Kempner function and Bangladesh University of Engineering and Technology: Difference between pages

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{{Infobox_University

'''The Smarandache function''' <math>S \colon \mathbb{Z}^+ \to \mathbb{Z}^+</math> in [[mathematics]] is defined for given ''n'' as the smallest [[integer]] ''S(n)'' such that the [[factorial]] ''S''(''n'') is divisible by''n''.<ref name="smf1">

|image_name= BUET logo.png

{{cite book

|

| author = Dumitrescu C, Popescu M, Seleacu V, Tilton H

|name=Bangladesh University of Engineering and Technology

| title = The Smarandache Function in Number Theory

|motto=

| publisher = Erhus University Press

|established=[[1962]] see history below |

| year = 1996

|type=[[Public university|Public]], [[Coeducational]]

| isbn = 1879585472

|chancellor=[[Dr. Fakhruddin Ahmed]] |

|vice_chancellor=Dr AMM Safiullah

|city=[[Dhaka]] |

|country=[[Bangladesh]]

|students=4000 |

|faculty=350 |

|address=Palashi,Dhaka-1000 |

|telephone=880.2.966 5650-80 |

|campus=[[urban area|Urban]], 76.85 acres (0.31 km²)|

|website=[http://www.buet.ac.bd/ www.buet.ac.bd]

}}

'''Bangladesh University of Engineering and Technology''' ([[Bengali]]: বাংলাদেশ প্রকৌশল বিশ্ববিদ্যালয় ''Bangladesh Prokoushol Bishshobiddalôe'') or '''BUET''' is a Government [[Engineering]] University in [[Bangladesh]]. It is the oldest Engineering institution in the region, and is regarded to be among the top universities for technical education.{{cn}} The medium of instructions in this top graded university{{cn}}is English. Many international students prefer this university for offering high standard of education and low expense.{{cn}}

</ref><ref name="smf2">

{{cite book

| author = Ashbacher C, Popescu M

| title = An Introduction to the Smarandache Function

| publisher = Erhus University Press

| year = 1995

| isbn = 1879585499

}}

</ref><ref name="smf3">

{{cite journal

| url = http://dx.doi.org/10.1109/ISPDC.2004.15

| author = Tabirca S, Tabirca T, Reynolds K, Yang LT

| title = Calculating Smarandache function in parallel

| journal = Parallel and Distributed Computing, 2004. Third International Symposium on Algorithms, Models and Tools for Parallel Computing on Heterogeneous Networks,

| pages = pp.79-82

| date = 2004-07-05

}}

</ref><ref name="smf4">

{{cite web

| author = Mehendale DP

| title = The Classical Smarandache Function and a Formula for Twin Primes

| url = http://arxiv.org/abs/math/0502384

| work = arXiv

| year = 2005

}}

</ref><ref name="math_world">{{MathWorld|title=Smarandache Constants|urlname=SmarandacheConstants}}

</ref><ref name="Smfunctions">

{{cite web

| url = http://www.gallup.unm.edu/~smarandache/CONSTANT.TXT

| title = Constants Involving the Smarandache Function

}}

</ref> For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!, therefore ''S''(8)=4. Historically, the function first was considered by Lucas in [[1883]]<ref name="history1">

{{cite journal

| url =

| author = Lucas E

| title = Question Nr. 288

| journal = Mathesis

| volume = 3

| pages = 232

| year = 1883

}}

</ref>, later by Neuberg in [[1887]]<ref name="history2">

{{cite journal

| url =

| author = Neuberg J

| title = Solutions de questions proposées, Question Nr. 288

| journal = Mathesis

| volume = 7

| pages = 68-69

| year = 1887

}}

</ref>, and Kempner in [[1918]]<ref name="history3">

{{cite journal

| url =

| author = Kempner AJ

| title = Miscellanea

| journal = Amer. Math. Monthly

| volume = 25

| pages = 201-210

| year = 1918

}}

</ref>, and subsequently rediscovered by [[Florentin Smarandache]] in [[1980]]<ref name="history4">

{{cite journal

| url =

| author = Smarandache F

| title = A Function in Number Theory

| journal = Analele Univ. Timisoara, Ser. St. Math.

| volume = 43

| pages = 79-88

| year = 1980

}}

</ref><ref>Jonathan Sondow and Eric Weisstein (2006) [http://mathworld.wolfram.com/SmarandacheFunction.html "Smarandache Function"] at [[MathWorld]].</ref>. A profound study of Smarandache function ''S''(''n'') would contribute to the study of [[prime number]]s in accordance with the following property: <blockquote>if ''p'' is a number greater than 4, then ''p'' is a prime [[iff|if and only if]] ''S''(''p'')=''p''<ref name="muller1990">

{{cite journal

| author = Muller R

| title = Editorial

| journal = Smarandache Function Journal

| url = http://www.gallup.unm.edu/~smarandache/SFJ1.pdf

| volume = 1

| issue =

| pages = 1

| year = 1990

}}

</ref></blockquote>

About five thousand [[student]]s are enrolled in [[undergraduate]] and [[postgraduate]] studies [[engineering]], [[architecture]], [[planning]] and [[science]] in this institution. The total number of [[teacher]]s is over 400. The University has continued to expand over the last three [[decade]]s with the construction of new academic buildings, [[auditorium]] complex, halls of residence etc.

A list of ''sixteen Smarandache constants'' denoted ''s''₁ to ''s''₁₆ have been defined with the use of ''S''(''n'') and they should not be confused with the [[Smarandache constant]], which is the smallest solution to the generalized [[Andrica's conjecture]].

==History==

[[Image:SmarandacheFunction.PNG|thumb|374px|right|The Smarandache function]]

[[Image:BUET monument 1.JPG|right|thumb|250px|Language movement monument in front of BUET]]

'''BUET''' was established in the late 19th century as a [[Cadastre|survey]] [[school]] for training [[Surveyor (surveying)|surveyor]]s. ''Dhaka Survey School'' was established at Nalgola, in Old Dhaka in 1876 to train surveyors for the then [[Government]] of [[Bengal]] of [[British India]]. Later, generous grants from Nawab Ahsanullah, a renowned [[Islam|Muslim]] patron of Education and member of the Nawab family of Dhaka enabled it to expand as a full fledged [[engineering]] school. ''Ahsanullah School of Engineering'' offered three-year [[diploma]] courses in [[Civil Engineering]], [[Electrical Engineering]] and [[Mechanical Engineering]]. In recognition of the generous financial contribution from the then [[Nawab]] of Dhaka, it was named after his father [[Khawja Ahsanullah]]. It moved to its present premises in 1912.

[[Image:Reg building.JPG|thumb|140px|left|Dr. M.A. Rashid Building]]

''The first Smarandache constant'' <ref name="cc1">

{{cite journal

| author = Cojocaru I, Cojocaru S

| title = The First Constant of Smarandache

| journal = Smarandache Notions Journal

| url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf

| volume = 7

| pages = 116-118

| year = 1996

}}

</ref> {{OEIS|id=A048799}} is defined as

After the [[partition of India]] in 1947, it was upgraded to ''Ahsanullah Engineering College'', as a [[Faculty (university)|Faculty]] of [[Engineering]] under the [[University of Dhaka]], offering four-year [[bachelor's]] courses in Civil, Electrical, Mechanical, [[chemical engineering|Chemical]] and [[metallurgy|Metallurgical Engineering]].

:<math>s_1=\sum_{n=2}^{\infty}\frac{1}{[S(n)]!}=1.09317 \ldots</math>

In 1962, it was renamed as ''East Pakistan University of Engineering and Technology'' (EPUET). A partnership with the ''Agricultural and Mechanical College of Texas'' (renamed [[Texas A&M University]] in 1963) was forged, and professors from A&M came to teach and to formulate the curriculum. During this period, EPUET offered courses in Mechanical, Electrical, Civil, and Chemical engineering, and Architecture.

''The second Smarandache constant'' {{OEIS|id=A048834}} is defined as

:<math>s_2=\sum_{n=2}^{\infty}\frac{S(n)}{n!}\approx 1.71400629359162 \ldots</math>

After the [[Bangladesh Liberation War|liberation war]] of 1971, [[Bangladesh]] became independent, and ''EPUET'' was renamed to ''Bangladesh University of Engineering and Technology (BUET)'''.

and it is an [[irrational number]].<ref name="cc2">

{{cite journal

| author = Cojocaru I, Cojocaru S

| title = The Second Constant of Smarandache

| journal = Smarandache Notions Journal

| url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf

| volume = 7

| pages = 119-120

| year = 1996

}}

</ref>

==Campus==

''The third Smarandache constant''<ref name="cc3">

The BUET campus is in the heart of the [[capital]] city of Dhaka. It has a compact campus with ''halls'' of residence within walking distances of the academic buildings. At present the campus occupies 76.85 [[acre]]s (311,000 [[metre|m²]]) of land. The academic area is confined in and around the old campus occupying 30.24 [[acre]]s (122,000 m²) of land defined by shahid sharani, Bakshi Bazar road and the [[Asian Highway]]. This area accommodates five faculties, two institutes, the administrative building, the central [[library]], main [[sports]] facilities, the [[auditorium]] complex, the BUET Club and eighty two units of residential accommodation of [[teacher]]s, staff and employees and the [[Vice-Chancellor]]'s bungalow.

{{cite journal

| author = Cojocaru I, Cojocaru S

| title = The Third and Fourth Constants of Smarandache

| journal = Smarandache Notions Journal

| url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf

| volume = 7

| pages = 121-126

| year = 1996

}}

</ref> {{OEIS|id=A048835}} is defined as

==Academics==

:<math>s_3=\sum_{n=2}^{\infty}\frac{1}{\prod_{i=2}^{n}S(i)}\approx 0.719960700043 \ldots</math>

===Faculties and departments===

[[Image:BUET EME Building.jpg|right|thumb|200px|Electrical and Mechanical Engineering Building at BUET]]

[[Image:Civil Engineering Building of BUET seen from EME Building.JPG|right|thumb|200px|Civil engineering building]]

[[Image:BUET archi front.JPG|right|thumb|200px|Architecture building]]

*Faculty of Architecture & Planning

**Dept. of Architecture (Arch)

**Dept. of Urban & Regional Planning (URP)

**Dept. of Humanities (Hum)

* Faculty of Civil Engineering

**Dept. of Civil Engineering (CE)

**Dept. of Water Resources Engineering (WRE)

* Faculty of Electrical & Electronic Engineering

**Dept. of Electrical & Electronic Engineering (EEE)

**Dept. of Computer Science & Engineering (CSE)

* Faculty of Engineering

**Dept. of Chemical Engineering (ChE)

**Dept. of Materials & Metallurgical Engineering (MME)

**Dept. of Chemistry (Chem)

**Dept. of Mathematics (Math)

**Dept. of Physics (Phys)

**Dept. of Petroleum & Mineral Resources Engineering (PMRE)

* Faculty of Mechanical Engineering

**Dept. of Mechanical Engineering (ME)

**Dept. of Naval Architecture & Marine Engineering (NAME)

**Dept. of Industrial & Production Engineering (IPE)

===Admissions===

''The [[Series_(mathematics)|series]] <math>s_4(\alpha)=\sum_{n=2}^{\infty}\frac{n^{\alpha}}{\prod_{i=2}^{n}S(i)}</math>

Undergraduate admission in BUET is very much competitive. After completion of higher secondary level(HSC) education, a student can submit his application for undergraduate admission if he fulfils the minimum requirement. Students with a minimum percentage in Mathematics, Physics, Chemistry and English of their higher secondary examination are allowed to appear in the admission test. The formidable screening process allows only 4500 - 5000 students to sit for the admission test out of approximately 40,000 applicants. After the intricate admission test, the best 825 students get the opportunity to study in this prestigious institution. However, for admission to M.S. and Ph. D. programs candidates are required to appear in interviews.

converges for a fixed real number &alpha; &ge; 1. Since ''s''₄ is a function of &alpha; it is not a single constant, but an infinite list of them. The values for small &alpha; have been computed

:<math>s_4(1) \approx 1.72875760530223 \ldots</math> {{OEIS|id=A048836}}

:<math>s_4(2) \approx 4.50251200619296 \ldots</math> {{OEIS|id=A048837}}

:<math>s_4(3) \approx 13.0111441949445 \ldots</math> {{OEIS|id=A048838}}

International students must submit satisfactory TOEFL or IELTS score as a proof of their English proficiency as a basic requirements for admission.

''The fifth Smarandache constant'' converges to an [[irrational number]].<ref name="s5">

{{cite journal

| author = Sandor J

| title = On The Irrationality Of Certain Alternative Smarandache Series

| journal = Smarandache Notions Journal

| url = http://www.gallup.unm.edu/~smarandache/SNJ8.pdf

| volume = 8

| pages = 143-144

| year = 1997

}}

</ref>

:<math>s_5=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}S(n)}{n!}</math>

==Student life==

Burton<ref name="burton1995">

===Halls of residence===

{{cite journal

[[Image:Sony Memorial Sculpture BUET by Ragib Hasan.jpg|right|thumb|Sabekunnahar Sony Memorial Sculpture, beside Titumir Hall.]]

| author = Burton E

Student dormitories (called ''halls'' in BUET) are important features in campus life. There are eight residential halls to provide housing for BUET students. The ''Shahid Smriti Hall'' is reserved for young [[teacher]]s who do not have an official quarter in the campus and for postgraduate students. The ''Ladies Hall'' is for female students, and the remaining six halls are for male students studying in the undergraduate level.

| title = On Some Series Involving the Smarandache Function

| journal = Smarandache Function Journal

| url = http://www.gallup.unm.edu/~smarandache/SFJ6.pdf

| volume = 6

| issue =

| pages = 13-15

| year = 1995

}}

</ref> showed that the series

:<math>s_6=\sum_{n=2}^{\infty}\frac{S(n)}{(n+1)!}</math>

converges and is bounded by 0.218282<''s''₆<0.5.

These halls were built in different times. Different architectural designs of different ages are prominent in the construction of these halls.

Dumitrescu and Seleacu<ref name="smf1"/> showed that the series

:<math>s_7(r)=\sum_{n=r}^{\infty}\frac{S(n)}{(n+r)!}</math>

and

:<math>s_8(r)=\sum_{n=r}^{\infty}\frac{S(n)}{(n-r)!}</math>

converge for <math>r \in \mathbb{Z}^+ </math>.

Administrative head of a hall is its [[provost (education)|provost]], usually chosen from the senior [[teacher]]s of different faculties. 3 Assistant Provosts are also in the authority of the Halls.

The same authors<ref name="smf1"/> show that the series

:<math>s_9=\sum_{n=2}^{\infty}\frac{1}{\sum_{i=2}^{n}\frac{S(i)}{i!}}</math>

is convergent.

Most of the halls are named after prominent personalities in the history of Bangladesh. These are listed below:

The series

:<math>s_{10}(\alpha)=\sum_{n=2}^{\infty}\frac{1}{[S(n)]^{\alpha}\sqrt{S(n)!}}</math>

*Ahsanullah Hall

and

*Ladies Hall

:<math>s_{11}(\alpha)=\sum_{n=2}^{\infty}\frac{1}{[S(n)]^{\alpha}\sqrt{[S(n)-1]!}}</math>

*Dr. M. A. Rashid Hall

converge for &alpha; > 1. <ref name="smf1"/><ref name="burton1995"/><ref>In the article published at [[MathWorld]] the series <math>s_{11}</math> is defined in wrong way using <math>[S(n)+1]!</math> in the [[denominator]] and it is obvious that if ''s''₁₀ converges with [''S''(''n'')]! in the denominator, then this will be true for [''S''(''n''+''k'')]!, where ''k// is any natural number.</ref>

*Nazrul Islam Hall

*Sher-e-Bangla Hall

*Suhrawardy Hall

*Titumir Hall

*Shahid Smriti Hall

===Activities===

*BUET participated in the [[ACM International Collegiate Programming Contest]] every year since 1998, achieving the 11th position in the whole world in 2000.

*BUET participated in the [http://www.buet.ac.bd/eee/news/news1/news.html IEEE Myron Zucker Student Design Contest] in 2001, placed first,[[Chicago]], [[USA]].

*BUET participated in [http://www.energychallenge.org/2003FEC.htm The 2003 International Future Energy Challenge] in 2003 and was awarded ''Honorable Mention'' award.

*BUET participated in the ''Asia-Pacific Robot Contest'' [[ABU ROBOCON 2005]] [[Beijing]] and was awarded the Panasonic Award.

*BUET participated in [http://www.energychallenge.org The 2005 International Future Energy Challenge] in 2005 and was awarded ''Honorable Mention'' award.

*BUET student won the 2006 IEEE[http://www.ieee.org] Region 10 Student Paper Contest.

*BUET won the [http://www.ieee.org/organizations/foundation/2007news.html Student Enterprise Award] in 2007 from IEEE[http://www.ieee.org]

==Institutes and Centers==

There are total nine institutes and centers in BUET.

* Institute of Information & Communication Technology (IICT)

* Institute of Appropriate Technology (IAT)

* Institute of Water and Flood Management (IWFM)

* International Training Network Centre (ITN)

* Centre for Energy Studies (CES)

* Center for Environmental & Resource Management (CERM)

* Center for Biomedical Engineering Research (CBER)

* Bureau of Research, Testing & Consultation (BRTC)

* Accident Research Centre (ARC)

==Distinguished Alumni==

A large number of BUET alumni are working in the industry and academia both in Bangladesh and outside Bangladesh. Famous BUET alumni and former faculty members include Structural Engineer [[Fazlur R. Khan|Fazlur Rahman Khan]] {{Fact|date=March 2007}}, who designed [[John Hancock Center]] (100-story) and [[Sears Tower]](108-story) , the tallest building in the United States since its completion in 1974.

==External links==

*[http://www.buet.ac.bd Official BUET Web Site]

If <math>f:\mathbb{N} \rightarrow \mathbb{R}</math> is a function satisfying the condition <math>f(t) \leq \frac{c}{t^{\alpha}d\left(t!\right)-d\left((t-1)!\right)}</math> for ''t'' a nonzero natural number, ''d''(''x'') the number of divisors of ''x'', and the given constants &alpha; > 1, ''c'' > 1. Then the series

:<math>s_{12}(f)= \sum_{n=1}^{\infty}f\left(S(n)\right)</math>

is convergent.<ref name="burton1996"/>

The series

:<math>s_{13}=\sum_{n=1}^{\infty} \left(\prod_{k=1}^{n}S(k)!\right)^{-\frac{1}{n}}</math>

is convergent.<ref name="burton1996"/>

The series

:<math>s_{14}(\alpha)=\sum_{n=1}^{\infty}\frac{1}{S(n)!\sqrt{S(n)!}[\log{S(n)}]^{\alpha}}</math>

is convergent for ''α''>1.

The series

:<math>s_{15}=\sum_{n=1}^{\infty}\frac{2^n}{S(2^{n})!}</math>

is convergent.<ref name="burton1996">

{{cite journal

| author = Burton E

| title = On Some Convergent Series

| journal = Smarandache Notions Journal

| url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf

| volume = 7

| issue = 1-3

| pages = 7-9

| year = 1996

}}

</ref>

{{Public Universities of Bangladesh}}

The series

:<math>s_{16}(\alpha)=\sum_{n=1}^{\infty}\frac{S(n)}{n^{1+ \alpha}}</math>

is convergent for ''α''>1.

[[Category:Public universities of Bangladesh]]

==References and notes==

[[Category: Educational institutions established in 1962]]

{{reflist|2}}

==See also==

*[[Florentin Smarandache]]

:{{Planetmath|id=9363|title=Smarandache Function}}

[[bn:বাংলাদেশ প্রকৌশল বিশ্ববিদ্যালয়]]

[[Category:Mathematics]]

[[de:Bangladesh University of Engineering and Technology]]

[[Category:Number theory]]

[[pl:BUET]]

[[Category:Prime numbers]]