Theta Functions and Brownian Motion

For arbitrarily small $varepsilon$, in a small enough neighborhood of any rational $frac pq$ (more precisely, for $z=varepsiloni(frac pqpmvarepsilon')$) one starts seeing regular behavior, so whether it is chaotic or not depends on the point of view.This is for $varepsilon=0.000001$ around $1/2$:And this for $varepsilon=.00000001$ around $0.506$:Turning things around - for a given $varepsilon'$ neighborhood of a number, you might need to go quite close to the real axis to start seeing chaos creeping in, especially if that number has very good rational approximations. For example, with $frac1pipm0. 000001$, things are still relatively quiet at $varepsilon=0.000000001$:(all graphs are for the modulus)

1. What is the difference between being manufactured/created and naturally occurring?

true but you are forgetting the origins... do things occur naturally because of evolution or because they were set in motion to occur that way by intelligent design? Your argument leads us no closer to the truth... Either theory can be supported by that argument

2. Not enough time to oppose a motion

Rule 12 of the Illinois Supreme Court says that this is sufficient time (4 days after mailing, if by US mail, which has the longest advance-notice requirement). From their perspective, this is timely notification. You can file a motion to continue, with the aid of this website. You could also file a motion to dismiss, if you can give the legal reasons to dismiss

3. Spiritually: You know electricity is invisiable right, and you believe in it right?

Because of electrical discharges... you can see it in motion and feel its effects. This is analogous to asking "why do not you believe in God because you can not see Him, but you believe in the wind?". Well, I can test both electricity and wind.

4. what are the differences between a wheel bearing and CV joint ? if any?

Both wheel bearings and CV joints turn as a vehicle is in motion, and either could emit a light ticking/clicking sound if they are dirty or damaged. A wheel bearing is donut shaped and fits between the axle spindle and the wheel hub. A Constant Velocity (CV) joint is part of an axle assembly that allows the tranmission to transmit rotation to the axle at various angles. The CV joint is actually a set of bearings encased in a rubber boot with accordian type baffles that permit flexibility. In either case, a rubber seal must be maintained to keep moisture and dirt from entering the bearings. If the seal is cracked of torn, the bearing will wear prematurely and fail. If either device is clicking, the damage has been done and the part will need to be replaced. A wheel bearing can cost as little as $10. A CV joint usually costs over $50. This is just for the part. The labor charge can exceed $100.

5. What did I do to my truck?

Mr. No, from your description it is apparent that you need lots of driving lessons. Driving lessons are beyond "driving" a car. Driving lessons are involve study of physics, mechanics and applied principles of motion. Friction and gravity are essential to drive a vehicle. Most people in cold climates know how slippery ice and snow roads can be. What they do not know is that for any condition of temperature, tire and surface material, there is a tire energy limit. Any type of car racing offers plenty of information as drivers lose control in otherwise perfect driving conditions. The ever popular donut victory maneuver shows exactly how to exceed the energy level of tires. The introductory "Okay so I live. .." marks you as a 16 year old new driver since your writing skills, or lack of them, point in that direction. A hill presents challenges for humans. We all know walking uphill, depending on the grade level, is not as easy as walking of flat terrain. The inclusion of "sharp turn" means slow speed. Combine the new driver, hill, sharp turn elements and the lack of knowledge and experience as to driving that the results you indicate are quite common. Add the final new driver, teen driver, hill, sharp turn, unidentified vehicle which in normal teenland these days means 10 to 20 years old vehicle with 200,000 miles or so and the scenario is complete. Here is the coup d'grace, the final touch, the critical element. "I kept trying and trying. " You do not know this but the classic definition of lunacy, a term indicated mental defect or bring crazy, is to repeat doing the same thing and expecting different results. For example, people get stuck on ice, mud or sad as the wheel or wheels loose traction. But, they keep gunning the engine and the wheels actually dig themselves deeper and deeper. Same action, same results. Others have offers guesses as to what you broke on this "truck" doing this maneuver. Only a mechanic can offer the correct analysis and proffer a repair estimate. The ultimate teen texting symbols closes this posting. "Thank you x" I do not who or where you learned to write a closing. But a simple, "Thank you." with the required period or punctuation mark at the was all that was needed. The ":x" is cryptic, mysterious, undefined and adds nothing to the word. Your grammar skills need polishing. The car needs to be towed to an auto repair shop. While you wait for the car, practice writing using sentences that have all of the needed elements. The long sentence that begins with "Anyways" an introductory word that requires a comma after it, has many sentences into one. Please edit and improve your writing skills. Once the truck is repaired I suggest driving lessons. Lots of them