Contents - GLLM Moodle
... so that the courses are enjoyable for all participants. It will enable students to progress to higher-level courses of mathematical studies. Following this linear course, learners could benefit from having a greater understanding of the links between subject areas, in particular graphical and algebr ...
... so that the courses are enjoyable for all participants. It will enable students to progress to higher-level courses of mathematical studies. Following this linear course, learners could benefit from having a greater understanding of the links between subject areas, in particular graphical and algebr ...
(pdf)
... p := {α ∈ k : |α| < 1} is obviously an ideal. Since it contains all non-unit elements, it is a maximal ideal, so that o/p is a field. We are interested in the case where the residue field o/p is finite. If we suppose further that | · | is discrete, then we can use the theory of discrete valuation ri ...
... p := {α ∈ k : |α| < 1} is obviously an ideal. Since it contains all non-unit elements, it is a maximal ideal, so that o/p is a field. We are interested in the case where the residue field o/p is finite. If we suppose further that | · | is discrete, then we can use the theory of discrete valuation ri ...
The Amazing Colors of Pascal`s Triangle
... Most of the ways to obtain a product of x involve taking a term from some “stuff,” which will be a multiple of 7. The only way not to obtain a multiple of 7 would be to use only the first and last terms from each factor. But this is like expressing 90 as the sum of powers of 7. Self-Similarity of Mu ...
... Most of the ways to obtain a product of x involve taking a term from some “stuff,” which will be a multiple of 7. The only way not to obtain a multiple of 7 would be to use only the first and last terms from each factor. But this is like expressing 90 as the sum of powers of 7. Self-Similarity of Mu ...
Discrete Mathematics - Harvard Mathematics Department
... Cardinalities Definition A set which is either finite or the same cardinality as the integers is called countable, and uncountable otherwise. When an infinite set is countable, we write its cardinality as ℵ0 . ...
... Cardinalities Definition A set which is either finite or the same cardinality as the integers is called countable, and uncountable otherwise. When an infinite set is countable, we write its cardinality as ℵ0 . ...
Projections in n-Dimensional Euclidean Space to Each Coordinates
... (19) Let a, b be real numbers, f be a map from ETn into R1 , and given i. Suppose that for every element p of the carrier of ETn holds f (p) = Proj(p, i). Then f −1 ({s : a < s ∧ s < b}) = {p; p ranges over elements of the carrier of ETn : a < Proj(p, i) ∧ Proj(p, i) < b}. (20) Let M be a metric spa ...
... (19) Let a, b be real numbers, f be a map from ETn into R1 , and given i. Suppose that for every element p of the carrier of ETn holds f (p) = Proj(p, i). Then f −1 ({s : a < s ∧ s < b}) = {p; p ranges over elements of the carrier of ETn : a < Proj(p, i) ∧ Proj(p, i) < b}. (20) Let M be a metric spa ...
ON TOPOLOGICAL NUMBERS OF GRAPHS 1. Introduction
... settled by Vijayakumar [20] in 2011. Motivated by this, the authors of the paper studied set-indexers of graphs in [14], [16] and [19]. Introducing the concept of topological set-indexers (t-set indexers) in [2], Acharya established a link between Graph Theory and Point Set Topology. He also propoun ...
... settled by Vijayakumar [20] in 2011. Motivated by this, the authors of the paper studied set-indexers of graphs in [14], [16] and [19]. Introducing the concept of topological set-indexers (t-set indexers) in [2], Acharya established a link between Graph Theory and Point Set Topology. He also propoun ...
Integer Functions - Books in the Mathematical Sciences
... [0] has the property that it contains the differences of every pair of numbers in the class. For example, since 18 and 54 belong to the class so do 18 - 54 = -36 and 54 - 18 = 36. We define a module as any class of numbers containing at least two numbers and containing the differences of every pair ...
... [0] has the property that it contains the differences of every pair of numbers in the class. For example, since 18 and 54 belong to the class so do 18 - 54 = -36 and 54 - 18 = 36. We define a module as any class of numbers containing at least two numbers and containing the differences of every pair ...
Logarithms and Exponentials - Florida Tech Department of
... The logarithm base e is called the natural logarithm. Using calculus it can be established that ln(1 + x ) ≈ x − ...
... The logarithm base e is called the natural logarithm. Using calculus it can be established that ln(1 + x ) ≈ x − ...
Year 8 - Portland Place School
... represent them in mappings or on a spreadsheet. 7.2 Express simple functions algebraically and represent them in mappings or on a spreadsheet. 7.3 Generate points in all four quadrants and plot the graphs of linear functions, where y is given explicitly in terms of x, on paper and using ICT. 7.4 Rec ...
... represent them in mappings or on a spreadsheet. 7.2 Express simple functions algebraically and represent them in mappings or on a spreadsheet. 7.3 Generate points in all four quadrants and plot the graphs of linear functions, where y is given explicitly in terms of x, on paper and using ICT. 7.4 Rec ...
Aneesh - Department Of Mathematics
... 7. Kulkarni, S. H.; Nair, M. T.; Namboodiri, M. N. N. An elementary proof for a characterization of $\sp *$-isomorphisms. Proc. Amer. Math. Soc. 134 (2006), no. 1, 229--234 8. Namboodiri, M. N. N.; Remadevi, S. A note on Szegö's theorem. J. Comput. Anal. Appl. 6 (2004), no. 2, 147--152. 9. Namboodir ...
... 7. Kulkarni, S. H.; Nair, M. T.; Namboodiri, M. N. N. An elementary proof for a characterization of $\sp *$-isomorphisms. Proc. Amer. Math. Soc. 134 (2006), no. 1, 229--234 8. Namboodiri, M. N. N.; Remadevi, S. A note on Szegö's theorem. J. Comput. Anal. Appl. 6 (2004), no. 2, 147--152. 9. Namboodir ...
Science- Kindergarten
... o concretely, pictorially, symbolically, and by using spoken or written language to express, describe, explain, justify, and apply mathematical ideas; may use technology such as screencasting apps, digital photos Reflect: o sharing the mathematical thinking of self and others, including evaluating s ...
... o concretely, pictorially, symbolically, and by using spoken or written language to express, describe, explain, justify, and apply mathematical ideas; may use technology such as screencasting apps, digital photos Reflect: o sharing the mathematical thinking of self and others, including evaluating s ...
to word - Warner School of Education
... NCTM 2012 1.A.3.8 Geometric constructions, axiomatic reasoning, and proof Geometric constructions, axiomatic reasoning, and proof NCTM 2012 1.A.3.9 Analytic geometry Analytic and coordinate geometry including algebraic proofs (e.g., the Pythagorean Theorem and its converse) and equations of lines an ...
... NCTM 2012 1.A.3.8 Geometric constructions, axiomatic reasoning, and proof Geometric constructions, axiomatic reasoning, and proof NCTM 2012 1.A.3.9 Analytic geometry Analytic and coordinate geometry including algebraic proofs (e.g., the Pythagorean Theorem and its converse) and equations of lines an ...
Primality Testing and Integer Factorization in Public
... Primality testing and integer factorization, as identified by Gauss in his Disquisitiones Arithmeticae, Article 329, in 1801, are the two most fundamental problems (as well as two most important research fields) in number theory. With the advent of modern computers, they have also been found unexpec ...
... Primality testing and integer factorization, as identified by Gauss in his Disquisitiones Arithmeticae, Article 329, in 1801, are the two most fundamental problems (as well as two most important research fields) in number theory. With the advent of modern computers, they have also been found unexpec ...
Evaluating the exact infinitesimal values of area of Sierpinski`s
... introduced in [14] and built using Cantor’s ideas. It is important to emphasize that our point of view on axiomatic systems is also more applied than the traditional one. Due to Postulate 2, mathematical objects are not define by axiomatic systems that just determine formal rules for operating with ...
... introduced in [14] and built using Cantor’s ideas. It is important to emphasize that our point of view on axiomatic systems is also more applied than the traditional one. Due to Postulate 2, mathematical objects are not define by axiomatic systems that just determine formal rules for operating with ...
Set Theory
... In a population of 100 people, 10 are rich, 5 are famous, and 3 are both rich and famous. How many persons are: ...
... In a population of 100 people, 10 are rich, 5 are famous, and 3 are both rich and famous. How many persons are: ...
MATH 210, Finite and Discrete Mathematics
... discrete methods in pure and applied mathematics; to supply an introduction or reintroduction to the art of very clear deductive explanation. Course Description: The terms combinatorics and discrete mathematics have similar meanings. The former refers to an area of pure mathematics concerned with ma ...
... discrete methods in pure and applied mathematics; to supply an introduction or reintroduction to the art of very clear deductive explanation. Course Description: The terms combinatorics and discrete mathematics have similar meanings. The former refers to an area of pure mathematics concerned with ma ...
Reduced decompositions of permutations in terms of star
... bracket sequences as an intermediary (Section 2). Many of these combinatorial objects have been studied earlier, which simpliÿes our task. Remark 1.2. The analogous problem has been studied for various other generating sets. See [10] for the case of adjacent transpositions and [2] for the case of al ...
... bracket sequences as an intermediary (Section 2). Many of these combinatorial objects have been studied earlier, which simpliÿes our task. Remark 1.2. The analogous problem has been studied for various other generating sets. See [10] for the case of adjacent transpositions and [2] for the case of al ...
Discrete Mathematics and Logic II. Formal Logic
... For some logics, there is no algorithm to tell whether an arbitrary formula is a theorem or not In fact, most logics that deal with interesting domains of discourse, like the integers, do not have decision procedures ...
... For some logics, there is no algorithm to tell whether an arbitrary formula is a theorem or not In fact, most logics that deal with interesting domains of discourse, like the integers, do not have decision procedures ...
Logical Inference and Mathematical Proof
... In fact, the process of logical inference is also the process of giving a formal proof. In mathematics, we need to do a lot of proofs. However, formal proofs are too long, since in each step we can only apply a simple inference rule. Formal proofs are also too hard to follow since we can be easily b ...
... In fact, the process of logical inference is also the process of giving a formal proof. In mathematics, we need to do a lot of proofs. However, formal proofs are too long, since in each step we can only apply a simple inference rule. Formal proofs are also too hard to follow since we can be easily b ...
1 - CS285
... Now we need to know: How many valid strings of length n are there, if the string ends with a 0? Valid strings of length n ending with a 0 must have a 1 as their (n – 1)st bit (otherwise they would end with 00 and would not be valid). And what is the number of valid strings of length (n – 1) that end ...
... Now we need to know: How many valid strings of length n are there, if the string ends with a 0? Valid strings of length n ending with a 0 must have a 1 as their (n – 1)st bit (otherwise they would end with 00 and would not be valid). And what is the number of valid strings of length (n – 1) that end ...
Report - Purdue Math
... classes of subsets of Rk – most notably semi-algebraic and semi-Pfaffian sets. The usual setting for proving these bounds is as follows. One considers a semi-algebraic (or semi-Pfaffian) set S ⊂ Rk defined by a Boolean formula whose atoms consists of P > 0, P = 0, P < 0, P ∈ P, where P is a set of p ...
... classes of subsets of Rk – most notably semi-algebraic and semi-Pfaffian sets. The usual setting for proving these bounds is as follows. One considers a semi-algebraic (or semi-Pfaffian) set S ⊂ Rk defined by a Boolean formula whose atoms consists of P > 0, P = 0, P < 0, P ∈ P, where P is a set of p ...
Logic and discrete mathematics (HKGAB4) http://www.ida.liu.se
... “persons x and y look similar to each other” “a car x is behind a car y” “a car x is close to a car y” “a car x is more comfortable than a car y” “a person x is inside of a car y” “a person x drives a car y” “a person x likes a car y” ...
... “persons x and y look similar to each other” “a car x is behind a car y” “a car x is close to a car y” “a car x is more comfortable than a car y” “a person x is inside of a car y” “a person x drives a car y” “a person x likes a car y” ...
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying ""smoothly"", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in ""continuous mathematics"" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term ""discrete mathematics."" Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.In the university curricula, ""Discrete Mathematics"" appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction to efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in freshmen; as such it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics it is sometimes seen a preparatory course, not unlike precalculus in this respect.The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.