#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS

Size: px
Start display at page:

Download "#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS"

Transcription

1 #A42 INTEGERS 11 (2011 ON THE CONDITIONED BINOMIAL COEFFICIENTS Liqun To Shool of Mthemtil Sienes, Luoyng Norml University, Luoyng, Chin Reeived: 12/24/10, Revised: 5/11/11, Aepted: 5/16/11, Pulished: 6/27/11 Astrt We nswer question on the onditioned inomil oeffiients rised in nd rtile of Brlotti nd Pnnone, thus giving n lterntive proof of n extension of Froenius generliztion of Sylow s theorem 1 Introdution In [2], Brlotti nd Pnnone proved the following extension of Sylow s theorem [6]: Theorem 1 Let G e finite group of order n, p prime dividing n, H sugroup of G of order p h Then for ny positive integer k > h suh tht n, the rdinlity of the set of ll the p-sugroups of G of order ontining H is ongruent to one modulo p In the speil se h 0, this result ws first proved y Froenius [3], nd redisovered y Krull [4] The proof in [2] is given y onsidering the olletion of ll the susets of G hving rdinlity nd ontining extly h right osets of H It is worth pointing out tht the ove result ws lso proved independently y Spiegel [5] using Möius inversion methods developed in Weisner s pper [7] As suggested nd finlly rised s question in [2], one should e le to show the result y onsidering the fmily of the susets of G hving order nd ontining t lest one right oset of H This leds to the following: Definition 2 ([2] Let,, e positive integers suh tht nd Let A e set of rdinlity prtitioned into susets ll of rdinlity The onditioned inomil oeffiient determined y, nd, denoted y is defined to e the numer of susets of A of rdinlity ontining t lest one omponent of the prtition,

2 INTEGERS: 11 ( The im of this pper is to nswer the question rised in [2], whih sks to prove the following: Theorem 3Let,, e positive integers suh tht nd Then (1 divides (2 If is power of prime p, then / 1 (mod p Remrk 4 If 1, the onditioned inomil oeffiient determined y, nd is just ( Then Theorem 3 follows from Lemms 5 nd 6 in the next setion Our method of proof of the min result is to express the onditioned inomil oeffiient expliitly s omintion of usul inomils nd then onsider the divisiility nd ongruene of these inomils (see (1 elow The method we use is very elementry We refer the reder to [1] (Chpters 2 nd 6, or other stndrd lger textooks for terminology nd nottions used in the pper 2 Proof of the Min Result We will need the following two results The first one n e verified diretly; the seond one is shown in [4], ut we still give proof here for the reder s onveniene Lemm 5 Let, e positive integers suh tht divides Then ( ( 1 1 Lemm 6 Let p e prime, s positive integer, nd g positive integer divisile y p s Then ( g 1 p s 1 1 (mod p Proof We prove the result y indution on s, the exponent of p For s 1, sine p g, we hve g i p i (mod p for 1 i < p, nd hene (g 1(g 2 (g (p ( 1 (p 1(p 2 1 (mod p As (p 1(p 2 1 is prime to p, we get g 1 p 1 (g 1(g 2 (g (p 1 (p 1(p (mod p Suppose tht the result holds for exponents less thn s By strightforwrd omputtion we otin the equlity ( g 1 p s 1 ( g p 1 p s 1 1 ps 1 1 j0 p 1 i1 g (jp+i p s (jp+i Then the result follows y the indution hypothesis nd the ove rgument for the se s 1

3 INTEGERS: 11 ( Proof of Theorem 3 Let A e set of rdinlity prtitioned into susets ll of rdinlity The numer of susets of A of rdinlity ontining no omponent of the prtition is 0 n i < n 1+n 2+ +n ( ( n 1 n 2 (, n whih is the oeffiient of the term x in ((1 + x x The oeffiient equls Then ( r0 r0 ( ( ( r ( ( ( r r r r r ( 1 r ( 1 r ( 1 r 1 A r, (1 r1 where A r ( r ( ( r r For 1 r, y Lemm 5, we hve A r ( 1 r r 1 the other hnd, y Lemm 5 gin, we hve A r ( ( ( r r r ( 1 ( r r 1 ( 1 ( ( r r 1 Thus, ( ra r Therefore we hve tht divides A r So we otin r r ( ( r 1 r 1 r r ( ( r 1 r 1, hene ra r On ( r,r (,r divides A r A fortiori, This ompletes the proof of the first prt of the theorem For the seond prt, in view of Remrk 4, we my ssume tht,, p h re positive integers, where p is prime nd k > h 1 From (1, we hve

4 INTEGERS: 11 ( p h ( k h p r0 where B r ( ( m (m rp h r rp, nd for 1 r < h, h ( 1 r B r, m ( h,r divides B m r Hene h 1 divides B r, whih implies tht B r / mph 0 (mod p, for 1 r < h By Lemms 5 nd 6, ( ( mp h mph 1, 1 hene, By the sme rgument, ( mp h/ mph 1 (mod p ( m / mph h 1 (mod p Thus whether p is n odd prime or not, we hve B 0 + B h( 1 pk h 0 (mod p Putting the ove fts together, we get / mph p h 1 (mod p Remrk As mentioned t the eginning of the pper, we my prove Theorem 1 y onsidering F H (pk, the fmily of the susets of G hving order nd ontining t lest one right oset of H Let S H ( e the set of ll the p-sugroups of G of order ontining H Consider the right-multiplition tion of G on F H (pk Then F H (pk is prtitioned s union of orits: F H (pk l i1 O Ui Note tht St(U i divides U i, nd St(U i if nd only if O Ui ontins (unique sugroup K of order, nd hene onsists of right osets of K Sine K ontins some right oset of H, tully K H Thus we hve F H (pk l i1 G St(U i G ( S H ( + multiple of p

5 INTEGERS: 11 ( Applying Theorem 3 to the se G,, H p h, we otin F H (pk / mph / mph 1 (mod p p h Thus s desired S H ( 1 (mod p Aknowledgement This work is prtilly supported y the Sientifi Reserh Foundtion for the Returned Overses Chinese Sholrs, Stte Edution Ministry nd Reserh Seed Fund of Luoyng Norml University for Provinil or Ministry level projet The uthor is very grteful to the referee for helpful omments nd suggestions Referenes [1] M Artin, Alger, Prentie-Hll, 1991 [2] M Brlotti nd V Pnnone, Sylow s theorem nd the rithmeti of inomil oeffiients, Note di Mtemti 22 (2003, [3] G Froenius, Verllgemeinerung des Sylowshen Stze, Berliner Sitz, (1895, [4] W Krull, Üer die p-untergruppen, Arhiv der Mth, 12 (1961, 1-6 [5] E Spiegel, Another Look t Sylow s Third Theorem, Mth Mg 77 (2004, [6] M L Sylow, Théoremès sur les groupes de sustitutions, Mth Ann 5 (1872, [7] L Weisner, Some properties of prime power groups, Trns Amer Mth So 38 (1935,

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment

More information

Hyers-Ulam stability of Pielou logistic difference equation

Hyers-Ulam stability of Pielou logistic difference equation vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

CS 491G Combinatorial Optimization Lecture Notes

CS 491G Combinatorial Optimization Lecture Notes CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,

More information

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4. Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix

More information

Co-ordinated s-convex Function in the First Sense with Some Hadamard-Type Inequalities

Co-ordinated s-convex Function in the First Sense with Some Hadamard-Type Inequalities Int. J. Contemp. Mth. Sienes, Vol. 3, 008, no. 3, 557-567 Co-ordinted s-convex Funtion in the First Sense with Some Hdmrd-Type Inequlities Mohmmd Alomri nd Mslin Drus Shool o Mthemtil Sienes Fulty o Siene

More information

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:

More information

ON LEFT(RIGHT) SEMI-REGULAR AND g-reguar po-semigroups. Sang Keun Lee

ON LEFT(RIGHT) SEMI-REGULAR AND g-reguar po-semigroups. Sang Keun Lee Kngweon-Kyungki Mth. Jour. 10 (2002), No. 2, pp. 117 122 ON LEFT(RIGHT) SEMI-REGULAR AND g-reguar po-semigroups Sng Keun Lee Astrt. In this pper, we give some properties of left(right) semi-regulr nd g-regulr

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

Symmetrical Components 1

Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

On Implicative and Strong Implicative Filters of Lattice Wajsberg Algebras

On Implicative and Strong Implicative Filters of Lattice Wajsberg Algebras Glol Journl of Mthemtil Sienes: Theory nd Prtil. ISSN 974-32 Volume 9, Numer 3 (27), pp. 387-397 Interntionl Reserh Pulition House http://www.irphouse.om On Implitive nd Strong Implitive Filters of Lttie

More information

A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS

A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS Commun. Koren Mth. So. 31 016, No. 1, pp. 65 94 http://dx.doi.org/10.4134/ckms.016.31.1.065 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS Hrsh Vrdhn Hrsh, Yong Sup Kim, Medht Ahmed Rkh, nd Arjun Kumr Rthie

More information

More Properties of the Riemann Integral

More Properties of the Riemann Integral More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their

More information

Discrete Structures Lecture 11

Discrete Structures Lecture 11 Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.

More information

CHENG Chun Chor Litwin The Hong Kong Institute of Education

CHENG Chun Chor Litwin The Hong Kong Institute of Education PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using

More information

MAT 403 NOTES 4. f + f =

MAT 403 NOTES 4. f + f = MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn

More information

Polynomials. Polynomials. Curriculum Ready ACMNA:

Polynomials. Polynomials. Curriculum Ready ACMNA: Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

Exercise sheet 6: Solutions

Exercise sheet 6: Solutions Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd

More information

EXTENSION OF THE GCD STAR OF DAVID THEOREM TO MORE THAN TWO GCDS CALVIN LONG AND EDWARD KORNTVED

EXTENSION OF THE GCD STAR OF DAVID THEOREM TO MORE THAN TWO GCDS CALVIN LONG AND EDWARD KORNTVED EXTENSION OF THE GCD STAR OF DAVID THEOREM TO MORE THAN TWO GCDS CALVIN LONG AND EDWARD KORNTVED Astrt. The GCD Str of Dvi Theorem n the numerous ppers relte to it hve lrgel een evote to shoing the equlit

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Mth 3329-Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then

More information

Section 2.3. Matrix Inverses

Section 2.3. Matrix Inverses Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue

More information

Ch. 2.3 Counting Sample Points. Cardinality of a Set

Ch. 2.3 Counting Sample Points. Cardinality of a Set Ch..3 Counting Smple Points CH 8 Crdinlity of Set Let S e set. If there re extly n distint elements in S, where n is nonnegtive integer, we sy S is finite set nd n is the rdinlity of S. The rdinlity of

More information

THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p

THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS PETE L CLARK Circ 1870, Zolotrev observed tht the Legendre symbol ( p ) cn be interpreted s the sign of multipliction by viewed s permuttion of the set Z/pZ

More information

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

More information

Bisimulation, Games & Hennessy Milner logic

Bisimulation, Games & Hennessy Milner logic Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory

More information

POSITIVE IMPLICATIVE AND ASSOCIATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS

POSITIVE IMPLICATIVE AND ASSOCIATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS Bull. Koren Mth. So. 35 (998), No., pp. 53 6 POSITIVE IMPLICATIVE AND ASSOCIATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS YOUNG BAE JUN*, YANG XU AND KEYUN QIN ABSTRACT. We introue the onepts of positive

More information

How many proofs of the Nebitt s Inequality?

How many proofs of the Nebitt s Inequality? How mny roofs of the Neitt s Inequlity? Co Minh Qung. Introdution On Mrh, 90, Nesitt roosed the following rolem () The ove inequlity is fmous rolem. There re mny eole interested in nd solved (). In this

More information

Figure 1. The left-handed and right-handed trefoils

Figure 1. The left-handed and right-handed trefoils The Knot Group A knot is n emedding of the irle into R 3 (or S 3 ), k : S 1 R 3. We shll ssume our knots re tme, mening the emedding n e extended to solid torus, K : S 1 D 2 R 3. The imge is lled tuulr

More information

CS 573 Automata Theory and Formal Languages

CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

Solutions to Homework 6. (b) (This was not part of Exercise 1, but should have been) Let A and B be groups with A B. Then B A.

Solutions to Homework 6. (b) (This was not part of Exercise 1, but should have been) Let A and B be groups with A B. Then B A. MTH 4- Astrct Alger II S7 Solutions to Homework 6 Exercise () Let A, B n C e groups with A B n B C Show tht A C () (This ws not prt of Exercise, ut shoul hve een) Let A n B e groups with A B Then B A ()

More information

Lecture 1 - Introduction and Basic Facts about PDEs

Lecture 1 - Introduction and Basic Facts about PDEs * 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

( ) { } [ ] { } [ ) { } ( ] { }

( ) { } [ ] { } [ ) { } ( ] { } Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or

More information

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1. Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

More information

The study of dual integral equations with generalized Legendre functions

The study of dual integral equations with generalized Legendre functions J. Mth. Anl. Appl. 34 (5) 75 733 www.elsevier.om/lote/jm The study of dul integrl equtions with generlized Legendre funtions B.M. Singh, J. Rokne,R.S.Dhliwl Deprtment of Mthemtis, The University of Clgry,

More information

Lecture Notes No. 10

Lecture Notes No. 10 2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite

More information

ON AN INEQUALITY FOR THE MEDIANS OF A TRIANGLE

ON AN INEQUALITY FOR THE MEDIANS OF A TRIANGLE Journl of Siene nd Arts Yer, No. (9), pp. 7-6, OIGINAL PAPE ON AN INEQUALITY FO THE MEDIANS OF A TIANGLE JIAN LIU Mnusript reeived:.5.; Aepted pper:.5.; Pulished online: 5.6.. Astrt. In this pper, we give

More information

On the Co-Ordinated Convex Functions

On the Co-Ordinated Convex Functions Appl. Mth. In. Si. 8, No. 3, 085-0 0 085 Applied Mthemtis & Inormtion Sienes An Interntionl Journl http://.doi.org/0.785/mis/08038 On the Co-Ordinted Convex Funtions M. Emin Özdemir, Çetin Yıldız, nd Ahmet

More information

12.4 Similarity in Right Triangles

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

42nd International Mathematical Olympiad

42nd International Mathematical Olympiad nd Interntionl Mthemticl Olympid Wshington, DC, United Sttes of Americ July 8 9, 001 Problems Ech problem is worth seven points. Problem 1 Let ABC be n cute-ngled tringle with circumcentre O. Let P on

More information

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

More information

Linear Algebra Introduction

Linear Algebra Introduction Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

Binding Number and Connected (g, f + 1)-Factors in Graphs

Binding Number and Connected (g, f + 1)-Factors in Graphs Binding Number nd Connected (g, f + 1)-Fctors in Grphs Jinsheng Ci, Guizhen Liu, nd Jinfeng Hou School of Mthemtics nd system science, Shndong University, Jinn 50100, P.R.Chin helthci@163.com Abstrct.

More information

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true. York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu

More information

MTH 505: Number Theory Spring 2017

MTH 505: Number Theory Spring 2017 MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c

More information

Inequalities of Olympiad Caliber. RSME Olympiad Committee BARCELONA TECH

Inequalities of Olympiad Caliber. RSME Olympiad Committee BARCELONA TECH Ineulities of Olymid Clier José Luis Díz-Brrero RSME Olymid Committee BARCELONA TECH José Luis Díz-Brrero RSME Olymi Committee UPC BARCELONA TECH jose.luis.diz@u.edu Bsi fts to rove ineulities Herefter,

More information

Lecture 8: Abstract Algebra

Lecture 8: Abstract Algebra Mth 94 Professor: Pri Brtlett Leture 8: Astrt Alger Week 8 UCSB 2015 This is the eighth week of the Mthemtis Sujet Test GRE prep ourse; here, we run very rough-n-tumle review of strt lger! As lwys, this

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw

More information

IN GAUSSIAN INTEGERS X 3 + Y 3 = Z 3 HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH

IN GAUSSIAN INTEGERS X 3 + Y 3 = Z 3 HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A2 IN GAUSSIAN INTEGERS X + Y = Z HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH Elis Lmpkis Lmpropoulou (Term), Kiprissi, T.K: 24500,

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

Parallel Projection Theorem (Midpoint Connector Theorem):

Parallel Projection Theorem (Midpoint Connector Theorem): rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length one-hlf the third side. onversely, If line isects

More information

#A29 INTEGERS 17 (2017) EQUALITY OF DEDEKIND SUMS MODULO 24Z

#A29 INTEGERS 17 (2017) EQUALITY OF DEDEKIND SUMS MODULO 24Z #A29 INTEGERS 17 (2017) EQUALITY OF DEDEKIND SUMS MODULO 24Z Kurt Girstmir Institut für Mthemtik, Universität Innsruck, Innsruck, Austri kurt.girstmir@uik.c.t Received: 10/4/16, Accepted: 7/3/17, Pulished:

More information

Duke Math Meet

Duke Math Meet Duke Mth Meet 01-14 Power Round Qudrtic Residues nd Prime Numers For integers nd, we write to indicte tht evenly divides, nd to indicte tht does not divide For exmle, 4 nd 4 Let e rime numer An integer

More information

Instructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.

Instructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting. ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s

More information

arxiv: v1 [math.ca] 21 Aug 2018

arxiv: v1 [math.ca] 21 Aug 2018 rxiv:1808.07159v1 [mth.ca] 1 Aug 018 Clulus on Dul Rel Numbers Keqin Liu Deprtment of Mthemtis The University of British Columbi Vnouver, BC Cnd, V6T 1Z Augest, 018 Abstrt We present the bsi theory of

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into

More information

Talen en Automaten Test 1, Mon 7 th Dec, h45 17h30

Talen en Automaten Test 1, Mon 7 th Dec, h45 17h30 Tlen en Automten Test 1, Mon 7 th Dec, 2015 15h45 17h30 This test consists of four exercises over 5 pges. Explin your pproch, nd write your nswer to ech exercise on seprte pge. You cn score mximum of 100

More information

Linear choosability of graphs

Linear choosability of graphs Liner hoosility of grphs Louis Esperet, Mikel Montssier, André Rspud To ite this version: Louis Esperet, Mikel Montssier, André Rspud. Liner hoosility of grphs. Stefn Felsner. 2005 Europen Conferene on

More information

arxiv: v1 [math.ca] 7 Mar 2012

arxiv: v1 [math.ca] 7 Mar 2012 rxiv:1203.1462v1 [mth.ca] 7 Mr 2012 A simple proof of the Fundmentl Theorem of Clculus for the Lebesgue integrl Mrch, 2012 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde

More information

Boolean Algebra cont. The digital abstraction

Boolean Algebra cont. The digital abstraction Boolen Alger ont The igitl strtion Theorem: Asorption Lw For every pir o elements B. + =. ( + ) = Proo: () Ientity Distriutivity Commuttivity Theorem: For ny B + = Ientity () ulity. Theorem: Assoitive

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Rectangular group congruences on an epigroup

Rectangular group congruences on an epigroup cholrs Journl of Engineering nd Technology (JET) ch J Eng Tech, 015; 3(9):73-736 cholrs Acdemic nd cientific Pulisher (An Interntionl Pulisher for Acdemic nd cientific Resources) wwwsspulishercom IN 31-435X

More information

An introduction to groups

An introduction to groups n introdution to groups syllusref efereneene ore topi: Introdution to groups In this h hpter Groups The terminology of groups Properties of groups Further exmples of groups trnsformtions 66 Mths Quest

More information

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A. 378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),

More information

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets. I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the

More information

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly

More information

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions

More information

ON THE LEAST PRIMITIVE ROOT EXPRESSIBLE AS A SUM OF TWO SQUARES

ON THE LEAST PRIMITIVE ROOT EXPRESSIBLE AS A SUM OF TWO SQUARES #A55 INTEGERS 3 (203) ON THE LEAST PRIMITIVE ROOT EPRESSIBLE AS A SUM OF TWO SQUARES Christopher Ambrose Mathematishes Institut, Georg-August Universität Göttingen, Göttingen, Deutshland ambrose@uni-math.gwdg.de

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

Electromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors

Electromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 1 Eletromgneti-ower-bsed Modl Clssifition Modl Expnsion nd Modl Deomposition for erfet Eletri Condutors Renzun Lin Abstrt Trditionlly

More information

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}

More information

On the degree of regularity of generalized van der Waerden triples

On the degree of regularity of generalized van der Waerden triples On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of

More information

ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs

ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,

More information

The Riemann-Stieltjes Integral

The Riemann-Stieltjes Integral Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0

More information

Dong-Myung Lee, Jeong-Gon Lee, and Ming-Gen Cui. 1. introduction

Dong-Myung Lee, Jeong-Gon Lee, and Ming-Gen Cui. 1. introduction J. Kore So. Mth. Edu. Ser. B: Pure Appl. Mth. ISSN 16-0657 Volume 11, Number My 004), Pges 133 138 REPRESENTATION OF SOLUTIONS OF FREDHOLM EQUATIONS IN W Ω) OF REPRODUCING KERNELS Dong-Myung Lee, Jeong-Gon

More information

Quadratic reciprocity

Quadratic reciprocity Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Math 4310 Solutions to homework 1 Due 9/1/16

Math 4310 Solutions to homework 1 Due 9/1/16 Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS

DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS Krgujev Journl of Mthemtis Volume 38() (204), Pges 35 49. DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS MOHAMMAD W. ALOMARI Abstrt. In this pper, severl bouns for the ifferene between two Riemn-

More information

Chapter 3. Vector Spaces. 3.1 Images and Image Arithmetic

Chapter 3. Vector Spaces. 3.1 Images and Image Arithmetic Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner

More information

MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM. May 30, 2007 MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

More information

s the set of onsequenes. Skeptil onsequenes re more roust in the sense tht they hold in ll possile relities desried y defult theory. All its desirle p

s the set of onsequenes. Skeptil onsequenes re more roust in the sense tht they hold in ll possile relities desried y defult theory. All its desirle p Skeptil Rtionl Extensions Artur Mikitiuk nd Miros lw Truszzynski University of Kentuky, Deprtment of Computer Siene, Lexington, KY 40506{0046, frtur mirekg@s.engr.uky.edu Astrt. In this pper we propose

More information

QUADRATIC RESIDUES MATH 372. FALL INSTRUCTOR: PROFESSOR AITKEN

QUADRATIC RESIDUES MATH 372. FALL INSTRUCTOR: PROFESSOR AITKEN QUADRATIC RESIDUES MATH 37 FALL 005 INSTRUCTOR: PROFESSOR AITKEN When is n integer sure modulo? When does udrtic eution hve roots modulo? These re the uestions tht will concern us in this hndout 1 The

More information

Special Numbers, Factors and Multiples

Special Numbers, Factors and Multiples Specil s, nd Student Book - Series H- + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book - Series H Contents Topics Topic - Odd, even, prime nd composite numers Topic - Divisiility tests

More information

TOPPER SAMPLE PAPER - 5 CLASS XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100

TOPPER SAMPLE PAPER - 5 CLASS XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100 TOPPER SAMPLE PAPER - 5 CLASS XI MATHEMATICS Questions Time Allowed : 3 Hrs Mximum Mrks: 100 1. All questions re compulsory.. The question pper consist of 9 questions divided into three sections A, B nd

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

Computing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt

Computing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2:

More information