Polynomial approximations to continuous functions and stochastic compositions

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f \in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k(n)$ to a polynomial $f$ when $k(n)/n$ tends to a constant.Comment: 21 pages, 5 figure

Sexual Citizenship, Incest, and the State: The Unseen of the Crime

This paper analyzes the colonial state’s role in manufacturing sexual violence. Deconstructing parallels between sexual violence in state detention centers and incestuous abuse of children, this paper examines theories of normativity, sexual citizenship, U.S. nationalism, and Marxist interpretations of the family unit. In identifying all citizenship as sexual citizenship—and identifying queer as all those who are denied sexual citizenship—I suggest that liberation from the state is crucial to queer liberation and the amelioration of sexual violence

Child\u27s Pose

Written from the perspective of my eight-year-old self, this poem illustrates the state-sanctioned power differentials between children and their legal guardians that give birth to incestuous violence. I hope for the reader to recall their own eight-year-old mind, life, and emotional world to further understand this severe power disparity

Sensitivity analysis for stationary and ergodic queues

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Age of Information for Small Buffer Systems

Consider a message processing system whose objective is to produce the most current information as measured by the quantity known as "age of information". We have argued in previous papers that if we are allowed to design the message processing policy ad libitum, we should keep a small buffer and operate according to a LIFO policy. In this small note we provide an analysis for the AoI of the P_m system which uses a buffer of size m, a single server, operating without service preemption and in a LIFO manner for stored messages. Analytical expressions for the mean (or even distribution) of the AoI in steady-state are possible but with the aid computer algebra. We explain the the analysis for m=3

The generating functions of stirling numbers of the second kind derived probabilistically

Stirling numbers of the second kind, S(n, r), denote the number of partitions of a finite set of size n into r disjoint nonempty subsets. The aim of this short article is to shed some light on the generating functions of these numbers by deriving them probabilistically. We do this by linking them to Markov chains related to the classical coupon collector problem; coupons are collected in discrete time (ordinary generating function) or in continuous time (exponential generating function). We also review the shortest possible combinatorial derivations of these generating functions

Relative Age of Information: Maintaining Freshness while Considering the Most Recently Generated Information

A queueing system handling a sequence of message arrivals is considered where each message obsoletes all previous messages. The objective is to assess the freshness of the latest message/information that has been successfully transmitted, i.e., "age of information" (AoI). We study a variation of traditional AoI, the "Relative AoI", here defined so as to account for the presence of newly arrived messages/information to the queue to be transmitted

A fully stochastic approach to limit theorems for iterates of Bernstein operators

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator taking a continuous function to a degree- polynomial when the number of iterations tends to infinity and is kept fixed or when tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright–Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright–Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of a number of times to a polynomial when tends to a constant