101.167.2.189: fixed a typo

2014-05-21T00:19:15Z

fixed a typo

← Older revision		Revision as of 02:19, 21 May 2014
Line 1:		Line 1:
	An '''addition-subtraction chain''', a generalization of [[addition chain]]s to include subtraction, is a [[sequence]] ''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... that satisfies

	~~:<math>a_0 = 1, \,</math>~~
	~~:<math>\text{for }k > 0,\ a_k = a_i \pm a_j\text{ for some }0 \leq i,j < k.</math>~~

	~~An addition-subtraction chain~~ for '~~'n''~~, of ~~length ''L'',~~ is an ~~addition-subtraction chain such~~ that <~~math~~>~~a_L = n~~</~~math>~~. ~~That is~~, ~~one~~ can ~~thereby compute ''n''~~ by '~~'L'' additions~~ and/~~or subtractions~~. ~~(Note~~ that ''n'' need ~~not be positive~~. In this ~~case~~, ~~one may also include '~~'a''<~~sub~~>-1<~~/sub~~>=0 in the ~~sequence, so~~ that ''n''=-1 can ~~be obtained by a chain of length 1~~.)		Do you have a heart for customer service? If so, you can make money online answering phone calls for businesses. Additionally, you can make money by chatting online with a business's customer to help them solve problems and answer their questions. There are many customer service sites available including: LiveOps, Working Solutions [http://www.comoganhardinheiro101.com/?p=73 como ganhar dinheiro] and ACD Direct. Write online to make money. There are quite a few legitimate companies that pay people to write articles online. You can find many different companies that will pay you to write different content with varying topics. If you enjoy writing, you should do research to look for companies that will pay you to write online. <br><br><br>Affiliate marketing can be a good way to earn money online if you have a lot of time and energy to spend on website setup and promotion. Affiliate marketers market the products of others in a number of ways. You can create a website and promote [http://www.comoganhardinheiro101.com/ferramentas/ ganhar dinheiro] it. Creating a blog or posting on the blogs of others in your niche is also an excellent way to share product information. Making smart use of social media and e mail marketing are also possibilities for successful affiliate marketing.<br><br>Several websites out there pay for people to do tasks that cannot be handled by a computer. It might be flagging inappropriate activity or finding the name of a business on a receipt. These tasks require human intelligence to find the answer. They are reasonably simple to perform and can provide you with some extra [http://www.comoganhardinheiro101.com/?p=18 ganhar dinheiro na internet] money. Translate documents if you are fluent in a second language and want to make money on the side. Check out the freelancing sites to find people who will need things altered into a different language. This can be anyone from a large corporation to an individual who wants to translate something for a friend.<br><br>Take advantage of paid per post if you have extra time and need to make money online. People will need you to make posts on their site to promote a product or give a good review, which will help them in the long run. In return, they will pay you a nice sum. Do you enjoy [http://www.comoganhardinheiro101.com/?p=16 como ficar rico] writing? If so, you can make cash by becoming a freelance writer. Just be aware that most freelance writing sites prefer writers with experience. But, there are other sites that will hire you if you pass a test. It's a bonus if you have knowledge on particular subjects.<br><br>If you want to do something creative, consider ways to create viral videos. Brainstorm ideas that you think are rip riotously hilarious, then start putting them into action. Post them on YouTube, turn on ads and, if they work, watch the money start rolling in! Learn about marketing to see how you can get them even more popular. [http://www.comoganhardinheiro101.com/?p=66 ganhe dinheiro na internet] If you know how to prepare taxes, you can earn money online. Apply for the necessary government [http://comoganhardinheironainternet.comoganhardinheiro101.com ganhe dinheiro] licenses and then begin offering your services online. Many of [http://Ganhardinheironainternet.Comoganhardinheiro101.com/ today's] top producers work from home with only their laptop and an email address. For best results, continually advertise your services throughout the year.<br><br>Make extra money on the side with email marketing. This is a great way to promote a company and use the subscribers that you have to your advantage. You will simply send a message or advertise a product to your large pool of subscribers and get paid money to do so. If you plan on making serious [http://www.comoganhardinheiro101.com/?p=66 ganhando dinheiro na internet] money online, make sure you've got the right equipment for the job. There's nothing more frustrating than trying to complete a job and your Internet times out, or is so slow that you miss a bid or other opportunity. Have the necessary equipment and make sure you're software is up to date too.<br><br>If you have the ability to type quickly, you may consider transcription jobs. You need to have good headphones and the ability to keep up with what is being said. Often, the companies will have their own formatting guidelines. Familiarize yourself with these prior to beginning. Once you have, you can begin transcribing. [http://www.comoganhardinheiro101.com/?p=71 ganhar dinheiro na internet] What do you do? Do it for pay! A number of websites offer you points when you search through them and/or make purchases of items that you were planning to buy anyway. Take the points you earn and trade them for gift cards, gifts and sometimes even cash. Sites thatprovide this opportunity include Swagbucks and Mypoints.<br><br>Do you have a heart for customer service? If so, you can make money online answering phone calls for businesses. Additionally, you can make money by chatting online with a business's [http://ganhedinheiro.comoganhardinheiro101.com/ como ganhar dinheiro] customer to help them solve problems and answer their questions. There are many customer service sites available including: LiveOps, Working Solutions [http://www.comoganhardinheiro101.com/ como ficar rico] and ACD Direct. Write online to make money. There are quite a few legitimate companies that pay people to write articles online. You can find many different [http://ganhardinheironainternet.comoganhardinheiro101.com/ companies] that will pay you to write different content with varying topics. If you enjoy writing, you should do research to look for companies that will pay you to write online.<br><br>Affiliate marketing can be a good way to earn money online if you have a lot of time and energy to spend on website setup and promotion. Affiliate marketers market the products of others in a number of ways. You can create a website and promote [http://www.comoganhardinheiro101.com/inicio/ ganhe dinheiro] it. Creating a blog or posting on the blogs of others in your niche is also an excellent way to share product information. Making smart use of social media and e mail marketing are also possibilities for successful affiliate marketing.<br><br>that's a lie. Don't fall prey to these types of websites and always read the reviews before joining. Talking to experts in their field can help you to use [http://www.comoganhardinheiro101.com como conseguir dinheiro] the Internet successfully. Try to network with people who do what you want to do. Doing these things can help give you an advantage over your competition.<br><br>To really make money online, consider setting up your own site and having all the control. You could sell things as an affiliate or offer professional services, among many other things. While working for others can earn you a nice income, there's nothing like raking in all the profits and calling [http://www.comoganhardinheiro101.com/?p=73 ganhar dinheiro] all the shots yourself! As stated above, there are lots of ways to increase your online income. Use these suggestions to help you get started. You will be amazed at how quickly the money can add up. Come back to this page and others like it to continue growing your online work portfolio as you go.<br><br>

	~~By definition, every addition chain is also an addition~~-~~subtraction chain~~, ~~but not vice-versa~~. ~~Therefore,~~ the ~~length of the ''shortest'' addition-subtraction chain for ''n'' is bounded above by the length of the shortest addition chain for ''n''~~. ~~In general, however, the determination of a minimal addition-subtraction chain (like the problem of determining a minimum addition chain) is a difficult problem for which no~~ efficient ~~algorithms are currently known. The related problem of finding an optimal [[addition sequence]] is [[NP-complete]] (Downey et al.~~, ~~1981), but it is not known for certain whether finding optimal addition or addition-subtraction chains is~~ [~~[NP-hard]].~~		Manage your time wisely when you are trying to make money online. You may have a real-world job, which means you have to budget time wisely. Set a schedule that will allow you enough time to balance the two. Also make sure that you are being efficient in that time, so you don't run into stress [http://www.comoganhardinheiro101.com/tag/ganhar-dinheiro/ como ganhar dinheiro] about working online. Becoming a ghost writer is a great way to earn money online. There are people who need content for their websites, but who lack good writing skills. You can write for them and earn money for what you write. Many people are earning a good income writing for others this way. If you are you looking for more info regarding [http://comoganhardinheiropelainternet.comoganhardinheiro101.com/ ganhar dinheiro] stop by comoganhardinheiropelainternet.comoganhardinheiro101.com/

	~~For example, one addition-subtraction chain is~~: ~~<math>a_0=1<~~/~~math>, <math>a_1=2=1+1<~~/~~math>, <math>a_2=4=2+2<~~/~~math>, <math>a_3=3=4~~-1</~~math>~~. ~~This~~ is ~~not~~ a ~~''minimal'' addition-subtraction chain for ''n''=3, however, because we could instead have chosen <math>a_2=3=2+1</math>~~. ~~The smallest ''n''~~ for ~~which an addition-subtraction chain is shorter than the minimal addition chain is ''n''=31~~, ~~which can be computed in only 6 additions (rather than 7 for the minimal addition chain):~~

	~~:<math>a_0=1,\ a_1=2=1+1,\ a_2=4=2+2,\ a_3=8=4+4,\ a_4=16=8+8,\ a_5=32=16+16,\ a_6=31=32-1~~.~~</math>~~

	~~Like an addition chain, an addition-subtraction chain~~ can ~~be used~~ for ~~[[addition-chain exponentiation]]: given the addition-subtraction chain of length ''L''~~ for ~~''n'', the power <math>x^n</math> can be computed by multiplying or dividing by ''x'' ''L'' times, where the subtractions correspond to divisions~~. ~~This is potentially efficient in problems where division is an inexpensive operation, most notably~~ for ~~exponentiation on [[elliptic curve]]s where division corresponds to a mere sign change (as proposed by Morain and Olivos, 1990)~~.

	~~Some [[multiplication ALU \| hardware multiplier]]s multiply by ''n'' using an addition chain described by n in binary:~~
	~~<pre>~~
	~~n = 31 = 0 0 0 1 1 1 1 1 (binary).~~
	~~</pre>~~
	~~Other hardware multipliers multiply by ''n'' using an addition-subtraction chain described by n in [[Booth encoding]]:~~
	~~<pre>~~
	~~n = 31 = 0 0 1 0 0 0 0 -1 (Booth encoding).~~
	~~</pre>~~


	~~==References==~~
	* Hugo Volger, "Some results on addition/subtraction chains," ''Information Processing Letters'' '''20''', pp. 155-160 (1985).
	* François Morain and Jorge Olivos, "[~~ftp~~://~~ftp~~.~~inria~~.fr/~~INRIA/publication/Theses/TU-0144/ch4.ps Speeding up the computations on an elliptic curve using addition-subtraction chains~~]~~," ''RAIRO Informatique théoretique et application'' '''24''', pp~~. ~~531-543 (1990)~~.
	* Peter Downey, Benton Leong, and Ravi Sethi, "Computing sequences with addition chains," ''SIAM J. Computing'' '''10''' (3), 638-646 (1981).
	* Sequence {{OEIS2C\|A128998}}, length of minimum addition-subtraction chain, ''[[The On-Line Encyclopedia of Integer Sequences]]''.

	~~[[Category:Addition chains]]~~

95.14.157.134 at 15:21, 6 December 2013

2013-12-06T15:21:54Z

← Older revision		Revision as of 17:21, 6 December 2013
Line 1:		Line 1:
	~~Livestock Producers Bert from Kahnawake~~, ~~has lots~~ of ~~pursuits including glowsticking~~, ~~ganhando dinheiro na internet~~ and ~~smoking pipes~~. ~~Finished~~ a ~~luxury cruise ship experience~~ that ~~consisted~~ of ~~passing~~ by ~~Stari Ras~~ and ~~Sopocani~~.<br><br>~~Also visit my blog~~: [~~http~~://~~comoganhardinheiropelainternet~~.~~comoganhardinheiro101~~.~~com~~/ ~~ganhar dinheiro~~]		An '''addition-subtraction chain''', a generalization of [[addition chain]]s to include subtraction, is a [[sequence]] ''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... that satisfies

			:<math>a_0 = 1, \,</math>
			:<math>\text{for }k > 0,\ a_k = a_i \pm a_j\text{ for some }0 \leq i,j < k.</math>

			An addition-subtraction chain for ''n'', of length ''L'', is an addition-subtraction chain such that <math>a_L = n</math>. That is, one can thereby compute ''n'' by ''L'' additions and/or subtractions. (Note that ''n'' need not be positive. In this case, one may also include ''a''<sub>-1</sub>=0 in the sequence, so that ''n''=-1 can be obtained by a chain of length 1.)

			By definition, every addition chain is also an addition-subtraction chain, but not vice-versa. Therefore, the length of the ''shortest'' addition-subtraction chain for ''n'' is bounded above by the length of the shortest addition chain for ''n''. In general, however, the determination of a minimal addition-subtraction chain (like the problem of determining a minimum addition chain) is a difficult problem for which no efficient algorithms are currently known. The related problem of finding an optimal [[addition sequence]] is [[NP-complete]] (Downey et al., 1981), but it is not known for certain whether finding optimal addition or addition-subtraction chains is [[NP-hard]].

			For example, one addition-subtraction chain is: <math>a_0=1</math>, <math>a_1=2=1+1</math>, <math>a_2=4=2+2</math>, <math>a_3=3=4-1</math>. This is not a ''minimal'' addition-subtraction chain for ''n''=3, however, because we could instead have chosen <math>a_2=3=2+1</math>. The smallest ''n'' for which an addition-subtraction chain is shorter than the minimal addition chain is ''n''=31, which can be computed in only 6 additions (rather than 7 for the minimal addition chain):

			:<math>a_0=1,\ a_1=2=1+1,\ a_2=4=2+2,\ a_3=8=4+4,\ a_4=16=8+8,\ a_5=32=16+16,\ a_6=31=32-1.</math>

			Like an addition chain, an addition-subtraction chain can be used for [[addition-chain exponentiation]]: given the addition-subtraction chain of length ''L'' for ''n'', the power <math>x^n</math> can be computed by multiplying or dividing by ''x'' ''L'' times, where the subtractions correspond to divisions. This is potentially efficient in problems where division is an inexpensive operation, most notably for exponentiation on [[elliptic curve]]s where division corresponds to a mere sign change (as proposed by Morain and Olivos, 1990).

			Some [[multiplication ALU \| hardware multiplier]]s multiply by ''n'' using an addition chain described by n in binary:
			<pre>
			n = 31 = 0 0 0 1 1 1 1 1 (binary).
			</pre>
			Other hardware multipliers multiply by ''n'' using an addition-subtraction chain described by n in [[Booth encoding]]:
			<pre>
			n = 31 = 0 0 1 0 0 0 0 -1 (Booth encoding).
			</pre>


			==References==
			* Hugo Volger, "Some results on addition/subtraction chains," ''Information Processing Letters'' '''20''', pp. 155-160 (1985).
			* François Morain and Jorge Olivos, "[ftp://ftp.inria.fr/INRIA/publication/Theses/TU-0144/ch4.ps Speeding up the computations on an elliptic curve using addition-subtraction chains]," ''RAIRO Informatique théoretique et application'' '''24''', pp. 531-543 (1990).
			* Peter Downey, Benton Leong, and Ravi Sethi, "Computing sequences with addition chains," ''SIAM J. Computing'' '''10''' (3), 638-646 (1981).
			* Sequence {{OEIS2C\|A128998}}, length of minimum addition-subtraction chain, ''[[The On-Line Encyclopedia of Integer Sequences]]''.

			[[Category:Addition chains]]

en>Michael Hardy: /* Statement */

2011-11-18T05:38:24Z

Statement

New page

Livestock Producers Bert from Kahnawake, has lots of pursuits including glowsticking, ganhando dinheiro na internet and smoking pipes. Finished a luxury cruise ship experience that consisted of passing by Stari Ras and Sopocani.<br><br>Also visit my blog: [http://comoganhardinheiropelainternet.comoganhardinheiro101.com/ ganhar dinheiro]

Limit comparison test - Revision history

101.167.2.189: fixed a typo

95.14.157.134 at 15:21, 6 December 2013

en>Michael Hardy: /* Statement */