Appreciate the process, by taking a random walk

2020 ◽  
pp. 241-244
Author(s):  
Susan D'Agostino
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Life ◽  
Mathematics Students ◽  
E Coli ◽  
Starting Point ◽  
Mathematical Formalization ◽  
Biased Random Walks

“Appreciate the process, by taking a random walk” offers a basic introduction to random walks—a mathematical formalization for a path as an object wanders away from a starting point—with a particular focus on biased random walks, including a real-life example of a random walk of an E. coli bacterium. The discussion is illustrated with hand-drawn sketches. Mathematics students and enthusiasts are encouraged to build in a productive bias in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Simple random walk statistics. Part I: Discrete time results

1996 ◽  
Vol 33 (2) ◽  
pp. 311-330 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Order Statistics ◽  
Discrete Time ◽  
Rank Order ◽  
Simple Random Walk ◽  
Alternative Method ◽  
Starting Point ◽  
Famous Paper ◽  
Rank Order Statistics

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

Simple random walk statistics. Part I: Discrete time results

1996 ◽  
Vol 33 (02) ◽  
pp. 311-330 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Order Statistics ◽  
Discrete Time ◽  
Rank Order ◽  
Simple Random Walk ◽  
Alternative Method ◽  
Starting Point ◽  
Famous Paper ◽  
Rank Order Statistics

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

THE SPEED OF RANDOM WALKS ON TREES AND ELECTRIC NETWORKS

2011 ◽  
Vol 26 (1) ◽  
pp. 105-116 ◽  
Author(s):  
Mokhtar Konsowa ◽  
Fahimah Al-Awadhi
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Trees ◽  
Hitting Times ◽  
Electric Networks ◽  
Starting Point ◽  
The Way

The speed of the random walk on a tree is the rate of escaping its starting point. It depends on the way that the branching occurs in the sense that if the average number of branching is large, the speed is more likely to be positive. The speed on some models of random trees is calculated via calculating the hitting times of the consecutive levels of the tree.

Activities for Students: Monod's Nightmare Problem

2007 ◽  
Vol 100 (9) ◽  
pp. 627-631
Author(s):  
R. Lee Collins
Keyword(s):  
Biological Sciences ◽  
Escherichia Coli ◽  
Real Life ◽  
Coli Cell ◽  
Mathematics Students ◽  
E Coli ◽  
Life Phenomena

Through problems that arise in contexts outside of mathematics, students are provided with rich opportunities to hypothesize, investigate, and analyze real-life phenomena. The particular connection in this activity is from the biological sciences to mathematics. Students will consider what would happen if Escherichia coli (E. coli) bacteria were allowed to grow unchecked in the absence of death. Starting with one E. coli cell, students will calculate the time, in hours, it would take to fill a room with the bacteria.

Identification of N-Substituted Triazolo-azetidines as Novel Antibacterials using pDualrep2 HTS Platform

2019 ◽  
Vol 22 (5) ◽  
pp. 346-354
Author(s):  
Yan A. Ivanenkov ◽  
Renat S. Yamidanov ◽  
Ilya A. Osterman ◽  
Petr V. Sergiev ◽  
Vladimir A. Aladinskiy ◽  
...  
Keyword(s):  
Antibacterial Activity ◽  
Large Scale ◽  
Underlying Mechanism ◽  
Luciferase Assay ◽  
Reporter System ◽  
E Coli ◽  
Starting Point ◽  
High Concentrations ◽  
Mic Value

Aim and Objective: Antibiotic resistance is a serious constraint to the development of new effective antibacterials. Therefore, the discovery of the new antibacterials remains one of the main challenges in modern medicinal chemistry. This study was undertaken to identify novel molecules with antibacterial activity. Materials and Methods: Using our unique double-reporter system, in-house large-scale HTS campaign was conducted for the identification of antibacterial potency of small-molecule compounds. The construction allows us to visually assess the underlying mechanism of action. After the initial HTS and rescreen procedure, luciferase assay, C14-test, determination of MIC value and PrestoBlue test were carried out. Results: HTS rounds and rescreen campaign have revealed the antibacterial activity of a series of Nsubstituted triazolo-azetidines and their isosteric derivatives that has not been reported previously. Primary hit-molecule demonstrated a MIC value of 12.5 µg/mL against E. coli Δ tolC with signs of translation blockage and no SOS-response. Translation inhibition (26%, luciferase assay) was achieved at high concentrations up to 160 µg/mL, while no activity was found using C14-test. The compound did not demonstrate cytotoxicity in the PrestoBlue assay against a panel of eukaryotic cells. Within a series of direct structural analogues bearing the same or bioisosteric scaffold, compound 2 was found to have an improved antibacterial potency (MIC=6.25 µg/mL) close to Erythromycin (MIC=2.5-5 µg/mL) against the same strain. In contrast to the parent hit, this compound was more active and selective, and provided a robust IP position. Conclusion: N-substituted triazolo-azetidine scaffold may be used as a versatile starting point for the development of novel active and selective antibacterial compounds.

Teaching Music Theory

2020 ◽  
Author(s):  
Jennifer Snodgrass
Keyword(s):  
Music Theory ◽  
Classroom Environment ◽  
Teaching And Learning ◽  
Undergraduate Research ◽  
Real Life ◽  
Aural Skills ◽  
New Faculty ◽  
Music Therapist ◽  
Starting Point ◽  
Teaching Music

Many innovative approaches to teaching are being used around the country, and there is an exciting energy about the scholarship of teaching and learning. But what is happening in the most effective music theory and aural skills classrooms? Based on 3 years of field study spanning 17 states, coupled with reflections from the author’s own teaching strategies, Teaching Music Theory: New Voices and Approaches highlights teaching approaches with substantial real-life examples from instructors across the country. The main premise of the text focuses on the question of “why.” Why do we assess in a particular way? Why are our curricula designed in a certain manner? Why should students master aural skills for their career as a performer, music educator, or music therapist? It is through the experiences shared in the text that many of these questions of “why” are answered. Along with answering some of the important questions of “why,” the book emphasizes topics such as classroom environment, undergraduate research and mentoring, assessment, and approaches to curriculum development. Teaching Music Theory: New Voices and Approaches is written in a conversational tone to provide a starting point of dialogue for students, new faculty members, and seasoned educators on any level. The pedagogical trends presented in this book provide a greater appreciation of outstanding teaching and thus an understanding of successful approaches in the classroom.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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