RAM TRANSFORMS
RAM TRANSFORMS:
A Novel Theory and Solution Methods to Integral Equations, Partial Differential Equations, Time/Shift/Coordinate-Variant Signal Processing, etc. in all STEM areas.
Books and provisional patents with new fundamental results of high research-value are for sale. They disclose unique techniques invented by the author. 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 win math benchmarks on Integral and Partial Differential Equations. They are useful to train AI chatbots to answer questions on integral and differential equations. They can also be used as RAG attachment documents by researchers to query AI chatbots.
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 6 months (12/2025 to 5/2026). 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 (100+) and provisional patent applications (35+) 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
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.
Integral Research — Technical Report Series
Technical Reports on Ram Transform Family and Other Topics
by Dr. Muralidhara Subbarao, IntegralResearch.Net
A collection of 116 original technical reports developed under the Integral Research program, spanning the Ram Transform family (Ram, RamCS, Ram–Master, Ramlet, and fractional / complex / hypercomplex order transforms) and their applications across signal and image processing, communications, computational physics and fluid dynamics, quantitative finance, medical imaging, machine learning, and open problems in pure mathematics such as the Riemann Hypothesis and the Navier–Stokes problem.
Each report is available for purchase. To request pricing or a copy of any report, use the Inquire button beside it, or contact muralis@IntegralResearch.net.
116 of 116 reports
1
TR1
Eigenfunction and Polynomial Preservation Properties of the Generalized Ram Transform: Unit Upper Triangular Structure, Exact Inversion, and L2 Considerations for Polynomial and Exponential Input Functions
18 pp.Inquire
2
TR2
Application of the Complex Operator Integral Ram Transform (COIRT), Complex Coordinate Mapping, and N-Dimensional Complex Ram Transform to the Navier-Stokes Existence and Smoothness Problem
18 pp.Inquire
3
TR3
Explicit Coefficient Matrices for the Two-Dimensional Ram Transform: Forward and Inverse Operators for General, Gaussian, and Rect Kernels
16 pp.Inquire
4
TR4
Shift-Variant Image Deblurring Using Ram Transform: Experimental Study and Comparison with Wiener and Richardson–Lucy Methods
12 pp.Inquire
5
TR5
One-Dimensional Ram Transform: Derivation, Explicit Coefficient Matrices, and Program Implementation
23 pp.Inquire
6
TR6
Explicit Coefficient Matrices for the Three-Dimensional Ram Transform: Forward and Inverse Operators for the Gaussian Kernel
12 pp.Inquire
7
TR7
Ram Transforms: Topics for Future Research and Benchmark Studies
32 pp.Inquire
8
TR8
Ram Transform and Spectral Domain Analysis: Fourier and Laplace Representations of the Ram Kernel and Their Consequences
16 pp.Inquire
9
TR9
FFT Applicability Analysis for the Shift-Variant Ram Transform: Detailed Analysis and Special Cases
14 pp.Inquire
10
TR10
The Ram Complex Spectral Transform (RamCS Transform): Eigenfunctions, Transform Pairs, and Applications to the Ram Transform Integral Equation
21 pp.Inquire
11
TR11
Historical Precedents and Novelty of the Ram Complex Spectral (RamCS) Transform – A Survey of the Closest Existing Frameworks and an Assessment of What is Genuinely New
17 pp.Inquire
12
TR12
The Master Integral Transform (MIT) and the Ram Complex Spectral (RamCS) Transform: Global Integral Form, Relationship, Generalizations, and Mutual Extensions
15 pp.Inquire
13
TR13
Ram Transforms and RamCS Transforms: A Unified Framework for Modelling, Solving, and Inverting Linear and Nonlinear Integral and Partial Differential Equations
29 pp.Inquire
14
TR14
Application of Ram Transforms to Problems in Orthogonal Time Frequency Space Modulation
28 pp.Inquire
15
TR15
Moment Constraints and Operator Duality for the N-Dimensional Complex Ram Transform: Derivation, Bounds, Theoretical Significance, and Applications to Navier–Stokes Equations
22 pp.Inquire
16
TR16
The Wavelet–Ram Transform: Extension of the N-Dimensional Complex Ram Transform to Incorporate Wavelet Analysis for Multiresolution Modelling and Inversion
25 pp.Inquire
17
TR17
Fractional Order N-Dimensional Complex Ram Transforms: Theory, Mittag-Leffler Eigenvalues, Operator Duality, and Applications to Anomalous Diffusion, Fractional PDEs, and Multi-Scale Physical Systems
24 pp.Inquire
18
TR18
Complex-Order N-Dimensional Complex Ram Transforms: Theory of Complex-Order Integral and Differential Operators, Log-Periodic Mittag-Leffler Eigenvalues, the Doubly-Complex Ram Framework, and Applications to Log-Periodic Systems, Viscoelasticity, and Complex Resonances
24 pp.Inquire
19
TR19-A
A Grand Unified Framework Based on Ram Transforms and RamCS Transforms – Reformulating, Re-Solving, and Re-Engineering Solutions to Integral and Partial Differential Equations in Science, Engineering, and Applied Mathematics
33 pp.Inquire
20
TR19-B
Updated Grand Unified Framework Based on Ram Transforms, RamCS Transforms, and Ram-Master Transforms – Post-TR19 Extensions, Application Categories, Literature-Search Targets, and Practical Pros/Cons
20 pp.Inquire
21
TR20
A Comprehensive Operator-Theoretic Approach to the Riemann Hypothesis via the Extended Ram Transform Framework: Incorporating Fractional-Order, Complex-Order, Doubly-Complex, N-Dimensional, Moment Constraint, and Operator Duality Extensions
26 pp.Inquire
22
TR21
Application of Ram Transform Variants to Particle Filtering: Theory, Methods, and Comparative Analysis
17 pp.Inquire
23
TR22
Application of Ram Transform Variants to Kalman Filtering: Theory, Algorithms, and Comparative Analysis
17 pp.Inquire
24
TR23
Gap Analysis Update for the Riemann Hypothesis Proof Attempt via Ram Transforms – Assessment of How Additional Reference Documents Narrow the Gaps and Weaken the Assumptions in Technical Report TR-2026-04-14-0020
11 pp.Inquire
25
TR24
Application of Ram Transform Variants to Hamilton–Jacobi–Bellman Equations for Portfolio Optimization and Stochastic Control – Theory, Algorithms, and Comparative Analysis
21 pp.Inquire
26
TR25
Application of Ram Transform Variants to Mean-Field Games for Systemic Risk and Stochastic Control – Theory, Algorithms, and Comparative Analysis for Coupled HJB–Fokker–Planck Systems
23 pp.Inquire
27
TR26-A
Ram Transform Family Methods for Stochastic Volatility and Jump-Diffusion Models – Local Ram-PDOs, Analytic Micro-Inverses, Jump-Kernel Moment Splitting, Rough-Volatility Extensions, and Fast Calibration
31 pp.Inquire
28
TR26-B
Addendum to TR26: Ram Transform Filtering, Monte Carlo, and Particle Methods for Stochastic Volatility and Jump-Diffusion Models
16 pp.Inquire
29
TR27
Ram Transform Family Methods for Nonlinear Black–Scholes Equations
28 pp.Inquire
30
TR28
Ram Transform Family Based Deterministic Alternatives and Enhancements for Monte Carlo Simulation Problems – Real-Time Derivative Pricing, Path-Dependent Options, Greeks, Particle Filtering, Sequential Bayesian Estimation, and General Simulation
31 pp.Inquire
31
TR29
Ram Transform Family Based Methods for Drug Discovery, Molecular Dynamics, Docking, Binding Free Energy, and Molecular Design – A Comprehensive Technical Report Addressing Section 8.1 of the Ram Transform Future-Research Report
22 pp.Inquire
32
TR30
Ram Transform Family Based Methods for Drug Release, Pharmacokinetics, Pharmacodynamics, Physiologically Based Modeling, and Personalized Dosing – A Comprehensive Technical Report Addressing Section 8.2 of the Ram Transform Future-Research Report
25 pp.Inquire
33
TR31
Ram Transform Family Based Methods for Enzyme Kinetics, Biochemical Reaction Networks, Systems Biology, and Multi-Scale Cellular Modeling – A Comprehensive Technical Report Addressing Section 8.3 of the Ram Transform Future-Research Report
24 pp.Inquire
34
TR32
Graph-Local Ram Operators, Brain Graphs, and Graph Neural Network Learning
23 pp.Inquire
35
TR33
Ram-Master and Wavelet-Ram Improvements for OTFS Delay-Doppler Communications – Verification of TR14 and Extension of Ram Transform Methods for the Problems in the Goldsmith–Hadani–Molisch–Calderbank OTFS Paper
27 pp.Inquire
36
TR34-A
Ram-Master Transform Extensions for LEO Satellite and High-Mobility Doubly Dispersive Communications
8 pp.Inquire
37
TR34-B
Ram-Master and Wavelet-Ram Methods for Automotive Radar, LiDAR, and Sonar Multipath Dereverberation
8 pp.Inquire
38
TR34-C
Ram-Master Transform Extensions for Electronic Dispersion and Nonlinear Compensation in Fiber Optic Receivers
8 pp.Inquire
39
TR34-D
Wavelet-Ram and Ram-Master Methods for Medical Ultrasound Phase-Aberration Correction and Adaptive Beamforming
8 pp.Inquire
40
TR34-E
Ram-Master Transform Methods for Underwater Acoustic Communication, Sonar Dereverberation, and Time-Varying Ocean Channels
8 pp.Inquire
41
TR34-F
Wavelet-Ram and Ram-Master Methods for Seismic Inversion, Full-Waveform Preconditioning, and Earth Profiling
8 pp.Inquire
42
TR35
Ram-Master Transform Methods for Deterministic Nonlinear State Estimation – Addressing Section 4.4 of TR7: Replacing and Enhancing Particle Filters and Monte Carlo Filtering
23 pp.Inquire
43
TR36
Ram-Master Transform Methods for Reaction–Diffusion Systems in Biology and Chemistry – Addressing Section 4.4 of TR7: GRLT Micro-Solvers, Nonlocal Terms, Gene Regulatory Networks, and Stiff Pattern-Forming Systems
21 pp.Inquire
44
TR37-A
Ram Transform and Analytic Mollifier-Ram Methods for High-Order Derivative Inverse PDE Learning – Applying the Ram Transform Family, Ram-Master Transform, Moment Constraints, and Local Inverse Operators to PDE Inverse Learning
19 pp.Inquire
45
TR37-B
Composite Ram-Mollifier Operators for Multi-Term Continuous PDE Layers – Extending the P15 Discrete Composite Operator Q and Complex Q(z,zeta) to Continuous Mollifier Layers, Inverse PDE Learning, Neural Operators, and Complex-Valued Fields
28 pp.Inquire
46
TR37-C
Appendix to TR38: Novelty and Prior-Art Assessment – Composite Ram-Mollifier Operators and Explicit Computational Speedup from Fused Operators
16 pp.Inquire
47
TR38-A
Potential Roles of Ram Transform Variants in LeWorldModel – Structured Encoders, Latent Dynamics, Inverse Diagnostics, Residual Surprise, and Self-Improving World Models
33 pp.Inquire
48
TR38-B
Graph Local Ram Transforms for LeWorldModel – Relevance of Graph-Local Forward, Inverse, and Adjoint Ram Operators to Joint-Embedding Pixel World Models
28 pp.Inquire
49
TR39
Cylindrical-PSF Rao/Ram Transform Shape Recovery – Explicit K'_1, K'_2 Matrices, Corrected PSF Moment, and Algebraic Solvability of Z_0, Z_X, Z_Y
9 pp.Inquire
50
TR40
Ram Transform Family for Forward Modeling, Inverse Recovery, Kernel Calibration, and Coupled Systems – Theory, Camera Shape Measurement, Industrial Calibration, and Ram-Master Extensions
19 pp.Inquire
51
TR41
Octonionic Ram Transforms for Post-Quantum Cryptography – Verification of the P12 ORT-PQC Claim, Strengthened Constructions, Algorithms, and Research Roadmap
25 pp.Inquire
52
TR42
New Ram Transform Improvements for Wireless Communications – Beyond Cohere US 11,470,485 B2 and the P33 Ram-Master OTFS Provisional
31 pp.Inquire
53
TR43
Deriving Forward and Inverse Ram Operators Directly from PDEs – Local Coordinate, Taylor-Jet, Boundary-Aware, and Nonlinear Algorithms
23 pp.Inquire
54
TR44
A Local Calculus for Global Science: Why Ram Transforms Deserve a Broad Reconsideration
11 pp.Inquire
55
TR45
Composite Forward and Inverse Ram Operators Q and Q': Discrete Filters, Continuous Mollifier Kernels, Ram-Master Transform Analysis, and Green-Kernel Connections
25 pp.Inquire
56
TR46
Bayesian Networks and Graph Local Ram Transforms – Representation, Inference, Inverse Diagnostics, and Opportunities Beyond Current BN Methods
35 pp.Inquire
57
TR47
Bayesian Neural Networks and Ram-Master Neural Networks – Bayesian RMNNs, Ram Moment Posterior Propagation, Inverse Uncertainty, and Practical Advantages
32 pp.Inquire
58
TR48
Ram Transforms: Complete Localization, Local Analytic Inversion, and a Transform Family for Kernel-Governed Problems Across Science, Engineering, Economics, and Finance
22 pp.Inquire
59
TR49
Reassessing the Relationship Between the Information Lattice Transform, Graph Local Ram Transforms, and Ram-Master Neural Networks – A Technical Report on Concept Lattices, Graph-Local Operator Calculus, LeWorld Models, and Hybrid ILT–GLRT Architectures
28 pp.Inquire
60
TR50
Mathematical Principles of the Forward and Inverse Graph-Local Ram Operator – Explanation of Equation (26) in Section 7.9 of the Ram Transform Enzyme Kinetics and Systems Biology Provisional Patent
18 pp.Inquire
61
TR51
Ram-Real and Ram-Sine Transforms Derived from Complex Ram Exponential Kernels – Eigenfunction Structure, Real/Imaginary Decomposition, Energy Compaction, and Applications in Signal/Image Processing, CFD, and Quantitative Finance
18 pp.Inquire
62
TR52
Magnetic Density Imaging and Field Image Tomography: A Critical Verification, Analysis, and Synthesis of US Patent 8,456,164 B2 and US Patent Application 2019/0041481 A1
14 pp.Inquire
63
TR53
A Handheld Field-Image-Tomography Instrument for 3D Imaging of Human Tissue: Theory, Algorithms, Scan-Time Analysis, and a Practical Engineering Design at the $2 M Cost Target – Combined MDI / Prepolarized-MRI Imaging Based on the Field Image Principle
18 pp.Inquire
64
TR54
A Cost-Optimised Handheld FIT-MDI-MRI Instrument with Stereo-Vision Registration, Polarization-Gradient Encoding, and Chunked Multi-Session Acquisition – Technical Report on the Calibration-Heavy / Hardware-Light Design Philosophy and the Comparison Against Augmentation of Existing Low-Field MRI Products
19 pp.Inquire
65
TR55
Ram Transform Variants for Uncertainty Quantification in Artificial Intelligence – Verification of a Prior Rao/Ram-UQ Report and a Technical Synthesis for GPLVMs, Bayesian AI, Neural Operators, and Safety-Critical Systems
21 pp.Inquire
66
TR56
Ram Transform Methods for Uncertainty Quantification in Artificial Intelligence Systems – A Comprehensive Theory, Algorithmic Framework, and Comparison with State-of-the-Art Uncertainty Quantification Methods
28 pp.Inquire
67
TR57
The Complex-Order Ram-Master Transform (C-RMT): A Six-Parameter Framework with Log-Periodic Atoms, Discrete Scale Invariance, and Applications to Critical Phenomena
26 pp.Inquire
68
TR58
The Fractional-Order Ram-Master Transform (F-RMT): A Five-Parameter Unified Framework with Mittag-Leffler Atoms, Fractional Ram–PDO Diagonalization, and Long-Memory Applications
25 pp.Inquire
69
TR59
The Octonionic-Order Ram-Master Transform (O-RMT): A Non-Associative Twelve-Parameter Framework, Cascade Cayley–Dickson Reduction, Seven-Dimensional Axis Selectivity, and the Hurwitz Terminal of the RMT Tower
22 pp.Inquire
70
TR60
The Quaternionic-Order Ram-Master Transform (Q-RMT): A Non-Commutative Eight-Parameter Framework with Axis-Selective Log-Periodic Atoms, Cayley–Dickson Reduction, and Applications to 3D Rotational, Polarimetric, and Spinor Signal Analysis
21 pp.Inquire
71
TR61
The Wavelet-Weighted Ramlet (Cross-Ramlet / "Double-Wavelet") Transform: Products of Classical Wavelet Kernels with the Ramlet Exponential Atom
20 pp.Inquire
72
TR62
The Ram-Master Transform (RMT): A Unified Framework Combining the Localized Ram Partial Differential Operator, Variable-Width Windowing, and the Cross-Ramlet Transform
25 pp.Inquire
73
TR63
The Ramlet Transform: A Structural Identification of the RamCS Transform as a Generalized Wavelet Transform with Exponential Mother Function
21 pp.Inquire
74
TR64
The Polynomial-Weighted (Hermite) Ramlet Transform: Theoretical and Practical Implications of Differentiating the Ramlet Kernel with Respect to the Spectral Variable
18 pp.Inquire
75
TR65
Proof of the Riemann Hypothesis via the Ram Transform Framework: A Comprehensive Synthesized Technical Report with Statement, Proof Steps, Verification, Explicit Assumptions, and Remaining Gaps
21 pp.Inquire
76
TR66
The Cross Wavelet–Ram-Real Transform and Cross Wavelet–Ram-Sine Transform – Global Real/Sine Multiresolution Extensions of WRRT and WRST for Transforming Functions, Operators, and Data
18 pp.Inquire
77
TR67
The Ram-Master Transform (RMT): A Unified Framework Combining the Localized Ram Partial Differential Operator, Variable-Width Windowing, Cross-Ramlet, Cross-WRRT, and Cross-WRST Transforms
16 pp.Inquire
78
TR68
Ram-Real Spectral and Ram-Sine Spectral Transforms: Real and Sine Global Spectral Extensions of the Ram Transform Framework
19 pp.Inquire
79
TR69
Ram-Real and Ram-Sine Transforms: A Unified Framework for Local Operators, Windowed Transforms, Global Spectral Transforms, and Applications
19 pp.Inquire
80
TR70
The Wavelet–Ram-Real Transform and Wavelet–Ram-Sine Transform: A Multiresolution Real/Sine Framework for Local and Spectral Ram Operators
18 pp.Inquire
81
TR71
Ram-Master Neural Networks (RMNN or RMT-NN): A Localized, Multiresolution, Algebra-Valued Operator-Learning Framework with Analytic Forward-Inverse Duality
21 pp.Inquire
82
TR72
Ram-Master Transform Kernels and Ram-Master Neural Networks for Feature Extraction, Tokenization, Recognition, Diagnosis, and Scientific AI – A Comprehensive Technical Report on RMT/RMNN Encoders for Images, Video, Medical Data, Engineering Streams, and Transformer/LLM Hybrid Systems
30 pp.Inquire
83
TR73
Adelic RamCS/Ram–Master Transform for Hecke and Dedekind L-Functions and Related Physical Models – A Development of the Adelic Ram Transform Program
24 pp.Inquire
84
TR74
Self-Improving Ram-Master Neural Networks (RMNNs) – Automated Discovery, Hyperparameter Optimization, Kernel-Moment Optimization, Multi-Agent Experimentation, and AGI-Relevant Research Directions
32 pp.Inquire
85
TR75
A Ram Transform and Ram-Master Transform Synthesis for the Three-Dimensional Navier–Stokes Existence and Smoothness Problem – Conditional Regularity, Moment Constraints, RMT Variants, Remaining Gaps, and a Realistic Clay-Millennium Assessment
18 pp.Inquire
86
TR76
Practical Computational Fluid Dynamics with Ram Transform Methods – A Technical Assessment of Theory, Algorithms, R&D Applications, Advantages, Limitations, and Future Directions
18 pp.Inquire
87
TR77
Resolving the Silent-Source Non-Uniqueness in MEG/MCG by Intracranial Magnetic Field Sensors: Theoretical Analysis, Quantitative Bounds, and Survey of Available Brain Implant Technologies
16 pp.Inquire
88
TR78
Critical Analysis and Verification of US Patent Application Publication No. 2011/0313274 A1 "Methods and Apparatuses for 3D Imaging in Magnetoencephalography and Magnetocardiography" (Subbarao, 2011)
19 pp.Inquire
89
TR79
Richardson–Lucy versus Residual Gradient-Descent for Shift-Variant Image Deblurring with a Known Kernel: Theory, Algorithms, Practical Implementation, and an Investigation of Catastrophic Failure
22 pp.Inquire
90
TR80
A Critical Technical Analysis of Ram Transform Methods for Wavefront Propagation, Electromagnetic Media, and Schrodinger-Type Quantum Problems – Errors, Assumptions, Corrected Mathematical Formulations, Pros and Cons, Accuracy, and Computational Speed Estimates
17 pp.Inquire
91
TR81
Technical Review and Corrected Framework for Applying Ram (Rao) Transforms to the Propagator-Hamiltonian Correspondence Problem
19 pp.Inquire
92
TR82
Addendum to the Technical Review of Ram (Rao) Transform Methods for the Propagator-Hamiltonian Correspondence – Analysis of the Generalized Rao Transform Limit Derivation and Historical Context
13 pp.Inquire
93
TR83
Ram Transform Formulation of Shift-Variant Affine Motion Blur – Forward Blurring, Local Differential Operators, Inverse Deblurring, Defocus Coupling, and Higher-Dimensional Extensions
27 pp.Inquire
94
TR84
A Unified Ram Transform Family Formulation of Known-Kernel Motion Blur and Deblurring – Coordinate Localization, Local Ram-PDOs, Analytic Inverse Operators, Windowed/Multi-Resolution Variants, and a Dirac-Delta Trajectory Kernel
23 pp.Inquire
95
TR85
A Critical Technical Analysis of Ram Transform Methods for Wavefront Propagation, Electromagnetic Media, and Schrodinger-Type Quantum Problems – Errors, Assumptions, Corrected Mathematical Formulations, Pros and Cons, Accuracy, and Computational Speed Estimates
19 pp.Inquire
96
TR86
Fokker–Planck–Ram Operators for Linear Time-Invariant Problems – Theory, Algorithms, and Practical Applications of the Ram Transform Family to Drift–Diffusion Operators
20 pp.Inquire
97
TR87
TFPR2: Fokker–Planck–Ram Operators for Linear Time-Variant and Coordinate-Variant Problems – Local Charts, Frozen-Kernel Parametrices, Conservative Algorithms, and Practical Applications
19 pp.Inquire
98
TR88
TFPR3: Nonlinear Fokker–Planck–Ram Operators – Generalized Ram Localization, Local Nonlinear Algebraic Solvers, Structure Preservation, and High-Impact Applications
21 pp.Inquire
99
TR89
TCM1: Coordinate-Mapped Ram Transforms for Linear Integral Equations and Linear PDEs – A Graduate-Level Technical Reconstruction of the Affine Coordinate Mapping Method in P5
19 pp.Inquire
100
TR90
TCM2: Coordinate-Mapped Ram Transforms for Nonlinear Integral Equations and Nonlinear PDEs – A Graduate-Level Sequel to TCM1 Based on the Nonlinear Coordinate-Mapping Content of P5
22 pp.Inquire
101
TR91
TCM3: Remaining Integral-Equation and PDE Problem Classes for Coordinate-Mapped Ram Transforms – Truncation in Source Jets, Kernel Geometry, and Nonlinear Powers Beyond TCM1 and TCM2
22 pp.Inquire
102
TR92
TCM4: Ram Master and Advanced Ram Transforms for Remaining Coordinate-Mapped Integral-Equation and PDE Problems – Multiresolution Windows, Fractional and Complex Orders, Hypercomplex Algebras, and Practical Solver Architectures Beyond TCM1–TCM3
30 pp.Inquire
103
TR93
RamDictionary: Dictionary and Taxonomy of Named Ram Transform Concepts – A Source-Derived Vocabulary for Ram/Rao Transforms, Operators, Matrices, Coefficients, Coordinate Maps, Algorithms, and Application Methods in res-rt
33 pp.Inquire
104
TR94
Ram Transform Family Paths Toward the Riemann Hypothesis – Beyond the Failed Hilbert-Polya Adjointness Ansatz
14 pp.Inquire
105
TR95
Relevance of the Ram-Master Transform and Its Variants to GRH for Broad Families of L-Functions – A Technical Investigation Based on Current Literature and the Ram Transform Corpus
16 pp.Inquire
106
TR96
Adelic RamCS/Ram–Master Transform for Hecke and Dedekind L-Functions and Related Physical Models – A Development of the Adelic Ram Transform Program
24 pp.Inquire
107
TR97
A Revised Conditional Proof Framework for the Riemann Hypothesis via Ram Transforms – Incorporating Weakened Assumptions, Corrected Operator-Theoretic Statements, and Explicit Remaining Gaps
14 pp.Inquire
108
TR98
The Navier–Stokes Millennium Problem via Ram Transforms: Applying the Gap C Closure to the Borel-Summability Gap, Updated Status, and Comprehensive Technical Assessment
22 pp.Inquire
109
TR99
Does the Chen–Hou Proof of 3D Euler Blowup and the Prospect of Navier–Stokes Blowup Impact the Riemann Hypothesis Proof Programme? – A Cross-Programme Analysis of the Ram Transform Structural Analogy between RH and NS
16 pp.Inquire
110
TR100
Hybrid Ram Transform Techniques with Optimization, Regularization, Multiresolution, and Multi-Interval Methods
27 pp.Inquire
111
TR101
Ram Transform Methods for Neural Integral Equation Systems – Part 1 of the P4 Neural Ram Transform Report Series: Theory, Algorithms, Glass-Box Identification, Analytic Inversion, and Comparison with Neural Integral Equations
23 pp.Inquire
112
TR102
Ram Transform Methods for Neural Integral Equation Systems – Updated Part 1: Coordinate-Mapped Localization, Ram-Master Extensions, Composite Q and Q' Operators, Glass-Box Identification, and Analytic Inversion
29 pp.Inquire
113
TR103
Ram Transform Methods for Attentional Neural Integral Equation Systems – Part 2: Rao/Ram-Corrected Attention, Local Moment Learning, Composite Q,Q' Operators, and Analytic Local Inversion
27 pp.Inquire
114
TR104
Ram Transform Methods for Attentional Neural Integral Equation Systems – Updated Part 2: Coordinate-Mapped Local Attention, Composite Q and Q' Operators, Ram-Master/RMNN Attention Features, Glass-Box Distillation, and Analytic Inversion
27 pp.Inquire
115
TR105
Ram Transform Methods for DeepONet and Physics-Informed Neural Operators – Part 3: Coordinate-Mapped Local Solution Operators, Complex-Valued Wave Problems, Ram-Master Features, and Analytic Inversion
24 pp.Inquire
116
TR106
The Ram-Master Transform (RMT): A Unified Framework Combining the Localized Ram Partial Differential Operator, Variable-Width Windowing, and the Cross-Ramlet Transform
21 pp.Inquire
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