Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability tha... more Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2,…, n} in π n is n 2 2 ( 1 - o ( 1 ) ) {{{n^2}} \over 2}\left( {1 - o\left( 1 \right)} \right) as n → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.
Preparing Teachers for Three-Dimensional Instruction, 2018
Book Summary: Its not enough for teachers to read through the Next Generation Science Standards (... more Book Summary: Its not enough for teachers to read through the Next Generation Science Standards (NGSS) and correlate their content to the established curriculum. Teachers must prepare to make the vision of the NGSS come alive in their classrooms. Editor Jack Rhoton maintains that the preparation will be most effective if it begins in undergraduate coursework and is sustained by ongoing professional development designed to bring about real change. The goal of Preparing Teachers for Three-Dimensional Instruction is to contribute to that preparation and that change. It showcases the many shifts that higher education science faculty, teacher education faculty, and others are already making to bring the standards to life.Preparing Teachers was written specifically for preservice science teachers, but science education faculty and practicing K 12 teachers can also benefit from it. The authors of the 18 chapters are outstanding classroom practitioners and science educators at all levels. Section I provides examples of teaching models that fulfill the intent of the NGSS. Section II describes approaches to professional development that can improve practice. Sections III and IV consider what can be done in both teacher preparation courses and undergraduate science courses for preservice science teachers. Section V explores ways to enlist the business community and other partners in support of the changes the standards can bring about. Rhoton calls the book a motivating resource for the science education community. Use it to achieve the ultimate goal of the NGSS: to move science education away from the formulaic classroom methods many students are now experiencing and instead support them in becoming true practitioners of science
We describe how a team approach that we developed as a mentoring strategy can be used to recruit,... more We describe how a team approach that we developed as a mentoring strategy can be used to recruit, advance, and guide students to be more interested in the interdisciplinary field of mathematical biology, and lead to success in undergraduate research in this field. Students are introduced to research in their first semester via lab rotations. Their participation in the research of four faculty members—two from biology and two from mathematics—gives them a first-hand overview of re-search in quantitative biology and also some initial experience in research itself. However, one of the primary goals of the lab rotation experience is that of developing teams of students and faculty that combine mathematics and statistics with biology and the life sciences, teams that subsequently mentor undergraduate research in genuine interdisciplinary environments. Thus, the team concept serves not only as a means of establishing interdisciplinary research, but also as a means of incorpo-rating new st...
The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare’s Hamlet an... more The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare’s Hamlet and any other works one may wish to add to the list will each be typed, not once, not twice, but infinitely often with a probability of 1. This dramatic fact is a simple consequence of the Borel-Cantelli lemma and will come as no surprise to anyone who has taken a graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary understanding of the concept of independence, together with the notion of limit superior and limit inferior of a sequence of sets.
Mathematicians have traditionally been a select group of academics that produce high-impact ideas... more Mathematicians have traditionally been a select group of academics that produce high-impact ideas allowing substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant rate of approximately 1% of bachelor degrees awarded in the United States. Even within STEM majors, mathematics or statistics only constitute about 6% of undergraduate degrees awarded nationally. However, the need for STEM professionals continues to grow and the list of needed occupational skills rests heavily in foundational concepts of mathematical modeling curricula, where the interplay of measurements, computer simulation and underlying theoretical frameworks takes center stage. It is not viable to expect a majority of these STEM undergraduates would pursue a double-major that includes mathematics. Here we present our solution, some early results of implementation, and a plan for nationwide adoption.
A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so... more A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so that each object appears exactly once as a set of consecutive elements in the cycle. In this paper, we show the power of de Bruijn's original theorem, namely that the cycles bearing his name exist for n-letter words on a k-letter alphabet for all values of k,n, to prove that we can create de Bruijn cycles for the assignment of elements of [n]={1,2,....,n} to the sets in any labeled subposet of the Boolean lattice; de Bruijn's theorem corresponds to the case when the subposet in question consists of a single ground element. The landmark work of Chung, Diaconis, and Graham extended the agenda of finding de Bruijn cycles to possibly the next most natural set of combinatorial objects, namely k-subsets of [n]. In this area, important contributions have been those of Hurlbert and Rudoy. Here we follow the direction of Blanca and Godbole, who proved that, in a suitable encoding, de Bru...
When considering binary strings, it's natural to wonder how many distinct subsequences might ... more When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct subsequences in a fixed string, we might next be interested in the expected number of distinct subsequences in random strings. This expected value is already known for random binary strings where each letter in the string is, independently, equally likely to be a 1 or a 0. We generalize this result to random strings where the letter 1 appears independently with probability $\alpha \in [0,1]$. Also, we make some progress in the case of random strings from an arbitrary alphabet as well as when the string is generated by a two-state Markov chain.
Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn wi... more Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn with replacement from a dichotomous population. The exact distribution of Nn,k has recently been obtained in closed algorithmically simple form; we discuss the programming of these algorithms for values of n that are large, but not so large that asymptotic results can be invoked. Using the conditional distribution of Nn,k we derive a test for randomness and compare it with standard procedures based on runs, ranks, and variances. The simulation results showed that the new test is significantly more powerful in detecting certain types of clustering. Applications in neurology and reliability are provided.
Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability tha... more Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2,…, n} in π n is n 2 2 ( 1 - o ( 1 ) ) {{{n^2}} \over 2}\left( {1 - o\left( 1 \right)} \right) as n → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.
Preparing Teachers for Three-Dimensional Instruction, 2018
Book Summary: Its not enough for teachers to read through the Next Generation Science Standards (... more Book Summary: Its not enough for teachers to read through the Next Generation Science Standards (NGSS) and correlate their content to the established curriculum. Teachers must prepare to make the vision of the NGSS come alive in their classrooms. Editor Jack Rhoton maintains that the preparation will be most effective if it begins in undergraduate coursework and is sustained by ongoing professional development designed to bring about real change. The goal of Preparing Teachers for Three-Dimensional Instruction is to contribute to that preparation and that change. It showcases the many shifts that higher education science faculty, teacher education faculty, and others are already making to bring the standards to life.Preparing Teachers was written specifically for preservice science teachers, but science education faculty and practicing K 12 teachers can also benefit from it. The authors of the 18 chapters are outstanding classroom practitioners and science educators at all levels. Section I provides examples of teaching models that fulfill the intent of the NGSS. Section II describes approaches to professional development that can improve practice. Sections III and IV consider what can be done in both teacher preparation courses and undergraduate science courses for preservice science teachers. Section V explores ways to enlist the business community and other partners in support of the changes the standards can bring about. Rhoton calls the book a motivating resource for the science education community. Use it to achieve the ultimate goal of the NGSS: to move science education away from the formulaic classroom methods many students are now experiencing and instead support them in becoming true practitioners of science
We describe how a team approach that we developed as a mentoring strategy can be used to recruit,... more We describe how a team approach that we developed as a mentoring strategy can be used to recruit, advance, and guide students to be more interested in the interdisciplinary field of mathematical biology, and lead to success in undergraduate research in this field. Students are introduced to research in their first semester via lab rotations. Their participation in the research of four faculty members—two from biology and two from mathematics—gives them a first-hand overview of re-search in quantitative biology and also some initial experience in research itself. However, one of the primary goals of the lab rotation experience is that of developing teams of students and faculty that combine mathematics and statistics with biology and the life sciences, teams that subsequently mentor undergraduate research in genuine interdisciplinary environments. Thus, the team concept serves not only as a means of establishing interdisciplinary research, but also as a means of incorpo-rating new st...
The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare’s Hamlet an... more The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare’s Hamlet and any other works one may wish to add to the list will each be typed, not once, not twice, but infinitely often with a probability of 1. This dramatic fact is a simple consequence of the Borel-Cantelli lemma and will come as no surprise to anyone who has taken a graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary understanding of the concept of independence, together with the notion of limit superior and limit inferior of a sequence of sets.
Mathematicians have traditionally been a select group of academics that produce high-impact ideas... more Mathematicians have traditionally been a select group of academics that produce high-impact ideas allowing substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant rate of approximately 1% of bachelor degrees awarded in the United States. Even within STEM majors, mathematics or statistics only constitute about 6% of undergraduate degrees awarded nationally. However, the need for STEM professionals continues to grow and the list of needed occupational skills rests heavily in foundational concepts of mathematical modeling curricula, where the interplay of measurements, computer simulation and underlying theoretical frameworks takes center stage. It is not viable to expect a majority of these STEM undergraduates would pursue a double-major that includes mathematics. Here we present our solution, some early results of implementation, and a plan for nationwide adoption.
A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so... more A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so that each object appears exactly once as a set of consecutive elements in the cycle. In this paper, we show the power of de Bruijn's original theorem, namely that the cycles bearing his name exist for n-letter words on a k-letter alphabet for all values of k,n, to prove that we can create de Bruijn cycles for the assignment of elements of [n]={1,2,....,n} to the sets in any labeled subposet of the Boolean lattice; de Bruijn's theorem corresponds to the case when the subposet in question consists of a single ground element. The landmark work of Chung, Diaconis, and Graham extended the agenda of finding de Bruijn cycles to possibly the next most natural set of combinatorial objects, namely k-subsets of [n]. In this area, important contributions have been those of Hurlbert and Rudoy. Here we follow the direction of Blanca and Godbole, who proved that, in a suitable encoding, de Bru...
When considering binary strings, it's natural to wonder how many distinct subsequences might ... more When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct subsequences in a fixed string, we might next be interested in the expected number of distinct subsequences in random strings. This expected value is already known for random binary strings where each letter in the string is, independently, equally likely to be a 1 or a 0. We generalize this result to random strings where the letter 1 appears independently with probability $\alpha \in [0,1]$. Also, we make some progress in the case of random strings from an arbitrary alphabet as well as when the string is generated by a two-state Markov chain.
Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn wi... more Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn with replacement from a dichotomous population. The exact distribution of Nn,k has recently been obtained in closed algorithmically simple form; we discuss the programming of these algorithms for values of n that are large, but not so large that asymptotic results can be invoked. Using the conditional distribution of Nn,k we derive a test for randomness and compare it with standard procedures based on runs, ranks, and variances. The simulation results showed that the new test is significantly more powerful in detecting certain types of clustering. Applications in neurology and reliability are provided.
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