Verknotete Strömungslinien führen zu Turbulenz und hydrodynamischer Instabilität. Eine in den “Annals of Mathematics” erscheinende Arbeit findet jetzt beliebig verknotete Strömungslinien für reibungsfreie, inkompressible Gase oder Flüssigkeiten.

Die Euler-Gleichungen beschreiben reibungsfreie Strömung.

Ganz allgemein wird die Strömung in Flüssigkeiten und Gasen durch die Navier-Stokes-Gleichungen beschreiben: sei ρ die Dichte, p der Druck und v die Geschwindigkeit eines Teilchens in der Strömung, dann ist
ρ (v’ + (v . ∇)v) = -∇ p + η Δ v + (λ + η) ∇ (∇ . v)+f
mit gewissen Viskositätskonstanten λ und η und der Volumenkraftdichte f. (Auf die Lösung der Navier-Stokes-Gleichungen, d.h. den Beweis ihrer Existenz und Glattheit, ist übrigens vom Clay-Institut eine Million Dollar ausgesetzt, genauere Beschreibung des Problems hier.)

Der Spezialfall reibungsfreier Flüssigkeiten, wo also Viskosität und Wärmeleitung vernachlässigt werden, gibt die Euler-Gleichungen
ρ (v’ + (v . ∇)v) = -∇ p

Eine stationäre Lösung der Euler-Geichung ist ein Vektorfeld v, daß diese Gleichung (zu einem Druck p) erfüllt. Die Trajektorien des Vektorfelds v sind die Strömungslinien.

Schon Lord Kelvin soll sich mit der Untersuchung verknoteter Strömungslinien befaßt haben, u.a. wegen ihrer Rolle für Turbulenz und hydrodynamische Instabilität. 1965 untersuchte V.I.Arnold in “Sur la topologie des écoulements stationnaires des fluides parfaits” die Topologie der Strömungslinien und stellte die “Topologische Hydrodynamik-Vermutung” auf, daß jeder Knoten sich als Strömungslinie der Euler-Gleichung realisieren läßt. Man weiß aus numerischen Berechnungen, daß Strömungslinien stark verknotet sein können, hatte aber bisher nur wenige theoretische Resultate. (Am bemerkenswertesten vielleicht die Konstruktion von Etnyre und Ghrist, veröffentlicht 2000 in TAMS, einer Lösung der Euler-Gleichung mit periodischen Strömungslinien in den Isotopieklassen aller möglichen Knoten. Allerdings betrachteten sie dabei nicht die Euler-Gleichung bzgl. der euklidischen Metrik, sondern zu nichteuklidischen Metriken auf der S3.)

In Knots and Links in steady solutions of the Euler equation (erschienen letzten Monat auf dem ArXiv und bereits zur Veröffentlichung in den “Annals of Mathematics” angenommen) beweisen Enciso und Peralta-Salas nun Arnold’s “Topologische Hydrodynamik-Vermutung”, daß es beliebig verknotete Strömungslinien für die Euler-Gleichung (im 3-dimensionalen euklidischen Raum, zu einer Druckfunktion p) gibt:

Theorem 1.1: Let L in R3 be a possibly unbounded, locally fi nite link. Then for any real constant λ ≠ 0 one can transform L by a Coo diffeomorphism Φ of R3 arbitrarily close to the identity in any Cr norm, so that Φ(L) is a set of stream lines of a Beltrami field u, which satis fies curl u = λu in R3.


Alberto Enciso, & Daniel Peralta-Salas (2010). Knots and links in steady solutions of the Euler equation Annals of Mathematics arXiv: 1003.3122v2

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  134. #134 Emmanuel Love
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    Would you believe that the Bible foretold of a global cashless society over 2000 years ago? In the book of Revelation we see this prophecy:

    “He causes all, both small and great, rich and poor, free and slave, to receive a mark on their right hand or on their foreheads, and that no one may buy or sell except one who has the mark or the name of the beast, or the number of his name. Here is wisdom. Let him who has understanding calculate the number of the beast, for it is the number of a man: His number is 666” (Revelation 13:16-18).

    Referring to the last days, this could only be speaking of a cashless society, which we have yet to see, but are heading towards. Otherwise we could still buy or sell without the mark among others if physical money was still currency. This mark could not be spiritual, because the word references two different physical locations.

    RFID microchip(see photo) technology will be the future of a one world cashless society being implanted in our right hand or forehead. Some time after the Antichrist takes the stage, Revelation 13:11-15 tells us that a false prophet will arise deceiving many to take the mark because of the miracles he does before men.

    This major Biblical prophecy is being fulfilled through the rapid growth of technology in our world. We can see how we have been conditioned through convenience in our every day life in different forms. More specifically by how we make payments…

    Since the ancient of times people have been using tangible currency to make payments among one another. Because of the rapid acceleration of technology in these recent generations, we have come to the digital age in which we are able to make virtual payments with such devices as debit and credit cards. We are now at the point where these cards have a chip contained with them, and some even allowing you to tap your card on a reader to make payment. Along with self checkout lines in our market places and the cryptocurrency industry making a rapid climb in such form as bitcoin, we can see that these are all stepping stones ultimately leading to a global cashless society.

    Carl Sanders sat in seventeen New World Order meetings with heads-of-state officials such as Henry Kissinger and Bob Gates of the C.I.A. to discuss plans on how to bring about a one-world system where everyone would have a RFID microchip implant…

    The government commissioned Carl Sanders to design a microchip for identifying and controlling the peoples of the world—a microchip that could be inserted under the skin with a hypodermic needle (a quick, convenient method that would be gradually accepted by society).

    Carl Sanders, with a team of engineers behind him, with U.S. grant monies supplied by tax dollars, took on this project and designed a microchip that is powered by a lithium battery, rechargeable through the temperature changes in our skin. Without the knowledge of the Bible(Brother Sanders was not a Christian at the time), these engineers spent approximately one-and-a-half-million dollars doing research on the best and most convenient place to have the microchip inserted.

    Guess what? These researchers found that the forehead and the back of the hand(the two places Revelation says the mark will go) are not just the most convenient places, but are also the only viable places for rapid, consistent temperature changes in the skin to recharge the lithium battery. The microchip is approximately seven millimeters in length, .75 millimeters in diameter, about the size of a grain of rice. It is capable of storing pages upon pages of information about you. All your general history, work history, crime record, health history, and financial data can be stored on this chip.

    Brother Sanders believes that this microchip, which he regretfully helped design, is the “mark” spoken about in Revelation 13:16–18. The original Greek word for “mark” is “charagma,” which means a “scratch or etching.” It is also interesting to note that the number 666 is actually a word in the original Greek. The word is “chi xi stigma,” with the last part, “stigma,” also meaning “to stick or prick.

    Mr. Sanders asked a doctor what would happen if the lithium contained within the RFID microchip leaked into the body. The doctor replied by saying a terrible sore would appear in that spot. This is what the book of Revelation says: “And the first(Angel) went, and poured out his vial on the earth; and there fell a noisome and grievous sore on the men which had the mark of the beast, and on them which worshipped his image” (Revelation 16:2).

    Visit http://WWW.BIBLEFREEDOM.COM to watch a video where Carl Sanders gives his testimony on how he helped design this RFID microchip and how it matches scripture in the Bible.

    Aaron Russo produced such movies as Mad as Hell, Trading Places, and America: Freedom to Fascism. He gained the attention of a man named Nick Rockefeller who told him their ultimate goal is to put an RFID microchip in everybody…

    Aaron and Nick(who is a lawyer) met through a female attorney of Aaron’s in the late 90’s. Nick had seen the movie Mad as Hell and knew Aaron was running for governor of Nevada. Knowing the impact Aaron was having on society, who being a freedom fighter standing for the constitution of the United States, Nick sought to befriend him to subtly recruit him into his secret society.

    Nick would come to visit Aaron’s home from time to time having dinner and sharing thoughts among one another. Nick had told Aaron 11 months before 9-11 occurred that there would be an event(though not sharing what that event would be), and from that event they would invade Afghanistan and Iraq to establish bases to help build the New World Order. And that there will be this war on terror where there is no real enemy, and the whole thing is a hoax; a tool used by the American Government to take over their own people through the propaganda of fear. Because you cannot define a winner, it can go on for as long as they desire.

    Aaron asked Nick what the purpose was for doing these things because they have all the money and power in the world. Nick replied by saying their ultimate goal is to put a microchip in everybody so they can have total control over the people.

    Visit http://WWW.BIBLEFREEDOM.COM to watch an excerpt of a historic interview with Aaron Russo where he reveals his conversations with Nick Rockefeller and more.

    In December of 2017 Donald Trump made history be recognizing Jerusalem as the capital of Israel. Why is this big news? Because by this the Jewish people are now able to push forward in bringing about their long awaited Third Temple prophesied in the Bible…

    Donald Trump is being compared by many to the Persian King Cyrus who brought about the decree for the Jewish people to return to their homeland Jerusalem and rebuild their Temple around 539 BC. An Israeli organisation has minted a coin bearing US President Donald Trump’s image(see photo above) to honor the US leader’s decision to move the American embassy to Jerusalem. This coin also bears the face of King Cyrus, as well as the Third Temple on the back. It is being sold to help raise funds to build this new Temple.

    The Jewish people have been without a temple for almost 2000 years since the destruction of the second temple in 70 AD by the invasion of the Roman army. This event was prophesied by Jesus in Matthew 24:1-2 where He told His disciples, “…Assuredly, I say to you, not one stone shall be left here upon another, that shall not be thrown down.” They are looking forward to not only having a temple once again, but to also receiving their messiah who they have been long awaiting for. In an interview, one Jewish man stated that this messiah will be a spiritual guide and political leader to all nations gathering all religions under the worship of one god.

    History tells us that the Orthodox Jewish people did not receive Jesus as their Messiah even though He fulfilled 300 Messianic prophecies concerning His birth place, details of His life, His mission, His nature, His death, and His resurrection.

    In the book of Isaiah chapter 53, prophet Isaiah gives a clear indication for recognizing the Messiah who was to come found in the Jewish peoples own Tanakh(also known as the Old Testament). He writes that the Messiah would be despised and rejected by men, He would be put to death for our transgressions; He was led as a lamb to the slaughter and His soul was an offering for sin. This chapter is clearly speaking of Jesus who is called the Lamb of God, who went to the cross to die for the sins of the world (John 3:16).

    Biblical prophecy tells us that this messiah who will seat himself in the Third Temple will be the Antichrist “who opposes and exalts himself above all that is called God or that is worshiped, so that he sits as God in the temple of God, showing himself that he is God” (2 Thessalonians 2:4).

    In the book of Daniel, prophet Daniel is given revelation from God telling us what the Antichrist will do in the last days: “Then he shall confirm a covenant with many for one week(known as 7 years); but in the middle of the week He shall bring an end to sacrifice and offering…” (Daniel 9:27). Jesus follows up with these verses and affirms to His disciples and readers about what is written in the book of Daniel by saying, “Therefore when you see the ‘abomination of desolation,’ spoken of by Daniel the prophet, standing in the holy place(temple) (Matthew 24:15)…then there will be great tribulation, such as has not been since the beginning of the world until this time, no, nor ever shall be” (Matthew 24:21).

    Visit http://WWW.BIBLEFREEDOM.COM to watch videos for proof that this Third Temple is in our near future, and not just a conspiracy.

    Astronomer(PhD) and best-selling author Hugh Ross travels the globe speaking on the compatibility of advancing scientific discoveries with the timeless truths of Christianity. His organization, Reasons to Believe (RTB), is dedicated to demonstrating, via a variety of resources and events, that science and biblical faith are allies, not enemies…

    Did you know that the 4 initial conditions and 10 creation events of how the universe and earth were formed written by Moses, in the book of Genesis, are not only all correct, but are also in the correct chronological order? Astronomer Hugh Ross calculated the probability that Moses could have known this by his own knowledge to be, conservatively, one in a quadrillion, quadrillion, quadrillion, quadrillion. Ask yourself, are you willing to bet your soul on them odds that Moses got it all correct by chance, or did God reveal it to him?

    Did you know that the Bible does not say the earth was formed in six 24-hour days, but rather six long finite periods of time? The universe is approximately 13.8 billion years old and the earth 4.5 billion. This does not contradict the Bible; it lines up with it. The Bible got it right from the beginning by concurring with the big bang saying, “In the beginning God created the heaven and the earth” (Genesis 1:1). And says our universe is expanding, thousands of years before scientists discovered these things.

    Earth resides in the darkest location within the Milky Way Galaxy where physical life is possible. And our Milky Way Galaxy lies in the darkest location in the universe where physical life is possible. Not only that, but so we are also able to observe all of the universe, from the cosmic beginning at the big bang, all the way through.

    Because we can observe the beginning at the big bang, we are able to confirm that the Bible had it right all along, that is, in the beginning God created the Heaven and Earth; as if God wanted us to discover Him through His creation and His word. The thought among scientists in years past was that the universe was eternal before they discovered a cosmic beginning. This discovery troubled some of them because they realized it favored the God of the Bible.

    The Bible, compared with the book of nature, has no contradictions, making ten times more creation claims than all other major “holy” books combined; while these other books contain scientific errors in them. Hugh Ross and his team have discovered these facts and many more. He has brought many scientists to faith in Jesus by showing how the Bible and the book of nature work together in harmony. His claims are based on published scientific documents. He has authored many books, backed even by atheist scientists.

    Hugh Ross is quoted to be saying, “Astronomers who do not draw theistic or deistic conclusions are becoming rare, and even the few dissenters hint that the tide is against them. Geoffrey Burbidge, of the University of California at San Diego, complains that his fellow astronomers are rushing off to join ‘the First Church of Christ of the Big Bang.”

    He founded and helps run an organization called Reasons to Believe that brings out the latest evidences for the God of the Bible on a weekly basis, only making stronger the case for the Biblical God as time goes on.

    Visit http://WWW.BIBLEFREEDOM.COM to watch a video where Dr. Hugh Ross goes over these facts and many more showing us that God has given us two books, the book of nature, and the Bible. And that these two books are allies, affirming to us that the Bible truly is the inspired word of God.

    Thousands of archaeological discoveries throughout the years have affirmed the history of people, places and events that are told within the Bible. Ron Wyatt(shown in photo) is well known for being a part of three big historical discoveries that many may call myths…

    In the year 1959, in Turkey was found a man-made ship buried, preserved underneath volcanic ash, at the elevation of 6,500 feet. The Bible says Noah’s Ark rested on the Mountains of Ararat after the floodwaters receded. Not only was this ship found on the Mountains of Ararat, it is also the exact same length God told Noah to build the Ark; that is, 515 feet, or 300 Royal Egyptian cubits. Ron Wyatt, along with scientists from around the world, lead the research of this artifact for many years.

    “Then the Lord rained upon Sodom and upon Gomorrah brimstone and fire from the Lord out of heaven; And he overthrew those cities, and all the plain, and all the inhabitants of the cities, and that which grew upon the ground” (Genesis 19:24-25).

    Archaeologists have discovered round balls of brimstone, or nearly pure sulfur(98), to be rained down on the cities of Sodom and Gomorrah turning it into ashes; unlike any other sulfur found on Earth. Geothermal activity near volcano’s produce about 40 sulfur, but is of a different kind. There is no geothermal activity around these cities of Sodom and Gomorrah.

    “…And the waters returned, and covered the chariots, and the horsemen, and all the host of Pharaoh that came into the sea after them; there remained not so much as one of them” (Exodus 14:21-28).

    Moses and the Israelites escaped the pursuing Egyptian army when God made a way by parting the waters of the Red Sea. The parted waters came crashing down upon the Egyptians, destroying all who pursued.

    Artifacts discovered across the bottom of the Red Sea include wheels, chariot bodies as well as human and horse bones. These chariot parts have been discovered to be from the Egyptian army during the era when the Exodus occurred.

    The Bible is more than just a book, it is a historical library consisting of 66 books written by 40 different authors over a span of 1,500 years. These authors lived in different times and regions, yet the Bible harmonizes itself beautifully across all books with no provable contradictions found within it. It contains God’s plan for the redemption of mankind from the cosmic beginning of creation, all the way to the end unto God’s new creation. Giving names, dates, places, full of prophecy, geography, history and science that can be put to test, and has, confirming the Bible to be true. There are over 25,000 ancient manuscripts of the New Testament lining up within 99.6 of each other, with just minor typos; As well as the Dead Sea scrolls found in 1947 affirming the Bible we have today has been preserved.

    Jesus came to pay a debt that we could not, to be our legal justifier to reconcile us back to a Holy God; only if we are willing to receive Him: “For the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord” (Romans 6:23).

    “Jesus answered and said to him, “Most assuredly, I say to you, unless one is born again, he cannot see the kingdom of God.”

    Nicodemus said to Him, “How can a man be born when he is old? Can he enter a second time into his mother’s womb and be born?”

    Jesus answered, “Most assuredly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God. That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. Do not marvel that I said to you, ‘You must be born again.’ The wind blows where it wishes, and you hear the sound of it, but cannot tell where it comes from and where it goes. So is everyone who is born of the Spirit” (John 3:3-8).

    We need to have a personal encounter with the living God to truly know Jesus Christ as our Lord and Savior. God is as close as we allow Him to be. It is sin that separates our hearts from Him. We need to come before Him in prayer, humbling ourselves with an open heart; confessing our sins and be willing to forsake them. We need to keep asking Him for the Holy Spirit so we may be born from above, so we may walk in the Spirit by living a holy life. He is willing to destroy the works of the devil in our lives making us feel like a child again, but we must be willing to truly receive Him by obeying his commands.

    When we receive Jesus, and His baptism, our body of sin is put to death, just as He was buried. And just as He was raised to newness of life on the third day, likewise His Spirit within us makes us alive again. The same Spirit that rose Jesus from the dead will dwell in us, so when the time comes, we may be resurrected into newness of life.

    God is more beautiful than anything we could ever imagine. He is the best Father we could ever have Who only wants the best for us. Jesus promises trouble in this life for those who follow Him, and we may wonder why He would allow us to go through certain trials. First start with this question, what is God doing hanging on a cross? It is not that God has kept Himself separate from the suffering of this world, but He himself came to be a part of it for the redemption of mankind. This life is a foreshadow to the eternal one to come, a refiners fire to test our souls to bring about a genuine end time harvest of the wicked and righteous. Without pain and suffering there can be no free will, and without free will there can be no true love.

    Then a third angel followed them, saying with a loud voice, “If anyone worships the beast and his image, and receives his mark on his forehead or on his hand, he himself shall also drink of the wine of the wrath of God, which is poured out full strength into the cup of His indignation. He shall be tormented with fire and brimstone in the presence of the holy angels and in the presence of the Lamb. And the smoke of their torment ascends forever and ever; and they have no rest day or night, who worship the beast and his image, and whoever receives the mark of his name” (Revelation 14:9-11).

    Visit http://WWW.BIBLEFREEDOM.COM to see proof of all of these things and much more!

    “…Holy, holy, holy, Lord God Almighty, Who was and is and is to come!” (Revelation 4:8).