This document provides an introduction to combinatorics and counting principles. It defines the fundamental counting principle, multiplication principle, and addition principle for counting the number of ways events can occur. Examples are provided to illustrate applications of these principles, including counting plate numbers, selecting class officers, and travel options. Permutations are also introduced with examples of evaluating permutation expressions. Homework problems are assigned from textbook chapters on combinatorics.
An Overview of Separation Axioms by Nearly Open Sets in Topology.IJERA Editor
Abstract: The aim of this paper is to exhibit the research on separation axioms in terms of nearly open sets viz
p-open, s-open, α-open & β-open sets. It contains the topological property carried by respective ℘ -Tk spaces (℘
= p, s, α & β; k = 0,1,2) under the suitable nearly open mappings . This paper also projects ℘ -R0 & ℘ -R1
spaces where ℘ = p, s, α & β and related properties at a glance. In general, the ℘ -symmetry of a topological
space for ℘ = p, s, α & β has been included with interesting examples & results.
Quantum optical models in noncommutative spacesSanjib Dey
Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
bT-Locally Closed Sets and bT-Locally Continuous Functions In Supra Topologic...IOSR Journals
The aim of this paper is to introduce a decompositions namely supra bT- locally closed sets and define supra bT-locally continuous functions. This paper also discussed some of their properties.
An Overview of Separation Axioms by Nearly Open Sets in Topology.IJERA Editor
Abstract: The aim of this paper is to exhibit the research on separation axioms in terms of nearly open sets viz
p-open, s-open, α-open & β-open sets. It contains the topological property carried by respective ℘ -Tk spaces (℘
= p, s, α & β; k = 0,1,2) under the suitable nearly open mappings . This paper also projects ℘ -R0 & ℘ -R1
spaces where ℘ = p, s, α & β and related properties at a glance. In general, the ℘ -symmetry of a topological
space for ℘ = p, s, α & β has been included with interesting examples & results.
Quantum optical models in noncommutative spacesSanjib Dey
Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
bT-Locally Closed Sets and bT-Locally Continuous Functions In Supra Topologic...IOSR Journals
The aim of this paper is to introduce a decompositions namely supra bT- locally closed sets and define supra bT-locally continuous functions. This paper also discussed some of their properties.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The field of designing subexponential time parameterized algorithms has gained a lot of momentum lately. While the subexponential time algorithm for Planar-k-Path (finding a path of length at least k on planar graphs) has been known for last 15 years. There was no such algorithms known on directed planar graphs. In this talk, I will survey this journey of designing subexponential time parameterized algorithms for finding a path of length at least k in planar undirected graphs to planar directed graphs; highlighting the new tools and techniques that got developed on the way.
Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite F...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Tim Maudlin: New Foundations for Physical GeometryArun Gupta
New Foundations for Physical Geometry
Original URL: http://www.unil.ch/webdav/site/philo/shared/summer_school_2013/NYU.ppt
Tim Maudlin
NYU
Physics & Philosophy of Time
July 25, 2013
Using ψ-Operator to Formulate a New Definition of Local Functionpaperpublications3
Abstract:In this paper, we use ψ-operator in order to get a new version of local function. The concepts of maps, dense, resolvable and Housdorff have been investigated in this paper, as well as modified to be useful in general الخلاصة: استخدمنافيهذاالبحثالعمليةابسايمنأجلالحصولعلىنسخهجديدهمنالدالةالمحلية.وقدتمالتحقيقفيمفاهيمالدوال, الكثافة,المجموعاتالقابلةللحلوفضاءالهاوسدورف,وكذلكتعديلهالتكونمفيدةبشكلعام. Keywords:ψ-operator, dense, resolvable and Housdorff.
Title:Using ψ-Operator to Formulate a New Definition of Local Function
Author:Dr.Luay A.A.AL-Swidi, Dr.Asaad M.A.AL-Hosainy, Reyam Q.M.AL-Feehan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Probability - Binomial distribution - Theorem of total probability - random variable probability distributions - Bayes' Theorem -Theorem of total probability - Independent Events - Multiplication Theorem on Probability - conditional probabilty - Independent Events
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The field of designing subexponential time parameterized algorithms has gained a lot of momentum lately. While the subexponential time algorithm for Planar-k-Path (finding a path of length at least k on planar graphs) has been known for last 15 years. There was no such algorithms known on directed planar graphs. In this talk, I will survey this journey of designing subexponential time parameterized algorithms for finding a path of length at least k in planar undirected graphs to planar directed graphs; highlighting the new tools and techniques that got developed on the way.
Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite F...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Tim Maudlin: New Foundations for Physical GeometryArun Gupta
New Foundations for Physical Geometry
Original URL: http://www.unil.ch/webdav/site/philo/shared/summer_school_2013/NYU.ppt
Tim Maudlin
NYU
Physics & Philosophy of Time
July 25, 2013
Using ψ-Operator to Formulate a New Definition of Local Functionpaperpublications3
Abstract:In this paper, we use ψ-operator in order to get a new version of local function. The concepts of maps, dense, resolvable and Housdorff have been investigated in this paper, as well as modified to be useful in general الخلاصة: استخدمنافيهذاالبحثالعمليةابسايمنأجلالحصولعلىنسخهجديدهمنالدالةالمحلية.وقدتمالتحقيقفيمفاهيمالدوال, الكثافة,المجموعاتالقابلةللحلوفضاءالهاوسدورف,وكذلكتعديلهالتكونمفيدةبشكلعام. Keywords:ψ-operator, dense, resolvable and Housdorff.
Title:Using ψ-Operator to Formulate a New Definition of Local Function
Author:Dr.Luay A.A.AL-Swidi, Dr.Asaad M.A.AL-Hosainy, Reyam Q.M.AL-Feehan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Probability - Binomial distribution - Theorem of total probability - random variable probability distributions - Bayes' Theorem -Theorem of total probability - Independent Events - Multiplication Theorem on Probability - conditional probabilty - Independent Events
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.
1. Assume that an algorithm to solve a problem takes f(n) microse.docxSONU61709
1. Assume that an algorithm to solve a problem takes f(n) microseconds for some function f of the input size n. For each time t labeled across the top, determine the exact largest value of n which can be solved in time f(n) where f(n) ≤ t. Use a calculator! You will find it helpful to convert the t values to microseconds, and you may find it helpful to insert a row for n. Note that “lg n” is the log2 n. Note that the only row you can’t write out the values for fully is the “lg n” row—only there may you write 2x for the appropriate value of x. Use the Windows built-in scientific calculator (under Accessories menu) as necessary. A couple values are filled in to get you started. Important: “exact values” means precisely that. Check your answers with values above and below!
Time t =
f(n) =
1 second
1 hour
1 day
1 month
=30 days
n2
1,609,968
lg n
n3
2n
n lg n
2,755,147,513
2. Use loop counting to give a O( ) characterization of each of the following loops basing each upon the size of its input:
a. Algorithm Loop1(n):
s ← 0
for i ← 1 to n do
s ← s + i
b. Algorithm Loop2(p):
p ← 1
for i ← 1 to 2n do
p ← p * i
c. Algorithm Loop3(n):
p ← 1
for j ← 1 to n2 do
p ← p * i
d. Algorithm Loop4(n):
s ← 0
for j ← 1 to 2n do
for k ← 1 to j do
s ← s + j
e. Algorithm Loop5(n):
k ← 0
for r ← 1 to n2 do
for s ← 1 to r do
k ← k + r
3. Order the following functions from smallest to largest by their big-O notation—you can use the letters in your answer rather than copying each formula. Be clear which is smallest and which is largest, and which functions are asymptotically equivalent. For example, if g, h, and m are all O(n lg n), you would write g = h = m = O(n lg n).
a. 562 log3 108
b. n3
c. 2n lg n
d. lg nn
e. n3 lg n
f. (n3 lg n3)/2
g. nn
h. 56n
i. log5 (n!)
j. ncos n
k. n / lg n
l. lg* n
m.
4. a. Which of these equations is true, and why?
b. Which of these is smaller for very large n?
Trisecting the Circle: A Case for Euclidean Geometry
Author(s): Alfred S. Posamentier
Source: The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006), pp. 414-418
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/27972006
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On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
Permutations and Combinations IIT JEE+Olympiad Lecture 3 Parth Nandedkar
Follows JEE Advanced syllabus, covering these topics:
Difference between Distinguishable and Indistinguishable objects,
Partitioning Indistinguishable Objects,
Counting Number of Groups formed,
No of ways of Choosing objects from a larger set,
Pascal's Triangle and nCr,
Using nCr multiplication to solve Road network problems.
Lecture #2 for course CS301: "Algorithmic Combinatorics I" for 3rd year students with a computer science major, Faculty of Science, Ain Shams University, Academic Year WS2014/2015.
Version 2: some improvements http://arxiv.org/abs/1505.04393.
We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring R having Z/2Z as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of R. We include a number illustrative examples and prove that the structure of a finite 3-field is not connected to any binary field.
1. Mathematics 4 Combinatorics Part 1
PSHS Main Campus - Fourth Quarter Mr. Fortunato A."facuboy III
Fundamerftcrl Countittg Prhciple or Multiplicntiou Prhtciple o.f Courttittg
IfotreuettttlcctLrintndifferentrnt1s,attdmttlthcreoent(ittdepetden't"f.omthefirsteuutt)cttttoccru,in
together the truo euents cnn occur ht nut different roarls.
Strppose thnt there sre k euents. lf the first euent ma1 ocuu' in f r0a1/s, the second euent irt ft2 7.0t1Lls, the thirtl in ft., rotuls and
so ort t'ntd so .fortl't, then the le euents together nlau lcrrLt i/t ntn2n3 .. .nl_ ((;)ays.
Ex1 Plate Nunrbers Hozo ntmttl ordhLnnl Philipphrc plntc nwnltet's Ltre ttunilable? (at ordinnnl plnte numbet' cortsists of
three letters follozued bt1 tlucc nunbers)
E*p The solution is called the Blank Method. We use a blank to repr'esent each component of the plate ntimber.
81 82 83 84 85 86
The English alphabet consists of 26 letters. So for the fir,st blank (B1), there are 26 choices; similarly there are
26 choices {or 82 ancl 83. Take note that for at plate nr-rmber, the letters can be repeated. As for the digits, there
are 10 digits (0-9). So for the fourth blank (B4), there are 10 choices; similarly there are 10 choiccs for 85 and
86. Jtrst like in the letters, the digits carn be repeated. Take note that in {iliirrg the blanks, it is seen as an
application of the Multiplication Principle. Therefore, the answer rs (261(26)(26) (10)(10)(10) = 17 576 000.
Ex2 Class Officers Corrsidcr ryour class roith 30 members. ln horu matlrl 'rurrys (N) cnn zoe selet't n prcsiderrt, uice'prasitTent,
secr et ar ry sr ttl tr e nst.Lrcr ?
E*p Usirrg the h'lank nrt'thod again.
PVPS T
Since there are 30 tnembers in the class aucl assuming there anyone is eligible to become an officer. So for the
president - first blank (P), there are 30 choices. For the second blatrl< (vice-president), take note that a student
cannot be elected to more than one position, therefore for VP there are 29 choices because the student elected
as president cantrot be elected as vice-president also. Similarly, there are 28 choicesfol S and 27 choices for T.
This is another application of the Multiplication Principle. Therefore, the answer is (30)(29)(28)(27 = 657 720.
Ex 3 Travel Optrons On n giuen tltu1, thert: are 3 differert nirlhrcs from Mnnila to Cclttt tuul 4 diffcrut sen uessels. Hozu
ttttnll/ waVS are there to go .front Mrmiltt to Cebu?
E*p Using tlre ['larrl< metlrod .rr:ain.
AS
Since there are 3 possibie airlines, for the first blarrk (A), there ale 3 choices. For the second blank (S), there are
4 choices. Take note now that to travel from Manila to Cebu, a person cannot take an airline and a sea vessel
at the same time. Therefore, the Mr-rltiplication Principle CANNOT be used. This is an application of the
Addition Principle.
Page I of3
2. Mathematics 4 Combinatoricp Part 1
PSHS Main Campus - Fourth Quarter Mr. Fortunato A. Tacuboy III
Adclitiorr Principle of Couttirtg
If one euent occur in m different roarls, and tutother auent can occw' irt n dit'ferent uays. If the truo eaents sccomplish the same
thing and cannot happen simultwreously ruiually exclusiae eaents)') then the ttuo eaents corr occur in m + n dift'erent ruaqs.
Suppose thnt there ctre k mutunlly arclusizte eautts. If Lhe first eucnt ntnrl occttr itr n1 Lpfrys, the secontl euerft in rt, roays, the
third in n3 T.Ltatls srrd so on and so forth, then the ntLmber of zoays any o17e of the k eaents can occur is nt + n2 + n3 +' . .* flL
)
TPntls.
E*p Continr-ring with the explanation, for this problem, the solution is called the Case Method.
Case 1: Airline (3 possible choices)
Case 2: Sea Vessel (4 possible choices)
Since this is an application of the Addition Principle, the answer is 3 + 4 = 7.
Ex 4 Plane Flights On s giuert clny, tlrcre rtre 5 clifferent clit'ect fliyqhts to Lord.on or tlrcrc nrc 2 JTi,gltts 7o Lft'nt! Kong ard 4
cornrcctirrg JTights J"rom Hong Kong to LondLm. Hrtrtt mantl rL)ar/s nrc there to go to London?
E*p Take note that the case method should be used.
Case l: direct flight ro London (5 possible choices)
Case 2: via Hong Kong
For Case 2, the Multiplication Principle should be used, there are two blanks: one for going to Hong Kong and
another from Hong Kor-rg to London. For Case 2, there are (2)(a) = 8 ways. So the answer is 5 + 8 = 13.
Ex 4 Subcommittee Au internntional committee corzsl>^f>^ of 5 representntiztes from the Philippittes, 6 from Malnysia nnd 4
from Thailnnd.
@) ht hout nTnrLy Llqys csn a subcommittee of three members be chosett, rro tzoo come from the sttme ctlunh'ry?
E*p Using the Multiplication Principle, the answer is (5)(6)(4) = 120.
'(B)
In horo matty 70aL/s can n sultcommittee of tzuo members frctm different cotnttries?
' E"p Using the case method and using the blank rnethod for each case:
Case 1: Phil-Mal pair, there are (5)(6) = 30 pairs
Case l: l)hil-Thai pair, there are (5)(4) = 20 pairs
-l:
Case Mal-Thai pair, there are (6)(4) = 24 pafts
Using the Addition Principle, the answer is 30 + 20 + 24 = 74.
HOME,WORK (due Monday)
COLE p.775-777 #s 76,24,26, 38, 42
Page 2 of 3
3. Mathematics 4 Combinatorics Part 1
PSHS Main Campus - Fourth Quarter Mr. Fortunato A. Taci-rboy III
Ex,rmple Recall:
Class Officers Consider yorLr class zoith 30 ntenrLtcrs. ln hozo tnafiV Taays (N) cnn tue sele ct n president, ttice-presidant, seuetarry
ntrd lrt'nsttt t't ?
To find the number of ways, it is possible that to first choose four people and etfter choosing these four people assign
their respective positions. This process is called a permutation.
Definition: Permutation - ordered arrangement, without repetition, of a set of objects
LetSbeasetof nelements and let 0<r <n. Thenumberof permutations of relements of Sis
,. P. = P(n, r) = t+ . ('The expression is read as "permtLtntiort rt takerr r" .)
{n-r}t
Take note that n! is read as n f:rctorial and is defined as: nl : n(n - 1)(n - 2) . . . (3X2)(1).
1!=1 3l=6 5!=120 7l=5040 9!=362880
2!=2 4t=24 6!=720 8l=40320 10!=3628800
By definition, 0! : 1.
MEMORIZE THE FACTORTAI, VAL,UES OF THE FIRST 10 NATURAL NUMBERS!
E*p For the Class Officers ploblem, since it is an application of a permutation (n=30, r :4),
30! 30! (30)(29)"'{?)(?)r1
P(30,4)= _
'(30-4)! 2bt --=(30X2eX28)(27)=657720
(26)(25) (3X2Xr)
To make the computation easier,
30t _ (30)(ze)(?8)(27)(26t)
: (30)(2e)(28 )(27) _ 657720
2tc! 26:
For exercise purposes, evaluate the following permutation expressions: P(5, 0), P(4, 1), P(9, 2, P(6, 6), P(5, 4), (7 , 3) ,
P(10,7), P(13,5). Use your scientific calculators to vcrify your answers.
Take note of the following properties of P(n, r):
P(n, 0) = 1 P(n, n) = n/
P(n, 1):ru P(n,n-7):n!
P(n' 2) = rt2 - rr
HOMF.WORK (due Monday)
VANCE p.282 #s 4, 6,8, p. 285 #s 6, B
Piece of Advice: Nofe thnt the nLrnilier of homeworh items is limited uthich menns that you do the
additionnl exercises 0n yutr lu)n. Ylu haue 3 textboolcs ruithloads of exercises and slme of the items
hsue ansrlers. YOU SHOULD HAVE'THE INITIATIVE'lO DO 1'HE EXERCISES ON YOUR
OWN IF YOU WAN1' ]'O END YOUR MATH 4 EXPERIENCE ON A POSI-]-IVE IJO1'E!
Page 3 of3