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'''Georg Scheffers''' was a German [[mathematician]] specializing in [[differential geometry]]. He was born on November 21, 1866 in the village of Altendorf near [[Holzminden]] (today incorporated into Holzminden). Scheffers began his university career at the [[University of Leipzig]] where he studied with [[Felix Klein]] and [[Sophus Lie]].  Scheffers was a coauthor with Lie for three of the earliest expressions of [[Lie theory]]:
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* ''Lectures on [[Differential equation]]s with known [[Infinitesimal transformation]]s'' (1893),
* ''Lectures on [[Continuous group]]s'' (1893), and
* ''Geometry of [[Contact geometry|Contact Transformations]]'' (1896).
 
All three are now available online through [[archive.org]]; see External links section below.
 
In 1896 Scheffers became [[docent]] at the [[Technical University of Darmstadt]], where he was raised to [[professor]] in 1900. From 1907 to 1935, when he retired, Scheffers was a professor at the [[Technical University of Berlin]].
 
In 1901 he published a German translation of the French textbook on analysis by [[Joseph Serret]]. The title was ''Anwendung der Differential- und Integralrechnung auf die Geometrie'' (application of differential and integral calculus to geometry). This [[textbook]] consisted of two volumes, one on [[curve]]s and the second on [[surface]]s. A second edition was published in 1910 (volume 2, 1913), and a third edition in 1922.
 
Another very successful book was prepared for students of science and technology: ''Lehrbuch der Mathematik'' (textbook of mathematics). It provided an introduction to [[analytic geometry]] as well as [[calculus]] of [[derivative]]s and [[integral]]s. In 1958 this book was republished for the fourteenth time.
 
Scheffers is known for an article on special transcendental curves (including [[W-curve]]s) which appeared in the [[Klein's encyclopedia|''Enzyklopädie der mathematischen Wissenschaften'']] in 1903: "Besondere transzendenten Kurven" (special transcendental curves). He wrote on translation surfaces for [[Acta Mathematica]] in 1904: "Das Abelsche und das Liesche Theorem über Translationsflächen" (the theorem of [[Niels Henrik Abel|Abel]] and [[Sophus Lie|Lie]] on translation surfaces).
 
Other books written by Scheffers are ''Lehrbuch der Darstellenden Geometrie'' (textbook on descriptive geometry) (1919), ''Allerhand aus der zeichnenden Geometrie'' (1930), and ''Wie findet und zeichnet man Gradnetze von Land- und Sternkarten?'' (1934).
 
Georg Scheffers died August 12, 1945, in [[Berlin]].
==Hypercomplex numbers==
{{main|hypercomplex number}}
In 1891 Georg Scheffers contributed his article "Zurück-führung komplexer Zahlensysteme auf typische formen" to [[Mathematische Annalen]] (39:293&ndash;390).
This article addressed a topic of considerable interest in the 1890s and contributed to the development of [[Abstract algebra|modern algebra]]. Scheffers distinguishes between a "Nichtquaternion system" (Nqss) and a Quaternion system (Qss). Scheffers characterizes the Qss as having three elements
<math>e_1,\ e_2,\ e_3</math> that satisfy (p 306)
:<math>e_1e_2-e_2e_1 = 2e_3, \quad e_2e_3-e_3e_2 = 2e_1, \quad e_3e_1-e_1e_3=2e_2.</math>
In today's language, Scheffers' Qss has the [[quaternion]] algebra as a [[subalgebra]].
 
Scheffers anticipates the concepts of [[direct product]] of algebras and [[direct sum of modules#Direct sum of algebras|direct sum of algebras]] with his section (p 317) on reducibility, addition, and multiplication of systems.
Thus Scheffers pioneered the structural approach to algebra.
 
Though the article covers new ground with its exploration of Nqss, it is also a [[literature review]] going back to the work of [[Hermann Hankel]]. In §14 (p 386) Scheffers reviews both German and English authors on hypercomplex numbers. In particular, he cites [[Eduard Study]]’s work of 1889. For volume 41 of ''Mathematische Annalen'' Scheffers contributed a further short note, this time including reference to 1867 work by [[Edmond Laguerre]] on linear systems, a rich source of hypercomplex numbers.
 
==References==
* Werner Burau (1975) "Georg Scheffers" [[Dictionary of Scientific Biography]]
* {{MacTutor Biography|id=Scheffers}}
* [http://genealogy.math.ndsu.nodak.edu/id.php?id=19534 Georg Scheffers] at [[Mathematics Genealogy Project]].
==External links==
* 1891: [http://www.archive.org/details/vorlesuberdiff00liesrich ''Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen''] from [[archive.org]]
** (lectures on [[differential equation]]s with known infinitesimal transformations)
* 1893: [http://www.archive.org/details/vorlescontingrup00liesrich ''Vorlesungen über continuerliche Gruppen'']
** (lectures on continuous [[group (mathematics)|groups]]), and
* 1896: [http://www.archive.org/details/geometriederber00liesuoft ''Geometrie der Berührungstransformationen'']
** (geometry of [[Contact geometry|contact transformations]])
 
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Scheffers, Georg
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Mathematician
| DATE OF BIRTH    = November 21, 1866
| PLACE OF BIRTH    = Altendorf bei Holzminden, [[Germany]]
| DATE OF DEATH    = August 12, 1945
| PLACE OF DEATH    = [[Berlin]], [[Germany]]
}}
{{DEFAULTSORT:Scheffers, Georg}}
[[Category:German mathematicians]]
[[Category:Differential geometers]]
[[Category:1866 births]]
[[Category:1945 deaths]]

Latest revision as of 17:31, 10 January 2015

We see the same scenery every single day in our houses, and that's why so many homeowners decide that it's time for a little remodeling. Painting the walls, rearranging the furniture, and going with new molding -- these can seem like easy enough tasks from the start. However, many people find that they're daunting. No matter what your goals are, check out these home- improvement tips.

If you have an older home, you may find that there are hardwood floors underneath any carpeting currently installed. Many homes prior to 1950s were required to have hard wood as their sub-floor. This means that there may be gold in those floorboards. They may need some refinishing, but not having to pay to have them installed would be a huge money saver.

A nice addition to any home is a nicely crafted fence. If your home is lacking a fence, you might decide it is time to build one. Fences look great in just about any home and can help raise the property value of your home. You can also feel a little more secure, as having a fence can help keep random strangers off your property.

When it comes to home improvement, consider replacing your windows and doors. This not only has a chance of greatly improving the value of the home, but may also severely decrease the amount of money required to keep your house warm and dry. You can also add extra security with new doors and windows.

Maybe the house or apartment you're living in is great, but the view to the outside is not. If you have any offending sights that you wish to conceal, consider putting some plants or flowers on your window sill or in a window box. If you would like to take drastic measures in concealing your offending view, plant some shrubs or large trees in front of your window. This can ensure that you have a nicer view.

Decorating your walls on a tight budget can be quite simple and requires no painting. Consider buying a large quilt or an interesting piece of tapestry and affixing it to your wall. This can be a nice conversation piece on your wall, and provide you with a nice piece of your personality to look at.

If your front lawn features a large, full tree, you can add upward-facing outdoor accent lighting to emphasize the beautiful texture of the leaves and trunk. Upward lighting is also a good choice to accentuate details like trellises or archways. Outdoor accent lighting also adds significant value to the home.

For a personalized and one-of-a-kind wall covering, consider designing and ordering your own customized wallpaper online or through a local design studio. New websites offer a variety of options for wallpaper design, including enlarged photographs, murals, or even your children's artwork delivered to your door in a matter of weeks.

Resurface the concrete walkways and driveways around your home for a new look. Replacing concrete can get incredibly expensive and doesn't add much to the curb appeal of your home. Consider resurfacing with cobblestone or brick which will cost a fraction of replacing concrete and will look like a million bucks.

Make your renovation plans, then stick to them. Contractors become gun-shy about doing work when the owner constantly changes and tweaks the plans. Even though a huge renovation project seems easier when you break it up, piece-mealing it slows things down because the only thing workers can depend on is the plans changing.

Something that can be done in the way of home improvement is to theme your rooms. For example you may want to have a teddy bear theme in one of your bathrooms and have a native Indian theme in your living room. Doing this will add a level of flare and comfort to your home.

Allow room for all the debris that comes from a renovation. The process of tearing down and rebuilding creates a whole lot of debris that needs to go somewhere. Plan ahead for how you will remove it from your property and safely dispose of it before getting your project started.

When you need to find someone to do some work on your home, but no one has any recommendations, be sure to do a thorough search on the company on the internet. You should check reviews from previous customers, as well as to check the better business bureau. This could save you a lot of money and a big hassle in the future.

Something that seems on the surface to be so simple can end up being incredibly complicated if you're not aware of how to handle a task. Painting can turn into a nightmare; adding more lights could leave you in the dark. The tips in this article will help you learn about home improvement, so put them to good use.

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