Equations, Probability & Algorithms

EQUATIONS, PROBABILITY
&
ALGORITHMS
PHD SCHOLAR: SANJAY SRIVASTAVA
B.ARCH, CECP, MUDP, CEPPM, AIIA, FISLE, FIBE.
GUIDE : PROF. DR ALOK SHARMA
MITS GWALIOR

• "SCIENCE IS NOT PERFECT. IT'S OFTEN MISUSED;
IT'S ONLY A TOOL, BUT IT'S THE BEST TOOL WE
HAVE. SELF-CORRECTING, EVER-CHANGING,
APPLICABLE TO EVERYTHING; WITH THIS TOOL,
WE VANQUISH THE IMPOSSIBLE.“
• CARL SAGAN
EQUATIONS, PROBABILITIES & ALGORITHMS
SANJAY SRIVASTAVA MITS
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“WE BEGIN WITH A BIT OF SCIENCE
OF
MATHEMATICS & COMPUTATION”
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IN DESIGNING, WHETHER IT IS ENGINEERING, ARCHITECTURE, URBAN PLANNING OR
ADMINISTRATION, WE NEED TO QUANTIFY AND COMPARE USING MATHEMATICS, ESTIMATE, PROJECT
OR PREDICT USING STATISTICS AND AS THESE ISSUES BECOME DEEPER AND DEEPER AND THE
SOLUTIONS BECOME MORE COMPLICATED WE USE COMPUTATIONAL TECHNIQUES BEGINNING WITH
ALGORITHMS, NEURAL NETWORKS , MACHINE LEARNING AND FINALLY ARTIFICIAL INTELLIGENCE.

EQUATIONS 1.1
• LINEAR EQUATIONS
• A LINEAR EQUATION IS ANY EQUATION THAT CAN
BE WRITTEN IN THE FORM
• AX+ B=0 WHERE A AND B ARE REAL NUMBERS
AND X IS A VARIABLE. THIS FORM IS SOMETIMES
CALLED THE STANDARD FORM OF A LINEAR
EQUATION.
• THE SLOPE-INTERCEPT FORM OF A LINEAR
EQUATION IS Y = MX + B. IN THE EQUATION, X
AND Y ARE THE VARIABLES. THE NUMBERS M AND
B GIVE THE SLOPE OF THE LINE (M) AND THE
VALUE OF Y WHEN X IS 0 IS (B) I.E. THE POINT
WHERE THE LINE CROSSES Y AXIS.
B
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EQUATIONS 1.2
• POLYNOMIAL EQUATIONS
• POLYNOMIAL COMES FROM POLY- (MEANING
"MANY") AND -NOMIAL (IN THIS CASE
MEANING "TERM") ... SO IT SAYS "MANY
TERMS“
• THAT CAN BE COMBINED USING ADDITION,
SUBTRACTION, MULTIPLICATION AND
DIVISION ...
• ... EXCEPT NOT DIVISION BY A VARIABLE... (SO
SOMETHING LIKE 2/X OR 2X^-2 IS RIGHT
OUT)
Polynomials can have no variable eg. 21 is a polynomial.
Or one variable: x4 − 2x2 + x has three terms, but one variable
(x)
Or two or more variables: xy4 − 5x2z has two terms & three
variables (x, y and z)
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“CUBIC , QUARTIC & QUINTIC FUNCTIONS”
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CUBIC FUNCTIONS
In mathematics, a cubic function is a function of the form
F(x)= ax3+bx2+cx+d
where the coefficients a, b, c, and d are real numbers, and the variable x takes
real values, and a ≠ 0. In other words, it is both a polynomial function of degree
three, and a real function. In particular, the domain and the co-domain are the
set of the real numbers.
Setting f(x) = 0 produces a cubic equation of the form
whose solutions are called roots of the function.
A cubic function has either one or three real roots (the existence of at least one
real root is true for all odd-degree polynomial functions).
The graph of a cubic function always has a single inflection point. It may have
two critical points, a local minimum and a local maximum.
Otherwise, a cubic function is monotonic.
The graph of a cubic function is symmetric with respect to its inflection point; that
is, it is invariant under a rotation of a half turn around this point. Up to an affine
transformation, there are only three possible graphs for cubic functions.
Cubic functions are fundamental for cubic interpolation.
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QUARTIC FUNCTIONS
• GRAPH OF A POLYNOMIAL OF DEGREE 4, QUARTIC FUNCTIONS
HAVE 3 CRITICAL POINTS AND FOUR REAL ROOTS (CROSSINGS OF
THE X AXIS) (AND THUS NO COMPLEX ROOTS). IF ONE OR THE
OTHER OF THE LOCAL MINIMA WERE ABOVE THE X AXIS, OR IF THE
LOCAL MAXIMUM WERE BELOW IT, OR IF THERE WERE NO LOCAL
MAXIMUM AND ONE MINIMUM BELOW THE X AXIS, THERE WOULD
ONLY BE TWO REAL ROOTS (AND TWO COMPLEX ROOTS). IF ALL
THREE LOCAL EXTREMA WERE ABOVE THE X AXIS, OR IF THERE
WERE NO LOCAL MAXIMUM AND ONE MINIMUM ABOVE THE X AXIS,
THERE WOULD BE NO REAL ROOT (AND FOUR COMPLEX ROOTS).
THE SAME REASONING APPLIES IN REVERSE TO POLYNOMIAL WITH
A NEGATIVE QUARTIC COEFFICIENT.
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QUINTIC FUNCTIONS
Fifth Degree Polynomials
Fifth degree polynomials are also known as quintic polynomials.
Quintics have these characteristics:
•One to five roots.
•Zero to four extrema.
•One to three inflection points.
•No general symmetry.
•It takes six points or six pieces of information to describe a quintic
function.
•Roots are not solvable by radicals (a fact established by Abel in 1820
and expanded upon by Galois in 1832).
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WRITING EQUATIONS FROM A GRAPH
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EQUATIONS 1.3
• EXPONENTIAL EQUATIONS
• IN ANY EXPONENTIAL EXPRESSION, B IS CALLED
THE BASE AND X IS CALLED THE EXPONENT.
AN EXAMPLE OF AN EXPONENTIAL FUNCTION IS
THE GROWTH OF BACTERIA. SOME BACTERIA
DOUBLE EVERY HOUR.
• EXPONENTIAL FUNCTIONS ARE USED TO MODEL
POPULATIONS, CARBON DATE ARTIFACTS, HELP
CORONERS DETERMINE TIME OF DEATH, COMPUTE
INVESTMENTS, AS WELL AS MANY
OTHER APPLICATIONS. WE DISCUSS HERE THREE
OF THE MOST COMMON APPLICATIONS:
POPULATION GROWTH, EXPONENTIAL DECAY, AND
COMPOUND INTEREST.
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VISUALISING BEHAVIOUR OF EQUATIONS -2D
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3D & ANIMATIONS
• 3 D GRAPHS AND ANIMATIONS ARE BEST
VISUAL TECHNIQUES TO UNDERSTAND THE
BEHAVIOUR OF EVENTS AND THEIR IMPACT
AND INTENSITY IN NEARBY AREAS.
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LOGARITHMS
• A LOGARITHM FUNCTION IS DEFINED WITH RESPECT TO A “BASE”, WHICH IS A POSITIVE NUMBER: IF B
DENOTES THE BASE NUMBER, THEN THE BASE-B LOGARITHM OF X IS, BY DEFINITION, THE NUMBER Y SUCH
THAT BY = X.
• FOR EXAMPLE, THE BASE-2 LOGARITHM OF 8 IS EQUAL TO 3, BECAUSE 23 = 8, THE BASE-10 LOGARITHM
OF 100 IS 2, BECAUSE 102 = 100.
• THERE ARE THREE KINDS OF LOGARITHMS IN STANDARD USE:
THE BASE-2 LOGARITHM (PREDOMINANTLY USED IN COMPUTER SCIENCE AND MUSIC THEORY),
THE BASE-10 LOGARITHM (PREDOMINANTLY USED IN ENGINEERING), AND
THE NATURAL LOGARITHM (USED IN MATHEMATICS AND PHYSICS AND IN ECONOMICS & BUSINESS).
• IN THE NATURAL LOG FUNCTION, THE BASE NUMBER IS THE TRANSCENDENTAL NUMBER “E” WHOSE
DECIMINAL EXPANSION IS 2.718282…, SO THE NATURAL LOG FUNCTION AND THE EXPONENTIAL FUNCTION
(EX) ARE INVERSES OF EACH OTHER. THE ONLY DIFFERENCES BETWEEN THESE THREE LOGARITHM
FUNCTIONS ARE MULTIPLICATIVE SCALING FACTORS, SO LOGICALLY THEY ARE EQUIVALENT FOR PURPOSES OF
MODELING, BUT THE CHOICE OF BASE IS IMPORTANT FOR REASONS OF CONVENIENCE AND CONVENTION,
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LOGARITHMS
• IN STANDARD MATHEMATICAL NOTATION, AND IN EXCEL AND MOST OTHER
ANALYTIC SOFTWARE, THE EXPRESSION LN(X) IS THE NATURAL LOG OF X,
AND
• EXP(X) IS THE EXPONENTIAL FUNCTION OF X,
• SO
EXP(LN(X)) = X AND LN(EXP(X)) = X.
• THIS MEANS THAT THE EXP FUNCTION CAN BE USED TO CONVERT NATURAL-
LOGGED FORECASTS (AND THEIR RESPECTIVE LOWER AND UPPER CONFIDENCE
LIMITS) BACK INTO REAL UNITS.
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LOGARITHMS
• LINEARIZATION OF EXPONENTIAL GROWTH AND INFLATION: THE LOGARITHM OF A PRODUCT EQUALS THE
SUM OF THE LOGARITHMS, I.E., LOG(XY) = LOG(X) + LOG(Y), REGARDLESS OF THE LOGARITHM BASE.
THEREFORE, LOGGING CONVERTS MULTIPLICATIVE RELATIONSHIPS TO ADDITIVE RELATIONSHIPS, AND BY THE
SAME TOKEN IT CONVERTS EXPONENTIAL (COMPOUND GROWTH) TRENDS TO LINEAR TRENDS. BY TAKING
LOGARITHMS OF VARIABLES WHICH ARE MULTIPLICATIVELY RELATED AND/OR GROWING EXPONENTIALLY
OVER TIME, WE CAN OFTEN EXPLAIN THEIR BEHAVIOR WITH LINEAR MODELS. FOR EXAMPLE, HERE IS A
GRAPH OF LOG(AUTOSALE). NOTICE THAT THE LOG TRANSFORMATION CONVERTS
THE EXPONENTIAL GROWTH PATTERN TO A LINEAR GROWTH PATTERN, AND IT SIMULTANEOUSLY CONVERTS
THE MULTIPLICATIVE (PROPORTIONAL-VARIANCE) SEASONAL PATTERN TO AN ADDITIVE (CONSTANT-
VARIANCE) SEASONAL PATTERN. (COMPARE THIS WITH THE ORIGINAL GRAPH OF AUTOSALE.) THESE
CONVERSIONS MAKE THE TRANSFORMED DATA MUCH MORE SUITABLE FOR FITTING WITH LINEAR/ADDITIVE
MODELS.
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LOGARITHMS
• Logging a series often has an effect very similar to deflating: it straightens out exponential
growth patterns and reduces heteroscedasticity (i.e., stabilizes variance).
• Logging is therefore a "poor man's deflator" which does not require any external data (or any
head-scratching about which price index to use).
• Logging is not exactly the same as deflating--it does not eliminate an upward trend in the
data--but it can straighten the trend out so that it can be better fitted by a linear
model. Deflation by itself will not straighten out an exponential growth curve if the growth is
partly real and only partly due to inflation.
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LOGARITHMS
• If we're going to log the data and then fit a model that implicitly or explicitly
uses differencing (e.g., a random walk, exponential smoothing, or ARIMA model),
then it is usually redundant to deflate by a price index, as long as the rate of
inflation changes only slowly: the percentage change measured in nominal dollars
will be nearly the same as the percentage change in constant dollars.
• In Statgraphics ( a software that helps see statistics and graphs )notation, this means that,
DIFF(LOG(Y/CPI)) is nearly identical to DIFF(LOG(Y)):
• the only difference between the two is a very faint amount of noise due to
fluctuations in the inflation rate. To demonstrate this point, here's a graph of the
first difference of logged auto sales, with and without deflation
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PROBABILITY
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BAYESIAN PROBABILITY
• BAYESIAN PROBABILITY REPRESENTS A LEVEL OF CERTAINTY RELATING TO A POTENTIAL OUTCOME OR IDEA.
THIS IS IN CONTRAST TO A FREQUENTIST PROBABILITY THAT REPRESENTS THE FREQUENCY WITH WHICH A
PARTICULAR OUTCOME WILL OCCUR OVER ANY NUMBER OF TRIALS.
• AN EVENT WITH BAYESIAN PROBABILITY OF .6 (OR 60%) SHOULD BE INTERPRETED AS STATING "WITH
CONFIDENCE 60%, THIS EVENT CONTAINS THE TRUE OUTCOME", WHEREAS A FREQUENTIST INTERPRETATION
WOULD VIEW IT AS STATING "OVER 100 TRIALS, WE SHOULD OBSERVE EVENT X APPROXIMATELY 60 TIMES."
• THE DIFFERENCE IS MORE APPARENT WHEN DISCUSSING IDEAS.
• A FREQUENTIST WILL NOT ASSIGN PROBABILITY TO AN IDEA; EITHER IT IS TRUE OR FALSE AND IT CANNOT BE
TRUE 6 TIMES OUT OF 10.
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BAYESIAN PROBABILITY
• WHEN WE GO TO THE VET, MY DOG SQUIRMS ON THE
SCALE. THAT MAKES IT HARD TO GET AN ACCURATE
READING.
• ON OUR LAST VISIT, WE GOT THREE MEASUREMENTS
BEFORE IT BECAME UNMANAGEABLE: 13.9 LB, 17.5 LB
AND 14.1 LB. THERE IS A STANDARD STATISTICAL
INTERPRETATION FOR THIS. WE CAN CALCULATE THE
MEAN, STANDARD DEVIATION AND STANDARD ERROR
FOR THIS SET OF NUMBERS AND CREATE A
DISTRIBUTION FOR ACTUAL WEIGHT. I REMEMBER LAST
TIME IT WEIGHED IN AT 14.2 POUNDS.
I BELIEVE THAT IT’S ABOUT 14.2 POUNDS BUT MIGHT
BE A POUND OR TWO HIGHER OR LOWER. TO
REPRESENT THIS, WE USE A NORMAL DISTRIBUTION
WITH A PEAK AT 14.2 POUNDS AND WITH A STANDARD
DEVIATION OF A HALF POUND. THE ADJACENT FIGURE
SHOWS BAYESIAN PROBABILITY OF DOGS WEIGHTSANJAY SRIVASTAVA MITS
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BAYESIAN DECISION THEORY
• BAYESIAN DECISION THEORY REFERS TO
A DECISION THEORY WHICH IS INFORMED
BY BAYESIAN PROBABILITY. IT IS A STATISTICAL
SYSTEM THAT TRIES TO QUANTIFY THE TRADEOFF
BETWEEN VARIOUS DECISIONS, MAKING USE OF
PROBABILITIES AND COSTS.
• AN AGENT OPERATING UNDER SUCH A DECISION
THEORY USES THE CONCEPTS OF BAYESIAN
STATISTICS TO ESTIMATE THE EXPECTED VALUE OF
ACTIONS, & UPDATES ITS EXPECTATIONS BASED
ON NEW INFORMATION. THESE AGENTS ARE
USUALLY REFERRED TO AS ESTIMATORS.
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BAYESIAN DECISION THEORY
• FROM THE PERSPECTIVE OF BAYESIAN DECISION
THEORY, ANY KIND OF PROBABILITY
DISTRIBUTION - SUCH AS THE DISTRIBUTION FOR
TOMORROW'S WEATHER - REPRESENTS
A PRIOR DISTRIBUTION. THAT IS, IT REPRESENTS
HOW WE EXPECT TODAY THE WEATHER IS GOING
TO BE TOMORROW.
• THIS CONTRASTS WITH FREQUENTIST INFERENCE,
THE CLASSICAL PROBABILITY INTERPRETATION,
WHERE CONCLUSIONS ABOUT AN EXPERIMENT
ARE DRAWN FROM A SET OF REPETITIONS OF
SUCH EXPERIENCE, EACH PRODUCING
STATISTICALLY INDEPENDENT RESULTS.
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ALGORITHMS
• FORMULAE HELP US RESOLVE AND PROVIDE A
DIMENSION TO PROBLEM.
• PROBABILITY HELPS IN FORECASTING THE
LIKELIHOOD OF OF A COMBINATION OF EVENTS
• ALGORITHMS IS THE NAME GIVEN TO SEQUENCES
OF STEPS USING THESE METHODS TO MOVE
FROM A STATE OF DISARRAY TO A COMPLETELY
NORMAL [SOLVED] STATE.
• IN URBAN PLANNING THESE WILL BE
CONSTRUCTED FROM A CONSIDERATION OF
MULTIPLE PARAMETERS
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• In mathematics and computer science, an algorithm is a sequence of instructions, typically to
solve a class of problems or perform a computation. Algorithms
are unambiguous specifications for performing calculation, data processing, automated
reasoning, and other tasks.
• As an effective method, an algorithm can be expressed within a finite amount of space and
time and in a well-defined formal language for calculating a function. Starting from an initial
state and initial input (perhaps empty), the instructions describe a computation that,
when executed, proceeds through a finite number of well-defined successive states, eventually
producing "output“ and terminating at a final ending state. The transition from one state to
the next is not necessarily deterministic; some algorithms, known as randomized
incorporate random input.
• The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in
the sieve of Eratosthenes for finding prime numbers, and the Euclidean algorithm for finding
the greatest common divisor of two numbers.
ALGORITHMS
SANJAY SRIVASTAVA
MITS GWALIOR
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• FORMAL VERSUS EMPIRICAL
• THE ANALYSIS, AND STUDY OF ALGORITHMS IS A DISCIPLINE OF COMPUTER SCIENCE, AND IS OFTEN
ABSTRACTLY WITHOUT THE USE OF A SPECIFIC PROGRAMMING LANGUAGE OR IMPLEMENTATION.
• IN THIS SENSE, ALGORITHM ANALYSIS RESEMBLES OTHER MATHEMATICAL DISCIPLINES IN THAT IT FOCUSES
ON THE UNDERLYING PROPERTIES OF THE ALGORITHM AND NOT ON THE SPECIFICS OF ANY PARTICULAR
IMPLEMENTATION.
• USUALLY PSEUDOCODE IS USED FOR ANALYSIS AS IT IS THE SIMPLEST AND MOST GENERAL
HOWEVER, ULTIMATELY, MOST ALGORITHMS ARE USUALLY IMPLEMENTED ON PARTICULAR
HARDWARE/SOFTWARE PLATFORMS AND THEIR ALGORITHMIC EFFICIENCY IS EVENTUALLY PUT TO THE TEST
USING REAL CODE.
• FOR THE SOLUTION OF A "ONE OFF" PROBLEM, THE EFFICIENCY OF A PARTICULAR ALGORITHM MAY NOT
HAVE SIGNIFICANT CONSEQUENCES BUT FOR ALGORITHMS DESIGNED FOR FAST INTERACTIVE, COMMERCIAL
LONG LIFE SCIENTIFIC USAGE, IT IS CRITICAL.
ALGORITHMS
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• SCALING FROM SMALL N TO LARGE N FREQUENTLY EXPOSES INEFFICIENT ALGORITHMS THAT ARE
OTHERWISE BENIGN.
• EMPIRICAL TESTING IS USEFUL BECAUSE IT MAY UNCOVER UNEXPECTED INTERACTIONS THAT AFFECT
PERFORMANCE.
• BENCHMARKS ARE USED TO COMPARE BEFORE/AFTER POTENTIAL IMPROVEMENTS TO AN ALGORITHM
AFTER PROGRAM OPTIMIZATION.
• EMPIRICAL TESTS CANNOT REPLACE FORMAL ANALYSIS, THOUGH, AND ARE NOT TRIVIAL TO PERFORM IN A
FAIR MANNER.
ALGORITHMS
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EXECUTION EFFICIENCY / ALGORITHMIC EFFICIENCY
• TO ILLUSTRATE THE POTENTIAL IMPROVEMENTS POSSIBLE EVEN IN WELL-ESTABLISHED ALGORITHMS, A
SIGNIFICANT INNOVATION, RELATING TO FAST FOURIER TRANSFORMS ALGORITHMS (FFT ARE USED HEAVILY
IN THE FIELD OF IMAGE PROCESSING), CAN DECREASE PROCESSING TIME UP TO 1,000 TIMES FOR
APPLICATIONS LIKE MEDICAL IMAGING.
• IN GENERAL, SPEED IMPROVEMENTS DEPEND ON SPECIAL PROPERTIES OF THE PROBLEM, WHICH ARE VERY
COMMON IN PRACTICAL APPLICATIONS.
• SPEEDUPS OF THIS MAGNITUDE ENABLE COMPUTING DEVICES, ( THAT MAKE EXTENSIVE USE OF IMAGE
PROCESSING LIKE DIGITAL CAMERAS AND MEDICAL EQUIPMENT) TO CONSUME LESS POWER.
ALGORITHMS
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TO SUM UP THE FOLLOWING PROPERTIES AND STEPS ARE ASSOCIATED WITH ALGORITHMS
1. STEP BY STEP SEQUENCE WHERE EACH STEP IS WELL DEFINED
2. CANNOT JUMP THE STEPS , LEAVE OR IGNORE THEM.
3. STEPS MAY BE LINEAR OR IN GROUPS OR RANDOMIZED
PROPERTIES OF ALGORITHMS
• FINITENESS
• PRECISELY DEFINED
• A GOOD INPUT
• A PRECISE OUTPUT
• EFFECTIVENESS
• UNAMBIGUOUS
ALGORITHMS
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EXAMPLES OF ALGORITHMS
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GWALIOR
EQUATIONS, PROBABILITIES & ALGORITHMS 36
WE END THIS
PRESENTATION WITH A
SMALL FILM ABOUT
NEURAL NETWORKS
FROM 3 BLUE 1 BROWN

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THANK YOU
SANJAY SRIVASTAVA
MITS GWALIOR
38

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