A note on diagonalization of integral quadratic forms modulo p m
|
|
|
- Janis Stokes
- 5 years ago
- Views:
Transcription
1 NNTDM 7 ( 3-36 A noe on diagonalizaion of inegral quadraic fors odulo Ali H Hakai Dearen of Maheaics King Khalid Universiy POo 94 Abha Posal Code: 643 Saudi Arabia E-ail: aalhakai@kkuedusa Absrac: Le be a osiive ineger be an odd rie and / ( be he ring of inegers odulo Le ( ( n be a nonsingular quadraic for wih ineger coefficiens In his aer we shall rove ha any nonsingular quadraic for ( over ( is equivalen o a diagonal quadraic for (odulo Keywords: Inegral quadraic for Nonsingular quadraic for Diagonalizaion quadraic for odulo rie AMS Classificaion: E8 Inroducion In his secion we sily enion he basic conces of quadraic fors which we shall need hroughou For deails he reader is referred o [] [] and [3] A quadraic for ( over is a olynoial of he ye ( ( a n ij i i j n wih a ij i j n We associae wih ( a syeric n n ari A A given by a a a3 L an a a a3 L a n A a3 a3 a33 L a3n M M M O M an an an3 L a nn Tha is a for i < j ij * * A [ aij ] n n aij a ji for i > j aii for i j Observe ha where ( A j 3
2 [ K n] M n Here denoes he ransose of he ari On he oher hand noe ha if he ari A is diagonal (An n n ari A is diagonal if a ij whenever i j hen he corresonding quadraic for has he diagonal reresenaion ( A a K annn ie he quadraic for will conain no "cross roduc" ers In he sae way we call a diagonal quadraic for (od for any rie ower if conains no "cross roduc" ers when read (od The deerinan of abbreviaed de is defined o be he deerinan of he ari A We say ha ( is nonsingular over if de Siilarly for any odd rie ower we say ( is nonsingular od if / de Again le ~ be an odd rie ower Le ( and ( be wo quadraic fors over wih associaed arices A A ~ resecively We now view he enries of hese arices as eleens of /( and regard / as he ulilicaive inverse of (od (Alernaively we can relace / wih ( / and regard A as having ineger enries We say ha ( is equivalen o ~ ~ ( (od wrien ( ~ ( (od if here is an inverible n n ari T over /( such ha ~ ( ( T(od ha is A~ T A T (od I is clear ha ~ is an equivalence relaion Noe ha ~ de de (det (od Eale: Le be any odd rie ower and ( where Tha is ( A A ( y aking he sile observaion ha [ ] { A Then ( ( (od 443 even 3
3 we can wrie ( A (od wih A M ( Noe ha since is odd he enries of A are all inegers Thus we ay assue ha A M ( when working wih congruences odulo odd ries Diagonalizaion of quadraic fors odulo Theore For any odd rie ower and nonsingular quadraic for ( over ( is equivalen o a diagonal quadraic for (odulo Proof We roceed by inducion on When i is well known (see [4] ha can be diagonalized over he finie field Say T A (od T D for soe T D M n n ( wih T nonsingular (od and D a diagonal ari Le us lif his o a soluion (od Le U T X where X [ ij ] is a ari of variables We wish o solve This is equivalen o ( T X U A U D D T AT 443 (od A ( T X D (od (od T AT T A X X AT D (od T AT D T A X X AT (od 3 where T A X and ( D T AT / Noe ha is a syeric ari wih ineger enries Le b b b b (od 3 b3 b 33 M M M O bn bn bn3 L b nn Then Thus we are lef wih solving he congruence T A X (od 3
4 Since T and A are nonsingular (od his equaion has a unique soluion X A ( T (od In he sae anner one can lif a soluion (od o (od for any Indeed roceeding as above suose ha T AT D (od for soe T D M n n ( wih T nonsingular (od and D a diagonal ari Le U T X where X is a ari of variables and solve This is equivalen o U AU D (od ( T X A ( T X D (od T AT T A X X AT D T AT D T A X X A T 443 (od D T AT (od 443 (od where T AX and ( D T AT / is a syeric ari wih ineger enries Le β β β β3 β3 β 33 (od M M M O βn βn βn3 L β nn Then (We noe ha he choice of is no unique Hence we are lef wih solving he congruence T AX (od As T and A are nonsingular (od his equaion has a unique soluion X A ( T (od This colees he inducion se Eales: Le 647 A 48 y ( [ y] y y y Noe ha (y is already a diagonal for when read (od We roceed o diagonalize (y (od 33
5 34 T I - D T AT Solve (od AX (od (od (od Check : U T X UAU (od Thus ( (od ~ y Le ( y [ ] y y A (od y Wha haens if A singular? Here A is no inverible so we canno direcly follow he ehod given in our roof Le us ry o solve (od AX T Firs we see ha T I since A is already diagonal (od Le ( a Then
6 35 A (od and he laer ari has ineger enries A D If we roceed as in he roof we would le (od Now solve X (od This is equivalen o (od (od which give us a conradicion ( and hence here is no soluion of his syse Ne le us ry he choice (od Then (od Solve X (od or equivalenly (od
7 Le Then obviously (od so ha X Hence i follows one can ake he change of variable a y y a y o diagonalize he quadraic for y ( (od Indeed y ~ y y y ( y y (od (od Our roof of Theore acually yields he sronger resul Corollary If is an odd rie ( is a quadraic for over nonsingular (od n and equivalen o diagonal for a (od hen ( is equivalen o he sae i i i i i n diagonal for a (od for any Noe: This fails for nonsingular fors Indeed ~ (od y bu y ~ / (od i References [] Larry J Gersein asic uadraic Fors Aerican Maheaical Sociey 8 [] G L Wason Inegral uadraic Fors Cabridge Universiy Press 96 [3] Michel Arin Algebra Prenice-Hall New Jersey 99 [4] R Lidl and H Niederreier Encycloedia of Maheaics and is Alicaions Finie Fields Addison-Wesley Publishing Coany
Riemann Hypothesis and Primorial Number. Choe Ryong Gil
Rieann Hyohesis Priorial Nuber Choe Ryong Gil Dearen of Maheaics Universiy of Sciences Gwahak- dong Unjong Disric Pyongyang DPRKorea Eail; ryonggilchoe@sar-conek Augus 8 5 Absrac; In his aer we consider
arxiv: v1 [math.fa] 12 Jul 2012
![arxiv: v1 [math.fa] 12 Jul 2012](/thumbs/89/97554371.jpg "arxiv: v1 [math.fa] 12 Jul 2012")
AN EXTENSION OF THE LÖWNER HEINZ INEQUALITY MOHAMMAD SAL MOSLEHIAN AND HAMED NAJAFI arxiv:27.2864v [ah.fa] 2 Jul 22 Absrac. We exend he celebraed Löwner Heinz inequaliy by showing ha if A, B are Hilber
23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
Planar Curves out of Their Curvatures in R
Planar Curves ou o Their Curvaures in R Tala Alkhouli Alied Science Dearen Aqaba College Al Balqa Alied Universiy Aqaba Jordan doi: 9/esj6vn6 URL:h://dxdoiorg/9/esj6vn6 Absrac This research ais o inroduce
THE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
M x t = K x F t x t = A x M 1 F t. M x t = K x cos t G 0. x t = A x cos t F 0
Forced oscillaions (sill undaped): If he forcing is sinusoidal, M = K F = A M F M = K cos G wih F = M G = A cos F Fro he fundaenal heore for linear ransforaions we now ha he general soluion o his inhoogeneous
Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
2 Some Property of Exponential Map of Matrix
Soluion Se for Exercise Session No8 Course: Mahemaical Aspecs of Symmeries in Physics, ICFP Maser Program for M 22nd, January 205, a Room 235A Lecure by Amir-Kian Kashani-Poor email: kashani@lpensfr Exercise
#A62 INTEGERS 16 (2016) REPRESENTATION OF INTEGERS BY TERNARY QUADRATIC FORMS: A GEOMETRIC APPROACH
#A6 INTEGERS 16 (016) REPRESENTATION OF INTEGERS BY TERNARY QUADRATIC FORMS: A GEOMETRIC APPROACH Gabriel Durha Deartent of Matheatics, University of Georgia, Athens, Georgia gjdurha@ugaedu Received: 9/11/15,
Chapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
Ayşe Alaca, Şaban Alaca and Kenneth S. Williams School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada. Abstract.
Journal of Cobinatorics and Nuber Theory Volue 6, Nuber,. 17 15 ISSN: 194-5600 c Nova Science Publishers, Inc. DOUBLE GAUSS SUMS Ayşe Alaca, Şaban Alaca and Kenneth S. Willias School of Matheatics and
Problem set 2 for the course on. Markov chains and mixing times
J. Seif T. Hirscher Soluions o Proble se for he course on Markov chains and ixing ies February 7, 04 Exercise 7 (Reversible chains). (i) Assue ha we have a Markov chain wih ransiion arix P, such ha here
Existence of Weak Solutions for Elliptic Nonlinear System in R N
Inernaional Journal of Parial Differenial Equaions and Alicaions, 4, Vol., o., 3-37 Available online a h://ubs.scieub.co/idea///3 Science and Educaion Publishing DOI:.69/idea---3 Exisence of Wea Soluions
GCD AND LCM-LIKE IDENTITIES FOR IDEALS IN COMMUTATIVE RINGS
GCD AND LCM-LIKE IDENTITIES FOR IDEALS IN COMMUTATIVE RINGS D. D. ANDERSON, SHUZO IZUMI, YASUO OHNO, AND MANABU OZAKI Absrac. Le A 1,..., A n n 2 be ideals of a commuaive ring R. Le Gk resp., Lk denoe
Exercises: Similarity Transformation
Exercises: Similariy Transformaion Problem. Diagonalize he following marix: A [ 2 4 Soluion. Marix A has wo eigenvalues λ 3 and λ 2 2. Since (i) A is a 2 2 marix and (ii) i has 2 disinc eigenvalues, we
EECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween
Bernoulli numbers. Francesco Chiatti, Matteo Pintonello. December 5, 2016
UNIVERSITÁ DEGLI STUDI DI PADOVA, DIPARTIMENTO DI MATEMATICA TULLIO LEVI-CIVITA Bernoulli numbers Francesco Chiai, Maeo Pinonello December 5, 206 During las lessons we have proved he Las Ferma Theorem
Distance Between Two Ellipses in 3D
Disance Beween Two Ellipses in 3D David Eberly Magic Sofware 6006 Meadow Run Cour Chapel Hill, NC 27516 eberly@magic-sofware.com 1 Inroducion An ellipse in 3D is represened by a cener C, uni lengh axes
The expectation value of the field operator.
The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining
ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
THE FINITE HAUSDORFF AND FRACTAL DIMENSIONS OF THE GLOBAL ATTRACTOR FOR A CLASS KIRCHHOFF-TYPE EQUATIONS
European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN 59-995 HE FINIE HAUSDORFF AND FRACAL DIMENSIONS OF HE GLOBAL ARACOR FOR A CLASS KIRCHHOFF-YPE EQUAIONS Guoguang Lin & Xiangshuang Xia Deparen
FREE ODD PERIODIC ACTIONS ON THE SOLID KLEIN BOTTLE
An-Najah J. Res. Vol. 1 ( 1989 ) Number 6 Fawas M. Abudiak FREE ODD PERIODIC ACTIONS ON THE SOLID LEIN BOTTLE ey words : Free acion, Periodic acion Solid lein Bole. Fawas M. Abudiak * V.' ZZ..).a11,L.A.;15TY1
Chapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
Chapter 3 Common Families of Distributions
Chaer 3 Common Families of Disribuions Secion 31 - Inroducion Purose of his Chaer: Caalog many of common saisical disribuions (families of disribuions ha are indeed by one or more arameers) Wha we should
arxiv: v1 [math.gm] 7 Nov 2017
![arxiv: v1 [math.gm] 7 Nov 2017](/thumbs/72/67237961.jpg "arxiv: v1 [math.gm] 7 Nov 2017")
A TOUR ON THE MASTER FUNCTION THEOPHILUS AGAMA arxiv:7.0665v [mah.gm] 7 Nov 07 Absrac. In his aer we sudy a funcion defined on naural numbers having eacly wo rime facors. Using his funcion, we esablish
THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX
J Korean Mah Soc 45 008, No, pp 479 49 THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX Gwang-yeon Lee and Seong-Hoon Cho Reprined from he Journal of he
23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
We just finished the Erdős-Stone Theorem, and ex(n, F ) (1 1/(χ(F ) 1)) ( n
Lecure 3 - Kövari-Sós-Turán Theorem Jacques Versraëe jacques@ucsd.edu We jus finished he Erdős-Sone Theorem, and ex(n, F ) ( /(χ(f ) )) ( n 2). So we have asympoics when χ(f ) 3 bu no when χ(f ) = 2 i.e.
SOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
Math 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
Mon Apr 9 EP 7.6 Convolutions and Laplace transforms. Announcements: Warm-up Exercise:
Mah 225-4 Week 3 April 9-3 EP 7.6 - convoluions; 6.-6.2 - eigenvalues, eigenvecors and diagonalizabiliy; 7. - sysems of differenial equaions. Mon Apr 9 EP 7.6 Convoluions and Laplace ransforms. Announcemens:
Homework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
arxiv: v1 [math.rt] 15 Dec 2017
![arxiv: v1 [math.rt] 15 Dec 2017](/thumbs/73/68347973.jpg "arxiv: v1 [math.rt] 15 Dec 2017")
POSITIVITY OF DENOMINATOR VECTORS OF CLUSTER ALGEBRAS PEIGEN CAO FANG LI arxiv:1712.06975v1 [mah.rt] 15 Dec 2017 Absrac. In his aer, we rove ha osiiviy of denominaor vecors holds for any sewsymmeric cluser
Higher Order Difference Schemes for Heat Equation
Available a p://pvau.edu/aa Appl. Appl. Ma. ISSN: 9-966 Vol., Issue (Deceber 009), pp. 6 7 (Previously, Vol., No. ) Applicaions and Applied Maeaics: An Inernaional Journal (AAM) Higer Order Difference
Math 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
An Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
Chapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
Synthesis of Supervisors Enforcing General Linear Vector Constraints in Petri Nets
The 00 Aerican onrol onference, May -0, Anchorage, Alaska. Synhesis of Suervisors Enforcing General Linear Vecor onsrains in Peri Nes Marian V. Iordache and Panos J. Ansaklis Absrac This aer considers
Linear Quadratic Regulator (LQR) - State Feedback Design
Linear Quadrai Regulaor (LQR) - Sae Feedbak Design A sysem is expressed in sae variable form as x = Ax + Bu n m wih x( ) R, u( ) R and he iniial ondiion x() = x A he sabilizaion problem using sae variable
Undetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
Representation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
Dirac s hole theory and the Pauli principle: clearing up the confusion.
Dirac s hole heory and he Pauli rincile: clearing u he conusion. Dan Solomon Rauland-Borg Cororaion 8 W. Cenral Road Moun Prosec IL 656 USA Email: dan.solomon@rauland.com Absrac. In Dirac s hole heory
3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
Fourier Series & The Fourier Transform. Joseph Fourier, our hero. Lord Kelvin on Fourier s theorem. What do we want from the Fourier Transform?
ourier Series & The ourier Transfor Wha is he ourier Transfor? Wha do we wan fro he ourier Transfor? We desire a easure of he frequencies presen in a wave. This will lead o a definiion of he er, he specru.
BOX-JENKINS MODEL NOTATION. The Box-Jenkins ARMA(p,q) model is denoted by the equation. pwhile the moving average (MA) part of the model is θ1at
BOX-JENKINS MODEL NOAION he Box-Jenkins ARMA(,q) model is denoed b he equaion + + L+ + a θ a L θ a 0 q q. () he auoregressive (AR) ar of he model is + L+ while he moving average (MA) ar of he model is
on the interval (x + 1) 0! x < ", where x represents feet from the first fence post. How many square feet of fence had to be painted?
Calculus II MAT 46 Improper Inegrals A mahemaician asked a fence painer o complee he unique ask of paining one side of a fence whose face could be described by he funcion y f (x on he inerval (x + x
10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
A study on Hermite-Hadamard type inequalities for s-convex functions via conformable fractional
Sud. Univ. Babeş-Bolyai Mah. 6(7), No. 3, 39 33 DOI:.493/subbmah.7.3.4 A sudy on Hermie-Hadamard ye inequaliies for s-convex funcions via conformable fracional inegrals Erhan Se and Abdurrahman Gözınar
The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
On Hankel type transform of generalized Mathieu series
Inernaional Journal of Saisika and Mahemaika ISSN: 77-79 E-ISSN: 49-865 Volume Issue 3 3- On Hankel ye ransform of generalized Mahieu series BBWahare MAEER s MIT ACSC Alandi Pune-45 Maharashra India Corresondence
Math 315: Linear Algebra Solutions to Assignment 6
Mah 35: Linear Algebra s o Assignmen 6 # Which of he following ses of vecors are bases for R 2? {2,, 3, }, {4,, 7, 8}, {,,, 3}, {3, 9, 4, 2}. Explain your answer. To generae he whole R 2, wo linearly independen
Finish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
ON THE DEGREES OF RATIONAL KNOTS
ON THE DEGREES OF RATIONAL KNOTS DONOVAN MCFERON, ALEXANDRA ZUSER Absrac. In his paper, we explore he issue of minimizing he degrees on raional knos. We se a bound on hese degrees using Bézou s heorem,
Convergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
VOL. 1, NO. 8, November 2011 ISSN ARPN Journal of Systems and Software AJSS Journal. All rights reserved
VOL., NO. 8, Noveber 0 ISSN -9833 ARPN Journal of Syses and Sofware 009-0 AJSS Journal. All righs reserved hp://www.scienific-journals.org Soe Fixed Poin Theores on Expansion Type Maps in Inuiionisic Fuzzy
F This leads to an unstable mode which is not observable at the output thus cannot be controlled by feeding back.
Lecure 8 Las ime: Semi-free configuraion design This is equivalen o: Noe ns, ener he sysem a he same place. is fixed. We design C (and perhaps B. We mus sabilize if i is given as unsable. Cs ( H( s = +
Hamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
Vehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
Solutions to the Olympiad Cayley Paper
Soluions o he Olympiad Cayley Paper C1. How many hree-digi muliples of 9 consis only of odd digis? Soluion We use he fac ha an ineger is a muliple of 9 when he sum of is digis is a muliple of 9, and no
DISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
ON UNIVERSAL LOCALIZATION AT SEMIPRIME GOLDIE IDEALS
ON UNIVERSAL LOCALIZATION AT SEMIPRIME GOLDIE IDEALS JOHN A. BEACHY Deparmen of Mahemaical Sciences Norhern Illinois Universiy DeKalb IL 6115 U.S.A. Absrac In his paper we consider an alernaive o Ore localizaion
Chapter 9 Sinusoidal Steady State Analysis
Chaper 9 Sinusoidal Seady Sae Analysis 9.-9. The Sinusoidal Source and Response 9.3 The Phasor 9.4 pedances of Passive Eleens 9.5-9.9 Circui Analysis Techniques in he Frequency Doain 9.0-9. The Transforer
Chapter Floating Point Representation
Chaper 01.05 Floaing Poin Represenaion Afer reading his chaper, you should be able o: 1. conver a base- number o a binary floaing poin represenaion,. conver a binary floaing poin number o is equivalen
INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE
INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE JAMES ALEXANDER, JONATHAN CUTLER, AND TIM MINK Absrac The enumeraion of independen ses in graphs wih various resricions has been a opic of much ineres
Radical Expressions. Terminology: A radical will have the following; a radical sign, a radicand, and an index.
Radical Epressions Wha are Radical Epressions? A radical epression is an algebraic epression ha conains a radical. The following are eamples of radical epressions + a Terminology: A radical will have he
MATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
![MATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.](/thumbs/86/93221008.jpg "MATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.")
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
Sections 2.2 & 2.3 Limit of a Function and Limit Laws
Mah 80 www.imeodare.com Secions. &. Limi of a Funcion and Limi Laws In secion. we saw how is arise when we wan o find he angen o a curve or he velociy of an objec. Now we urn our aenion o is in general
Hamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
fakultät für informatik informatik 12 technische universität dortmund Petri Nets Peter Marwedel TU Dortmund, Informatik /10/10
2 Peri Nes Peer Marwedel TU Dormund, Informaik 2 2008/0/0 Grahics: Alexandra Nole, Gesine Marwedel, 2003 Generalizaion of daa flow: Comuaional grahs Examle: Peri nes Inroduced in 962 by Carl Adam Peri
Congruences involving Bernoulli and Euler numbers Zhi-Hong Sun
The aer will aear in Journal of Nuber Theory. Congruences involving Bernoulli Euler nubers Zhi-Hong Sun Deartent of Matheatics, Huaiyin Teachers College, Huaian, Jiangsu 300, PR China Received January
INDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
fakultät für informatik informatik 12 technische universität dortmund Petri Nets Peter Marwedel TU Dortmund, Informatik /11/09
2 Peri Nes Peer Marwedel TU Dormund, Informaik 2 2009//09 Grahics: Alexandra Nole, Gesine Marwedel, 2003 Models of comuaion considered in his course Communicaion/ local comuaions Undefined comonens Communicaing
EXPONENTIAL PROBABILITY DISTRIBUTION
MTH/STA 56 EXPONENTIAL PROBABILITY DISTRIBUTION As discussed in Exaple (of Secion of Unifor Probabili Disribuion), in a Poisson process, evens are occurring independenl a rando and a a unifor rae per uni
An Excursion into Set Theory using a Constructivist Approach
An Excursion ino Se Theory using a Consrucivis Approach Miderm Repor Nihil Pail under supervision of Ksenija Simic Fall 2005 Absrac Consrucive logic is an alernaive o he heory of classical logic ha draws
WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
Two Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
Chapter #1 EEE8013 EEE3001. Linear Controller Design and State Space Analysis
Chaper EEE83 EEE3 Chaper # EEE83 EEE3 Linear Conroller Design and Sae Space Analysis Ordinary Differenial Equaions.... Inroducion.... Firs Order ODEs... 3. Second Order ODEs... 7 3. General Maerial...
On a problem of Graham By E. ERDŐS and E. SZEMERÉDI (Budapest) GRAHAM stated the following conjecture : Let p be a prime and a 1,..., ap p non-zero re
On a roblem of Graham By E. ERDŐS and E. SZEMERÉDI (Budaes) GRAHAM saed he following conjecure : Le be a rime and a 1,..., a non-zero residues (mod ). Assume ha if ' a i a i, ei=0 or 1 (no all e i=0) is
Solutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
Inequality measures for intersecting Lorenz curves: an alternative weak ordering
h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 Absrac Inequaliy measures for
1 Inroducion A (1 + ") loss-resilien code encodes essages consising of sybols ino an encoding consising of c sybols, c > 1, such ha he essage sybols c
A Lower Bound for a Class of Grah Based Loss-Resilien Codes Johannes Bloer Bea Trachsler 1 May 5, 1998 Absrac. Recenly, Luby e al. consruced inforaion-heoreically alos oial loss-resilien codes. The consrucion
A Generalization of Student s t-distribution from the Viewpoint of Special Functions
A Generalizaion of Suden s -disribuion fro he Viewpoin of Special Funcions WOLFRAM KOEPF and MOHAMMAD MASJED-JAMEI Deparen of Maheaics, Universiy of Kassel, Heinrich-Ple-Sr. 4, D-343 Kassel, Gerany Deparen
Comments on Window-Constrained Scheduling
Commens on Window-Consrained Scheduling Richard Wes Member, IEEE and Yuing Zhang Absrac This shor repor clarifies he behavior of DWCS wih respec o Theorem 3 in our previously published paper [1], and describes
Solutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
The Miki-type identity for the Apostol-Bernoulli numbers
Annales Mahemaicae e Informaicae 46 6 pp. 97 4 hp://ami.ef.hu The Mii-ype ideniy for he Aposol-Bernoulli numbers Orli Herscovici, Toufi Mansour Deparmen of Mahemaics, Universiy of Haifa, 3498838 Haifa,
R.#W.#Erickson# Department#of#Electrical,#Computer,#and#Energy#Engineering# University#of#Colorado,#Boulder#
.#W.#Erickson# Deparmen#of#Elecrical,#Compuer,#and#Energy#Engineering# Universiy#of#Colorado,#Boulder# Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance,
Note on oscillation conditions for first-order delay differential equations
Elecronic Journal of Qualiaive Theory of Differenial Equaions 2016, No. 2, 1 10; doi: 10.14232/ejqde.2016.1.2 hp://www.ah.u-szeged.hu/ejqde/ Noe on oscillaion condiions for firs-order delay differenial
4 Sequences of measurable functions
4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences
Conservation of Momentum. The purpose of this experiment is to verify the conservation of momentum in two dimensions.
Conseraion of Moenu Purose The urose of his exerien is o erify he conseraion of oenu in wo diensions. Inroducion and Theory The oenu of a body ( ) is defined as he roduc of is ass () and elociy ( ): When
CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR
Annales Academiæ Scieniarum Fennicæ Mahemaica Volumen 31, 2006, 39 46 CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR Joaquim Marín and Javier
On the Infinitude of Covering Systems with Least Modulus Equal to 2
Annals of Pure and Applied Mahemaics Vol. 4, No. 2, 207, 307-32 ISSN: 2279-087X (P), 2279-0888(online) Published on 23 Sepember 207 www.researchmahsci.org DOI: hp://dx.doi.org/0.22457/apam.v4n2a3 Annals
) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha