Equivalence relations and Counting

... We start by finding [0]R – the equivalence class containing 0 – so need to find all the integers x such that (0, x) ∈ R, in other words all the integers satisfying |0| = |x|. A moment’s thought tells us that x = 0 is the only integer with absolute value 0, and so ...

... We start by finding [0]R – the equivalence class containing 0 – so need to find all the integers x such that (0, x) ∈ R, in other words all the integers satisfying |0| = |x|. A moment’s thought tells us that x = 0 is the only integer with absolute value 0, and so ...

Necessary conditions on minimal system configuration for general

... expense of a larger number of fuzzy sets and fuzzy rules is not desirable both in theory and in practice. One always wants to approximate a given function with as a simple system configuration as possible. Thus, the central and practical question is: What are the necessary conditions under which fuz ...

... expense of a larger number of fuzzy sets and fuzzy rules is not desirable both in theory and in practice. One always wants to approximate a given function with as a simple system configuration as possible. Thus, the central and practical question is: What are the necessary conditions under which fuz ...

Equivalent Fractions

... that is called the simplest form of the fraction. A fraction is in simplest form when the GCF of the numerator and the denominator is 1. ...

... that is called the simplest form of the fraction. A fraction is in simplest form when the GCF of the numerator and the denominator is 1. ...

Continued Fractions

... few as 5000 people may have been polled, with 2143 of them responding “Yes.” However, none of these conclusions are correct, because they ignore the fact that statistics are rounded off. What the newspaper is really saying, therefore, is that the percentage of Yes responses may not have been exactly ...

... few as 5000 people may have been polled, with 2143 of them responding “Yes.” However, none of these conclusions are correct, because they ignore the fact that statistics are rounded off. What the newspaper is really saying, therefore, is that the percentage of Yes responses may not have been exactly ...

Slides Week 5 Modular Arithmetic

... • Two numbers r and s are said to be “congruent mod m” if r mod m = s mod m • In this case we write r s mod m • The difference between r and s will be a multiple of m So r-s = km for some value of k • E.g. ...

... • Two numbers r and s are said to be “congruent mod m” if r mod m = s mod m • In this case we write r s mod m • The difference between r and s will be a multiple of m So r-s = km for some value of k • E.g. ...

The Mathematical Universe

... lack the common understanding of concepts that we humans have evolved, e.g., “particle”, “observation” or indeed any other English words. Put differently, such a description must be expressible in a form that is devoid of human “baggage”. ...

... lack the common understanding of concepts that we humans have evolved, e.g., “particle”, “observation” or indeed any other English words. Put differently, such a description must be expressible in a form that is devoid of human “baggage”. ...

The Formulation and Justification of Mathematical

... We of course assume that it is important to describe and understand the natural world. Thus, if the preformal idea is valuable or expected to be valuable for describing and understanding the natural world, natural-world justification is reasonable. Clearly, there can be debates about what makes a pr ...

... We of course assume that it is important to describe and understand the natural world. Thus, if the preformal idea is valuable or expected to be valuable for describing and understanding the natural world, natural-world justification is reasonable. Clearly, there can be debates about what makes a pr ...

Ordered and Unordered Factorizations of Integers

... Canfield, Erdös, and Pomerance [8] remarked in their paper that it is not particularly easy to compute PHnL. They mention that even showing PH1800L = 137 takes some work. Their approach was based on a tree traversal algorithm. With our recursion this computation presents no problem. In[45]:= Out[45] ...

... Canfield, Erdös, and Pomerance [8] remarked in their paper that it is not particularly easy to compute PHnL. They mention that even showing PH1800L = 137 takes some work. Their approach was based on a tree traversal algorithm. With our recursion this computation presents no problem. In[45]:= Out[45] ...

Adventures in Flatland

... Everyone assumed that the group had inadvertently traveled in a large circular route instead of staying on a straight line. It didn’t help that Flatlanders used the words “Euclidean 2-space” and “Flatland” interchangeably. Anyone talking about Flatland having a different configuration would be stuck ...

... Everyone assumed that the group had inadvertently traveled in a large circular route instead of staying on a straight line. It didn’t help that Flatlanders used the words “Euclidean 2-space” and “Flatland” interchangeably. Anyone talking about Flatland having a different configuration would be stuck ...

Brent Revisited - Institut für Mathematik

... • Foundation of scientific and engineering computation • Inadequacy of standard computability/complexity theory • Two current schools of thought ∗ Algebraic School (Blum-Shub-Smale, . . . ∗ Analytic School (Turing (1936), Grzegorczyk (1955), Weihrauch, Ko,. . . ...

... • Foundation of scientific and engineering computation • Inadequacy of standard computability/complexity theory • Two current schools of thought ∗ Algebraic School (Blum-Shub-Smale, . . . ∗ Analytic School (Turing (1936), Grzegorczyk (1955), Weihrauch, Ko,. . . ...

mass problems associated with effectively closed sets

... the specific, natural, unsolvable problems which were discovered during this period are: the Entscheidungsproblem for logical validity in the predicate calculus, the triviality problem for finitely presented groups, Hilbert’s Tenth Problem in number theory [39], the domino problem, the homeomorphis ...

... the specific, natural, unsolvable problems which were discovered during this period are: the Entscheidungsproblem for logical validity in the predicate calculus, the triviality problem for finitely presented groups, Hilbert’s Tenth Problem in number theory [39], the domino problem, the homeomorphis ...

COMBINATORICS OF NORMAL SEQUENCES OF BRAIDS

... braids in many recent developments, in particular those of algorithmic or cryptographical nature [20, 22]. The question we address here is to count the number of braids with a normal form of a given length. It was addressed by P. Xu in [26], and by R. Charney in [9]: she observed that, because norma ...

... braids in many recent developments, in particular those of algorithmic or cryptographical nature [20, 22]. The question we address here is to count the number of braids with a normal form of a given length. It was addressed by P. Xu in [26], and by R. Charney in [9]: she observed that, because norma ...

Simple random sampling with over-replacement

... There are several methods for drawing a sample, different goals, and different situations that require different sampling designs. The most basic sampling procedures are simple random sampling with and without replacement. In this paper we show that there are several ways to select units with replac ...

... There are several methods for drawing a sample, different goals, and different situations that require different sampling designs. The most basic sampling procedures are simple random sampling with and without replacement. In this paper we show that there are several ways to select units with replac ...

pdf

... to find a program that satisfies a mathematical model (i.e., a required set of properties) that is correct-by-construction. The synthesis problem has mainly been studied in two contexts: synthesizing programs from specification, where the entire specification is given, and synthesizing programs from ...

... to find a program that satisfies a mathematical model (i.e., a required set of properties) that is correct-by-construction. The synthesis problem has mainly been studied in two contexts: synthesizing programs from specification, where the entire specification is given, and synthesizing programs from ...

PDF

... existing M TL properties. Since our algorithm is complete, if it fails to synthesize a program then it informs the designer a more comprehensive (and expensive) approach must be used. Moreover, since the complexity of our algorithm is comparable with that of model checking, the algorithm has the pot ...

... existing M TL properties. Since our algorithm is complete, if it fails to synthesize a program then it informs the designer a more comprehensive (and expensive) approach must be used. Moreover, since the complexity of our algorithm is comparable with that of model checking, the algorithm has the pot ...

A bibliography for the development of an intelligent mathematical

... in 1977 at the U.S. Federal Energy Administration. This evolved into ANALYZE (1978), which itself has evolved several levels, including a rule-based intelligent support component. In the late 1970's, Alex Meeraus, in collaborations with Jan Bisschop, Anthony Brooke, and David Kendrick, developed the ...

... in 1977 at the U.S. Federal Energy Administration. This evolved into ANALYZE (1978), which itself has evolved several levels, including a rule-based intelligent support component. In the late 1970's, Alex Meeraus, in collaborations with Jan Bisschop, Anthony Brooke, and David Kendrick, developed the ...

Introduction to "Mathematical Foundations for Software Engineering"

... He went on to prove that there was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide, in general, algorithmically whether a given Turing machine will ever halt. (This proof depended on the notion of a Un ...

... He went on to prove that there was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide, in general, algorithmically whether a given Turing machine will ever halt. (This proof depended on the notion of a Un ...

The complexity of numeral systems

... course of millennia through a sequence of higher and higher abstractions into a complex system of symbols for the abstract numbers and then for words and sentences (See Fig.1 and relative caption). It is astonishing the variety of numeral systems accross languages and cultures in the world. Many lan ...

... course of millennia through a sequence of higher and higher abstractions into a complex system of symbols for the abstract numbers and then for words and sentences (See Fig.1 and relative caption). It is astonishing the variety of numeral systems accross languages and cultures in the world. Many lan ...

Quantifier-Free Linear Arithmetic

... 4. Calculate the new values for the others rows. New row numbers = numbers in old row - ( Number in old row above or below pivot number * Corresponding number in the new row) 5. Calculate the Cj and the Cj – Zj values for this tableau. If there are any Cj – Zj numbers greater than zero, return to st ...

... 4. Calculate the new values for the others rows. New row numbers = numbers in old row - ( Number in old row above or below pivot number * Corresponding number in the new row) 5. Calculate the Cj and the Cj – Zj values for this tableau. If there are any Cj – Zj numbers greater than zero, return to st ...

slides

... image with OMP (error is 1.1833), (c) recovered image with Lasso (error is 0.7075) and (d) recovered image with StructOMP (error is 0.1203) ...

... image with OMP (error is 1.1833), (c) recovered image with Lasso (error is 0.7075) and (d) recovered image with StructOMP (error is 0.1203) ...

evolcomp - Centre for Policy Modelling

... • The complexity of things seems to vary depending on which aspect you are concerned with; • Any “real” complexity is irrelevant, as we never interact with the total systems, only certain aspects of them at any one time; • The complexity of an object is only revealed though interaction with the comp ...

... • The complexity of things seems to vary depending on which aspect you are concerned with; • Any “real” complexity is irrelevant, as we never interact with the total systems, only certain aspects of them at any one time; • The complexity of an object is only revealed though interaction with the comp ...

A Geometric Perspective on the Riemann Zeta Function`s Partial Sums

... 1. The Importance of the Riemann Zeta Function Very complex mathematical ideas often spring from the investigation of questions that are simple to understand. The subject of this article - the behavior of the Riemann Zeta Function - is one such complex mathematical object. However, the study of the ...

... 1. The Importance of the Riemann Zeta Function Very complex mathematical ideas often spring from the investigation of questions that are simple to understand. The subject of this article - the behavior of the Riemann Zeta Function - is one such complex mathematical object. However, the study of the ...

Year 8 - Portland Place School

... symbols; represent them in mappings. 1.5 Use letter symbols to represent unknown numbers or variables. 1.6 Identify the necessary information to understand or simplify a context or problem; represent problems, making correct use of symbols, words, diagrams, tables and graphs; use appropriate procedu ...

... symbols; represent them in mappings. 1.5 Use letter symbols to represent unknown numbers or variables. 1.6 Identify the necessary information to understand or simplify a context or problem; represent problems, making correct use of symbols, words, diagrams, tables and graphs; use appropriate procedu ...

Computational Complexity of Fractal Sets 1

... In this paper, we consider both the eciency of approximation and the time of computation. Although the former has been investigated, e.g. [9], [7], the latter has never been studied. This is the motivation for investigating the time to compute an approximation of box-counting dimension as one of t ...

... In this paper, we consider both the eciency of approximation and the time of computation. Although the former has been investigated, e.g. [9], [7], the latter has never been studied. This is the motivation for investigating the time to compute an approximation of box-counting dimension as one of t ...

VITAE Jesús Antonio De Loera

... Dr. Matthias Köppe (now at UC Davis, full professor). Dr. Fu Liu (now at UC Davis, associate professor) Dr. Peter Malkin (now at Billington Inc. Australia) Dr. Steve Klee (now at Seattle University, tenure-track) Dr. Steffen Borgwardt (currently at UC Davis) Dr. Elisabeth Finhold (currently at UC D ...

... Dr. Matthias Köppe (now at UC Davis, full professor). Dr. Fu Liu (now at UC Davis, associate professor) Dr. Peter Malkin (now at Billington Inc. Australia) Dr. Steve Klee (now at Seattle University, tenure-track) Dr. Steffen Borgwardt (currently at UC Davis) Dr. Elisabeth Finhold (currently at UC D ...