RAM CALCULUS, RAM TRANSFORMS
HISTORIC RESEARCH BREAKTHROUGH IN THE
CALCULUS OF INTEGRAL AND DIFFERENTIAL EQUATIONS
SIGNAL/IMAGE PROCESSING, GRAPH/NETWORK OPERATIONS
MASS SCALE IMPACT ON SCIENCE, MATHEMATICS, ENGINEERING, AND
QUANTITATIVE FINANCE
A Novel Theory and Solution Methods to Integral Equations, Partial Differential Equations, Time/Shift/Coordinate-Variant Signal/Image Processing, Physics-Informed Neural Nets, Graph Neural Nets, Graph and Network Operators, etc., in all areas of Science, Mathematics, Engineering, and Quantitative Finance.
A NOVEL AND PROFOUND MATHEMATICAL RESULT WITH A BROAD AND DEEP IMPACT ON BOTH THEORETICAL AND PRACTICAL APPLICATIONS IN ALL AREAS OF SCIENCE, TECHNOLOGY, ENGINEERING, and MATHEMATICS (STEM). THE IDEA IS RELATIVELY SIMPLE BUT THE THEORETICAL AND PRACTICAL IMPACTS ARE HUGE. ADVANTAGES OF THIS TECHNOLOGY ARE A UNIFIED FRAMEWORK, SIGNIFICANT COMPUTATIONAL SPEEDUP, IMPROVED ACCURACY, THEORETICAL TRANSPARENCY, CONTROLLABILITY, EXTENDABILITY, AND ADAPTABILITY.
Ram Calculus theory and methods have been used to extend/improve the solutions to 10 open problems in Mathematics and Computer Science obtained by OpenAI's Astra AI model. See this on github for a summary: muralisubbarao2/Improvments-and-extensions-to-solutions-of-10-open-problems-by-OpenAI: Ram Calculus has been used to improve and extend solutions to 10 open problems in Math and Computer Science derived by the Astra AI model of OpenAI in August 2026.
Ram Calculus has also been used to improve a result related to Riemann Hypothesis obtained by Mythos AI model of Anthropic. See this on github: muralisubbarao2/Riemann-Hypothesis-Result.
Ram Calculus has been used to make progress on solving unsolved problems; substantial progress has been made on 40 problems and modest progress on 252 problems. See here for details:
I am not joikng. I guarantee your money back and more as explained below.
If you find a substantive error in the core idea described in any item that you bought here, or if you find an alternative approach that is better than the one provided by the Ram Transform techniques and tools to solve a relevant problem (without training data that AI systems need), then you can get a full refund. The first person to report the error will be acknowledged and may be paid a small reward of $25. The correctness of the mathematics and algorithms described in the items below are easily verified using advanced AI chatbots, often both symbolically and numerically. If you skip using the methods described in the items below to solve a problem that can be solved by them but use an alternative approach in the current literature, your results will be improved later by someone who will use the methods described in the items offered here for sale/licensing.
ITEMS FOR SALE:
Hardcopies of books, over 3,000 technical reports, and over 2,500 provisional patent applications with some key applications already submitted (license to practice is separate and not included) with new fundamental mathematical and computational results of high research-value are for sale. They disclose unique and novel techniques invented by the author Dr. Muralidhara Subbarao. They are useful for research and development teams in all STEM areas and in Quantitative Finance. Licensing of this technology by major AI companies will help them to train their AI models and chatbots to answer questions on integral and partial differential equations as well as time/shift/coordinate-variant signal/image processing problems. The AI models trained on this technology will beat other AI models that are not trained on this technology in mathematical problems related to integral equations, partial differential equations, signal/image processing problems, Physics-Informed Neural Nets, Graph Neural Nets, Graph and Network Operators, etc., all of which arise in all areas of STEM (Science, Technology, Engineering, Mathematics). Therefore, this technology is indispensable for training frontier AI models on topics addressed by Ram Calculus. The books and technical reports on sale here can also be used as RAG attachment documents by researchers and R&D engineers to query AI chatbots to obtain optimal answers for their queries on relevant topics.
CONDITIONS OF SALE OF HARDCOPY DOCUMENTS:
ONE HARD COPY OF THE DOCUMENT WILL BE SENT BY MAIL WITH TRACKING ID.
THE BUYER AGREES TO:
NO DUPLICATION IN ANY FORM- HARDCOPY OR SOFTCOPY,
NO DIGITIZATION OR SCANNING/PHOTOGRAPHING TO CREATE A SOFTCOPY,
NO SHARING OF THE HARD COPY WITH MORE THAN ONE OTHER PERSON WHO WILL NOT SHARE IT WITH ANYONE.
ALL COPYRIGHT LAWS APPLY TO THE FULL EXTENT.
Here is an extremely valuable research and development resource to researchers in industry and academia in all STEM areas, Artificial Intelligence, and Quantitative Finance. Ram Calculus is a novel theory and a set of mathematical tools and techniques invented by Dr. Muralidhara Subbarao in 2026. These new results are a huge expansion and generalization of an original idea invented and published by Dr. Subbarao over 20 years ago. Ram Calculus provides a new general, unified, and elegant theory, and computationally efficient, transparent, controllable, and fully parallel algorithms to a large class of mathematical problems. The class of problems include linear and nonlinear integral equations (IE), partial differential equations (PDE), time/shift/coordinate-variant signal/image processing problems, physics-informed neural nets (PINN), Graphs and Networks Operators, etc. Access to this resource can help to accelerate research work and improve products, invent new technologies, publish papers, prepare grant proposals, teach the most useful mathematical/computational techniques, etc. Dr. Subbarao has developed the corpus of confidential technical reports (3000+) and provisional patent applications (2500+) offered here for sale based on the Ram Calculus/Transform theory, techniques, and tools. This research technical report library can be used as reference documents in a folder for an advanced Chatbot, and other relevant recent research papers or patents can be attached as RAG documents, and the results and methods in those papers/patents can be improved, sometimes substantially, through intuitive, insightful, and thoughtful prompting and follow-up prompting sequences, by experts in the field related to the papers. It is expected that there is potential for obtaining new research results on hundreds of research topics by exploring relevant topics in all STEM areas. The results obtained by one set of chatbots can be verified crudely by other set of chatbots and later by expert researchers in the field.
Ideally, AI chatbot companies like Google, OpenAI, Anthropic, Meta etc. should license this technology and train their chatbots on the confidential library here. These companies are welcome to contact me at muralis@integralresearch.net regarding licensing this technology.
USE AI TOOLS TO APPLY RAM CALCULUS/TRANSFORM THEORY, TOOLS, AND TECHNIQUES TO
RE-SEARCH CURRENT PROBLEMS
RE-VISIT IMPORTANT PROBLEMS
RE-ANALYZE SELECTED PROBLEMS
RE-FORMULATE THE APPLICABLE PROBLEMS
RE-SOLVE THE PROBLEMS
RE-ENGINEER PROCESSES AND SYSTEMS
RE-BUILD THE SYSTEMS
RE-DEPLOY FAR SUPERIOR SOLUTIONS
RE-TEACH THE NEW METHODS TO COLLEGE STUDENTS
PRICE OF EVERY PROVISIONAL PATENT APPLICATION IS US $1000. This does not include license to practice rights which is to be negotiated separately. SOME ARE LISTED BELOW AND OTHERS ARE FOUND IN THE FILES BY CLICKING ON THE DOWNLOAD BUTTONS ABOVE ON THIS PAGE.
PRICE OF EVERY TECHNICAL REPORT IS US $300. THEY ARE LISTED ON WEBPAGES WITH LINKS ON THE TOP MARGIN OF THIS PAGE. This does not include license to practice rights on overlapping material covered in a related provisional patent application which is to be negotiated separately.
PREVIOUS BOOK OF 2007 INFORMATION
SEE LATER FOR INFORMATION ON 2026 TECHNOLOGY
RAO TRANSFORMS RENAMED AS RAM TRANSFORMS
REPLACE ALL OCCURRENCES OF "RAO TRANSFORM" WITH "RAM TRANSFORM" IN THIS ENTIRE WEBSITE.
RAO TRANSFORMS: A New Approach to Integral and Differential Equations
Dr. Muralidhara SubbaRao (Rao), Professor of Electrical and Computer Engineering.
Second Edition 2007 is on sale now.
Third Edition 2026 is under preparation for sale in 2026.
This book presents a novel technique that was invented in 2005 by this author. It has remained unnoticed by others until now (Dec. 2025) due to a lack of demonstration of its applications to important practical problems. Such demonstration has now been found in preliminary studies for some very important practical problems (Dec. 2025). It is found to have great advantages in medical image deblurring and other applications. The contents of this book can be simply attached/uploaded to popular chatbots (e.g. ChatGPT, Gemini, Grok) to get working programs for use in actual applications in many areas including the deblurring of images in MRI, CT, PET, SPECT, Ultrasound, digital photos and videos, etc. Many new theoretical and practical applications are yet to be discovered for the results in this book and therefore it serves as a valuable resource in research and development in industry and academia. The methods in this book are likely to be included in college textbooks on image processing and applied mathematics.
EXPERT COMMENTS ON THIS RESEARCH RESULTS:
“In summary the proposed research appears to be valid and
has applications. It is guaranteed to produce doctoral dissertations."
-An expert researcher in the field in his review of this research for a US federal research funding agency.
"... Congratulations. It does seem that you have a novel and powerful method for solving integral equations. ... .
I admire what appears to be a brand new and promisingly important advancement to spatial signal processing."
--A Distinguished Professor of Engineering in his comments on this research.
"In Mathematics, sometimes, the simplest results are the most useful results."
-- A Professor of Mathematics in his comments on this research.
Background
Rao Transforms (RTs) provide a novel approach to the centuries old problem of integral equations. Differential equations are solved by first converting them to integral equations by incorporating boundary conditions, and then solving the resulting integral equations. Since many fundamental laws of physics are stated using differential equations, RTs are expected to have wide applications in scientific, engineering, and medical applications. RTs are based on a breakthrough strategy of “Localize, Solve, and Synthesize” using the simple equation L(u,v)=G(u+v,u) to change a global form integration kernel G to a local form kernel L. This apparently simple idea seems to have eluded researchers until now. RTs were invented by this author while doing research on shift-variant image deblurring which involves solving a Fredholm integral equation of the First Kind, and extending the results of that work to general integral equations. RTs are an extension of the Spatial-Domain Convolution/Deconvolution Transform (S Transform) invented by this author in 1989 related to convolution integral equations. S transform has been successfully used in computer vision and image processing applications such as depth-from-defocus and image restoration, and RTs are expected to be similarly useful. RTs provide both symbolic and numerical solutions. The solution is fully localized and therefore offers significant computational savings and permits extremely fine-grained parallel implementation on a computer. The approach is simple as it is easy to implement and comprehend, and unified as a common framework solves a large class of diverse problems. Therefore RTs have both computational and theoretical advantages in comparison with existing techniques. RTs can be naturally extended from the case of one dimensional problems to multidimensional cases. The basic theory of RTs, and their application to two practical problems are presented.
The Third Edition of this Book is under preparation for publication in 2026. The Rao Transforms will be renamed in the Third Edition in 2026. The naming rights for this Transform is on sale by this author. Contact the author at rao@integralresearch.net to negotiate a mutually agreeable price. The interest in renaming this new transform is expected to be high and therefore the price will be high. The high price is justified by the fact that Rao Transform is expected to be included in college text books on image/signal processing, calculus, and applied mathematics, in the future.
Contact:
muralis@outlook.com
rao@integralresearch.net
6email: muralis@outlook.com31