Weierstrass functions Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere". Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy. Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr.
The Memory Palace The Non Prophets: An Austin-based Internet Radio Show Fairy Tales Can Come True Here's a "photo story" based on the book, Fairy Tales Can Come True (Just Not Every Day!), published by Shake It! Books. Falling in love is the easy part. "This relationship stuff. Try to change your attitude a bit. "Okay, I'm taking notes!" Tip #1 Make a list. "Hmmm... Pretty soon, you'll find yourself thinking... "Say, he's not so bad after all!" And making that list goes for both of you —to help you both remember the good. "That's what we do. Click to continue 10 Tips for Spiritual Growth Spiritual growth is the process of inner awakening, and becoming conscious of our inner being. It means the rising of the consciousness beyond the ordinary existence, and awakening to some Universal truths. It means going beyond the mind and the ego and realizing who you really are. Spiritual growth is a process of shedding our wrong and unreal conceptions, thoughts, beliefs and ideas, and becoming more conscious and aware of our inner being. By discovering who we really are we take a different approach to life. You can walk on the path of spiritual growth, and at the same time live the same kind of life as everyone else. A balanced life requires that we take care not only of the necessities of the body, feelings and mind, but also of the spirit, and this is the role of spiritual growth. 10 tips for spiritual growth: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
6174 (number) 6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property: Take any four-digit number, using at least two different digits. 9990 – 0999 = 8991 (rather than 999 – 999 = 0) 9831 reaches 6174 after 7 iterations: 8820 – 0288 = 8532 (rather than 882 – 288 = 594) 8774, 8477, 8747, 7748, 7487, 7847, 7784, 4877, 4787, and 4778 reach 6174 after 4 iterations: Note that in each iteration of Kaprekar's routine, the two numbers being subtracted one from the other have the same digit sum and hence the same remainder modulo 9. Sequence of Kaprekar transformations ending in 6174 Sequence of three digit Kaprekar transformations ending in 495 Kaprekar number Bowley, Rover. "6174 is Kaprekar's Constant".
Money Girl Laura D. Adams is a personal finance expert and award-winning author of multiple books, including Money Girl's Smart Moves to Grow Rich. She’s frequently quoted in the media and has been featured in NBC News, CBS News, FOX News, FOX Business, CBS Radio, ABC Radio, NPR, USA Today, US News and World Report, Consumer Reports, Kiplinger's, Huffington Post, and many other radio, print, and online outlets. In addition to hosting the top-rated Money Girl podcast, Laura is a writer, speaker, and spokesperson. She has a passion for making money easy to understand. Laura received an MBA from the University of Florida. Awards 2011 Excellence in Financial Literacy Education (EIFLE) Award: "Book of the Year for Adult Money Management" for Money Girl’s Smart Moves to Grow Rich -- Institute for Financial Literacy 2010 Nominee Best Business Podcast -- Podcast Awards 2008 Nominee Best Business Podcast -- Podcast Awards 2007 Nominee Best Business Podcast – Podcast Awards Publications “Subscribe, ASAP!
The Atheist Experience TV Show What Does Love Mean? When my grandmother got arthritis, she couldn’t bend over and paint her toenails anymore. So my grandfather does it for her all the time, even when his hands got arthritis too. That’s love. 2. “When someone loves you, the way they say your name is different. 3. 4. 5. Terri – age 4 6. Danny – age 7 7. Emily – age 8 8. Bobby – age 7 9. Nikka – age 6 10. Noelle – age 7 11. Tommy – age 6 12. Cindy – age 8 13. Clare – age 6 14. Elaine-age 5 15. Chris – age 7 16. Mary Ann – age 4 17. Lauren – age 4 18. Karen – age 7 19. Mark – age 6 20. Jessica – age 8 21. The purpose of the contest was to find the most caring child. The winner was a four year old child whose next door neighbor was an elderly gentleman who had recently lost his wife. Upon seeing the man cry, the little boy went into the old gentleman’s yard, climbed onto his lap, and just sat there.
Top Ten Powerful Law of Attraction Tips: Simple Secrets to Attract What You Want I have gathered some of the most potent tips to create your reality that I know of. The Secret and the Law of Attraction teach us to visualize and feel our goals. This is a start, but if we really want to manifest more of our desires, more wealth, better and more conscious relationships, and all the goals in our heart that make us want to celebrate life, we need to relinquish EVERYTHING which does not serve us and step into a realm of true magic where our attraction power is natural and effortless. I’ve gathered ten of the best tips I know to support this inner power. Let’s explore them now: 1. Change comes with flow and contentment. Your only job is to be in the Flow of Life by consciously experiencing your joy and your clarity about your deepest mission and Higher Vision. This will activate the Law of Attraction and Universal Flow, and it will improve the circumstances of your life immensely. 2. Try this: Make an intention right now to do something very simple and easy. 3. 4. 5. 6. 7.
Collatz conjecture The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers. Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO) indefinitely. Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems In 1972, J. Statement of the problem In notation: (that is: applied to