Proceedings - NCUR VII. (1993), Vol. II, pp. 793-796. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction We introduce remodulization and use it to characterize the complementary sets of systems of congruences. The following is an excerpt of a continuing effort to characterize systems of congruences.
Proceedings - NCUR VIII. (1994), Vol. II, pp. 794-798. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah In this paper we present some preliminary results on a conjecture by Paul Erdos concerning covering sets of congruences. A covering set consists of a finite system of congruences with distinct moduli, such that every integer satisfies as a minimum one of the congruences. An interesting consequence of this conjecture is the dependence of the solution on abundant numbers; an abundant number is an integer whose sum of its proper divisors exceeds the integer.
Proceedings - NCUR VI. (1992), Vol. II, pp. 1036-1041.
Jeffrey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Don H. Tucker
Department of Mathematics
University of Utah
Introduction
Remodulization introduces a new method applied to congruences and systems of congruences. We prove the Chinese Remainder Theorem using the remodulization method and establish an efficient method to solve linear congruences. The following is an excerpt of Remodulization of Congruences and its Applications [2].
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Proceedings - NCUR VIII. (1994), Vol. II, pp. 794-798. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah In this paper we present some preliminary results on a conjecture by Paul Erdos concerning covering sets of congruences. A covering set consists of a finite system of congruences with distinct moduli, such that every integer satisfies as a minimum one of the congruences. An interesting consequence of this conjecture is the dependence of the solution on abundant numbers; an abundant number is an integer whose sum of its proper divisors exceeds the integer.
Proceedings - NCUR VI. (1992), Vol. II, pp. 1036-1041.
Jeffrey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Don H. Tucker
Department of Mathematics
University of Utah
Introduction
Remodulization introduces a new method applied to congruences and systems of congruences. We prove the Chinese Remainder Theorem using the remodulization method and establish an efficient method to solve linear congruences. The following is an excerpt of Remodulization of Congruences and its Applications [2].
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Journal Molecular Crystals and Liquid Crystals (1994), Vol. 256, pp. 555-561.
S. A. Jeglinski, M. E. Hollier, J. F. Gold, and Z. V. Vardeny
University of Utah, Physics Department, Salt Lake City, UT 84112
Y. Ding and T. Barton
Iowa State University, Chemistry Department, Ames, IA 50011
Abstract
A diode has been fabricated with poly(phenylene acetylene) [PPA] as the electroluminescent polymer. The device exhibited an unusual symmetric (positive and negative bias) I-V characteristic and electroluminescent output. These experimental results are discussed in terms of tunneling of electrons and holes via localized states.
Is Gravitation A Result Of Asymmetric Coulomb Charge Interactions?Jeffrey Gold
Journal of Undergraduate Research (JUR), University of Utah (1992), Vol. 3, No. 1, pp. 56-61.
Jeffrey F. Gold
Department of Physics, Department of Mathematics
University of Utah
Abstract
Many attempts have been made to equate gravitational forces with manifestations of other phenomena. In these remarks we explore the consequences of formulating gravitational forces as asymmetric Coulomb charge interactions. This is contrary to some established theories, for the model predicts differential accelerations dependent on the elemental composition of the test mass. The
predicted di erentials of acceleration of various elemental masses are compared to those differentials that have been obtained experimentally. Although the model turns out to fail, the construction of this model is a useful intellectual and pedagogical exercise.
A Characterization of Twin Prime PairsJeffrey Gold
Proceedings - NCUR V. (1991), Vol. I, pp. 362-366. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah The basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs (p; p+ 2) in which both p and p + 2 are prime integers.
A Novel Solution Of Linear CongruencesJeffrey Gold
Proceedings - NCUR IX. (1995), Vol. II, pp. 708-712
Jeffrey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Salt Lake City, Utah 84112
Don H. Tucker
Department of Mathematics
University of Utah
Salt Lake City, Utah 84112
Introduction
Although the solutions of linear congruences have been of interest for a very long time, they still remain somewhat pedagogically di cult. Because of the importance of linear congruences in fields such as public-key cryptosystems, new and innovative approaches are needed both to attract interest and to make them more accessible. While the potential for new ideas used in future research
is difficult to assess, some use may be found here. In this paper, the authors make use of the remodulization method developed in [1] as a vehicle to characterize the conditions under which solutions exist and then determine the solution space. The method is more efficient than those cited in the standard references. This novel approach relates the solution space of cx = a mod b to the Euler totient function for c rather than that of b, which allows one to develop an alternative and somewhat more efficient
approach to the problem of creating enciphering and deciphering keys in public-key cryptosystems.
Vector Products Revisited: A New and Efficient Method of Proving Vector Ident...Jeffrey Gold
Proceedings - NCUR X. (1996), Vol. II, pp. 994-998 Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction The purpose of these remarks is to introduce a variation on a theme of the scalar (inner, dot) product and establish multiplication in R^{n}. If a = (a_{1},...,a_{n}) and b = (b_{1},...,b_{n}), we define the product ab = (a_{1}b_{1},...,a_{n}b_{n }). The product is the multiplication of corresponding vector components as in the scalar product; however, instead of summing the vector components, the product preserves them in vector form. We define the inner sum (or trace) of a vector a = (a_{1},...,a_{n}) by (a) = a_{1}+...+a_{n}. If taken together with an additional definition of cyclic permutations of vectors, we are able to prove complicated vector products (combinations of dot and cross products) extremely efficiently, without appealing to the traditional (and cumbersome) epsilon-ijk proofs. When applied to determinants, this method hints at the rudiments of Galois theory.
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This topic on matrix theory and linear algebra is fundamental. The focus is on subjects like systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices that will be helpful in other fields.
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LINEAR RECURRENCE RELATIONS WITH CONSTANT COEFFICIENTSAartiMajumdar1
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PROJECT (PPT) ON PAIR OF LINEAR EQUATIONS IN TWO VARIABLES - CLASS 10mayank78610
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This talk concerning The Importance of Arts in Education was delivered at Westminster College on March 23, 2009. My personal favorite part of the talk was revealing the percentage of GNP the arts represent---a real sabot for that certain kind of philistine social (and economic) conservative who wants to crush arts funding in schools. For that individual, here's an economic argument that almost sounds like an artifact of some capitalist Utopia. Considering we don't produce much in the U.S. anymore---the legacy of Bretton-Woods---at least we still export the arts.
Short Lifetimes of Light Emitting PolymersJeffrey Gold
Jeffrey Frederick Gold
University of Cambridge
Microelectronics Research Centre
Cavendish Laboratory
Cambridge CB3 0HE
Introduction
The following manuscript was submitted as part of the MPhil program in Microelectronic Engineering and Semiconductor Physics at the Microelectronics Research Centre (MRC) of the Cavendish Laboratory at the University of Cambridge. The manuscript is a literature survey undertaken by the author as part of the MESP program during Lent term 1997 and is the basis of a talk of the same
title given at the MRC on January 20, 1997.
Presentation about Story Structure given April 20, 2009 at Westminster College. This presentation discusses the intrinsic three act structure and uses The Shawshank Redemption to show how two overlapping three-act structures gives rise to a four-act structure.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
1. Chapter 4
Complementary Sets Of
Systems Of Congruences
Proceedings|NCUR VII. 1993, Vol. II, pp. 793 796.
Je rey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Don H. Tucker
Department of Mathematics
University of Utah
Introduction
We introduce remodulization and use it to characterize the complementary sets
of systems of congruences. The following is an excerpt of a continuing e ort to
characterize systems of congruences.
Remodulization
De nition 1 If a and b are integers, then
a mod b = fa; a b; a 2b; : : :g :
We will write x a mod b, meaning that x is an element of the set a mod b.
The symbol is not an equality symbol, but rather a specialized form of the
2 or is an element of : : : symbol. The common terminology is to say that x
1
2. CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 2
is congruent to a modulo b. These sets are also frequently called residue classes
since they consist of those integers which, upon division by b, leave a remainder
residue of a. It is customary to write a as the least non-negative residue.
De nition 2 If a1; a2; : : : ; an,b 2 Z, then
n
a1 ; a2 ; : : : ; an mod b = a1 mod b a2 mod b an mod b = ai mod b :
i=1
Theorem 1 Remodulization Theorem Suppose a, b, and c 2 Z and c 0,
then
a mod b = a; a + b; : : : ; a + bc , 1 mod cb :
Proof. We write
a mod b = f : : : a , cb; a , c , 1b; : : : a , b;
a; a + b; : : : a + c , 1b;
a + cb; a + c + 1b; : : : a + 2c , 1b; : : : g
and upon rewriting the columns,
a mod b = f : : : a , cb; a + b , cb; : : : a + c , 1b , cb;
a; a + b; : : : a + c , 1b;
a + cb; a + b + cb; : : : a + c , 1b + cb; : : : g :
Then, forming unions on the extended columns, the result follows.
We refer to this process as remodulization by a factor c.
Complementary Sets
Using De nition 2, the complementary set of a mod b is given by
a mod b c = Z n fa mod bg = 0; 1; 2; : : :; a , 1; a + 1; : : : ; b , 1 mod b :
In this case, the complementary set consists of b , 1 congruences modulo b. We
will always refer to the size of a complementary set with respect to a speci c
modulus.
The following represents a system of congruences:
8
x a1 mod b1
x a2 mod b2
.. 4.1
.
: x a mod b
n n
3. CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 3
Characterizing the size of the complementary set of a system of congruences
4.1 is equivalent to counting those integers which are not elements of any of the
given congruences. Our method will be to remove the set of numbers satisfying
the system of congruences 4.1 from the set of integers Z in a systematic way.
Note that Z can be written as a complete residue class, 1; 2; : : :; b mod b, for
all b 1.
All numbers satisfying the rst congruence in 4.1 will be removed from Z,
leaving the complementary set for that congruence. Then we iterate the process
by removing all integers satisfying the second congruence from this remaining
set, and so on.
Stated another way, we are interested in determining the size of
n
a1 mod b1 a2 mod b2 an mod bn = Z n c ai mod bi ;
i=1
where the bi are pairwise relatively prime. Our method is to determine the
number of remaining congruences needed to characterize this complementary
set. Using the remodulization method, the set of congruences can be expressed
Q
in terms of the common modulus, mod n=1 bi and the integers, Z, can be ex-
i
pressed in terms of a complete residue class of the same modulus.
To illustrate this procedure, suppose we have the following system of con-
gruences
x a1 mod b1
x a2 mod b2
where gcdb1 ; b2 = 1. By the Chinese Remainder Theorem 1,2 , these
intersect in a unique residue class modulo b1 b2 .
Remodulizing the congruences by b2 and b1 , respectively, this system can be
expressed as
a1 mod b1 = a1 ; a1 + b1 ; : : : ; a1 + b1 b2 , 1 mod b1 b2
a2 mod b2 = a2 ; a2 + b2 ; : : : ; a2 + b2 b1 , 1 mod b1 b2
The rst congruence, after remodulization, consists of b2 congruences mod b1 b2
while the second remodulized congruence consists of b1 congruences mod b1 b2 ;
furthermore, Z consists of b1 b2 congruences modulo b1 b2 . Therefore, subtract-
ing b1 and b2 from b1 b2 and adding one to this sum the unique intersection of
the two congruences was removed twice from b1 b2 , we obtain
b1 b2 , b1 , b2 + 1
remaining congruences mod b1 b2 in the complementary set
a1 mod b1 a2 mod b2 c :
4. CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 4
However, b1 b2 , b1 , b2 + 1 can be rewritten as b1 , 1b2 , 1; this is typical.
Theorem 2 The complementary set of fnSn=1 ai mod big, where the bi arenpair-
Q i Q
wise relatively prime, contains exactly i=1 bi , 1 congruences modulo i=1 bi .
Proof. Suppose we have a system of congruences 4.1 where the bi are pair-
wise relatively prime. We have already found that the complementary sets of
a1 mod b1 and a1 mod b1 a2 mod b2 consist of b1 , 1 congruences modulo b1
and b2 , 1b1 , 1 congruences modulo b1 b2 , respectively.
For the induction argument, suppose we have found the complementary set
Q
up to kth congruence of 4.1 to consist of the union of k=1 bi , 1 congru-
Qk i
ences modulo i=1 bi ; then we remove from it the set of numbers congruent
fak+1 mod bk+1 g. The complementary set of the latter congruence is comprised
of bk+1 , 1 congruences modulo bk+1 . Each of these congruences shares a
unique intersection with each Q the congruences in the remaining complemen-
of Q +1
tary set; there are bk+1 , 1 k=1 bi , 1 such intersections modulo k=1 bi .
i i
Therefore, the remaining complementary set, after removing the k + 1st con-
gruence from the remaining complementary set, consists of
k+1
Y
bi , 1
i=1
k+1
Y
congruences modulo bi .
i=1
If the moduli bi are primes, then we may use Euler's phi function 3,4 , or
totient, to formulate the complementary set of a system of congruences. The
totient m counts the number of integers not exceeding m which are relatively
prime to m. For any prime p, p = p , 1; moreover, because the totient is
Q
multiplicative, p1 p2 pn = p1 p2 pn = n=1 pi , 1.
i
Corollary 1 Suppose we have a system of congruences where the moduli pi are
primes and pi = pj , for i 6= j . Then the complementary set of
6
8
x a1 mod p1
x a2 mod p2
.
.
:
.
x an mod pn
n ! n
Y Y
consists of pi congruences modulo pi .
i=1 i=1
5. CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 5
De nition 3 The density of the complementary set of a system of congruences
6
4.1, where gcdbi ; bj = 1 for i = j , with respect to the set Z, is
Y
6. bi , 1
n
n = bi :
i=1
As an illustration, we calculate the size and density of the complementary
set of the following system:
8
x 1 mod 3
x 2 mod 5
: x 3 mod 7
The complement of the rst congruence, 1 mod 3, is 2; 3 mod 3, a union of
two congruences modulo 3. The complement of 2 mod 5 is 1; 3; 4; 5 mod 5.
Each of the congruences in the complementary set modulo 3 shares a unique
intersection with the congruences of the complementary set modulo 5; there are
3 , 15 , 1 = 8 remaining congruences modulo 15. Finally, removing all num-
bers satisfying 3 mod 7 from these 8 remaining congruences, the complementary
set consists of 3 , 15 , 17 , 1 = 48 congruences modulo 105. The density
of the complementary set with respect to the set Z at each step in the process
is 2=3, 8=15, and 48=105, respectively.
If a system consists of bi congruences of the same modulus bi , for each bi ,
we have the following extension of Theorem 2. Suppose
8
x a1;1 ; a1;2 ; : : : ; a1; b1 mod b1
x a2;1 ; a2;2 ; : : : ; a2; mod b2
...
b2
4.2
:
x an;1 ; an;2 ; : : : ; an; bn mod bn
where the bi are pairwise relatively prime, and ai;j 6= ai;k for all j = k, and 6
bi bi .
The complementary set of the rst congruence is the union of b1 , b1 con-
gruences. Likewise, the complementary set of the second congruence contains
b2 , b2 congruences; their intersection contains b2 , b2 b1 , b1 congruences
modulo b1 b2 . Iterating the process, Qn complementary set of 4.2 consists of
Qn the
i=1 bi , bi congruences modulo i=1 bi . At each step, however, it is neces-
sary to insure that bi , bi 0; otherwise, if bi = bi , the complementary set
vanishes altogether, since for that particular value of i, ai;1 ; ai;2 ; : : : ; ai; bi is a
complete residue class modulo bi , i.e., the entire set Z.
n o
Theorem 3 The complementary set of Sn=1 ai;1 ; : : : ; ai; mod bi , where
i bi
ai;j = ai;k for j 6= k, and b bi , and the bi are pairwise relatively prime,
6 i
7. CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 6
Q Qn
contains exactly n=1 bi , bi congruences modulo i=1 bi . The density of the
i Qn
complementary set is n = i=1 bi ,i bi .
b
References
1 Gold, Je rey F. and Don H. Tucker, Remodulization of Congruences and Its
Applications. To be submitted.
2 Gold, Je rey F. and Don H. Tucker, Remodulization of Congruences, Pro-
ceedings - National Conference on Undergraduate Research, University of North
Carolina Press, Asheville, North Carolina, 1992, Vol. II, pp. 1036 41.
3 David M. Burton, Elementary Number Theory Wm. C. Brown Publishers,
Iowa, 1989, Second Edition, pp. 156 160.
4 Oystein Ore, Number Theory and Its History Dover Publications, Inc., New
York, 1988, pp. 109 115.