ACE ENGINEERING ACADEMY

THE ORIGINAL ACE ENGINEERING ACADEMY

**(ON-LINE EDUCATION, OUTSOURCING HRD & GATE/IES)**

gateguru@gateguru.org

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gateguru

**THE ROO-POOH-TIGGER STUDIES**

**1st March 2018**

NOT IN THE GATE SYLLABUS

*[The P=NP problem, NP-completeness, intractabililty and perhaps the whole of Computational Complexity has been deleted from the GATE syllabus. The present study is purely a hobbyist study and has yielded unexpected, interesting results.]*

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POOH 'ROPES' IN PENGUIN ON HIS 91st BIRTHDAY TO TACKLE

**INT****RAC TABILITY**

**AND**

**SPEED-UP THE ROO-POO-TIGGER STUDIES**

A HONORARY CONTRACT FOR NINE YEARS TILL POOH'S HUNDREDTH BIRTHDAY

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**PENGUIN'S DISCOURSE**

**( CHRISTMAS 2017-EASTER 2018)**

*THE FOUNDATION*

(The resonant humps/pillars of pairs of (a composite integer, a factor) in the Harmonic series/Euler zeta functions/Riemann zeta function)

Everyone with an average high school literacy or at least eighth grade literacy in Mathematics is expected to tackle for some time in ones life the ancient centuries old problems of integer factorisaton, tractability of NP-completeness and the Riemann Hypothesis. These easily stated and understood problems have evaded a simple practical, common sense explanation. These problems are ideally suited to the timeless, remote areas of the Global Village.

A problem of basic fundamental concern is the representation of integers by space requirements that are superior to the Arabic Positional Notation. This will pave the way for unlimted lossless data compression of data beyond a threshold in size. A method obtained requires a solution to the sum of subsets problem to recover the data from the compressed format. The method of data compression followed is to pass the given string through a deterministic controlled channel where the Shannon Limit does not apply. The string is bloated up enormously using error control coding techniques. Then it is collapsed to the representation of a refereed NP-complete problem.

The current GATE examination syllabus and examination pattern which concentrate on fundamental tools, concepts and methods are excellenct stimulants to study and tackle these problems with little or no resources and expenditure. The problems are so fundamental that solutions to them will help many branches of Science, Technology, Engineering and Mathematics (& Medicine)(STEM).

Over the last decade 2008-2017++, Pooh and his friends who are the most important citizens of New York and are also leading world citizens favouring simplicity and common sense in all things, have studied these problems from an unconventional, non-traditonal angle and have come to some surprising conclusions.

**To cut a long story short what Pooh and his friends say is that we can take a unified view of problems like integer factorsation, many cryptographic problems and in general many combinatorial decision/optimisation problems like NP-completeness. It is in effect the discovery of a SILVER BULLET. The solutions to an instance of these problems are the zeroes of a function known as the Lumpy function which is constructed by choosing a subset of the zeroes of the Riemann zeta function. These zeroes can be obtained by repeated use of binary search from the Lumpy function.**

The Lumpy function is based on a recasting in Arithmetic of the famous Fourier Transform. It uses weighted periodic integer factorisation functions rather than the traditional weighted periodic sine and cosine functions to represent periodic functions. The concept however remains the same with resonance occuring at the zeores which are the solutions to the problem being considered. A simple common sense explanation is that in a simulated Pooh-Tigger cake distribution race mandated by the Queen of Gaul, Pooh and Tigger distribute cakes to the houses in cities as per the Harmonic series. Tigger is always trying to be ahead of Pooh, in his usual aggressive behaviour. When Tigger is exactly(sic) half a house ahead of Pooh resonance occurs and a solution to the problem in terms of a zero of the Lumpy function occurs.

The basic magic is to use exhaustive search in the heap of exponential data of candidates for a solution to the combinatorial problem and reduce the problem to polynomial complexity by using the magic of the mysterious Euler gamma constant. The magic of the Euler gamma constant is that it allows the sum of an exponental number of terms of the Harmonic series to be determined in the polynomial time taken to compute natural logarithms. This was shown by Euler. The magic in exhaustive search being tamed lends itself to a simple common sense explanation. We are to find a steel needle in a haystack. The needle is heavy and the components of the haystack are light. We partition the haystack and weigh the partitions. The abnormally heavy partittion contains the needle. We repeat the process with this partition.

There is a simple and trivial way of explaining the Riemann Hypothesis. For every pair (composite integer, one of its factors) there exists a zero of the Riemann zeta function and it occurs when Tigger can balance himself exactly half a house ahead of Pooh in the cake distribution races.The 1/2 house comes in from the cumulative effect and cancelling effect of residues that occur as an integer is divided by an increasing/decreasing sequence of divisors exhaustively. Tigger should be ahead of Pooh by the same amount whether we use increasing divisors or decreasing divisors. So by symmetry Tigger will be exactly(sic) 1/2 a house ahead of Pooh at the factor or the zero of the zeta function. The basic idea is an exhaustive search for the factor based on an Arithmetic variant of the Fourier Transform that uses weighted Harmonic Series/Euler zeta functions/Riemann zeta function. Since there are an infinite number of pairs of composite integers and their factors we have an infinite number of zeroes of the Riemann zeta function. As integers become larger and larger the primes become sparser and sparser and the zeroes of the Riemann zeta function occur closer and closer to each other. When we shift from Analysis to Arithmetic and cosider the Harmonic Series/Euler zeta functions rather than the Riemann zeta function the same behaviour is repeated and exploited by Pooh and his friends to render NP-completeness tractable.

The crucial point to note is that there exist many smooth, continuous and monotonic paths to and from the zeroes of the Riemann zeta function and this monotonocity is what enables an efficient polynomial time complexity algorithm by the repeated use of binary search.

One may ask as to what computational platforms are needed by Pooh and his friends. Traditional methods to tackle the problems considered have used up fractions of the internet capacity, even many huge server farms, massively parallel wierd architectures. massive cloud computing, and fanciful little understood new computational platforms have been suggested. This is beacuse exponential or subexponential algorithms on traditional computers have been used.

Pooh and his friends fall back on simplicity and common sense and are satisfied with an engineering marvel like an used and junked by Corporates T-400 lenovo laptop with free software like a 32/64 operating system with Java. This can be obtained today at a very low price anywhere in the world. The 'blazingly fast' software needed like 'apfloat' can be downloaded free from the internet.

The major hurdle is ensuring uninterrupted power. In many parts of the world one can acquire and stay in a million dollar house but there will always be severe p0wer outages. One solution is solar based power. Another alternative is to use a car battery based gadget that takes over when there is power outage and gets recharged when power is present. Such a gadget can work reliably for 3-5 years.

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The polynomial hierarchy goes

Roo, Pooh & Tigger need not even subscribe to Cantor and his infinities they can make do with Kronecker's views of integers. They need not worry about the formal Turing Machine but only the halting variety. Somr cultures around the world find Cantor's 'bandying' about infinities almost a sacrilege!!!! The integer factorisation problem can be solved using elementary eight grade arithmetic given the Euler-Mascheroni constant!

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**— Edsger W. Dijkstra**

All the problems shown to be NP-complete have tractable solutions using the Pooh-Tigger cake distribution races. A NP-compete problem is basically a statement of Godel's lost letter. It turns out that to obtain tractable solutions to NP-complete problem we do not need Logic, we do not need Cantor and we do not need the Turing Machine. We can make do with Kronecker's view of the integers. We construct for each instance of a NP-complete problem a selector Lumpy function which has zeroes selected from the zeroes of the Riemann Zeta function. (In fact we do not need something as complex as the Riemann zeta function as the plain vanilla Harmoic series suffices. It has a behaviour of humps and resonances occuring which are similar to those found with the Riemann zeta function). A solution occurs whenever Tigger is exactly(sic) 1/2 a house ahead of Pooh and this is a zero of the Lumpy function and the corresponding Riemann zeta function. The zeroes of the Lumpy function can be obtained by iterative binary search and this makes NP-completeness tractable.

The existence of closed form formulas for Harmonic numbers allows exponetial time complexity to the reduced to polynomial time complexity. In a way, loosely speaking, the exponential and polynomial seem separted asymptotically by a constant i.e. the famous mighty Euler gamma constant!!!

This allows Pooh and his frieds to study combinatorial decision problems without using Permutations and Combinations as using them leads to severe circular reasoning, unending Tom and Jerry shows and exponential time complexity. Pooh and friends prefer to use exhaustive search and find 'the needles in the haystack'.

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The smooth, continuous and monotonic paths to the zeroes of the Riemann zeta fuction and all the Lumpy selector functions.

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The smooth, continuous and monotonic paths to the zeroes of the Riemann zeta fuction and all the Lumpy selector functions.

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**WHAT HAPPENED TO INTEGER FACTORISATION AND NP-COMPLETENESS OVER THE LAST**

**50 YEARS?**

THEY HAVE BEEN VOCIFEROURSLY, CONSTANTLY AND CONSISTENTLY DECLARED TO BE IN 'GOEBBELS STYLE'

INTRACTABLE!!!

**STRATEGY TO DEMOSTRATE TRACTABILITY**

*REDO PURE COMBINATORIAL DECISION/OPTIMISATION PROBLEMS WITHOUT PERMUTATIONS AND COMBINATIONS BUT USING ONLY EXHAUSTIVE SEARCH BASED ON 'ITERATIVE' BINARY SEARCH AND 'HEY PRESTO' INTEGER FACTORISATION AND NP-COMPLETENESS BECOME EASILY TRACTABLE!!!!*

*FOR EXHAUSTIVE SEARCH USE THE 'VANILLA' HARMONIC SERIES, EULER ZETA FUNCTIONS OR THE RIEMANN ZETA FUNCTION!!! THE MAGIC OCCURS WHEN HARMONIC NUMBERS GIVING THE SUM OF AN EXPONENTIAL NUMBER OF**TERMS OF A SERIES CAN BE EVALUATED IN POLYNOMIAL TIME AS IN THE CASE OF THE MIGHTY EULER GAMMA CONSTANT OR APERY'S CONSTANT!!!!!*

*THE REST OF THE MAGIC IS BASED ON THE VALIDITY OF THE RIEMANN HYPOTHESIS BASED ON THE POOH-TIGGER CAKE DISTRIBUTION RACES USING THE 'VANILLA' HARMONIC SERIES ONLY RATHER THAN THE RIEMANN ZETA FUNCTION.*

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"EXTRACT FROM THE MEMEX"

"Methods to calculate the sum of the first n positive integers, the sum of the squares and of the cubes of the first n positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity which considered this problem were: Pythagoras (c. 572–497 BCE, Greece), Archimedes (287–212 BCE, Italy), Aryabhata (b. 476, India), Abu Bakr al-Karaji (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham (965–1039, Iraq).

The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B0, B1, B2,… which provide a uniform formula for all sums of powers (Knuth 1993).

Bernoulli's result was published posthumously in *Ars Conjectandi* in 1713. Seki Kōwaindependently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712. However, Seki did not present his method as a formula based on a sequence of constants."

"END OF EXTRACT FROM THE MEMEX"

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For the purposes of integer factorisation and tractability of NP-completeness it is to be noted that the great Newton suspected the existence of the Euler gamma constant but it was formalised a century later by Euler. The existence of the Euler ganmma constant combined with the validity of the Riemann Hypothesis leads to the tractability of integer factorisation and NP-completeness.

Lorenzo Mascheroni

Italian mathematician

Ernesto Cesàro

Italian mathematician

Euler

NOTE

Following Kronecker rather than Cantor Pooh and his friends restrict the precision of the numbers they consider given an integer n by stating and enforcing a bound. This restricts all the above formulas to have only a finite number of terms. Also the algorithms turn out to be merely exercises in Numerical Methods. In particular arithmetic involving gigantic numbers is to be computed to obtain a vanishingly small number in the range 0..1/2 giving the position of Tigger closest to the peak of a hump and standard Numerical Methods techniques are used effectively. Also the required solutions come approximated digit by digit from the most significat digit to the least significant digit with the residual fraction discarded.

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**THE MYSTERIOUS CONSTANT THAT RELATES THE EXPONENTIAL TO THE POLYNOMIAL .**

THE MYSTERIOUS CONSTANT THAT IN THE UNIVERSE SEPARATES MATTER GIVEN BY THE PAIR(COMPOSITE INTEGER, ONE OF ITS FACTORS) FROM SPACE(PRIMES)

Euler-Mascheroni Constant = 0.5772156649015328606065120900824024310421593359399235988057672348848677267776\ 646709369470632917467495146314472498070824809605040144865428362241739976449235\ 362535003337429373377376739427925952582470949160087352039481656708532331517766\ 115286211995015079847937450857057400299213547861466940296043254215190587755352\ 673313992540129674205137541395491116851028079842348775872050384310939973613725\ 530608893312676001724795378367592713515772261027349291394079843010341777177808\ 815495706610750101619166334015227893586796549725203621287922655595366962817638\ 879272680132431010476505963703947394957638906572967929601009015125195950922243\ 501409349871228247949747195646976318506676129063811051824197444867836380861749\ 455169892792301877391072945781554316005002182844096053772434203285478367015177\ 394398700302370339518328690001558193988042707411542227819716523011073565833967\ 348717650491941812300040654693142999297779569303100503086303418569803231083691\ 640025892970890985486825777364288253954925873629596133298574739302373438847070\ 370284412920166417850248733379080562754998434590761643167103146710722370021810\ 745044418664759134803669025532458625442225345181387912434573501361297782278288\ 148945909863846006293169471887149587525492366493520473243641097268276160877595\ 088095126208404544477992299157248292516251278427659657083214610298214617951957\ 959095922704208989627971255363217948873764210660607065982561990102880756125199\ 137511678217643619057058440783573501580056077457934213144988500786415171615194\ 565706170432450750081687052307890937046143066848179164968425491504967243121837\ 838753564894950868454102340601622508515583867234944187880440940770106883795111\ 307872023426395226920971608856908382511378712836820491178925944784861991185293\ 910293099059255266917274468920443869711147174571574573203935209122316085086827\ 558890109451681181016874975470969366671210206304827165895049327314860874940207\ 006742590918248759621373842311442653135029230317517225722162832488381124589574\ 386239870375766285513033143929995401853134141586212788648076110030152119657800\ 681177737635016818389733896639868957932991456388644310370608078174489957958324\ 579418962026049841043922507860460362527726022919682995860988339013787171422691\ 788381952984456079160519727973604759102510995779133515791772251502549293246325\ 028747677948421584050759929040185576459901862692677643726605711768133655908815\ 548107470000623363725288949554636971433012007913085552639595497823023144039149\ 740494746825947320846185246058776694882879530104063491722921858008706770690427\ 926743284446968514971825678095841654491851457533196406331199373821573450874988\ 325560888873528019019155089688554682592454445277281730573010806061770113637731\ 824629246600812771621018677446849595142817901451119489342288344825307531187018\ 609761224623176749775564124619838564014841235871772495542248201615176579940806\ 296834242890572594739269638633838743805471319676429268372490760875073785283702\ 304686503490512034227217436689792848629729088926789777032624623912261888765300\ 577862743606094443603928097708133836934235508583941126709218734414512187803276\ 150509478055466300586845563152454605315113252818891079231491311032344302450933\ 450003076558648742229717700331784539150566940159988492916091140029486902088485\ 381697009551566347055445221764035862939828658131238701325358800625686626926997\ 767737730683226900916085104515002261071802554659284938949277595897540761559933\ 782648241979506418681437881718508854080367996314239540091964388750078900000627\ 997942809886372992591977765040409922037940427616817837156686530669398309165243\ 227059553041766736640116792959012930537449718308004275848635083808042466735093\ 559832324116969214860649892763624432958854873789701489713343538448002890466650\ 902845376896223983048814062730540879591189670574938544324786914808533770264067\ 758081275458731117636478787430739206642011251352727499617545053085582356683068\ 322917676677041035231535032510124656386156706449847132695969330167866138333333\ 441657900605867497103646895174569597181553764078377650184278345991842015995431\ 449047725552306147670165993416390660912054005322158902091340802782251533852899\ 511665452245869185993671220132150144801424230986254604488672569343148870491593\ 044640189164502022405495386291847586293077889350643771596606909604681243702305\ 465703160679992587166675247219409777980186362625633582526279422393254860132693\ 530701388937436923842878938512764740856548650281563067740442203064403756826309\ 102917514572234441050369317711452170888907446416048688701083862311426128441425\ 960956370400619200579335034155242624026206465693543061258526583452192121497771\ 878069586608516334922104836737994592594340379560002192785418379417760203365594\ 673078879838084816314678241492354649148876683368407492893865281863048589820354\ 818624383848175997635849075180791480634943916284705482200754945348986133827235\ 730922190030740096800337666844932505567654937530318112516410552492384077645149\ 842395762012781552322944928854557853820248918942441857095919558208100071578384\ 039627479985817880888865716830699436060735990421068511427913169699596792300828\ 988156097538338059109360341252998656790389568795673455083362907823862638563490\ 747319275278740166557531190111543470018186256971261120126852923129937161403906\ 965112224816615082353643982396620532633322248505191593682690715004315589871802\ 783353845448309107249498057880961717996337167036554180041464667538719586948483\ 331543583330641935929487420951478832347748481418149776871694413640056645156936\ 116524161555734141935424721373067468333849054426626038372788217552709930958141\ 026136979500786465876771608630804460749802801576962675913897794772214337515470\ 829345879123898433055067223474969984942486706721502569273529585065869588997486\ 535562186958043997125168976654169862653862891977542187721939605817001104236414\ 158780810386172101557551923711160049880682291618097732421958328974869227183979\ 190467716542668138893379296036815457939611339621922245430151580631743708405608\ 536416031384982969518566952612822123716939368130321296561939718710207098007948\ 833910197535104307441823448833331796978277332091143324514305086573457500687391\ 475470777577559918467118308583660159437193718449039061770232536567977596744475\ 747511584195746700997345002454428406585024508585646392791246119879093693072019\ 804029303603738838430742162821201635386466226097198958436799430572030149638050\ 832232365825557724534237187737439818333306454662906993311125973721950274646899\ 065457155440303917835419756434315739034883866750542742161831050060550464223545\ 708427393549359051762717479299472398908632970101905610107742690926475235740304\ 630159243442464900834188630859320685522507790910195858895314328799817570981916\ 829315940453005632543314488517357302698256937253469964013440871580108145287865\ 790408663637945071108505104241797691911292615132010316363498086606948624407800\ 668400671696221463718114777268341846646364242734053003138077349611998146861768\ 585463120816316479893796426373835661893831371098328956490521148813402974238886\ 863154313297876579912545424333856347200268129048994955042698088213026726358153\ 248067538790323057421040330149788786752377860705468861472100992632942510887801\ 970284117922402591091466584809257857192786282147667074087863519714256292427867\ 028407703241437569931883243331559002433304769111009247979118006286202213707800\ 621725732904735994398883139279927969397063567628116694054128859081982023838277\ 035483496879734048888293016736770941584654400954862465146101353913496855912040\ 236361872150992980651905861682815302875042754525860533196343259577747881343723\ 939499124380614375449859068607518563142725525564259396701498041425981823785257\ 682943639596562438852065654807103884546394453770191784571874101186223227802525\ 194362657438242256093567692582387749116073775945140144703190224153559112506138\ 178297421264982641618724606313340891926702359795802365841631755679233566210123\ 133584549459059006998420067226025116774384736482438571540714626594564239112717\ 078030637141692638644010057131095896063264963755295676936468941051795200061645\ 202188435340473018243930514881984593076296404445687762416528716207276731860632\ 540801428874571198657307471701886603687970364770854852871670003622928528837468\ 246605881411754047446061676354303739923756596593696708792316774468569310838210\ 783048315919643002144125970228906320317410114936648095290301171633453191792293\ 924242877283787234956992923213609223494722645824375509451533552011761289751733\ 951371782933287158609438662701179184155458726489825139255594379519731786744876\ 992532617942338129994127939860264424519600605436818664670986594436593015437629\ 148697959769499653352721002002096791043948254724411334224487005463765684086676\ 215337362746159120547008629057169825735370523976123123841256434941178949861582\ 859702097109703919635216812258224756271951938272828520914718237534365525402074\ 620306673047695247009441381456178282666319613967359367257264603388494642489472\ 448978481596715306153846712236987128231978476931105716623637326207595971180401\ 514389623170601958570982381392466613527913695637655904861676205129740998314965\ 597860253410129454399867288300624408440186181511750687088764671609799292017769\ 624963301575829259618849485872002292248060628812177873383145882551293953651088\ 191865120044923154983947731473578688973142047310945381464822632102331079439597\ 462852354129575191063558792300195618131294612030157576340159785681751749842377\ 882447375398247457599686770740854557432942402678193848220354096210606072192499\ 082510485400318496319986322156908909761405310716511312932685349852440286448293\ 470591608586995981988903959955918110764134466588145254256588153545028884739975\ 32432780902152256952184414529865083551298398022

6238264974881811581525034599749..............+++++++++++++++

GATEGURU.ORG

HYDERABAD, TELANGANA 500076

ph: 914027174000

alt: 09052345550

gateguru