Here - Dartmouth Math Home
... problems, although on our way we will encounter algebraic and structural questions. The first thing we learn about maths is to count, but as we shall see counting can become quite tricky and requires techniques. The course has two parts. The first one introduces the basic objects, ideas, and princip ...
... problems, although on our way we will encounter algebraic and structural questions. The first thing we learn about maths is to count, but as we shall see counting can become quite tricky and requires techniques. The course has two parts. The first one introduces the basic objects, ideas, and princip ...
Unitary Amicable Numbers - American Mathematical Society
... Remark A. To my knowledge, no results analogous to Theorems 4 and 5 are known concerning even amicable numbers. However, the referee has pointed out that many "isomeric" pairs of odd amicable numbers exist (for example, see the pairs 60 and 61, 65 and 66, 138 and 139 in [3]), and that theorems simil ...
... Remark A. To my knowledge, no results analogous to Theorems 4 and 5 are known concerning even amicable numbers. However, the referee has pointed out that many "isomeric" pairs of odd amicable numbers exist (for example, see the pairs 60 and 61, 65 and 66, 138 and 139 in [3]), and that theorems simil ...
A STUDY OF EULERIAN NUMBERS FOR PERMUTATIONS IN THE
... ascent number. Therefore, the identities for Dn,k presented here correspond to those in [1] that are obtained by replacing k with n − k − 1. In order to study these numbers, we make use of an operator on permutations in [n], which was introduced in [9]. The operator σ is defined by adding one to all ...
... ascent number. Therefore, the identities for Dn,k presented here correspond to those in [1] that are obtained by replacing k with n − k − 1. In order to study these numbers, we make use of an operator on permutations in [n], which was introduced in [9]. The operator σ is defined by adding one to all ...
A Combinatorial Miscellany
... The study of partition enumeration was begun by Euler and is very active to this day. We will exposit some parts of this theory. All along the way there are interesting connections with algebra, but these are unfortunately too sophisticated to go into details here. We will, however, give a few hints ...
... The study of partition enumeration was begun by Euler and is very active to this day. We will exposit some parts of this theory. All along the way there are interesting connections with algebra, but these are unfortunately too sophisticated to go into details here. We will, however, give a few hints ...
MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets
... In fact, this intuition is very misguided. Consider the following definition. Definition 1.2. Consider a set S. We will say that S is: 1. finite if the elements of S may be put in one-to-one correspondence with {1, . . . , N } for some N ∈ N; 2. countably infinite if S is not finite but the elements ...
... In fact, this intuition is very misguided. Consider the following definition. Definition 1.2. Consider a set S. We will say that S is: 1. finite if the elements of S may be put in one-to-one correspondence with {1, . . . , N } for some N ∈ N; 2. countably infinite if S is not finite but the elements ...
On the expansions of a real number to several integer bases Yann
... lies in the interval bn · Iw ⊂ [0, bn ). Since the length of bn · Iw ⊂ [0, bn ) is at least equal to bn · a−L ≥ bn · b−n+`b = b`b , there exists an integer in {0, 1, . . . , bn − 1} which is congruent to c modulo b`b and belongs to bn · Iw . On the other hand, the subgroup of (Z/bn Z)∗ generated by ...
... lies in the interval bn · Iw ⊂ [0, bn ). Since the length of bn · Iw ⊂ [0, bn ) is at least equal to bn · a−L ≥ bn · b−n+`b = b`b , there exists an integer in {0, 1, . . . , bn − 1} which is congruent to c modulo b`b and belongs to bn · Iw . On the other hand, the subgroup of (Z/bn Z)∗ generated by ...
A square from similar rectangles
... deleting a vertex whose degree is 1 without deleting the terminal to an edge, which connects two terminals. Choose the left most and the right most vertices to be terminals, then it’s easy to see that there aren’t any vertices whose degree is 1 except the terminals, so deleting a vertex whose degree ...
... deleting a vertex whose degree is 1 without deleting the terminal to an edge, which connects two terminals. Choose the left most and the right most vertices to be terminals, then it’s easy to see that there aren’t any vertices whose degree is 1 except the terminals, so deleting a vertex whose degree ...
On the Number of Markoff Numbers Below a Given Bound
... The work described in this paper was done during a visit to the Istituto di Matemática in Pisa in 1979, with support from the Centro Nazionale di Ricerca and the Sonderforschungsbereich Theoretische Mathematik of the University of Bonn. I would like to thank the members of the Institute at Pisa, and ...
... The work described in this paper was done during a visit to the Istituto di Matemática in Pisa in 1979, with support from the Centro Nazionale di Ricerca and the Sonderforschungsbereich Theoretische Mathematik of the University of Bonn. I would like to thank the members of the Institute at Pisa, and ...
real analysis - Atlantic International University
... If for every 0 there is a corresponding integer N such that an L where n If lim a a n exists, we say the sequence converges (of is convergent). Otherwise, we say the sequence diverges (or s divergent). ...
... If for every 0 there is a corresponding integer N such that an L where n If lim a a n exists, we say the sequence converges (of is convergent). Otherwise, we say the sequence diverges (or s divergent). ...
Combinatorial formulas connected to diagonal
... The theory of symmetric functions arises in various areas of mathematics such as algebraic combinatorics, representation theory, Lie algebras, algebraic geometry, and special function theory. In 1988, Macdonald introduced a unique family of symmetric functions with two parameters characterized by ce ...
... The theory of symmetric functions arises in various areas of mathematics such as algebraic combinatorics, representation theory, Lie algebras, algebraic geometry, and special function theory. In 1988, Macdonald introduced a unique family of symmetric functions with two parameters characterized by ce ...
DOC
... Theorem 2. The Second Principle of Mathematical Induction: A set of positive integers that has the property that for every integer k, if it contains all the integers 1 through k then it contains k + 1 and if it contains 1 then it must be the set of all positive integers. More generally, a property c ...
... Theorem 2. The Second Principle of Mathematical Induction: A set of positive integers that has the property that for every integer k, if it contains all the integers 1 through k then it contains k + 1 and if it contains 1 then it must be the set of all positive integers. More generally, a property c ...
An Introductory Course in Elementary Number Theory
... Theorem 2. The Second Principle of Mathematical Induction: A set of positive integers that has the property that for every integer k, if it contains all the integers 1 through k then it contains k + 1 and if it contains 1 then it must be the set of all positive integers. More generally, a property c ...
... Theorem 2. The Second Principle of Mathematical Induction: A set of positive integers that has the property that for every integer k, if it contains all the integers 1 through k then it contains k + 1 and if it contains 1 then it must be the set of all positive integers. More generally, a property c ...
An Introduction to Higher Mathematics
... of discourse we substitute for the various variables, then we say they are equivalent. The advantage of equivalent formulas is that they say the same thing but in a different way. For example, algebraic manipulations such as replacing x2 − 2x = 12 with (x − 1)2 = 13 fit into this category. It is alw ...
... of discourse we substitute for the various variables, then we say they are equivalent. The advantage of equivalent formulas is that they say the same thing but in a different way. For example, algebraic manipulations such as replacing x2 − 2x = 12 with (x − 1)2 = 13 fit into this category. It is alw ...
Chapter 3 Elementary Number Theory The expression lcm(m,n
... The expression lcm(m,n) stands for the least integer which is a multiple of both the integers m and n. The expression gcd(m,n) stands for the biggest integer that divides both m and n. Find lcm and gcd in the TI-85 CATALOG and place them into your custom catalog. Following the procedures of the firs ...
... The expression lcm(m,n) stands for the least integer which is a multiple of both the integers m and n. The expression gcd(m,n) stands for the biggest integer that divides both m and n. Find lcm and gcd in the TI-85 CATALOG and place them into your custom catalog. Following the procedures of the firs ...
p-adic Num b ers
... and we do not know whether there are elements of Q which are not equivalent to elements of Q . There are more things we can realize about Q . First of all, Q is a eld. Because it seems rather intuitive, the proof is omitted. For a proof, see [Vladimirov 94]. Also, in order for in nitely long p-adic ...
... and we do not know whether there are elements of Q which are not equivalent to elements of Q . There are more things we can realize about Q . First of all, Q is a eld. Because it seems rather intuitive, the proof is omitted. For a proof, see [Vladimirov 94]. Also, in order for in nitely long p-adic ...