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Com’e possible che la Mattematica pottrebe aiutaci prevedere i futuro?

Periodic – con periodo 2 2, 5, 2, 5, 2, 5 Prevedibile

Convergent e– arriva al LIMITO che e 0 1 , 1 2 , 1 4 , 1 8 , Cos’e l ‘ultimo termina nel questo seria numerico che arriveremo se continarsi per sempre? Prevedibile

Cos’e il prossimo numero? 1, 5, 9, 13, Divergente – non convergera al limito Cos’e il prossimo numero? Cos’e il modello? E predictbili? Cos’e il 100th numbero? Cos’e il formula (o nth termino)? 4n - 3

Livello 10 mattematica 𝑈 1 , 𝑈 2 , 𝑈 3 , 𝑈 4 1, 5, 9, 13, Possiamo scrivere questo serie in un modo pui facile Prossima numero nel sequenza = quello actuale + 4 𝑈 𝑛+1 = 𝑈 𝑛 +4 𝑈 2 = 𝑈 1 +4 𝑒𝑡𝑐….

Pottete scrivere I primi quattro termini per questa sequenza 𝑈 𝑛+1 = 𝑈 𝑛 +4 Bisognati una punta per iniziare !

Eco uaa sequenza famosa Come si chiama? 𝑈 𝑛+1 = 𝑈 𝑛 + 𝑈 𝑛−1 Dove 𝑈 1 =1 𝑒 𝑈 2 =2 Fibonacci Sequence!

𝑈 𝑛+1 = 6−𝑈 𝑛 , 𝑈 1 =1 𝑈 𝑛+1 = 𝑛 𝑛+1 𝑈 𝑛 = 5 −1 𝑛 ScrivateVi il primo quatro termini per n1 per ogni di questi sequenze e dire se cisono periodici, convergente o divergente 𝑈 𝑛+1 = 6−𝑈 𝑛 , 𝑈 1 =1 𝑈 𝑛+1 = 𝑛 𝑛+1 Adesso metteteVi in Excel 𝑈 𝑛 = 5 −1 𝑛 𝑈 𝑛+1 = 100 𝑈 𝑛 2 , 𝑈 1 =1

C’e cos’e Chaos???

Automotive Turbulenza Front spoiler to break up vortices forming at the front of the car Rear diffuser to break up vortices at rear of car to reduce drag

Treni di Alta Velocita Reduzione di rumore “Drag” Che raggioni abbiamo per ridottere il turbulenza? Reduzione di rumore “Drag”

ANSYS http://www.ansys.org/Products/Simulation+Technology/Fluid+Dynamics/Fluid+Dynamics+Products/ANSYS+Fluent#/0/10

Ecco nostra Amazonian Velenoso Raganella Vidremo a una equazione che fa una modello per il crescita dello populazione di nostra rana in via di estinzione. 𝑈 𝑛+1 = 𝑎𝑈 𝑛 (1− 𝑈 𝑛) Dove 𝒂 e nostra constanta di populazione constant una numero intera!! Questo numbero rappresenta una cambio nel clima/ambiente. Nostra populazione sara una numero decimale. Tra 0 e 1 (alora 0 significa estinzione e 1 significa saturazione, troppi!!)

Here is our Amazonian Poison Tree frog Iniziamo con 𝑼 𝟏 = 0.2 come populazione, comunque 20% di tutti rani sono questi. 𝑈 𝑛+1 = 𝑎𝑈 𝑛 (1− 𝑈 𝑛) UsateVi Excel per indagare il previdibile di nostra populazione di rane inizio di a = 2 e aumentare di 0.1 ogni volta. Molto picoli cambi succedono e spaventoso.

Secondary 2D States Rolls Hexagons Pentagons Quadrilaterals Above are rolls from free boundary conditions. Heated from below. Heat up a pan of oil slowly and put corn flour on the surface and you can see cells forming. Busse 1964 (PhD) said that rolls become stable first then hexagons because hexagons are formed dependent on how the heat is applied. Quadrilaterals are unstable.

This summer we took a trip to the Giant’s Causway on the Antrim Coast. Mostly hexagons, but some are 4, 5, 7 and 8 sided. Caused when highly fluid molten basalt intruded through chalk beds to form an extensive lava plateau. As the lava cooled rapidly these structures were formed. Hence convection formed these.

Interesting gravity waves in the low level clouds and mid-level "aspect rolls" / horizontal convective rolls (HCRs) above them (upper-right in picture). The view is from about 25,000 feet in an aircraft (Kansas City to Florida) over Missouri

Heated From Above As the previous video shows here we have another picture of the stabalizing effects of bouyancy

Heated From Below Destabalized

Tornadoes A 3D structures caused by changing boundary conditions. Hot below cold above high local pressure difference leads to turbulence – vortex.

Aircraft Turbulence Deadly Wake. The turbulence in the wake of an airplane can be extremely dangerous to other aircraft. According to computer simulations, a pair of spinning rollers (bottom) can counteract the vortices that normally form in a fast-moving object's wake (middle). Wake vortices are a by-product of lift generated by aircraft. Turbulence associated with these wakes poses a potential hazard to other aircraft, especially lighter aircraft following at low altitude. This risk is mitigated by air traffic control standards that require increased aircraft separation when wake turbulence avoidance is a concern. American Airlines Flight 587, an Airbus A300, crashed into the Belle Harbor neighborhood of Queens; a borough of New York City in New York, shortly after takeoff from John F. Kennedy International Airport on November 12, 2001. After investigation they found that a contributing factor was the wake turbulence left by a 747 that took off a few minutes earlier. The National Transportation Safety Board, attributed the disaster to the first officer's overuse of rudder controls compensating for the turbulence. Rudder in a large passenger aircraft not intented for use in upset recovery. Used to a balance a turn (as aircraft turns it can slip and skid if you roll the aircraft it will roll and slip into the turn hence lose height correct but rudder control can put you into a balanced turn, have to lift nose also called kicking the ball. Smaller aircraft don’t have such a problem with this cos bigger rudder. To deal with turbulence better at slower speed and this is where envelope comes in.

Mathematics Vs. Nature Mathematics enables us to see ALL structures that are in equilibrium. We have to perform stability analysis to see which structures will survive in Nature. Is the structure stable to very small disturbances that will naturally occur in nature? How does the flow change as we increase the internal heating and/or the wavelength of disturbance

Eloquency in nature DNA Snowflakes Honeycomb Courtesy of NASA

Navier – Stokes Equations Continuity Incompressibility condition Bouyancy force Motion Corolis force (rotation) Lin. vel x ang. vel Advection term Y ou now F = ma from physics Energy Pressure Term The Applied Dynamic Pressure on the particles