Sums of Digits and the Distribution of Generalized Thue
... which we call the generalized Thue-Morse sequence. Here sq (k) is the digit sum of k in base q. We give an explicit formulation of the exact minimal value of M such that every M consecutive terms in tq,n cover the residue system of n, i.e., {0, 1, . . . , n − 1}. Also, we prove some stronger related ...
... which we call the generalized Thue-Morse sequence. Here sq (k) is the digit sum of k in base q. We give an explicit formulation of the exact minimal value of M such that every M consecutive terms in tq,n cover the residue system of n, i.e., {0, 1, . . . , n − 1}. Also, we prove some stronger related ...
Document
... Irreflexive and anti-symmetric relations, partially ordered relations, Composite relations, Inverse relation, Functions: Definition of Function, Domain, Co domain and Range, Well defined function. ...
... Irreflexive and anti-symmetric relations, partially ordered relations, Composite relations, Inverse relation, Functions: Definition of Function, Domain, Co domain and Range, Well defined function. ...
![[Part 1]](http://s1.studyres.com/store/data/008795717_1-da61206028950a8b76c72065c95ca070-300x300.png)
[Part 1]
... Again, equality at this stage implies the result. Otherwise, we continue the process, and in general, n, ...
... Again, equality at this stage implies the result. Otherwise, we continue the process, and in general, n, ...
Completeness of the real numbers
... above, let L = sup({xn ; n ≥ 0}). It is easy to show that xn → L. The converse is true! Theorem. Let F be an odered field with the monotone convergence property. Then if A ⊂ F is non-empty and bounded above, there exists L ∈ F so that L = sup(A). Proof. (i) Let a1 ∈ A be arbitrary, M1 ∈ F an upper b ...
... above, let L = sup({xn ; n ≥ 0}). It is easy to show that xn → L. The converse is true! Theorem. Let F be an odered field with the monotone convergence property. Then if A ⊂ F is non-empty and bounded above, there exists L ∈ F so that L = sup(A). Proof. (i) Let a1 ∈ A be arbitrary, M1 ∈ F an upper b ...
Infinite Sequences
... on in…nite sequences. Here are some general facts about sequences studied in this class: Every number an in the sequence has a successor an+1 ; the sequence never stops since we study in…nite sequences. Think of the index of a particular element as indicating the position of the element in the list. ...
... on in…nite sequences. Here are some general facts about sequences studied in this class: Every number an in the sequence has a successor an+1 ; the sequence never stops since we study in…nite sequences. Think of the index of a particular element as indicating the position of the element in the list. ...
Lecture 8 Notes
... A function is primitive recursive iff it can be defined by a sequence of applications of these five operations. What is a good name for (5a )? Note, there is a natural way of extending the sequence a0 (x, y), a1 (x, y), a2 (x, y) Successor addition multiplication What is next? How to define it? How ...
... A function is primitive recursive iff it can be defined by a sequence of applications of these five operations. What is a good name for (5a )? Note, there is a natural way of extending the sequence a0 (x, y), a1 (x, y), a2 (x, y) Successor addition multiplication What is next? How to define it? How ...
these
... axis and the graph of a piecewise constant function equal to f (x0j ) on the interval Ij . This function is a good approximation for f when (adjacent) points of P are close to each other. For this reason, we would like to define the integral of f (i.e., the area between the horizontal axis and the g ...
... axis and the graph of a piecewise constant function equal to f (x0j ) on the interval Ij . This function is a good approximation for f when (adjacent) points of P are close to each other. For this reason, we would like to define the integral of f (i.e., the area between the horizontal axis and the g ...
mplications of Cantorian Transfinite Set Theory
... http://www.buzzle.com/editorials/9-9-2002-26002.asp ...
... http://www.buzzle.com/editorials/9-9-2002-26002.asp ...
Finite Calculus: A Tutorial - Purdue University :: Computer Science
... Definition (Falling Power). The expression x to the m falling is denoted xm .3 The value of xm = x(x − 1)(x − 2) · · · (x − (m − 1)). Using the falling powers, we’ll now prove that falling powers are analogous to regular powers in finite calculus. Theorem 3.1. The discrete derivative of a falling po ...
... Definition (Falling Power). The expression x to the m falling is denoted xm .3 The value of xm = x(x − 1)(x − 2) · · · (x − (m − 1)). Using the falling powers, we’ll now prove that falling powers are analogous to regular powers in finite calculus. Theorem 3.1. The discrete derivative of a falling po ...
Functions I - Australian Mathematical Sciences Institute
... Union and intersection are familiar operations on sets. Another useful operation on sets is set difference. For two sets A and B , we define A \ B = { x ∈ A | x ∉ B }. That is, A \ B is the set of all elements of A that are not in B . We can read A \ B as ‘A take away B ’ or ‘A minus B ’. For exampl ...
... Union and intersection are familiar operations on sets. Another useful operation on sets is set difference. For two sets A and B , we define A \ B = { x ∈ A | x ∉ B }. That is, A \ B is the set of all elements of A that are not in B . We can read A \ B as ‘A take away B ’ or ‘A minus B ’. For exampl ...
Insertion Sort – Worst Case Analysis
... in our sequence). We have 100 sums (as many as the numbers in our sequence) of 101 for twice the sequence. Therefore, the sum of the numbers in [1, 100] is: 100 * 101 / 2. In general, we can always compute the sum of all n elements in a given arithmetic progression as n times the arithmetic mean of ...
... in our sequence). We have 100 sums (as many as the numbers in our sequence) of 101 for twice the sequence. Therefore, the sum of the numbers in [1, 100] is: 100 * 101 / 2. In general, we can always compute the sum of all n elements in a given arithmetic progression as n times the arithmetic mean of ...
History of Solving Polynomial Equations
... • The proofs of d’Alembert, Euler, and Foncenex all make implicit use of the FTA or invalid assumptions. • All the pre-Gaussian proofs of the FTA assumed the existence of the zeros and attempted to show that the zeros are complex numbers. • Gauss’s was the first to avoid this assumption of the exist ...
... • The proofs of d’Alembert, Euler, and Foncenex all make implicit use of the FTA or invalid assumptions. • All the pre-Gaussian proofs of the FTA assumed the existence of the zeros and attempted to show that the zeros are complex numbers. • Gauss’s was the first to avoid this assumption of the exist ...
Fibonacci Numbers
... sequence using Excel is here. The Fibonacci sequence exhibits a certain numerical pattern which originated as the answer to an exercise in the first ever high school algebra text. This pattern turned out to have an interest and importance far beyond what its creator imagined. It can be used to model ...
... sequence using Excel is here. The Fibonacci sequence exhibits a certain numerical pattern which originated as the answer to an exercise in the first ever high school algebra text. This pattern turned out to have an interest and importance far beyond what its creator imagined. It can be used to model ...