National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 6485
MMRC, AMSS, Academia, Sinica, Beijing
No. 22, December 2003
Generalized Stewart Platforms and their Direct
Kinematics1)
XiaoShan Gao and Deli Lei
Key Laboratory of Mathema
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
SM261.1011
Test 2
25 October 2006
Name: Solutions
1. Suppose that L = (v1, , v p ) is a list of p vectors in Rn.
a. Assuming that 0 < p < n, describe completely but succinctly a
specific procedure, em
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Triangular Decomposition of SemiAlgebraic Systems
Changbo Chen
James H. Davenport
University of Western Ontario
University of Bath
[email protected]
Marc Moreno Maza
[email protected]
Bican
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 194230
MMRC, AMSS, Academia, Sinica, Beijing
No. 22, December 2003
A Symbolic Approach to Polyhedral Scene
Analysis by Parametric Calotte Propagation1)
Hongbo Li, Lina Zhao
Math
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 1428
MMRC, AMSS, Academia, Sinica, Beijing
No. 22, December 2003
Constructing Blending Surfaces for Two Arbitrary
Surfaces
Jinsan Cheng and XiaoShan Gao1)
Institute of System S
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 86106
MMRC, AMSS, Academia, Sinica, Beijing
No. 22, December 2003
Rational Quadratic Approximation to Real Plane
Algebraic Curves1)
XiaoShan Gao and Ming Li
Key Lab of Mathemat
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 134147
MMRC, AMSS, Academia, Sinica, Beijing
No. 22, December 2003
On the Probability of the Number of Solutions for
the Perspective n Point Problem1)
XiaoShan Gao and Jianlian
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 186193
MMRC, AMSS, Academia, Sinica, Beijing
No. 22, December 2003
On a Problem of Steinhaus
DeLi Lei and Hong Du
Key Lab of Mathematics Mechanization
Institute of Systems Scien
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 264274
MMRC, AMSS, Academia Sinica, Beijing
No. 22, December 2003
An improved algebra method and its applications
in nonlinear wave equations
Zhenya Yan
Key Lab of Mathematics M
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
MM Research Preprints, 275284
MMRC, AMSS, Academia Sinica, Beijing
No. 22, December 2003
275
The Riccati equation with variable coefficients
expansion algorithm to find more exact solutions of
nonline
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Special Problem E Solution.
Let X and Y be the 33 matrices for rotation by /2 about the x and yaxes,
respectively, in R3. Angles are regarded as positive if they are counterclockwise
when viewed by
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Notes
Sept. 14: Definition and Examples of Rings
Geoffrey Scott
Rings: Motivation and Definition
Throughout your education, youve encountered many different mathematical objects that can
be ad
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Homework 3
Due by the start of class on Wednesday Oct. 14
The problems with the asterisks might need material from Wednesdays (or next Wednesdays)
class.
Problem 1: In Z3 [x], show that x4 + x
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
What is a Vector Space?
Geoffrey Scott
These are informal notes designed to motivate the abstract definition of a vector space to
my MAT185 students. I had trouble understanding abstract vector spaces
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Notes
Oct. 26: Constructible Numbers
Geoffrey Scott
Were finally ready for our second application of ring theory (the first was the Chinese
remainder theorem). This week, well show how the the
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Homework 7
Due by the start of class on Mon Dec. 7
Problem 1:
1. Prove that every subgroup of a solvable group is solvable.
2. Use the fact that S5 is not solvable to prove that Sn is not solv
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Homework 5
Due by the start of class on Wednesday Nov. 4
Problem 1: Find two different field extensions of Z3 , one with 9 elements and another with
27 elements, in which the polynomial f (x)
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Homework 2
Due by the start of class on Wednesday Sept. 30
The problems with the asterisks might need material from Wednesdays class.
Problem 1: List all of the ideals in Z12 . For each ideal
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Homework 4
Due by the start of class on Wednesday Oct. 28
Problem 1: Show that Q(4 i) = Q(1 + i), where i =
1 C.
Problem 2: Let p Z be any prime number, and n 2
1. Find a basis for Q( n p) as
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Homework 6
Due by the start of class on Wednesday Nov. 25
Problem 1:
1. Find the minimal polynomial for
5 in Q[x].
2. Find the minimal polynomial for 3 + 5 in Q( 5)[x].
3. Find the minimal pol
National College of Commerce & Computer Science Gilgit
Polynomial equation and fields
MATH 401

Fall 2015
Algebra Homework 1
Due by the start of class on Wednesday Sept. 23
The problems with the asterisks might need material from Wednesdays class.
Problem 1: Below are two possible addition and multiplicat
National College of Commerce & Computer Science Gilgit
Intro Statistics
MATH 2000

Fall 2015
Solutions to 18.781 Problem Set 3  Fall 2008
Due Tuesday, Sep. 30 at 1:00
1. (Niven 2.3.3) Solve the congruences x 1 (mod 4), x 0 (mod 3), x 5 (mod 7).
First we note that 4, 3, and 7 are pairwise rel
National College of Commerce & Computer Science Gilgit
Intro Statistics
MATH 2000

Fall 2015
16
Physiology of Woody Plants
than sun leaves. Leaves with deep lobes characteristic
of the upper and outer crown positions are more efficient energy exchangers than shallowly lobed leaves
(Chapter 12
National College of Commerce & Computer Science Gilgit
Intro Statistics
MATH 2000

Fall 2015
2
Physiology of Woody Plants
longevity. Equally important are hereditary differences
in capacity to tolerate or avoid environmental stresses;
phenology and growth patterns; and yield of useful
product
National College of Commerce & Computer Science Gilgit
Intro Statistics
MATH 2000

Fall 2015
Preface
This book expands and updates major portions of
the 1997 book on Physiology of Woody Plants
(Second edition) by Theodore T. Kozlowski and
Stephen G. Pallardy, published by Academic Press.
Sinc
National College of Commerce & Computer Science Gilgit
Intro Statistics
MATH 2000

Fall 2015
6
Physiology of Woody Plants
large footprints, allowing ecosystemscale sampling
(Baldocchi, 2003). Carefully designed sampling and
analysis of stable isotopes of carbon, hydrogen, and
oxygen has prov
National College of Commerce & Computer Science Gilgit
Intro Statistics
MATH 2000

Fall 2015
xi
Contents
Introduction 287
Importance of Water 287
Cell Water Relations 288
Cell Structure 288
Water Status Quantification and Terminology 288
Water Movement 290
Measurement of Water Potential and
I
National College of Commerce & Computer Science Gilgit
Intro Statistics
MATH 2000

Fall 2015
4
Physiology of Woody Plants
of wood per unit of land area and in the shortest time
possible. They routinely deal with trees growing in
plant communities and with factors affecting competition among t