6:34 PM (2 minutes ago) to me Someone just viewed: “Re: Someone just viewed: LESSONS IN NUMBER THEORY REGARDS 324368521 AND 6433361467 ETCETRATA ND” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: iPhone Location: Quincy, WA

FIRST

LESSONS IN NUMBER THEORY REGARDS 324368521 AND 6433361467 ETCETRATA ND

LAST

LESSONS IN NUMBER THEORY REGARDS 324368521 AND 6433361467 ETCETRATA NDCLICK LINK BELOW

https://www.google.com/search?q=%22LESSONS+IN+NUMBER+THEORY+REGARDS+324368521+AND+6433361467+ETCETRATA+ND%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjR9dPx4vLrAhUTeysKHR-zDXkQ_AUoBHoECAUQBg&biw=1536&bih=758

4:40 PM (1 hour ago) to me Someone just viewed: “324368521 OLGA CLINTON SHAKESPEARE VERSUS KALIDAS MOVIE MEANS ?” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: Unknown Device Location: Boydton, VA

324368521 OLGA CLINTON SHAKESPEARE VERSUS KALIDAS MOVIE MEANS

CLICK LINK BELOW

https://www.google.com/search?q=:+%22324368521+OLGA+CLINTON+SHAKESPEARE+VERSUS+KALIDAS+MOVIE+MEANS+%3F%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwiv6bzk2fLrAhVObn0KHbfjCu8Q_AUoAnoECAQQBA&biw=1536&bih=692

5:14 PM (7 minutes ago) to me Someone just viewed: “NAMASTE MOSCOW -> PRIVET -> THIS EQUAL TO HAS – RUSHSIA – ON THE TOP RIGHT SEE -> https://i2.wp.com/yoninetanyahu.com/wp-content/uploads/2018/01/442da-68aad-olga2bshulamn2blednichenko2bajay2bmishra2b3243685212bmovie.jpg” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: PC Location: Chicago, IL

PRIVET MSCOW HWZ 6 AND 9

NAMASTE MOSCOW PRIVET THIS EQUAL TO HAS - RUSHSIA - ON THE TOP RIGHT SEE -> https://i2.wp.com/yoninetanyahu.com/wp-content/uploads/2018/01/442da-68aad-olga2bshulamn2blednichenko2bajay2bmishra2b3243685212bmovie.jpg

[A]

NAMASTE MOSCOW PRIVET THIS EQUAL TO HAS - RUSHSIA - ON THE TOP RIGHT SEE

[B]

NAMASTE MOSCOW PRIVET THIS EQUAL TO HAS - RUSHSIA - ON THE TOP RIGHT SEE

[C]

NAMASTE MOSCOW PRIVET THIS EQUAL TO HAS - RUSHSIA - ON THE TOP RIGHT SEE -> https://i2.wp.com/yoninetanyahu.com/wp-content/uploads/2018/01/442da-68aad-olga2bshulamn2blednichenko2bajay2bmishra2b3243685212bmovie.jpg

[D]

NAMASTE MOSCOW PRIVET THIS EQUAL TO HAS - RUSHSIA - ON THE TOP RIGHT SEE -> https://i2.wp.com/yoninetanyahu.com/wp-content/uploads/2018/01/442da-68aad-olga2bshulamn2blednichenko2bajay2bmishra2b3243685212bmovie.jpg

[5]

NAMASTE MOSCOW PRIVET THIS EQUAL TO HAS - RUSHSIA - ON THE TOP RIGHT SEE -> https://i2.wp.com/yoninetanyahu.com/wp-content/uploads/2018/01/442da-68aad-olga2bshulamn2blednichenko2bajay2bmishra2b3243685212bmovie.jpg`CLICK LINK BELOW

https://www.google.com/search?q=NAMASTE+MOSCOW+-%3E+PRIVET+-%3E+THIS+EQUAL+TO+HAS+-+RUSHSIA+-+ON+THE+TOP+RIGHT+SEE+-%3E+https://i2.wp.com/yoninetanyahu.com/wp-content/uploads/2018/01/442da-68aad-olga2bshulamn2blednichenko2bajay2bmishra2b3243685212bmovie.jpg&source=lnms&tbm=isch&sa=X&ved=2ahUKEwi-naG80fLrAhU4wjgGHYv_AGIQ_AUoAnoECAwQBA

 

 

PRIVET MSCOW HWZ 6 AND 9

Someone just viewed: Fwd: Google Alert – international cricket stadium lucknow Inbox x BIG SHOT FROM CHICAGO x Streak 3:46 PM (0 minutes ago) to me Someone just viewed: “Fwd: Google Alert – international cricket stadium lucknow” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: PC Location: Chicago, IL

INTERNATIONAL CRICKET STADIUM. LUCKNOWCLICK LINK BELOW

https://www.google.com/search?q=%22%3A+Google+Alert+-+international+cricket+stadium+lucknow%22&tbm=isch&ved=2ahUKEwj9-NCXvfLrAhUPQn0KHSR5CwUQ2-cCegQIABAA&oq=%22%3A+Google+Alert+-+international+cricket+stadium+lucknow%22&gs_lcp=CgNpbWcQA1CcG1jgHWDPImgAcAB4AIABkQGIAZsDkgEDMC4zmAEAoAEBqgELZ3dzLXdpei1pbWfAAQE&sclient=img&ei=fYlkX73GMo-E9QOk8q0o&bih=758&biw=1536

3:29 PM (16 minutes ago) to me Someone just viewed: “Fwd: Someone just viewed: 643361467 SO BUBBA HOW IS SAAD HARIRI ?” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: PC Location: Chicago, IL

643361467 SO BUBBA HOW IS SAAD HARIRI --CLICK LINK BELOW

https://www.google.com/search?q=%22643361467+SO+BUBBA+HOW+IS+SAAD+HARIRI+%3F%22&tbm=isch&ved=2ahUKEwiN1duUvPLrAhUFV30KHYgADhsQ2-cCegQIABAA&oq=%22643361467+SO+BUBBA+HOW+IS+SAAD+HARIRI+%3F%22&gs_lcp=CgNpbWcQA1DbNVjbNWDPOmgAcAB4AIABcYgBcZIBAzAuMZgBAKABAaoBC2d3cy13aXotaW1nwAEB&sclient=img&ei=a4hkX83zEIWu9QOIgbjYAQ&bih=758&biw=1536

1:43 PM (5 minutes ago) to me Someone just viewed: “Fwd: Someone just viewed: Fwd: Someone just viewed: Fwd: Someone just viewed: FROM SANJAY DUTT TO OLGA SHULMAN LEDNICHENKO -> ORANGE BACKLESS CHOLI IN RUSSIA AND SHORT RED AND WHITE DRESS IN AFGHAN MEANS ?” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: iPhone Location: Quincy, WA

FIRST

FROM SANJAY DUTT TO OLGA SHULMAN LEDNICHENKO ORANGE BACKLESS CHOLI IN RUSSIA AND SHORT RED AND WHITE DRESS IN AFGHAN MEANS

MIDDLE

FROM SANJAY DUTT TO OLGA SHULMAN LEDNICHENKO ORANGE BACKLESS CHOLI IN RUSSIA AND SHORT RED AND WHITE DRESS IN AFGHAN MEANS

LAST

FROM SANJAY DUTT TO OLGA SHULMAN LEDNICHENKO ORANGE BACKLESS CHOLI IN RUSSIA AND SHORT RED AND WHITE DRESS IN AFGHAN MEANSCLICK LINK BELOW

https://www.google.com/search?q=%22FROM+SANJAY+DUTT+TO+OLGA+SHULMAN+LEDNICHENKO+-%3E+ORANGE+BACKLESS+CHOLI+IN+RUSSIA+AND+SHORT+RED+AND+WHITE+DRESS+IN+AFGHAN+MEANS+%3F%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjbkMPLofLrAhXOF3IKHcdEAREQ_AUoAXoECAQQAw&biw=1536&bih=758

1:08 PM (26 minutes ago) to me Someone just viewed: “OLGA AND OBAMA DINNNER WITH GANDHI – YEAH BARACK SAID – IT MIGHT BE GANDHI – HE DIDNT SAY IT SHOULD OR WOULD BE GANDHI” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: Unknown Device Location: Boydton, VA

DINNER WITH GANDHIOLGA AND OBAMA DINNNER WITH GANDHI - YEAH BARACK SAID - IT MIGHT BE GANDHI - HE DIDNT SAY IT SHOULD OR WOULD BE GANDHI

https://www.google.com/search?q=OLGA+AND+OBAMA+DINNER+WITH+GANDHI+-+YEAH+BARACK+SAID+-+IT+MIGHT+BE+GANDHI+-+HE+DIDN%27T+SAY+IT+SHOULD+OR+WOULD+BE+GANDHI%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwiz3Kupn_LrAhXEAnIKHVOlBI4Q_AUoAXoECA4QAw&biw=1536&bih=758

Streak 12:47 PM (39 minutes ago) to me Someone just viewed: “MORPHOLOGY OF OLGA AND THAT NUMBER 3243686521” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: Android Tablet Location: Tel aviv-yafo, TA

CLICK LINK BELOW

https://www.google.com/search?q=:+%22MORPHOLOGY+OF+OLGA+AND+THAT+NUMBER+3243686521%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjhkvX6iPPqAhVz4jgGHYxcAxcQ_AUoAnoECAQQBA&biw=1536&bih=758

12:19 PM (58 minutes ago) to me Someone just viewed: “SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING – NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY : -> S FOR SOUL OR S FOR STRATEGY ? OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTIN ” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: iPhone Location: Quincy, WA

[1]

SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING - NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY   S FOR SOUL OR S FOR STRATEGY  OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTIN

[2]SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING - NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY   S FOR SOUL OR S FOR STRATEGY  OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTIN

[3]SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING - NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY   S FOR SOUL OR S FOR STRATEGY  OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTIN

[4]

SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING - NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY   S FOR SOUL OR S FOR STRATEGY  OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTIN

[5]

SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING - NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY   S FOR SOUL OR S FOR STRATEGY  OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTIN

[6]

SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING - NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY   S FOR SOUL OR S FOR STRATEGY  OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTIN

[7]

SINCE THIS IS THE DIWLAI SEASONA DN HANUKKAH IS COMING - NOT JUST BBIRTHDAYAND X MAS AND NEW YEAR AND BIRTHDAY   S FOR SOUL OR S FOR STRATEGY  OR S FOR SPIRITUAL S FOR SANSKIRT : I MEAN I ASK BECAUSE I AM A 3RD CWORLD CALL CENTER SERVIVE PROVIDER SHIT FROM A DEPICIT CPLACE CALELD LUCKNOW WHO EVEN WENT TO HOPE NOT JUSTS NEW YORK , MOSCOW AND AUSTINCLICK LINK BELOW

https://www.google.com/search?q=SINCE+THIS+IS+THE+DIWLAI+SEASONA+DN+HANUKKAH+IS+COMING+-+NOT+JUST+BBIRTHDAYAND+X+MAS+AND+NEW+YEAR+AND+BIRTHDAY+:+-%3E+S+FOR+SOUL+OR+S+FOR+STRATEGY+%3F+OR+S+FOR+SPIRITUAL+S+FOR+SANSKIRT+:+I+MEAN+I+ASK+BECAUSE+I+AM+A+3RD+CWORLD+CALL+CENTER+SERVIVE+PROVIDER+SHIT+FROM+A+DEPICIT+CPLACE+CALELD+LUCKNOW+WHO+EVEN+WENT+TO+HOPE+NOT+JUSTS+NEW+YORK+,+MOSCOW+AND+AUSTIN+&source=lnms&tbm=isch&sa=X&ved=2ahUKEwijlpDLmPLrAhWWe30KHRLvBgIQ_AUoAXoECA4QAw&biw=1536&bih=758

12:52 PM (5 minutes ago) to me Someone just viewed: “Fwd: A life you can’t make up:” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: PC Location: Chicago, IL

A LIFE YOU CANT MAKE UPCLICK LINK BELOW

https://www.google.com/search?q=%22Fwd:+A+life+you+can%27t+make+up:%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwiBm7ymlvLrAhXVeisKHY60BBsQ_AUoAXoECAQQAw&biw=1536&bih=758

10:19 PM (2 minutes ago) to me Someone just viewed: “C FOR COMLEECATED C FOR COMPLECATED CASABLACA NUMBERS C FOR CLINTON TRINAGLE CIRCLES AND COMLEIACTED GEMORETY WHICH MEANS NOT ALGEBRA NBUT NUMBERS – MENAS AROTHEMTIC – REGARDS AJY FROM INSEAD IVEC CAHTEAU POLEONE, EN PAIRS, EN + 33 OL IL DE FRANCEZ: CC: EMANUEL MACRON AND MEANS ALOS EAMNUEL MICHAI THE ATTOENY – OF AJAY MISHRA – FROM INSEAD, PARIS, AND PARIS IS IN FRANCE MFRANCE MEANS + 33 AND PRIS MEANS 06” People on thread: 208 Westhaven Drive 78746 Blog Post By Email Device: PC Location: Chicago, IL

LETS SAY U WANT TO BE HIP AND COOL.. U KNOW HOW CLASSES EXIST, THAT WE CANT ADMIT, BECAUSE IT SUND LIKE HINDU CASTE SYSTEM, BUT THE FACT IS THAT CLASSES DO EXIST – FR EXAMPLE ? WELL FR EXAMPLE , LETS SAY – U WANT TO SHMOOZE WITH THE INTELELCTUALS – CLASS, THEN – WHATS THE TOPCIS ? ATTHSOE COCKTAILS – NO ITSNT KIM JARADAHSIANS OR THE KAHSN – BUT ITS SMETHING LIKE PICASSOO, MONET, KANIDSKY, BRANCUSSI – AND STUFF FOR ONE CLASS – AND SIMULATEMTAOUSLY TOPCIS LIKE MONOTHEISM, POLYTHEISM, MNISM, – U CAN FIND BEING TALKED IN – THOSE ELTE COCTAIL PARTIES, AND – YES, OF COUSRE, – BRACUSSSI AND – CHOPIN AND – MARKISKY AND LA OPERA AND – JEAN PAUL SATRRE, MC CCHAGALL, AND SO ON- SMILIALRY UMAY HEAR STUFF – LIKE – ARB SPING, KOHN LOCKE, – FUKUYANA HNTINGTON, JOHN MERSHEMIER, AND SO ON- ANDIN SE PARTIES U MAY HEAR FIRERICH HAYEK – JOHN MAYYARD KEYNES – PAULKRUGMAN – GSUNDAR PICHAI , GOGLE APPLE, IBM DELL AND GREG MANKIW, AND SIN SOME CIRCLES, U MAY HEAR WORDS LIKE CLITON – REGULATION DEREGULATIN AND REREGULATIN – FOLWOEED – BY FDR – AND – REAGAN AND – ENNEDY AND KRUSCHEV NAD IN SOME CIRCLES U MAY HEAR – WORDS, LIKE BUDDHISM, GANDHI -GANJA, GOA, – AND IN SOME CIRCLES U MAY HEAR BAMA MODI, TRUMP – AND PUTIN AND NETANYAHU – AND IRON DOME- BTW HOW IS IRON DOME NOW ? AND IN SOME CIRCLES U MAY HEAR – AND THERE – CIRCLES BTW ARE HIGEHR LEVELS OF MASLWO – THEY DONTHAVE TO WORRY ABOUT WHOS GONNA PAY THE MORTGAGA EBABY – SO, IN SE CIRCLES, U MAY HEAR – JOHN PAUL SATRE, – NEZIZHE, KASNIDKSY, CHOPIN – TSCHOKOPSCY, AND NAM CHMOMSLY :) ND ? U KNOW – DADISM, AND IMRESSIONISM AND DOCLHE GANAFBNA AND GUCCI – AND SIN SOME CIRCLES U MAY HEAR – WORDS LIKE SAND HILL, ROAD, VENCTURE CAPITAL – RETURNS, – AND VINOD KHOSLA, – AND SERGIE BRIN AND LARY PAGE AND – FRED – WHATS HIS ANEM? I FOREGOT – HIS ANEM- – ITS BEEN A WHILE- AND YEAH – KELINERPERKINS – AND STANFRD AND PALO ALT – AND YURI MILNER – BTW VALDIMRIIS YURI MILNER D STILL HAS FALTIN PETERSBURH -? AND IN SME CIRCLES, U MAY HEAR – FAMILY VALUES, TONY PERKINS, NETANYAHU, ISRAEL, BIBLE, AND THE 10 COMAMNDMENTS AND – BAR ILAN UNIEVRSITY AND SO ON – ANDIN SOE CIRCLES,U MAY HEAR – GOD, GUNDS AND GLORSY AND IN SOME CRCLES, UMAY HEAR WORDS – LIEK AKSPAROV – AND HERISTAGE FUNDATE AND ADELSON AND MACAU AND – SO ON -? – K- LET ME – INTRIDUCCE U TO AWORLD, WHICH SPEAKS LESS – BUT – FOR THIS WORLD, LIEK – I MENA FR EXAMPLE IF U MEET SUNDAR PICHAI R ANJALI PICHAI AND VINDO KHOSLA WIFE AND DAVID AND HIS WIFE – AND – THIS CIRCLE DOES EXIST AND PAULKRUGMAN MAY GET THIS ONE -? BTW HOW IS U ? ->SO – ? OTHERTHAN KUN LAW AND PHILIPS, CURCE AND DOCTR FACUCI AND RIBONUCLEAIC AID – AND MICHICHAN PURAHSING INDEX, – THERS S EXISTS – THIS CIRCLE – AS WELL -> THIS -> What is Riemannian Geometry? A description for the nonmathematician.

What is Riemannian Geometry? A description for the nonmathematician.

Euclidean Geometry is the study of flat space. Between every pair of points there is a unique line segment which is the shortest curve between those two points. These line segments can be extended to lines. Lines are infinitely long in both directions and for every pair of points on the line, the segment of the line between them is the shortest curve that can be drawn between them. Furthermore, if you have a line and a point which isn’t on the line, there is a second line running through the point, which is parallel to the first line (never hits it). All of these ideas can be described by drawing on a flat piece of paper. From the laws of Euclidean Geometry, we get the famous theorems like Pythagorus’ Theorem and all the formulas you learn in trigonometry, like the law of cosines. In geometry you also learned how to find the circumference and area of a circle.

Now, suppose instead of having a flat piece of paper, you have a curved piece of paper. You might have a cylinder, or a sphere. You can use a cardboard paper towel roll to study a cylinder and a globe to study a sphere. A shortest curve between any pair of points on such a curved surface is called a minimal geodesic. You can find a minimal geodesic between two points by stretching a rubber band between them. The first thing that you will notice is that sometimes there is more than one minimal geodesic between two points. There are many minimal geodesics between the north and south poles of a globe. We can also look for lines, which are curves like the ones in Euclidean space such that between every pair of points on the line, the segment between them is a minimal geodesic. There are no lines on a sphere! Every time you try to extend a minimal geodesic it starts to wrap around and it isn’t a minimal geodesic anymore. On a cylinder, some minimal geodesics can be extended to lines but most of them start to wrap around the cylinder and cannot be extended. Surfaces like these are harder to study than flat surfaces but there are still theorems which can be used to estimate the length of the hypotenuse of a triangle, the circumference of a circle and the area inside the circle. These estimates depend on the amount that the surface is curved or bent. One of the basic topics in Riemannian Geometry is the study of curved surfaces.

An important tool used to measure how much a surface is curved is called the sectional curvature or Gauss curvature. It can be computed precisely if you know Vector Calculus and is related to the second partial derivatives of the function used to describe a surface. To study the sectional curvature of a surface at a given point, you first find the tangent plane to the surface at that point. If you can find a small piece of the surface around the given point which only touches the tangent plane at that point, then the surface has positive or zero sectional curvature there. For example, a paraboloid or a sphere has positive sectional curvature at every point. If it is not possible to find a small piece of the surface which fits on one side of the tangent plane, then the surface has negative or zero curvature at the given point. This happens around the neck of a one-sheeted hyperboloid and on points where the surface looks like a saddle. If you use the precise formula to compute the sectional curvature of a point on a plane or a cylinder, then you will discover that these surfaces have exactly zero curvature everywhere.

In Vector Calculus you are also taught how to measure surface area using double integrals. Sometimes when you compute double integrals you use a change of variables and a Jacobian. These techniques are used regularly by Riemannian Geometers.

Riemannian Geometers also study higher dimensional spaces. The universe can be described as a three dimensional space. Near the earth, the universe looks roughly like three dimensional Euclidean space. However, near very heavy stars and black holes, the space is curved and bent. There are pairs of points in the universe which have more than one minimal geodesic between them. The Hubble Telescope has discovered points which have more than one minimal geodesic between them and the point where the telescope is located. This is called gravitational lensing. The amount that space is curved can be estimated by using theorems from Riemannian Geometry and measurements taken by astronomers. Physicists believe that the curvature of space is related to the gravitational field of a star according to a partial differential equation called Einstein’s Equation. So using the results from the theorems in Riemannian Geometry they can estimate the mass of the star or black hole which causes the gravitational lensing.

Like most mathematicians, Riemannian Geometers look for theorems even when there are no practical applications. The theorems that can be used to study gravitational lensing are much older than Einstein’s Equation and the Hubble telescope. We expect that practical applications of our theorems will be discovered some day in the future. Without having mathematical theorems sitting around for them to apply, physicists would have trouble discovering new theories and describing them. Einstein, for example, studied Riemannian Geometry before he developed his theories. His equation involves a special curvature called Ricci curvature, which was defined first by mathematicians and was very useful for his work. Ricci curvature is a kind of average curvature used in dimensions 3 and up. In Linear Algebra you are taught how to take the trace of a matrix. Ricci curvature is a trace of a matrix made out of sectional curvatures.

One kind of theorem Riemannian Geometers are looking for today is a relationship between the curvature of a space and its shape. For example, there are many different shapes that surfaces can take. They can be cylinders, or spheres or paraboloids or tori, to name a few. A torus is the surface of a bagel and it has a hole in it. You could also stick together two bagels and get a surface with two holes. How many holes can you get? Certainly, as many as you want. If you string together infinitely many bagels then you will get a surface with infinitely many holes in it. Now suppose you make a rule about how the surface is allowed to bend. If a surface must always bend in a rounded way (like a sphere) at every point, then we say it has positive curvature. A paraboloid has positive curvature and so does a sphere. A cylinder doesn’t and neither does a torus (look inside the hole to see it bends more like a saddle). There is a theorem which says that if a surface has positive curvature then it cannot have any holes.

A conjecture is a suggestion of a possible theorem which has not yet been proven. In 1969, Milnor stated a conjecture about spaces with positive Ricci curvature. He conjectured that such a space can only have finitely many holes. I am working on trying to find a proof for this conjecture and so are many other Riemannian Geometers. So far there are some partial results. Professors Schoen and Yau showed that 3 dimensional spaces with positive Ricci curvature have no holes at all. On the other hand, Professor Wei has constructed higher dimensional spaces with positive Ricci curvature and many holes, just not infinitely many holes. She doesn’t actually build a model with her hands; she describes the spaces explicitly with formulas similar to the way one can describe a globe with an atlas full of maps. The distances between the grid lines are described with formulas and then she does a lot of calculus to compute the Ricci curvature and make sure it is positive. Professors Li, Anderson, and Wilking also have theorems which are related to this conjecture but don’t quite prove the conjecture itself. I have proven one theorem which is related to the conjecture. Our theorems can be used as building blocks to find a proof for the whole conjecture but there are still some very important pieces missing. It is almost as if we have put together the outer edge of the puzzle and now we have to fill in the middle. Filling in the middle might be impossible.

 

TO MAHAN MITRA AKA MAHAN MJ FROM JHABRU AKA AJAY MISHRA - AKA - THE TIGER - AND AKA THE MISHRA MONK

 


written by Professor Sormani, CUNY Graduate Center and Lehman College, April 2002.

Prof. Sormani’s research is partially supported by NSF Grant: DMS-0102279.

7:51 PM (1 hour ago) to me Someone just viewed: “FROM NUMBER 324368521 SO WHY ARE JEWS WEAK IN MATHS ? WELL NUMBER 643361467 SAYS ITS BECASUE MATHS IS ABSTARCT AND JEWS TUENE IN TO CHANNEL, THATS REAL. MATHS IS NOT REAL, PHYSICS IS – CHEMISTRY IS – MATHS ALSO MEANS – WHATS IN IT FOR ME IS HARD AND JEWS LOVE TO MAKE MONEY . NOW THIS IS ONLY HALF MEANS 50 % THE ANSWER – WHATS THE OTHER 50 % – WELL THE OTEHR 50 % MEANS WHY ARE IHIDNOOS SO GOOD AT MATHS AND HERE IS WHY – BECAUSE MATHS DOES NOT NEED EXPIRMENST WITH MATERIAL OBJECTS BUT ONLY WITH MIND, HINDOOS ARE POOR, WELL POORER COMPARED TO JEWS – THATS WHY MEANS THATS BECAUE – AND WHAT ARE THE IMPLACTION S OF THIS – WHY DEOSNT USA AND HILELL, THINK AND WHY DOESNT MODI AND OUTIN THINK AND BTW RUSSIANS ARE ALSO BETETR AT MATHS – NBUT NOT AS GODO AS IS HINDOSO – WHY ?” People on thread: Yoni Netanyahu Olga Blog Post By Email Device: Unknown Device Location: Boydton, VA

"FROM NUMBER 324368521 SO WHY ARE JEWS WEAK IN MATHS ? WELL NUMBER 643361467 SAYS ITS BECASUE MATHS IS ABSTARCT AND JEWS TUENE IN TO CHANNEL, THATS REAL. MATHS IS NOT REAL, PHYSICS IS - CHEMISTRY IS - MATHS ALSO MEANS - WHATS IN IT FOR ME IS HARD AND JEWS LOVE TO MAKE MONEY . NOW THIS IS ONLY HALF MEANS 50 % THE ANSWER - WHATS THE OTHER 50 % - WELL THE OTEHR 50 % MEANS WHY ARE IHIDNOOS SO GOOD AT MATHS AND HERE IS WHY - BECAUSE MATHS DOES NOT NEED EXPIRMENST WITH MATERIAL OBJECTS BUT ONLY WITH MIND, HINDOOS ARE POOR, WELL POORER COMPARED TO JEWS - THATS WHY MEANS THATS BECAUE - AND WHAT ARE THE IMPLACTION S OF THIS - WHY DEOSNT USA AND HILELL, THINK AND WHY DOESNT MODI AND OUTIN THINK AND BTW RUSSIANS ARE ALSO BETETR AT MATHS - NBUT NOT AS GODO AS IS HINDOSO - WHY ?"

AND ? WELL AND U KNOW HINDUS, FRENCH AND RUSSIANS ARE VERY GOOD AT MATHS S GOOD THAT THEY CAN EVEN COMPETE AND IN SOME CASES DEFEAT THE JEWS :)

HERE IS MY THEORY –

[A] THERE ARE- ACTUALLY – WERE – 3 FACTORS OF PRODUCTIONS – – LAND -LABOR – AND CAPITAL.

[B] LAND IS FIXED – AND LABOR ISA CLAIM ON CAPITAL – BUT JEWS FROM AGES, HAD – CAPITAL – PERHAPS BECAUSE – THEY SAY – THAT JEWS WERE BARRED FROM OWING LAND AND BIBLE SAID UURY – MEANING INTERREST – ON CAPITALWAS BARRED FOR CHRISTIANS – WELL SO THEY SAY – SO, WERE INVENTED MERCHANTAND I THINK INVESTMENT BANKING – PERHAPS THEY HAD SOME RESTRCTIONS IN THAT BUSINESS AS WELL.

[C] NOW, USUALLY THE MATH REQUIRED FOR FINANCE WAS NOT THATHARD,IMEAN JEWS -KNEW ENUGH MATHS – TO HANDLE FINANCE.. FOR ALL WE KNOW – THEY SAY – MEDICIC – FAMILY – THE INVENTOR OF DOUBLE ENTRY ACCUNTING WAS ALSO -WELL AJEW :) – EVERYWERE I GO I SEE JEWS :) AND RUSSIONS ? WELL RUSSHIONSARE VERY VERY GOOD AT MATHS

SO, [1] NOW A DAYS ITS ABOUT DATA – I MEAN CHEMSTIRY IS RUSSHIAN – AND EVEN BILOLOGY NOW IS ABOUT DATA – ABOUT MATHS – THAT KIND OF MATHS IS HARD – AND WLEL, WE NOW HAVE A NEW MEANS 4TH FACTOR OF PRODUCTION – CALELD – HUMNA CAPITAL – THAT COMBINED WITH THE INFORMATION AND DATA – ECONOMY SPECIALLY AI – MEANS JEWS SHOOULD USE THEIR CAPITAL LEVERAGE AND MAKE ALLIANECS WITH HINDOOS AND RUSSIANS .. AND ITS NO SUPRISE THAT IBM CEO, MICROSOFT CEO AND GOOGLE CEO ISALL HINDOOS ANDTHAT 2 – 66.66 % OF THE 3 CES IS IIT TRIBE.. NOW, CAN SOMENE HIRE ME -? IMEAN I AM IIT AND HINDOOS :)

COME ON JEWS – BUY ME.. ME IS MEANS I WILL WORK, HARD, I WONT LIE, I WONT CHEAT – I WONT STEAL AND ? WELL, AND I WILL TRY TO MEET YOUR STANDARDS ..

BTW -SERIOUSLY TELLING U – I HAVE BEEN – SINCE – LIKE MY CHILDGOOD DAYS WODNERING HOW COME – IN PHSYICS, AND CHEMSTIRY ITS ALWAYS RUSSIANS AND JEWS – AND SOME CHRISTIANS -BUT WHY MATHS – IS KIND OF NOT THAT DIFFCULT FOR HINDUS TO BE – NUMEBR 1 IN ? – I MEAN – ITS ALL RELATIVE.. THIS IS NOT TO SAY THAT JEWS ARENT GOOD AT MATHS – IT IS TO SAY -THAT – MATHS -IS ONE SUBJECT – WHERE -ITS NOTDOMINATED BY JEWS – AS UCAN SEE – RUSSIANS AND HINDUS – REALLY REALLY GOOD AT MATHS – AND YESH FRENCH TOO – ALTHUGH – I DONT KNOW WHATS GOING ON WITH PARIS – AND YES THEIR ECOLE- THSE GRAND ECOLES ARE GOOD AND YES. ABOUT 50 – 70 YEARS AGO FRENCH MATHEMATICIANS WERE REALLY GREAT – I THINK CAUCHY IS FRENCH

I THINK IN MATHS -THE HEIRARCHY IS SORTA LIKE

RAMANUJAN

CAUCHY

GAUSS

AND REIMMAAN

REIMAN I THINK WAS GERMAN.. I MAY BE WRONG – OK REIMMAAN WAS GERMAN AND GAUSS WAS GERMAN TOO BUT I THINK CAUCHY WAS FRENCH

BASCIALLY EUROPE HAD REALLY GREAT MATHEMATICANS IN THE PAST.. BUT TAHTS PAST ..

I THINK RUSSIANS ARE VERY GOOD AT MATHS .. AND U KNOW THEY DID THE CORONA VACCINE.. – ONE F THEM IS VECTOR BASED AND THE SECOND OONE IS COMING TOO.. AND FORGET CORONA. THEY DISD THE SPUTNIK..

SO.

?

WELL, SO

[A] JEWS – STILL – HAVE CAPITAL ADVANTAGE – KIND OF LIKE GOGLE FOUNDERS ARE JEWS – AND TEH CEO – THE WORKER IS HINDOOS

[B] JEWS CAN MAKE ALLIANCES – AND INVEST IN AI – GIGS – AI IS VERY MUCH ABOUT MATHS – WELL, A LOT OF STATS, I KNOW BECAUSE I AM THE FOUNDER OF OLINTEL.. ITS NOT LIKE ROCKET SCIENCE AI – BUT – U KNOW MY GIGS ARE SUDEENLY TAKEN OFF – SHELF, BY THE POWERSTHAT BE – AND THEN – OF COUSRE -I HAVE SOME REAL LIFE ISSUES, TO RECKON WITH –

OK I GOT INTO IIT – ND ITHAT YEAR IIT EXAM WAS VERY HARD. LIKE REALLY HARD, SPECIALLY THE MATHS – PAPER WAS BLEEDING HARD. I WAS OK WITH MATHS – I MEAN EVEN BY IIT AND JEWS STANDARDS.. AND NOW? W ELL, NW I AM LEARNING MATHS AGAIN

:)

SO BUY ME JEWS

U CANT LOOSE WITH ME ..

REGARDS

ME

PS:U KNOW JEWS, THERE IS – A LAND CALELD ISRAEL – THEY ARE SMALL IN LAND, AND LOW IN NUMBERS, BUT THEY HAVE TECHNINION – WHICH IS THE IIT OF ISRAEL AND – THEN? WELL, THEN THEY HAVE THE HEBREW UNIVERISTY AND EINSTEIN SCHOOL OF MATHS I THINK ? –

SO? WELL, SO, IN THIS DAY AND AGE – MY DEAR RUSSIANS AND JEWS – INDIANS – ARE MALELABLE – ENGINEERING ISHARDER THAN MST OF THE STUFF THAT MAKES MONEY – BUT – ENGINEERING PAYS , LESS SO, WE GET INTO WHATEVER – MAKES MONEY,.. LUCKILY NOW, A NEW FIELD IS EMERGING WHICH ISNT ENGINEERING – PER SE – BUT ITS CALELD DATA SCIENCE OR – AI – AND ML – AI MEANS ARTIFICIAL INTELLIGENCE AND ML MEANS MAHCINE LEARNING.. THERE IS WORDS LIKE BAYSEAIN – PORBABILITY UPDATES AND BAYSEAIN STATISTCIS – ETC – WHICH IS A FIELD IIN PROBABILITY –
..WHICH IS A FIELD IN MATHS – BUT YES, FO COSUSRE – AS ANDREW NG SAID – WHAT MAKES MONEY IN AI – IS NT WHATS SEXY OR HARD –

BUT – HEEARD ITEM SONG – KABHI KISI KO MUKAMMAL JAHAN NAHIN MILTA ?

SO, MY ADVSIE TO JEWS AND RUSSIANS ?

[A] MAKE ALLIANECS WITH HINDUS

[B] BUY ME

[C] U SELL – THEY MAKE – SOMETHING LIKE THAT -AND I SAY SO, BECAUSE SELLING – AND MARKETS – ISNT PART OF THE SOUL OF HINDUS -I MEAN – ITS NOT PART OF RUSSIAN SOUL EITEHR – BUT JEWS -SOMEHOW, ARE GOOD AT SELLING ALSO,.. WELL, THE REASON WHY SELLING CULTURE ISNT HINDU OR SA;VIC CULTURE AHS ALOT TO DO WITH – THE PAST -BEING – MOSTLY – NOT ONLY BUT MOSTLY – GOVERNMENT BASED COCULTURE – OR – U CAN SAY – SCINECE AND ENGINEERING NOT MARLETING AND SELLING BASED SOCIETIES –

SO

WHO WANTS TO OWN ME ?

COMEON – U CANT LOOSE – MY DEAR JEWS AND RUSSIANS – JUST BUY ME – SIT, BACK AND RELAX –

THERE IS A SONG CALED PAISA PHEKO TAMASHA DEKHO

AND BTW HOW IS U MODI JI ? – WOH BLUE MEIN 6 – AAPA NUMBER HAI

"FROM NUMBER 324368521 SO WHY ARE JEWS WEAK IN MATHS ? WELL NUMBER 643361467 SAYS ITS BECASUE MATHS IS ABSTARCT AND JEWS TUENE IN TO CHANNEL, THATS REAL. MATHS IS NOT REAL, PHYSICS IS - CHEMISTRY IS - MATHS ALSO MEANS - WHATS IN IT FOR ME IS HARD AND JEWS LOVE TO MAKE MONEY . NOW THIS IS ONLY HALF MEANS 50 % THE ANSWER - WHATS THE OTHER 50 % - WELL THE OTEHR 50 % MEANS WHY ARE IHIDNOOS SO GOOD AT MATHS AND HERE IS WHY - BECAUSE MATHS DOES NOT NEED EXPIRMENST WITH MATERIAL OBJECTS BUT ONLY WITH MIND, HINDOOS ARE POOR, WELL POORER COMPARED TO JEWS - THATS WHY MEANS THATS BECAUE - AND WHAT ARE THE IMPLACTION S OF THIS - WHY DEOSNT USA AND HILELL, THINK AND WHY DOESNT MODI AND OUTIN THINK AND BTW RUSSIANS ARE ALSO BETETR AT MATHS - NBUT NOT AS GODO AS IS HINDOSO - WHY ?"

“FROM NUMBER 324368521 SO WHY ARE JEWS WEAK IN MATHS ? WELL NUMBER 643361467 SAYS ITS BECASUE MATHS IS ABSTARCT AND JEWS TUENE IN TO CHANNEL, THATS REAL. MATHS IS NOT REAL, PHYSICS IS – CHEMISTRY IS – MATHS ALSO MEANS – WHATS IN IT FOR ME IS HARD AND JEWS LOVE TO MAKE MONEY . NOW THIS IS ONLY HALF MEANS 50 % THE ANSWER – WHATS THE OTHER 50 % – WELL THE OTEHR 50 % MEANS WHY ARE IHIDNOOS SO GOOD AT MATHS AND HERE IS WHY – BECAUSE MATHS DOES NOT NEED EXPIRMENST WITH MATERIAL OBJECTS BUT ONLY WITH MIND, HINDOOS ARE POOR, WELL POORER COMPARED TO JEWS – THATS WHY MEANS THATS BECAUE – AND WHAT ARE THE IMPLACTION S OF THIS – WHY DEOSNT USA AND HILELL, THINK AND WHY DOESNT MODI AND OUTIN THINK AND BTW RUSSIANS ARE ALSO BETETR AT MATHS – NBUT NOT AS GODO AS IS HINDOSO – WHY ?”

 

CLICK LINK BELOW

https://www.google.com/search?q=%22FROM+NUMBER+324368521+SO+WHY+ARE+JEWS+WEAK+IN+MATHS+%3F+WELL+NUMBER+643361467+SAYS+ITS+BECASUE+MATHS+IS+ABSTARCT+AND+JEWS+TUENE+IN+TO+CHANNEL,+THATS+REAL.+MATHS+IS+NOT+REAL,+PHYSICS+IS+-+CHEMISTRY+IS+-+MATHS+ALSO+MEANS+-+WHATS+IN+IT+FOR+ME+IS+HARD+AND+JEWS+LOVE+TO+MAKE+MONEY+.+NOW+THIS+IS+ONLY+HALF+MEANS+50+%25+THE+ANSWER+-+WHATS+THE+OTHER+50+%25+-+WELL+THE+OTEHR+50+%25+MEANS+WHY+ARE+IHIDNOOS+SO+GOOD+AT+MATHS+AND+HERE+IS+WHY+-+BECAUSE+MATHS+DOES+NOT+NEED+EXPIRMENST+WITH+MATERIAL+OBJECTS+BUT+ONLY+WITH+MIND,+HINDOOS+ARE+POOR,+WELL+POORER+COMPARED+TO+JEWS+-+THATS+WHY+MEANS+THATS+BECAUE+-+AND+WHAT+ARE+THE+IMPLACTION+S+OF+THIS+-+WHY+DEOSNT+USA+AND+HILELL,+THINK+AND+WHY+DOESNT+MODI+AND+OUTIN+THINK+AND+BTW+RUSSIANS+ARE+ALSO+BETETR+AT+MATHS+-+NBUT+NOT+AS+GODO+AS+IS+HINDOSO+-+WHY+%3F%22&source=lnms&tbm=isch&sa=X&ved=2ahUKEwintaiMwPDrAhWbbysKHc_kCjgQ_AUoAXoECAQQAw&biw=1536&bih=758

 

 

 

REGARDS
AJAY