mathematic

For the first one, you need to count how many terms in the inner sum ((n-1) - (i+1) + 1). The outer sum is then an arithmetic series. The second problem is a geometric series. I suggest you look up both types of series on Wikipedia - this will give a thorough understanding.

There is a much simpler explanation, based on special relativity (not general) which ignores acceleration. Example: twin 1 stays on earth and twin 2 goes to alpha centauri (4.3 light years away) and back. Assume twin 2 can get to near the speed of light very fast, travel to the star, stop fast, turn around, get up to near the speed of light, and stop fast upon returning to earth. In twin 1's reference frame, twin 2 has been gone over 8.6 years, so twin 1 has aged over 8.6 years. Twin 2's experience is quite different, ignoring the acceleration and deceleration, he has traveled to the star and back at near the speed of light. Applying Lorentz transformation to the distance and the time of flight, the distance will be close to zero and so will the travel time, so he will have aged very little.

Matter - antimatter reaction is always between particle and its anti-particle producing gamma rays. Any other matter might interact with these gamma rays.

Both dark matter and dark energy have one thing in common, which is why they are called dark. Neither of the can be "seen" by any sort of e-m radiation. The evidence for their existence is gravitational.

The announcement from CERN will be about results of experiments carried out at LHC. The data has be subject to detailed analysis by two teams, who are responsible for two different detectors. The anticipated announcement concerns the possible existence of the Higgs boson. It has absolutely nothing to do with dark energy.

DevelopmentMain article: Development of Windows 7Originally, a version of Windows codenamed Blackcomb was planned as the successor to Windows XP (codename Whistler) and Windows Server 2003. Major features were planned for Blackcomb, including an emphasis on searching and querying data and an advanced storage system named WinFS to enable such scenarios. However, an interim, minor release, codenamed "Longhorn," was announced for 2003, delaying the development of Blackcomb.[44] By the middle of 2003, however, Longhorn had acquired some of the features originally intended for Blackcomb. After three major viruses exploited flaws in Windows operating systems within a short time period in 2003, Microsoft changed its development priorities, putting some of Longhorn's major development work on hold while developing new service packs for Windows XP and Windows Server 2003. Development of Longhorn (Windows Vista) was also restarted, and thus delayed, in August 2004. A number of features were cut from Longhorn.[45] Blackcomb was renamed Vienna in early 2006[46] and again renamed Windows 7 in 2007.[47] In 2008, it was announced that Windows 7 would also be the official name of the operating system.[48][49] There has been some confusion over naming the product Windows 7,[50] while versioning it as 6.1 to indicate its similar build to Vista and increase compatibility with applications that only check major version numbers, similar to Windows 2000 and Windows XP both having 5.x version numbers.[51] ------------------------------------------------------ The above from Wikipeida. Note that the name was introduced in 2007.

"Could dark energy just be an illusion from light traveling faster in-between galaxies?" No - speed of light is constant in vacuum. "Or could dark energy be something way more complex like other smaller universes attaching to our own in the low gravitational areas between galaxies?" These is no evidence of such things.

Anything that goes on at Planck length is "theoretically possible", since we have no idea.

Volume is 3-d. Area is 2-d. Surface area refers to area of a specified surface.

There is a point that is being missed. It is not just linking or unifying. In a stuation where both need to apply at the same time (inside a black hole), you will end up with mathematical nonsense. This implies that it will be necessary to modify one or both theories to make them work together.

Is there any evidence that it exists?

I am as skeptical about string theory as you are. Unfortunately string theory and loop quantum gravity seem to be the only games in town at the moment. I hope that someday physicists will either come up with something better, or else develop one of the above to the point where it can be subject to testing.

http://en.wikipedia.org/wiki/Planck_length I suggest you do a Wikipedia search for your questions to get a better understanding of neutrinos, photons, etc.

Let Y = (X-mu)/sigma. (X and Y random variables). Write down P(Y < y) (standard normal) and then substitute x = mu + y(sigma).

As a I understand it, string theory is also an attempt develop a general theory to include Gen. Rel. and Quantum theory.

There is something called a Planck length. It is much smaller than any length associate with particle (electron, etc.) processes that are considered under quantum theory. Photons are completely different from neutrinoes. There are (as far as it is known) exactly 3 kinds of neutrinoes. Photons come in all sizes (energies) radio waves to gamma rays and everything in between.

"mathematic, the definition of continuity is that the function is continuous everywhere on it's domain, and so the limit everywhere on the domain exists" It sounds like the definition is that a function is continuous if it is continuous. Somethings missing!!

http://en.wikipedia.org/wiki/Theory_of_relativity http://www.dummies.com/how-to/content/einsteins-special-relativity.html http://www.dummies.com/how-to/content/einsteins-general-relativity-theory-gravity-as-geo.html You can start with the above.

Often the statement you are trying to prove is the definition of continuity. What is the definition of continuity yiou are given?

Electrons have wave properties. For example, the double slit experiment using electrons gives the same kind of pattern one gets from using light.

"I am curious about an issue raised in another thread of this forum: namely, that of the interpretation of atomic orbitals in light of the fact that electrons are particles, not waves. " Fact??? Electrons have wave and particle properties. http://en.wikipedia.org/wiki/Electron_configuration

In general electrons in different atoms would not have the same quantum state, more precisely, would not have the same wave function. The Pauli principle is usually used to describe the states of electrons in the same atom.

If you mean by the mathematical definition of irrational (not a ratio of integers), then the question is to some extent pointless. The energy is defined in terms of the units that it is measured in joules, electron-volts, etc. The number of units is measured to a certain precision, i.e. within an error interval. In that interval there are an infinite number of rational numbers and an infinite number of irrational numbers, any of which can be the actual energy.

How about (2/3)(4/5)(6/7)..... instead of doing the numerator and denominator separately?

0/0 etc. are questions that belong in mathematics. They have nothing to do with the physics of the universe.