Updated index.rst

index.rst

@@ -374,7 +374,7 @@ isolate the meanings of common pictogram B in three ways by 1) Set intersection
         52. Integer Partitions and String complexity measures are related - Every string is encoded in some alphabet (ASCII or Unicode) having a numeric value and thus every string is a histogram set partition whose bins have sizes equal to ASCII or Unicode values of alphabets which partition the sum of ASCII or Unicode values of constituent alphabets of a string. This enables partition distance (a kind of earth mover distance - e.g. Optimal transport and integer partitions - https://arxiv.org/pdf/1704.01666.pdf) between string histograms as a distance measure between strings apart from usual edit distance measures.
         53. Byzantine Fault Tolerance (BFT) has theoretical implications for mitigating faults including cybercrimes in electronic networks and containment of pandemics in social networks modelled by Cellular automaton graphs.
         54. Economic Merit - fluctuations in economy and stock markets are modelled by Chaotic multifractals wherein single exponent is not sufficient and behaviour around any point is defined by a local exponent. NeuronRain envisages Collatz conjecture model of market vagaries which is a 2-colored pseudorandom sequence of odd and even integers always ending in 1.
-	55. Graph theory originated from an urban sprawl analytics problem - Euler circuit and Closed trail of Seven bridges of Konigsberg. Variety of Urban sprawl metrics could be derived from FaceGraph of segmented GIS imagery - Built-up area (impervious surface (IS) land cover derived from satellite imagery), Urbanized area (built-up area + urbanized open space (OS)), Urbanized OS (non-IS pixels in which more than 50% of the neighborhood is built-up), Buildable (does not contain water or excessive slope), Urban footprint (built-up area + urbanized open space + peripheral open space), Peripheral OS (non-IS pixels that are within 100 meters of the built-up area), Open space (OS - the sum of the urbanized and peripheral OS) (from URBAN SPRAWL METRICS: AN ANALYSIS OF GLOBAL URBAN EXPANSION USING GIS - [by Shlomo Angel , Jason Parent , Daniel Civco] - Table 3 of metrics for measuring urban extent - https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.586.5052). Polya Urn Urban Growth Model (which simulates the growth of new segments along periphery of urban sprawl facegraph by Rich-get-richer preferential attachment) and DNFSAT MARA arithmetic progression n-segment coloring of all-pair walks in facegraph (which is a Constraint Satisfaction Problem for interior of urban sprawl facegraph) together stipulate a stringent condition for exterior and interior of urban area (and often less satisfiable theoretical fancy because DNFSAT MARA mandates that the segments of same color are equally spaced-out by arithmetic progression hops as interior segments change and new segments along urban sprawl periphery are attached by Polya Urn process dynamically) for equitable and sustainable urban growth - for instance arithmetic progression 4-coloring of facegraph all-pair walks is disturbed when interior segments develop and new segment faces from Polya Urn growth are added to periphery (Example: openness, urbanness, infill, extension, leapfrog, reflectance, urban fringe, ribbon development, scatter development segmentation of Bangkok metro area - https://proceedings.esri.com/library/userconf/proc08/papers/papers/pap_1692.pdf) of urban sprawl facegraph and 4-coloring has to be recomputed causing re-classification of few segments. Earlier instance of DNFSAT MARA by arithmetic progression coloring of facegraph walks is just one example of how MARA could be implemented in urban sprawls and DNFSAT could be derived from any other arbitrary constraints on graph complexity measures that are satisfiable in practice. Reallife problems of urban areas are solvable by vertex cover, edge cover, maxflow-mincut, leaky bucket model of traffic, strongly connected components, dense subgraphs of transportation network graphs aiding efficient drone navigation. NeuronRain implements ranking of Urban Sprawls from segmented Contour polynomial areas bounding urban sprawls based on NASA VIIRS NightLights imagery which is a polynomial variant of Space filling problem usually limited to Packing by Circles and Chaotic Mandelbrot set curves. Delaunay triangulation graph of SEDAC and VIIRS Urban sprawl contours approximates transportation graph and could be an estimator of Euler Circuit and Hamiltonian for efficient drone navigation. Design of Urban transportation networks could be formulated by a reduction from Global wiring and Detailed wiring 0-1 Integer Linear Programming NP-Hard problems mostly used to solve layouts in logic gate arrays in chip design - [Randomized Algorithms - Rajeev Motwani and Prabhakar Raghavan - Section 4.3 - Pages 79-81 - simplified for nets containing at most one 90 degree turn and each net is an optimal path-to-be-found connecting two logic gates]: Edges of urban transportation networks are carriageways (wires) connecting two urban centres or two suburbs within an urban centre (logic gate vertices) and number of turns in each carriageway and their angles are indefinite. Finding optimal layout of transportation network carriageways for unrestricted turns is harder than one 90 degree turn version, in which case multi-turn carriageway could be approximated by multiple one 90 degree turn segments and encoded as 0-1 linear program variables - monotonic walks on lattice grids (random walk from bottom-left to top-right on m*n grid) are of multiple 90 degree turns which could approximate unrestricted-angle-and-multiple-turn carriageways connecting 2 urban centres. Optimal alignment of a carriageway could be its monotonic random walk (multiple 90 degree turn) approximation of least root mean square error between turn points on lattice walk and straightline connecting 2 urban centres (Illustration: https://www.statisticshowto.com/probability-and-statistics/regression-analysis/rmse-root-mean-square-error/).
+	55. Graph theory originated from an urban sprawl analytics problem - Euler circuit and Closed trail of Seven bridges of Konigsberg. Variety of Urban sprawl metrics could be derived from FaceGraph of segmented GIS imagery - Built-up area (impervious surface (IS) land cover derived from satellite imagery), Urbanized area (built-up area + urbanized open space (OS)), Urbanized OS (non-IS pixels in which more than 50% of the neighborhood is built-up), Buildable (does not contain water or excessive slope), Urban footprint (built-up area + urbanized open space + peripheral open space), Peripheral OS (non-IS pixels that are within 100 meters of the built-up area), Open space (OS - the sum of the urbanized and peripheral OS) (from URBAN SPRAWL METRICS: AN ANALYSIS OF GLOBAL URBAN EXPANSION USING GIS - [by Shlomo Angel , Jason Parent , Daniel Civco] - Table 3 of metrics for measuring urban extent - https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.586.5052). Polya Urn Urban Growth Model (which simulates the growth of new segments along periphery of urban sprawl facegraph by Rich-get-richer preferential attachment) and DNFSAT MARA arithmetic progression n-segment coloring of all-pair walks in facegraph (which is a Constraint Satisfaction Problem for interior of urban sprawl facegraph) together stipulate a stringent condition for exterior and interior of urban area (and often less satisfiable theoretical fancy because DNFSAT MARA mandates that the segments of same color are equally spaced-out by arithmetic progression hops as interior segments change and new segments along urban sprawl periphery are attached by Polya Urn process dynamically) for equitable and sustainable urban growth - for instance arithmetic progression 4-coloring of facegraph all-pair walks is disturbed when interior segments develop and new segment faces from Polya Urn growth are added to periphery (Example: openness, urbanness, infill, extension, leapfrog, reflectance, urban fringe, ribbon development, scatter development segmentation of Bangkok metro area - https://proceedings.esri.com/library/userconf/proc08/papers/papers/pap_1692.pdf) of urban sprawl facegraph and 4-coloring has to be recomputed causing re-classification of few segments. Earlier instance of DNFSAT MARA by arithmetic progression coloring of facegraph walks is just one example of how MARA could be implemented in urban sprawls and DNFSAT could be derived from any other arbitrary constraints on graph complexity measures that are satisfiable in practice. Reallife problems of urban areas are solvable by vertex cover, edge cover, maxflow-mincut, leaky bucket model of traffic, strongly connected components, dense subgraphs of transportation network graphs aiding efficient drone navigation. NeuronRain implements ranking of Urban Sprawls from segmented Contour polynomial areas bounding urban sprawls based on NASA VIIRS NightLights imagery which is a polynomial variant of Space filling problem usually limited to Packing by Circles and Chaotic Mandelbrot set curves. Delaunay triangulation graph of SEDAC and VIIRS Urban sprawl contours approximates transportation graph and could be an estimator of Euler Circuit and Hamiltonian for efficient drone navigation. Design of Urban transportation networks could be formulated by a reduction from Global wiring and Detailed wiring 0-1 Integer Linear Programming NP-Hard problems mostly used to solve layouts in logic gate arrays in chip design - [Randomized Algorithms - Rajeev Motwani and Prabhakar Raghavan - Section 4.3 - Pages 79-81 - simplified for nets containing at most one 90 degree turn and each net is an optimal path-to-be-found connecting two logic gates]: Edges of urban transportation networks are carriageways (wires) connecting two urban centres or two suburbs within an urban centre (logic gate vertices) and number of turns in each carriageway and their angles are indefinite. Finding optimal layout of transportation network carriageways for unrestricted turns is harder than one 90 degree turn version, in which case multi-turn carriageway could be approximated by multiple one 90 degree turn segments and encoded as 0-1 linear program variables - monotonic walks on lattice grids (random walk from bottom-left to top-right on m*n grid) are of multiple 90 degree turns which could approximate unrestricted-angle-and-multiple-turn carriageways connecting 2 urban centres. Optimal alignment of a carriageway could be its monotonic random walk (multiple 90 degree turn) approximation of least root mean square error between turn points on lattice walk and straightline connecting 2 urban centres (Illustration: https://www.statisticshowto.com/probability-and-statistics/regression-analysis/rmse-root-mean-square-error/). Public transit data (daily transport patterns) are available through OpenStreetMap PTNA - https://ptna.openstreetmap.de/ . On the other hand, pedestrian networks are as important as transportation networks. There have been some recent Pedestrian network construction algorithms based on inputs from GPS Traces feature of OpenStreetMap (https://www.openstreetmap.org/traces) for inferring pedestrian geometric patterns - https://www.sciencedirect.com/science/article/abs/pii/S0968090X12001179
 	56. Quantum Circuits in Deustch Model could be translated to classical Parallel RAMs by Memory Peripheral Model which maps quantum circuits to PRAM instruction set - https://uwspace.uwaterloo.ca/bitstream/handle/10012/16060/Schanck_John.pdf. This bridges a missing link between Quantum and Classical computations which might resolve lot of conflicts involving derandomization of Shor BQP Factorization to P and NC, Classical PRAM-NC-BSP and Quantum NC Computational Geometric factorizations described and implemented in NeuronRain.
 	57. In NeuronRain complement implementation, Complement Diophantines are learnt by Least Squares and Lagrangian interpolations which are total functions while Lagrange Four Square Theorem complement map is a partial surjective function (not all domain tuples are mapped to a point in complementary set)
 	58. MapReduce cloud parallel computing framework (on which Hadoop, Spark are based) has separate complexity class MRC (MapReduce Class) defined for itself: