RAM TRANSFORMS, RAM CALCULUS
A Novel Theory and Solution Methods to Integral Equations, Partial Differential Equations, Time/Shift/Coordinate-Variant Signal Processing, etc. in all STEM areas.
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.
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 plus $100. 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 two books, over 200 technical reports, and over 75 provisional patent applications (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, and signal/image processing problems, all of which arise in all areas of STEM (Science, Technology, Engineering, Mathematics) .This technology is also relevant to Artificial Intelligence and Machine Learning systems through Physics-Informed Neural Nets, Bayesian Neural Nets, Graph Neural Nets, and similar topics. 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 Transforms (RT) and related mathematical tools and techniques are novel research results obtained by Dr. Muralidhara Subbarao in the last 8 months (12/2025 to 7/2026). These new results are a huge expansion and generalization of an original idea invented and published by Dr. Subbarao over 20 years ago. RTs provide 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), 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 (200+) and provisional patent applications (75+) offered here for sale based on the Ram 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 expertes 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 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 BELOW IS US $1000. This does not include license to practice rights which is to be negotiated separately.
PRICE OF EVERY TECHNICAL REPORT BELOW IS US $300. This does not include license to practice rights on overlapping material covered in a related provisional patent application which is to be negotiated separately.
Integral Research — U.S. Provisional Patent Applications
Provisional Patent Applications (Jan. 2026 to May 2026)
on Ram Transform Family and Other Topics
by Dr. Muralidhara Subbarao, IntegralResearch.Net
An itemized list of 37 U.S. provisional patent applications filed by Dr. Muralidhara Subbarao between January 2026 and May 2026, covering the Ram / Rao Transform family (including RamCS, Ram–Master, graph-local, and complex / quaternionic / octonionic transforms) and their applications across signal and image processing, communications, computational physics and fluid dynamics, quantitative finance, drug discovery, medical imaging, and artificial intelligence.
Each application is available for sale, licensing, or assignment. To request terms or further information on any application, use the Inquire button beside it, or contact muralis@IntegralResearch.net. Each entry shows the internal patent label, the title of invention, the USPTO application number, the reception (filing) date, and the number of specification pages.
PRICE OF EVERY PROVISIONAL PATENT APPLICATION BELOW IS US $1000. This does not include license to practice rights which is to be negotiated separately.
37 of 37 applications
1
P1
Rao Localization Transforms (RLT) and Rao Transform (RT) Extensions for Fast Localized Computation and Inversion of Linear and Nonlinear Shift-Variant Integral Operators
App. No. 63/967,229Filed January 24, 2026390 spec. pp.
2
P2
Methods and Systems for Solving Nonlinear Integral Equations and Nonlinear Partial Differential Equations Using Rao Transforms with Applications in Physics, Finance, and STEM Areas
App. No. 63/978,857Filed February 9, 202697 spec. pp.
3
P3
Systems and Methods for Solving Shift-Variant Integral Equations and Partial Differential Equations Using Rao Localization Transform Based Hybrid Inversion and Optimization
App. No. 63/978,414Filed February 9, 202695 spec. pp.
4
P4
Rao Transforms Based Local Neural Integral Equation Systems and Methods for Machine Learning and Artificial Intelligence
App. No. 63/978,056Filed February 9, 2026100 spec. pp.
5
P5
Systems and Methods Based on Coordinate Mappings and Generalized Rao Transforms for Solving Integral and Partial Differential Equations in Science and Finance
App. No. 63/984,989Filed February 17, 202699 spec. pp.
6
P6
Systems and Methods for Real-Time Computational Structural Mechanics and Neural Operator Learning Using Generalized Rao Transforms
App. No. 63/985,770Filed February 18, 2026100 spec. pp.
7
P7
Systems and Methods for Solving Integral and Differential Equations in Quantitative Finance Using Hybrid Rao Transforms
App. No. 63/985,915Filed February 18, 202696 spec. pp.
8
P8
Systems and Methods for Solving Integral and Differential Equations in Real-Time Quantitative Finance Applications Using Rao Transforms
App. No. 63/985,928Filed February 19, 202699 spec. pp.
9
P9
Systems and Methods for Drug Discovery and Molecular Dynamics Optimization in Pharmaceutical and Chemical Industry Using Generalized Rao Transforms
App. No. 63/986,635Filed February 19, 202694 spec. pp.
10
P10
System and Method for Continuous Doppler Equalization and Phase-Tracking in Low Earth Orbit Satellite Communications Using Rao Transforms
App. No. 63/987,975Filed February 22, 202698 spec. pp.
11
P11
Methods and Systems for Computing the Behavior of Physical Systems and Processes Modeled by Integral and Differential Equations Using Rao Transforms
App. No. 63/995,869Filed March 3, 2026100 spec. pp.
12
P12
Systems and Methods for High-Dimensional Physical System Analysis and Quantum Error Correction Using Complex, Quaternionic, and Octonionic Rao Transforms
App. No. 63/999,498Filed March 8, 202698 spec. pp.
13
P13
Systems and Methods for Deterministic Signal Processing and State Estimation Using Generalized Rao Transforms as Analytic Alternatives to Stochastic Monte Carlo and Particle Filtering
App. No. 64/005,479Filed March 13, 202651 spec. pp.
14
P14
Agentic AI System and Methods for Automated Mathematical Discovery and Patent Generation Based on Generalized Rao Transforms
App. No. 64/005,979Filed March 15, 202616 spec. pp.
15
P15
Systems and Methods for Computing the Behavior of Physical and Informational Systems Using N-Dimensional Complex Rao Transforms
App. No. 64/007,567Filed March 16, 202627 spec. pp.
16
P16
Systems and Methods for Physics-Consistent PDE Solving Using Localized Analytic Rao Transform Operators
App. No. 64/008,078Filed March 17, 202626 spec. pp.
17
P17
System and Method for Automated Research Integration and Intellectual Property Generation via an Agentic Artificial Intelligence Architecture
App. No. 64/010,855Filed March 19, 202625 spec. pp.
18
P18
Systems and Methods Based on Ram and RamCS Transforms for Reformulating and Solving Integral and Partial Differential Equations
App. No. 64/037,105Filed April 12, 2026128 spec. pp.
19
P19
Systems and Methods for Solving Hamilton-Jacobi-Bellman Equations for Portfolio Optimization and Mean-Field Game Equations for Systemic Risk Analysis and Multi-Agent Optimization Based on Ram Transforms
App. No. 64/041,759Filed April 16, 2026120 spec. pp.
20
P20
Systems and Methods for Multi-Parameter Signal Analysis, Multi-Resolution Processing, and Shift-Variant Inverse-Problem Solution Using the Ram-Master Transform
App. No. 64/043,813Filed April 19, 2026126 spec. pp.
21
P21
Methods and Systems for Uncertainty Quantification in Artificial Intelligence Systems Using Ram Transform Based Methods
App. No. 64/049,833Filed April 26, 202687 spec. pp.
22
P22
Systems and Methods for Computational Fluid Dynamics, Aeronautical Design, Turbulent Flow Control, and Digital-Twin Modeling Using Ram Transforms
App. No. 64/049,975Filed April 26, 2026100 spec. pp.
23
P23
Systems and Methods for Ram-Master Neural Networks Using Trainable Ram-Master Transform Kernels
App. No. 64/051,098Filed April 28, 2026109 spec. pp.
24
P24
Artificial Intelligence Systems and Methods Based on Ram-Master Neural Networks and Ram-Master Transforms
App. No. 64/052,115Filed April 29, 202659 spec. pp.
25
P25
Self-Improving Artificial Intelligence Systems and Methods Based on Ram-Master Neural Networks and Ram-Master Transforms
App. No. 64/052,450Filed April 29, 2026119 spec. pp.
26
P26
Systems and Methods for Solving Stochastic Volatility, Jump-Diffusion Pricing, and Nonlinear Black-Scholes Equations, in Quantitative Finance Using Ram Transforms
App. No. 64/055,864Filed May 3, 2026128 spec. pp.
27
P27
Systems and Methods for Monte Carlo Simulation and Drug Discovery Using Ram Transforms
App. No. 64/056,233Filed May 4, 2026106 spec. pp.
28
P28
Systems and Methods for Drug Release, Pharmacokinetics, and Enzyme Kinetics Using Ram Transforms
App. No. 64/056,241Filed May 4, 2026115 spec. pp.
29
P29
Systems and Methods for Orthogonal Time Frequency Delay-Doppler and Satellite Communication Using Ram Transforms
App. No. 64/056,254Filed May 4, 2026118 spec. pp.
30
P34
Systems and Methods for Designing Reaction-Diffusion Systems in Chemical, Pharmaceutical, and Materials Manufacturing Industry Using Ram Transforms
App. No. 64/056,344Filed May 4, 2026131 spec. pp.
31
P37
Systems and Methods for Solving High-Order Derivative Inverse Partial Derivative Equation Problems Using Ram Transforms Based Analytic Mollifier Operators
App. No. 64/059,275Filed May 6, 2026116 spec. pp.
32
P38
Systems and Methods for Ram Transform Structured Components Inside Latent World Models and Joint-Embedding Predictive Architectures
App. No. 64/061,688Filed May 9, 2026114 spec. pp.
33
P39
Apparatus and Methods for Hand-Held 3D Magnetic Resonance Imaging, Magnetic Density Imaging, Magnetoencephalography, and Magnetocardiography Based on Field-Image-Tomography
App. No. 64/063,162Filed May 11, 2026116 spec. pp.
34
P42
Systems and Methods for Ram Transform Based Wireless Communication
App. No. 64/068,931Filed May 18, 2026121 spec. pp.
35
P44
Systems and Methods for Solving Partial Differential Equations and Hybrid Information Lattice Transform, Graph Local Ram Transform, and Ram-Master Neural Network Architectures
App. No. 64/069,949Filed May 19, 2026121 spec. pp.
36
P45
Systems and Methods for Solving Partial Differential Equations and Bayes Networks for Decision Support Using Ram and Graph Local Ram Operators
App. No. 64/071,689Filed May 21, 2026115 spec. pp.
37
P47
Systems and Methods for Bayesian Ram-Master Neural Networks and Bayesian Network Inference Using Graph Local Ram Transforms
App. No. 64/071,849Filed May 22, 2026128 spec. pp.
SEE BELOW FOR MORE ITEMS (SECOND BATCH)
LIST CONTINUED: SECOND BATCH
Integral Research — U.S. Provisional Patent Applications
Provisional Patent Applications (June 2026)
on Ram Transform Family and Other Topics
by Dr. Muralidhara Subbarao, IntegralResearch.Net
An itemized list of 25 U.S. provisional patent applications filed by Dr. Muralidhara Subbarao in June 2026 (continuing the earlier January–May 2026 series), covering the panel and multimodal Ram Transform framework and Ram-operator methods across 5G/6G wireless and integrated sensing, quantitative finance, computational lithography, battery and physiological inverse problems, radar/lidar, quantum computing, AI weather and world models, industrial CAE, and the calibration of shift-variant kernels and partial-differential-equation coefficients.
Each application is available for sale, licensing, or assignment. To request terms or further information on any application, use the Inquire button beside it, or contact muralis@IntegralResearch.net. Each entry shows the internal patent label, the title of invention, the USPTO application number, the reception (filing) date, and the number of specification pages.
Each application is available for sale, licensing, or assignment. To request terms or further information on any application, use the Inquire button beside it, or contact muralis@IntegralResearch.net. Each entry shows the internal patent label, the title of invention, the USPTO application number, the reception (filing) date, and the number of specification pages.
PRICE OF EVERY PROVISIONAL PATENT APPLICATION BELOW IS US $1000. This does not include license to practice rights which is to be negotiated separately.
25 of 25 applications
1
P117
Adaptive Advantage-Aware and Cost-Aware Ram Transform Systems and Methods for 5G and 6G Wireless Communications
App. No. 64/084,083Filed June 6, 2026132 spec. pp.
2
P121
Systems and Methods for Option Pricing and Local Volatility Analysis in Quantitative Finance based on Ram Transforms
App. No. 64/089,452Filed June 12, 202697 spec. pp.
3
P123
SYSTEMS AND METHODS FOR TRANSITION PROBABILITY DENSITY GENERATION AND OPTION PRICING BASED ON RAM TRANSFORMS
App. No. 64/089,545Filed June 12, 2026107 spec. pp.
4
P125
SYSTEMS AND METHODS FOR TENSOR NEURAL NETWORKS AND DERIVATIVE MODEL CALIBRATION IN QUANTITATIVE FINANCE BASED ON RAM TRANSFORMS
App. No. 64/089,655Filed June 12, 2026111 spec. pp.
5
P127
SYSTEMS AND METHODS FOR NEURAL NETWORK BASED FINANCIAL RISK DETECTION AND DERIVATIVE PRICING USING RAM TRANSFORMS
App. No. 64/089,694Filed June 12, 2026104 spec. pp.
6
P135
SYSTEMS AND METHODS FOR NEURAL NETWORK UNCERTAINTY ESTIMATION AND FEDERATED NEURAL GRAPHICAL MODELS BASED ON GRAPH LOCAL RAM TRNSFORMS
App. No. 64/092,025Filed June 16, 2026128 spec. pp.
7
P137
SYSTEMS AND METHODS FOR SYNTHESIZING INVERTIBLE DELAY-DOPPLER AND CHIRP-DOMAIN WAVEFORMS FOR INTEGRATED SENSING AND COMMUNICATION USING RAM-MASTER ATOMS
App. No. 64/094,292Filed June 18, 2026127 spec. pp.
8
P138
SYSTEMS AND METHODS FOR LOCALIZED INVERSION OF THE PARTIALLY-COHERENT HOPKINS IMAGING OPERATOR FOR COMPUTATIONAL LITHOGRAPHY USING LOCAL HOPKINS-RAM TAYLOR-JET TENSORS
App. No. 64/094,300Filed June 18, 2026131 spec. pp.
9
P139
SYSTEMS AND METHODS FOR INVERSION OF THE DISTRIBUTION-OF-RELAXATION-TIMES IMPEDANCE KERNEL AND CERTIFIABLE BATTERY STATE ESTIMATION USING RAM COMPLEX-SPECTRAL SYMBOLS
App. No. 64/094,311Filed June 18, 2026130 spec. pp.
10
P140
SYSTEMS AND METHODS FOR INVERTIBLE PHYSIOLOGICAL TRANSFER OPERATORS FOR WEARABLE SENSING BASED ON LOCAL RAM INVERSION
App. No. 64/094,327Filed June 18, 2026124 spec. pp.
11
P141
SYSTEMS AND METHODS FOR CLOSED-FORM RAM FORWARD AND INVERSE OPERATORS IN QUANTITATIVE FINANCE PROBLEMS WITH NOISY DATA AND WIDE KERNELS
App. No. 64/094,937Filed June 19, 2026127 spec. pp.
12
P142
SYSTEMS AND METHODS FOR CLOSED-FORM EXACT-INVERSE RAM OPERATORS IN FREQUENCY-MODULATED CONTINUOUS-WAVE RADAR AND LIDAR APPLICATIONS
App. No. 64/094,944Filed June 19, 2026127 spec. pp.
13
P143
SYSTEMS AND METHODS FOR CLOSED-FORM RAM OPERATORS IN QUANTUM COMPUTING APPLICATIONS
App. No. 64/094,952Filed June 19, 2026126 spec. pp.
14
P144
SYSTEMS AND METHODS FOR CLOSED-FORM INVERTIBLE RAM OPERATORS AS CERTIFIED MIDDLEWARE FOR MACHINE-LEARNING WEATHER AND CLIMATE STACKS
App. No. 64/094,969Filed June 20, 2026125 spec. pp.
15
P145
SYSTEMS AND METHODS FOR EXACTLY INVERTIBLE RAM TOKENIZERS AND CLOSED-FORM INVERSE LATENT-DYNAMICS OPERATORS AS RETRAINING-FREE MIDDLEWARE FOR WORLD MODELS AND AUTONOMOUS-DRIVING DATA ENGINES
App. No. 64/094,974Filed June 20, 2026118 spec. pp.
16
P146
SYSTEMS AND METHODS FOR ANALYTIC TRAINING-FREE INVERTIBLE RAM OPERATORS FOR INDUSTRIAL COMPUTER-AIDED ENGINEERING APPLICATIONS
App. No. 64/094,979Filed June 20, 2026118 spec. pp.
17
P152
SYSTEMS AND METHODS FOR FORWARD AND INVERSE MODELING OF A SHIFT-VARIANT LINEAR SYSTEM BASED ON PANEL AND MULTIMODAL RAM TRANSFORM
App. No. 64/093,256Filed June 17, 2026114 spec. pp.
18
P153
SYSTEMS AND METHODS FOR SHIFT-VARIANT IMAGE AND VOLUME DEBLURRING BY A TWO- AND THREE-DIMENSIONAL MULTI-PANEL RAM TRANSFORM
App. No. 64/093,381Filed June 18, 2026115 spec. pp.
19
P154
SYSTEMS AND METHODS FOR FOKKER-PLANCK RAM OPERATORS PROVIDING SOLUTION OF NONLINEAR INTEGRAL EQUATIONS
App. No. 64/093,409Filed June 18, 2026110 spec. pp.
20
P155
SYSTEMS AND METHODS FOR IDENTIFICATION AND CALIBRATION OF MULTI-PANEL RAM TRANSFORM KERNELS
App. No. 64/093,448Filed June 18, 2026120 spec. pp.
21
P156
SYSTEMS AND METHODS FOR SOLVING LINEAR PARTIAL DIFFERENTIAL EQUATIONS BASED ON MULTI-PANEL RAM TRANSFORM OPERATORS
App. No. 64/093,513Filed June 18, 2026115 spec. pp.
22
P157
SYSTEMS AND METHODS FOR NONLINEAR AND INVERSE MULTI-PANEL RAM OPERATORS FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
App. No. 64/093,557Filed June 18, 2026125 spec. pp.
23
P157-revised
SYSTEMS AND METHODS FOR NONLINEAR AND INVERSE MULTI-PANEL RAM OPERATORS FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND CROSS-CENTER IDENTIFICATION
App. No. 64/094,064Filed June 18, 2026131 spec. pp.
24
P158
SYSTEMS AND METHODS FOR CALIBRATING SHIFT-VARIANT COEFFICIENTS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS BY LOCAL RAM-OPERATOR JETS
App. No. 64/094,209Filed June 18, 2026126 spec. pp.
25
P159
SYSTEMS AND METHODS FOR CALIBRATING COEFFICIENTS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY LOCAL RAM-OPERATOR JETS
App. No. 64/094,227Filed June 18, 2026131 spec. pp.
Notes
A total of 25 distinct provisional patent applications are listed, comprising 3,019 pages of specifications, all filed in June 2026. The patent labels (e.g., P117, P121, …) are taken from the prefixes of the corresponding internal folders and USPTO Electronic Acknowledgement Receipts; gaps in the label sequence correspond to drafts that were filed jointly under a companion application's receipt or not filed separately.
Titles, application numbers, reception dates, and specification page counts are reproduced as recorded on the USPTO Electronic Acknowledgement Receipts. "P157-revised" is a separately filed, expanded version of P157 with its own application number. A duplicate (empty) acknowledgement receipt for application 64/094,952 (P143) was disregarded.
© Integral Research · Dr. Muralidhara Subbarao · www.IntegralResearch.net
PRICE OF EVERY TECHNICAL REPORT BELOW IS US $300. This does not include license to practice rights if the same material is covered in a provisional patent application above. Also, the price of the Technical Books Volume 1 to 6 below are $3000 each.
For more information, use the Inquire button beside it, or contact muralis@IntegralResearch.net.
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