Public Articles
Serendipity
and 4 collaborators
Modular Arithmetic
Blog Post 6
1 $\underline{\text{Card Shuffling}}$ In discrete mathematics, a riffle shuffle is a shuffling of cards in which the top half is placed in one hand and the other half lies in the opposite hand. The cards are then alternatively interlaced with one another. There are two different types of riffle shuffles that will be the topic of discussion: the out-shuffle and the in-shuffle. An out-shuffle keeps the top card on top and the bottom card on bottom. When the top card is, instead, placed in the second position we have an in-shuffle.
Consider a typical 52 card deck. If we name and order the cards 0, 1, 2, 3, 4..., then after an out-shuffle we receive the order 0, 26, 1, 27, 2, 28.... After an in-shuffle, the cards appear as 26, 0, 27, 1, 28, 2.... Note that the first card is in the 0 position. To return to the original order, we must make 8 out-shuffles. Interestingly enough, however, it takes 52 in-shuffles.
Fermat's Little Theorem
Enabling a muscle-based gesture interface for an Air Guitar game.
With the advancement of technology and the decline of manual labor, humans try to improve their quality of life using any innovation that they can think of. Technology makes our lives easier and more efficient. In turn, efficiency implies that we can allot the saved time for other tasks. For example, the evolution of buttons to touch screen. This evolution saved precious time by introducing dynamic menus, faster input, and a lot of flexibility. However, evolution does not stop there. With the introduction of electromyography (EMG), or the technique of evaluating and recording muscle activity through either a needle (intramuscular) or electrodes on muscles (surface), another form of input was made. It’s unclear if it provides a substantial increase in efficiency compared to traditional touch screens. However, having different options for certain situations are favorable. Other forms of input are: speech recognition, eye gaze trackers, and computer vision. These three other forms of input however, are unfavorable in certain situations; electromyography may be used for those situations. Surface EMG may be preferred over intramuscular EMG because it’s too professional and expensive. One device that’s able to conduct surface EMG is the Myo armband.
Recollimation boundary layers as X-ray sources in young stellar jets
and 2 collaborators
Young stars accrete mass from circumstellar disks and in many cases, the accretion coincides with a phase of massive outflows, which can be highly collimated. Those jets emit predominantly in the optical and IR wavelength range. However, in several cases X-ray and UV observations reveal a weak but highly energetic component in those jets. X-rays are observed both from stationary regions close to the star and from knots in the jet several hundred AU from the star. In this article we show semi-analytically that a fast stellar wind which is recollimated by the pressure from a slower, more massive disk wind can have the right properties to power stationary X-ray emission. The size of the shocked regions is compatible with observational constraints. Our calculations support a wind-wind interaction scenario for the high energy emission near the base of YSO jets. For the specific case of DG Tau, a stellar wind with a mass loss rate of 5 ⋅ 10−10 M⊙ yr−1 and a wind speed of 800 km s−1 reproduces the observed X-ray spectrum. We conclude that a stellar wind recollimation shock is a viable scenario to power stationary X-ray emission close to the jet launching point.
[section] [section] [section] [section] [section] Optimal Predictors: A Bayesian Notion of Approximation Algorithm
Modular Arithmetic
Blog Post 5
1 $\underline{\text{Binary Notation}}$
We define binary numbers as the powers of two that lay the foundation for the additive building blocks of positive integers. Note that the word binary comes from “Bi” meaning two. In this system, integers are expressed in terms of only 0s and 1s. The values that represent each integer are calculated by finding the sum of the powers of two that make up the given number. We pull out the amount of times that each power of two occurs. For example, the decimal number ten is written as “1010” because it is $\underline{1}$ ⋅ $2^3+ \underline{0} $ ⋅ $2^2 + \underline{1} \cdot 2^1 + \underline{0} \cdot 2^0 $. Notice that this starts with the largest power of two. We read “1010” as “one-zero-one-zero” as opposed to one thousand and ten. The binary representation of the first few natural numbers are shown in the table below.
Euclidean Algorithm
The Euclidean Algorithm, also known as Euclid’s Algorithm, is
used in discrete mathematics to find the greatest common divisor of two natural
numbers, namely \(a\) and \(b\). The greatest common divisor is typically denoted as \(gcd\left(a,b\right).\) In general, the Euclidean Algorithm is used
within numerous applications like solving Diophantine equations, constructing
continued fractions, and is even used when dividing in modular arithmetic.
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On the evolution of Cold Classical Kuiper Belt Objects
At a region between 39.4 AU and 47.8 AU, the Kuiper Belt contains planetesimals thought to remnants of the protoplanetary disk in the early Solar System. There are 3 types of object which occupy the Kuiper Belt: Scattered Objects, Hot Classicals, and Cold Classicals.
Scattered Objects have highly eccentric and inclined orbits, not constrained to the 40-50 AU region, believed to be due to Neptune’s early migration outwards when Jupiter and Saturn entered a 2:1 resonance, derived from the Nice Model. This migration scattered the primordial Kuiper Belt which had occupied the region where Neptune orbits today.
Hot Classicals are objects which were believed to have been scattered by Neptune but not to the extent of the Scattered Objects. These objects would’ve had their eccentricities and inclinations increased, but ultimately would find stable orbits within the region bounded by 2:3 and 1:2 resonances with Neptune, thus being deemed “Classical”
Cold Classicals are objects which are believed to be unaffected by Neptune’s early migration, being formed between 39.4 AU and 47.8 AU with low eccentricities and inclinations (with e ≈ 0.02, i ≈ 0.02), and remaining stable within the region bounded by 2:3 and 1:2 resonances with Neptune, with perturbations in orbits being due to self-scattering.
Since the Cold Classical Kuiper Belt Objects (CCKBOs) presumably are only affected by self-scattering, by understanding that process it would be possible to simulate these objects over the age of the Solar System to see if that alone can reproduce the observed orbital properties of CCKBOs. The process in question is the combined effect of Viscous Stirring and Dynamical Friction.
Viscous Stirring involves multiple encounters of multiple objects over time, which lead to a wider spread of velocity distribution, which leads to an increase of eccentricity and inclination of objects. Dynamical Friction involves gravitational encounters with masses of different velocities. Large mass objects are dragged by low mass objects, while low mass objects are gravitationally assisted by large mass objects. This leads to a general decrease of velocity for large masses and an increase for low masses.
Ohtsuki & Stewart (2002, Icarus 155, 436-453, hereafter O&S) obtained evolution rates of eccentricity and inclination using Viscous Stirring and Dynamical Friction rates where the coordinates are referred to a reference point that moves in a circular orbit with semimajor axis a0 at the Keplerian angular velocity Ω = (GM⊙/a03)1/2, and described them for a system of 2 planetesimal components, with object masses m1 and m2
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Growth of 48 Built Environment Bacterial Isolates on Board the International Space Station (ISS)
Abstract
Background: While significant attention has been paid to the potential risk of pathogenic microbes aboard crewed spacecraft, the non-pathogenic microbes in these habitats have received less consideration. Preliminary work has demonstrated that the interior of the International Space Station (ISS) has a microbial community resembling those of built environments on earth. Here we report results of sending 48 bacterial strains, collected from built environments on earth, for a growth experiment on the ISS. This project was a component of Project MERCCURI (Microbial Ecology Research Combining Citizen and University Researchers on ISS).
Results: Of the 48 strains sent to the ISS, 45 of them showed similar growth in space and on earth using a relative growth measurement adapted for microgravity. The vast majority of species tested in this experiment have also been found in culture-independent surveys of the ISS. Only one bacterial strain showed significantly different growth in space. Bacillus safensis JPL-MERTA-8-2 grew 60% better in space than on earth.
Conclusions: The majority of bacteria tested were not affected by conditions aboard the ISS in this experiment (e.g., microgravity, cosmic radiation). Further work on Bacillus safensis could lead to interesting insights on why this strain grew so much better in space.
Requiring Teaching Items
Fibonacci Numbers
Fibonacci numbers in discrete mathematics are a sequence of
numbers that satisfy a linear recurrence relation. The first few Fibonacci numbers are
\(0,1,1,2,3,5,8,13,21,...\) By definition, the first two Fibonacci numbers are either \(0,1\) or \(1,1.\)
As you can see, by adding the two previous consecutive numbers
in the sequence, one can generate the next numbers in this sequence using this method. The general rule of the Fibonacci sequence satisfies the
second-order recurrence relation pictured below:
Blog Post 4
1 $\underline{\text{Recurrence Relations}}$ Throughout one’s education and as one progresses through grade levels, a common math problem that students are faced with are puzzles in which they have to determine what number comes next in a sequence. For example, a teacher may ask a student to fill in the missing number given the following:
\[ 1,4,5,9,14 \underline{\hspace{1cm}}, ... \]
A student may then notice a pattern of adding the two terms before to get to the next. This idea, at its very core, is an illustration of a recurrence relation.
Defined mathematically, a recurrence relation is an equation in which the next term in the sequence is dependent upon or acts as a function of the previous term(s).
Using the case above, and applying it to recurrence relations, we would write the relation as: an = an − 1 + an − 2 where the initial conditions are a0 = 1 and a1 = 4.
Some of the more well known examples from the study of discrete mathematics include the Fibonacci sequence as well as binomial coefficients. While the Fibonacci sequence defined by Fn = Fn − 1 + Fn − 2 where F1 = F2 = 1 is fairly easy to recognize as a recurrence relation, binomial coefficients are not primarily described in this way. However, we see from the following illustration that they can, indeed, be accepted as following a recurrence relation. \[ {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k} \]
A game that has been derived from the idea of recurrence relations that some may find entertaining is called the Tower of Hanoi. This was invented by Edouard Lucas in 1883. If this name sounds familiar, it is because Edouard Lucas is more famously known for his recurrence relation called the Lucas Numbers, which follow the same relation as the Fibonacci sequence. In this game, there are a certain number of disks and three “towers.” The objective is to move all the disks stacked on one tower over to another without placing larger disks on smaller ones. An illustration can be seen as such:
Flexible On-Stack Replacement in LLVM
and 1 collaborator
On-Stack Replacement (OSR) is a technique for dynamically transferring execution between different versions of a function at run time. OSR is typically used in virtual machines to interrupt a long-running function and recompile it at a higher optimization level, or to replace it with a different one when a speculative assumption made during its compilation no longer holds.
In this paper we present a framework for OSR that introduces novel ideas and combines features of existing techniques that no previous solution provided simultaneously. New features include OSR with compensation code to adjust the program state during a transition and the ability to fire an OSR from arbitrary locations in the code. Our approach is platform-independent as the OSR machinery is entirely encoded at a compiler’s intermediate representation level.
We implement and evaluate our technique in the LLVM compiler infrastructure, which is gaining popularity as Just-In-Time (JIT) compiler in virtual machines for dynamic languages such as Javascript, MATLAB, Python, and Ruby. As a case study of our approach, we show how to improve the state of the art in the optimization of the instruction, a performance-critical construct of the MATLAB language.
EBH Notes