Web (differential geometry) - Revision history

en>Rjwilmsi: Added 2 dois to journal cites using AWB (10159)

2014-05-14T14:15:25Z

Added 2 dois to journal cites using AWB (10159)

en>Mgvongoeden: added |authorlink=Gregorio Ricci-Curbastro

2013-07-20T16:47:54Z

added |authorlink=Gregorio Ricci-Curbastro

← Older revision		Revision as of 18:47, 20 July 2013
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	~~For everyone that owns a home~~, ~~home improvement~~ is ~~an interesting and engaging topic. Although most~~ of ~~us want to live in the nicest place possible, it can be scary to think about home improvement~~. ~~The following paragraphs will introduce you~~ to ~~a few basic home improvement tasks.~~<br><br>Replace your air conditioning filter from time to time. If the filter gets clogged up, the air conditioning unit will have to work harder to cool your house, using up more energy. This will also cost you money on the time it stays on. You need to replace the filter once a month before it gets clogged.<br><br>~~If you can afford it, go for hard wood floors~~ and ~~stay away from laminate. Laminate floors may look like real wood~~ and ~~be easier to clean, however~~ they ~~cannot be refinished~~. ~~Having real wood floors means that in a decade or two, you~~ or the ~~new home owner will have the option~~ of ~~refinishing instead~~ of ~~replacing the floor.~~<br><br>~~An over-the-range microwave~~ is a ~~great way~~ to ~~save space in an already cramped kitchen~~. ~~Replacing~~ the ~~range hood~~, ~~these microwave ovens come in~~ a ~~wide range of prices~~, ~~and some have special features such~~ as ~~convection cooking. Most~~ of ~~these have a filter that recirculates air~~ as ~~opposed to having~~ a ~~vent~~. ~~This means they~~ are ~~better for people who do not need a vent.~~<br><br>Seal cracks and openings to save on heating bills, as well as preventing bugs from entering your home. Add sealants or caulk around windows, doors and baseboards. When caulk dries, it will help prevent bugs from entering and air from escaping.<br><br>~~Think outside~~ the ~~box~~ in ~~terms of artwork. Instead of expensive objects of art~~, ~~try using unconventional items such as 3D tiles~~. ~~Use three dimensional tiles in two colors such as green and white and arrange them on~~ the ~~wall (rotating colors) behind your couch for an original kind of art.~~<br><br>~~Anyone who owns~~ a ~~home can fix their own faucets which leak~~. ~~If you can locate the leaks yourself and quickly fix them~~, ~~you can save yourself some money~~ and ~~a whole lot of water. The savings you get from this will add up since your water bill will go down.~~<br><br>~~Homes that have a nice view tend to sell for a lot more than similar houses without a view. There are some instances~~ where ~~the extra expense for~~ a ~~good view might not be worth it. Potential buyers may not care for the changes you make and may not want to pay more money for your home.~~<br><br>~~You can reduce the costs associated with~~ a ~~project by putting in cabinets without hardware in your kitchen. Thankfully~~, ~~it's possible to buy handles, drawer pulls, and cabinet knobs at a home improvement store or online. If you want something that you can install~~ in ~~only a few minutes~~, ~~you can select modern knobs, designer-made accents, and pulls with a vintage look.~~<br><br>~~Choose your color scheme based on something already present~~ in the ~~room. Pay attention to the drapes~~, ~~rugs~~ or ~~upholstery and use these elements to create pleasant color patterns. For example,~~ a ~~green rug with hints~~ of ~~yellow~~ and ~~red can be a great basis for your color scheme. The other two shades should be secondary and used as accent colors.~~<br><br>~~A real estate slump~~ is the perfect time for homeowners to invest in renovations and additions. When homes aren't selling, it often means that contractors are not as busy as they would like, and they may discount their services. Remember that ~~home improvements don't carry~~ an ~~expiration date. When you renovate your home during a slump, you save money and your home's worth will grow in the future.~~<br><br>When looking for assistance with your home improvement projects, look for friends that have already been through their own projects, so that they can offer help and support if necessary. Always be prepared for the worst. This also holds true in this case.<br><br>~~A stainless steel taping knife can be a priceless tool to have~~ on ~~hand for~~ a ~~sheetrock taping project. Knives~~ of ~~this sort make clean up easier and they also last longer. Clean this stainless steel knife well~~, ~~and then store~~ it ~~properly. You'll~~ be ~~able to use it forever.~~<br><br>~~You don't have to let~~ a ~~home improvement project overpower you~~. The ~~advice~~ in ~~this article is designed to assist you from start~~ to ~~finish with any home improvement plans you have in mind~~, ~~whether it's a DIY project or a task for a professional contractor.~~<br><br>~~When you have virtually any inquiries about where by~~ and ~~how you can use show homes ideas ([http:~~//~~www~~.~~homeimprovementdaily.com http~~:~~//www.homeimprovementdaily.com/~~]~~), you possibly can contact us on the web-site.~~		{{Transformation rules}}

			In [[propositional logic]], '''tautology''' is one of two commonly used [[rule of replacement\|rules of replacement]].<ref>{{cite book \|title=A Concise Introduction to Logic 4th edition \|last=Hurley \|first=Patrick \|authorlink= \|coauthors= \|year=1991 \|publisher=Wadsworth Publishing \|location= \|isbn= \|page= \|pages=364–5 \|url= \|accessdate=}}</ref><ref>Copi and Cohen</ref><ref>Moore and Parker</ref> The rules are used to eliminate redundancy in [[logical disjunction\|disjunctions]] and [[Logical conjunction\|conjunctions]] when they occur in [[formal proof\|logical proofs]]. They are:

			The principle of '''idempotency of disjunction''':

			: (''P'' {{or-}} ''P'') <math>\Leftrightarrow</math> ''P''

			and the principle of '''idempotency of conjunction''':
			: (''P'' {{and}} ''P'') <math>\Leftrightarrow</math> ''P''

			Where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)\|symbol]] representing "can be replaced in a logical proof with."

			== Relation to tautology ==

			The rule gets its name from the fact that the concept of the rule is the same as the [[Tautology (logic)\|tautologous statement]]s ''If "p and p" is true then "p" is true.'' and ''If "p or p" is true then "p" is true.'' This type of tautology is called [[idempotency]]. Although the rule is the expression of a particular tautology, this is a bit misleading, as every rule of inference can be expressed as a tautology and vice-versa.

			== Formal notation ==

			[[Theorem]]s are those [[Well-formed formula\|logical formulas]] <math>\phi</math> where <math>\vdash \phi</math> is the conclusion of a valid proof,<ref>Logic in Computer Science, ''p. 13''</ref> while the equivalent [[semantic consequence]] <math>\models \phi</math> indicates a tautology.

			The ''tautology'' rule may be expressed as a [[sequent]]:

			: <math>P \or P \vdash P \, </math>

			and

			: <math>P \and P \vdash P \, </math>

			where <math>\vdash</math> is a metalogical symbol meaning that <math>P</math> is a [[logical consequence\|syntactic consequence]] of <math>P \or P</math>, in the one case, <math>P \and P</math> in the other, in some [[formal system\|logical system]];

			or as a [[rule of inference]]:

			:<math>\frac{P \or P}{\therefore P}</math>

			and

			:<math>\frac{P \and P}{\therefore P}</math>

			where the rule is that wherever an instance of "<math>P \or P</math>" or "<math>P \and P</math>" appears on a line of a proof, it can be replaced with "<math>P</math>";

			or as the statement of a truth-functional tautology or [[theorem]] of propositional logic. The principle was stated as a theorem of propositional logic by [[Bertrand Russell\|Russell]] and [[Alfred Whitehead\|Whitehead]] in ''[[Principia Mathematica]]'' as:

			:<math>(P \or P) \to P \, </math>

			and

			: <math>(P \and P) \to P \, </math>

			where <math>P</math> is a [[proposition]] expressed in some logical system.

			==References==
			{{reflist}}

			[[Category:Rules of inference]]
			[[Category:Theorems in propositional logic]]

en>Mgvongoeden: /* References */ ref added

2012-07-01T11:25:27Z

References: ref added

New page

For everyone that owns a home, home improvement is an interesting and engaging topic. Although most of us want to live in the nicest place possible, it can be scary to think about home improvement. The following paragraphs will introduce you to a few basic home improvement tasks. Replace your air conditioning filter from time to time. If the filter gets clogged up, the air conditioning unit will have to work harder to cool your house, using up more energy. This will also cost you money on the time it stays on. You need to replace the filter once a month before it gets clogged. If you can afford it, go for hard wood floors and stay away from laminate. Laminate floors may look like real wood and be easier to clean, however they cannot be refinished. Having real wood floors means that in a decade or two, you or the new home owner will have the option of refinishing instead of replacing the floor. An over-the-range microwave is a great way to save space in an already cramped kitchen. Replacing the range hood, these microwave ovens come in a wide range of prices, and some have special features such as convection cooking. Most of these have a filter that recirculates air as opposed to having a vent. This means they are better for people who do not need a vent. Seal cracks and openings to save on heating bills, as well as preventing bugs from entering your home. Add sealants or caulk around windows, doors and baseboards. When caulk dries, it will help prevent bugs from entering and air from escaping. Think outside the box in terms of artwork. Instead of expensive objects of art, try using unconventional items such as 3D tiles. Use three dimensional tiles in two colors such as green and white and arrange them on the wall (rotating colors) behind your couch for an original kind of art. Anyone who owns a home can fix their own faucets which leak. If you can locate the leaks yourself and quickly fix them, you can save yourself some money and a whole lot of water. The savings you get from this will add up since your water bill will go down. Homes that have a nice view tend to sell for a lot more than similar houses without a view. There are some instances where the extra expense for a good view might not be worth it. Potential buyers may not care for the changes you make and may not want to pay more money for your home. You can reduce the costs associated with a project by putting in cabinets without hardware in your kitchen. Thankfully, it's possible to buy handles, drawer pulls, and cabinet knobs at a home improvement store or online. If you want something that you can install in only a few minutes, you can select modern knobs, designer-made accents, and pulls with a vintage look. Choose your color scheme based on something already present in the room. Pay attention to the drapes, rugs or upholstery and use these elements to create pleasant color patterns. For example, a green rug with hints of yellow and red can be a great basis for your color scheme. The other two shades should be secondary and used as accent colors. A real estate slump is the perfect time for homeowners to invest in renovations and additions. When homes aren't selling, it often means that contractors are not as busy as they would like, and they may discount their services. Remember that home improvements don't carry an expiration date. When you renovate your home during a slump, you save money and your home's worth will grow in the future. When looking for assistance with your home improvement projects, look for friends that have already been through their own projects, so that they can offer help and support if necessary. Always be prepared for the worst. This also holds true in this case. A stainless steel taping knife can be a priceless tool to have on hand for a sheetrock taping project. Knives of this sort make clean up easier and they also last longer. Clean this stainless steel knife well, and then store it properly. You'll be able to use it forever. You don't have to let a home improvement project overpower you. The advice in this article is designed to assist you from start to finish with any home improvement plans you have in mind, whether it's a DIY project or a task for a professional contractor. When you have virtually any inquiries about where by and how you can use show homes ideas ([http://www.homeimprovementdaily.com http://www.homeimprovementdaily.com/]), you possibly can contact us on the web-site.