436.


  1. 2017-present: Artificial Intelligence, Neural Networks, Deep Learning, Explainable AI.
  2. 2000: LOCR - An Optical Character Recognition Program for Linux. Department of Mathematics, Northwestern University.
  3. 1996-1998: Analytic Number Theory. Development of new analytic methods for the study of discrepancy of sequences. Extremal functions in Fourier Analysis. Department of Mathematics, University of Texas at Austin.
  4. 1994-1996: Algebraic Number Theory. Applications of p-adic analysis to distribution of sequences modulo 1. Department of Mathematics, University of Texas at Austin.
  5. 1995: Didactic computer projects. Development of documents in emacs bookmode implementing mathematical problems: Peano's space filling curves, Urysohn's lemma, topological operations, primality tests, RSA algorithm, and permutation matrices. Department of Mathematics, University of Texas at Austin.
  6. 1992-1993: Development of computer tools. Proyecto Lope (Lope project), School of Computer Science, Universidad Politécnica de Madrid. Designing of a meta-environment for automatic generation of integrated environments for developing programming projects. The prototype environment, called LOPE, would allow editing the whole set of linked descriptions for a given piece of software, in a uniform manner through all phases of its development. LOPE would provide a useful set of consistent views of these descriptions, including documents, diagrams, source code, etc.
  7. 1991-1992: Computational Geometry. Extension to the n-dimensional space of the winding number algorithm for polygon inclusion, with applications to polyhedron and polytope inclusion. School of Computer Science, Universidad Politécnica de Madrid.
  8. 1989-1991: Neural Networks. New heuristic and constructive methods for fitting neural networks. School of Computer Science, Universidad Politécnica de Madrid. New training algorithms for feedforward multilayer neural networks were developed. They were based on considering the task of fitting the weights in all the layers of a neural network as a coupled set of linear optimization problems.