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{{expert-subject|mathematics|date=February 2009}}
{{New Testament manuscript infobox
'''Interval boundary element method''' is classical [[boundary element method]] with the interval parameters.<BR>
| form  = Papyrus
Boundary element method is based on the following integral equation
| number = <math>\mathfrak{P}</math><sup>16</sup>
| image  =
| isize  =
| caption=
| name  = [[Oxyrhynchus Papyri|P. Oxy.]] 1009
| sign  =
| text  = [[Epistle to the Philippians|Philippians]] 3-4 †
| script = [[Greek language|Greek]]
| date   = 3rd/4th century
| found  = [[Egypt]]
| now at = [[Egyptian Museum]]
| cite  = [[Bernard Pyne Grenfell|B. P. Grenfell]] & [[Arthur Surridge Hunt|A. S. Hunt]], ''Oxyrhynchus Papyri'' VII, (London 1910), pp. 8-11
| size  =
| type  = [[Alexandrian text-type]]
| cat    = I
| hand  =
| note  =
}}


<math> c\cdot u=\int\limits_{\partial \Omega}\left(G\frac{\partial u}{\partial n} - \frac{\partial G}{\partial n}u\right)dS </math>
'''Papyrus 16''' (in the [[Biblical manuscript#Gregory-Aland|Gregory-Aland]] numbering), designated by <math>\mathfrak{P}</math><sup>16</sup>, is an early copy of the [[New Testament]] in [[Greek language|Greek]]. It was originally a [[papyrus]] [[Biblical manuscript|manuscript]] of the Pauline Corpus of letters, but now only contains [[Epistle to the Philippians|Philippians]] 3:10-17; 4:2-8.<ref name = philip1>{{Cite book
| last = Comfort
| first = Philip W.
| authorlink = Philip Comfort
| author2 = David P. Barrett
| title = The Text of the Earliest New Testament Greek Manuscripts
| publisher = Tyndale House Publishers
| year = 2001
| location = Wheaton, Illinois
| page = 93
| url =
| isbn = 978-0-8423-5265-9}} </ref> The manuscript has been [[Palaeography|paleographically]] assigned to the late 3rd century.<ref name = philip1/>  


The exact interval solution on the boundary can be defined in the following way:
== Description ==


<math> \tilde{u}(x)=\{u(x,p):c(p)\cdot u(p)=\int\limits_{\partial \Omega}\left(G(p)\frac{\partial u(p)}{\partial n} - \frac{\partial G(p)}{\partial n}u(p)\right)dS, p\in\hat{p} \}</math>
The manuscript is written in a documentary hand.<ref name = philip1/> There is about 37-38 lines per page.<ref name = philip1/> Grenfeld and Hunt conjectured that [[Papyrus 15|<math>\mathfrak{P}</math><sup>15</sup>]] and <math>\mathfrak{P}</math><sup>16</sup> might have been part of the same manuscript. Both manuscripts have the same formation of letters, line space, and punctuation.<ref name = philip1/>


In practice we are interested in the smallest interval which contain the exact solution set
The [[nomina sacra]] are written in an abbreviated way. The text was not corrected.<ref>Klaus Wachtel, Klaus Witte, [http://books.google.pl/books?id=gEzctvICPS0C&pg=RA1-PA1&lpg=RA1-PA1&dq= ''Das Neue Testament auf Papyrus: Gal., Eph., Phil., Kol., 1. u. 2. Thess., 1. u. 2 Tim., Tit., Phlm., Hebr''], Walter de Gruyter, 1994, p. XLI.</ref>


<math> \hat{u}(x)=hull \ \tilde {u}(x)=hull \{u(x,p):c(p)\cdot u(p)=\int\limits_{\partial \Omega}\left(G(p)\frac{\partial u(p)}{\partial n} - \frac{\partial G(p)}{\partial n}u(p)\right)dS, p\in\hat{p} \}</math>
The Greek text of this codex is a representative of the [[Alexandrian text-type]] (rather proto-Alexandrian). [[Kurt Aland|Aland]] placed it in [[Categories of New Testament manuscripts#Category I|Category I]].<ref name = Aland>{{Cite book
|last=Aland
|first=Kurt
|authorlink=Kurt Aland
|coauthors=[[Barbara Aland]]; Erroll F. Rhodes (trans.)
|title=The Text of the New Testament: An Introduction to the Critical Editions and to the Theory and Practice of Modern Textual Criticism
|publisher=[[William B. Eerdmans Publishing Company]]
|year=1995
|location=Grand Rapids
|page=97
|url=http://books.google.pl/books?id=2pYDsAhUOxAC&pg=PA97&lpg=PA97&dq=#v=onepage&q&f=false
|isbn=978-0-8028-4098-1}}</ref> This manuscript staying in great agreement with [[Codex Sinaiticus]] and [[Codex Vaticanus]].


In similar way it is possible to calculate the interval solution inside the boundary <math> \Omega </math>.
It is currently housed at the [[Egyptian Museum]] (JE 47424) in [[Cairo]].<ref name = Aland/><ref>{{Cite web|url=http://intf.uni-muenster.de/vmr/NTVMR/ListeHandschriften.php?ObjID=10016|title=Handschriftenliste|publisher=Institute for New Testament Textual Research|accessdate=23 August 2011|location=Münster}}</ref>


==References==
== See also ==
* T. Burczyński and J. Skrzypczyk, The fuzzy boundary element method: a new solution concept, Proceedings of XII Polish conference on computer methods in mechanics, Warsaw-Zegrze, Poland (1995), pp. 65–66.
* T. Burczynski, J. Skrzypczyk J. The fuzzy boundary element method: a new methodology. Series Civil Eng, Vol. 83. Gliwice: Sci Fasc of Silesian Tech Univ; 1995, pp. 25–42.
* J. Skrzypczyk, A Note on Interval Fredholm Integral Equations. Zeszyty Naukowe Politechniki Śląskiej, Seria Budownictwo, Z.85, pp.75-83, 1998
* T. Burczynski, J. Skrzypczyk, Fuzzy aspects of the boundary element method, Engineering Analysis with Boundary Elements, Vol.19, No.3, pp.209-216, 1997
* H. Witek, Boundary element method in analysis of civil engineering structures with uncertain parameters. Ph.D. Dissertation, Silesian University of Technology, Faculty of Civil Engineering, Poland, 2005
* B.F. Zalewski, R.L. Mullen, R.L. Muhanna, “Boundary Element Analysis of Systems Using Interval Methods”, Proceedings of the NSF Workshop on Modeling Errors and Uncertainty in Engineering Computations, Georgia Tech Savannah, February 2006.
* B.F. Zalewski and R.L. Mullen, "Interval Bounds on the Local Discretization Error in Boundary Element Analysis for Domains with Singular Flux", SAE 2008 Reliability and Robust Design in Automotive Engineering, SP-2170, Pages 237-246, 2008.
* B.F. Zalewski and R.L. Mullen, "Discretization Error in Boundary Element Analysis using Interval Methods", SAE 2007 International Transactions Journal of Passenger Cars: Mechanical Systems, Volume 116, Issue 6, Pages 1353-1361, 2008.
* B.F. Zalewski and R.L. Mullen, "Point-wise Discretization Errors in Boundary Element Method for Elasticity Problem", Third NSF Workshop on Reliable Engineering Computing, Pages 429-457, February 2008.
* B.F. Zalewski, “Uncertainties in the Solutions to Boundary Element Method: An Interval Approach”, Case Western Reserve University, Ph.D. Dissertation, © 2008.
* B.F. Zalewski and R.L. Mullen, “Local Discretization Error Bounds Using Interval Boundary Element Method”, International Journal for Numerical Methods in Engineering, Volume 78, Issue 4, April 2009, Pages 403-428.
* A. Piasecka Belkhayat, Interval boundary element method for 2D transient diffusion problem, Engineering Analysis with Boundary Elements, Volume 32, Issue 5, May 2008, Pages 424-430
* B.F. Zalewski, R.L. Mullena, and R.L. Muhanna, "Interval Boundary Element Method in the Presence of Uncertain Boundary Conditions, Integration Errors, and Truncation Errors", Engineering Analysis with Boundary Elements, Volume 33, Issue 4, April 2009, Pages 508-513. [http://dx.doi.org/10.1016/j.enganabound.2008.08.006]
* B.F. Zalewski, R.L. Mullen, and R.L. Muhanna, “Fuzzy Boundary Element Method for Geometric Uncertainty in Elasticity Problem”, SAE 2009 International Journal of Materials and Manufacturing, Volume 2, Issue 1, Pages 310-316, 2009.
* B.F. Zalewski and R.L. Mullen, “Worst Case Bounds on the Point-wise Discretization Error in Boundary Element Method for the Elasticity Problem”, Computer Methods in Applied Mechanics and Engineering, Volume 198, Issue 37-40, Pages 2996-3005, 2009.
* B.F. Zalewski and R.L. Mullen, “Worst Case Point-wise Discretization Error Bounds for Systems with Geometrically Induced Singular Flux Solutions Using Interval Boundary Element Method”, ASCE Journal of Engineering Mechanics, Volume 136, Issue 6, Pages 710-720, 2010.
* B.F. Zalewski, "Fuzzy Boundary Element Method for Material Uncertainty in Steady State Heat Conduction", SAE 2010 International Journal of Materials and Manufacturing, Volume 3, Issue 1, Pages 372-379, 2010.
* B.F. Zalewski and W.B. Dial, "Fuzzy Boundary Element Method with Uncertain Shear Modulus in Linear Plane Strain Elasticity", SAE 2011 International Journal of Materials and Manufacturing, Volume 4, Issue 1, Pages 947-956, 2011.
* A. Piasecka-Belkhayat, "Interval Boundary Element Method for Transient Diffusion Problem in Two-Layered Domain", Journal of Theoretical and Applied Mechanics, Volume 49, Issue 1, Pages 265-276, 2011.
* A. Piasecka-Belkhayat, "Interval Boundary Element Method for 2D Transient Diffusion Problem Using the Directed Interval Arithmetic", Engineering Analysis with Boundary Elements, Volume 35, Issue 3, Pages 259-263, 2011.


==See also==
* [[List of New Testament papyri]]
[[Interval finite element]]


[[Category:Numerical differential equations]]
== References ==
{{Reflist}}
 
== Further reading ==
 
* {{Cite book
| last = B. P.
| first = Grenfell
| authorlink = Bernard Pyne Grenfell
| coauthors = [[Arthur Surridge Hunt|A. S. Hunt]]
| title = Oxyrhynchus Papyri VII
| publisher =
| year = 1910
| location = London
| pages = 8–11
| url = http://www.archive.org/stream/pt7oxyrhynchuspa00grenuoft#page/8/mode/2up
| doi =
| id =
| isbn = }}
 
== External links ==
* [http://163.1.169.40/cgi-bin/library?e=d-000-00---0POxy--00-0-0--0prompt-10---4----ded--0-1l--1-en-50---20-about-1708--00031-001-1-0utfZz-8-00&a=d&c=POxy&cl=CL5.1.7&d=HASHf4150af583a6c27fa50c76 P. Oxy. VII 1009]
 
{{Grenfell and Hunt}}
 
{{DEFAULTSORT:Papyrus 0016}}
[[Category:New Testament papyri]]
[[Category:3rd-century biblical manuscripts]]
[[Category:Early Greek manuscripts of the New Testament]]
 
[[de:Papyrus 16]]
[[es:Papiro 16]]
[[it:Papiro 16]]
[[nl:Papyrus 16]]
[[pl:Papirus 16]]
[[pt:Papiro 16]]

Revision as of 15:49, 17 August 2014

Template:New Testament manuscript infobox

Papyrus 16 (in the Gregory-Aland numbering), designated by P16, is an early copy of the New Testament in Greek. It was originally a papyrus manuscript of the Pauline Corpus of letters, but now only contains Philippians 3:10-17; 4:2-8.[1] The manuscript has been paleographically assigned to the late 3rd century.[1]

Description

The manuscript is written in a documentary hand.[1] There is about 37-38 lines per page.[1] Grenfeld and Hunt conjectured that P15 and P16 might have been part of the same manuscript. Both manuscripts have the same formation of letters, line space, and punctuation.[1]

The nomina sacra are written in an abbreviated way. The text was not corrected.[2]

The Greek text of this codex is a representative of the Alexandrian text-type (rather proto-Alexandrian). Aland placed it in Category I.[3] This manuscript staying in great agreement with Codex Sinaiticus and Codex Vaticanus.

It is currently housed at the Egyptian Museum (JE 47424) in Cairo.[3][4]

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

Template:Grenfell and Hunt

de:Papyrus 16 es:Papiro 16 it:Papiro 16 nl:Papyrus 16 pl:Papirus 16 pt:Papiro 16

  1. ↑ 1.0 1.1 1.2 1.3 1.4 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. ↑ Klaus Wachtel, Klaus Witte, Das Neue Testament auf Papyrus: Gal., Eph., Phil., Kol., 1. u. 2. Thess., 1. u. 2 Tim., Tit., Phlm., Hebr, Walter de Gruyter, 1994, p. XLI.
  3. ↑ 3.0 3.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. ↑ Template:Cite web
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