Renyi Parking Problem Essay

   At dinner recently, a friend mentioned the following problem:

    There is a street of length x (not necessarily an integer) on which cars of length 1 wish to park.  However, instead of parking in a nice organized way, they park at random, picking uniformly from the availible positions to park (they are apparently jerks).  Assuming no cars leave, and continue to arrive until no more can fit, what is the expected number of cars that will fit?

     For instance, if the street is length 2, then the first car will almost always park so that no other can fit,  and so the expected number of cars is 1.  If the street is length 3, the first car can never prevent a second from fitting; but a third car almost never will fit, and so the expected value is 2.  The trend of simple answers is broken at street length 4, where the expected number of cars is .  This came up as a fun, back-of-the-envelope type problem, but the answer is actually an iterated integral that becomes prohibitively difficult after a few steps.  Perhaps a intrepid reader can solve more steps than I was able to.

    The name of the game is finding the function , the expected value of the number of cars for a street of length x.  We can find a relation this function has to satisfy in the following way.   

     Lets say the first car parks so its front is at , dividing the previously empty street into two pieces of length and .  Hence, the expected number of cars that park given the exact place the first car parks is .  We can get the overall expected value (that is, without fixed starting location) by averaging over all possible starting locations:

Of course, the integral splits apart into three pieces:

Notice that the second integral becomes the first after the change of variables , and so we get:


We should note that this formula is only valid for , since we implicitly assumed the first car could park.

    The reason this equation is useful is because of its inductive flavor.  To compute for some fixed x requires only knowledge of for , and we can start things with the simple observation that for .

     Computationally, this amounts to plugging equation (*) into itself over and over again.  Lets see how this works for one step.

Note that the lower limit of integration become 1, because equation (*) was only valid for ; similarly, the above equation is only valid for .

    We can repeat this process as many times as we like (always being careful with the lower limit of integration and the validity of x).  After every step, the only term with a reference to will be the last one, a big nested integral of .  Ah, but must be at least 1 less than , etc, and so is less than .  Therefore, if , then , and so the big nested integral will be zero.  Hence, we can get an expression for which involves no references to itself (as long as we know what integers is between).

    Heres the result (for :

Now all the trickiness is hidden in evaluating the integral.  Its easy for the first few steps.

Thus, we can see that for ,

    However, the next term of has resisted all of my attempts to integrate, and I haven’t found it in any of the integration tables I looked up.  Its possible its not solvable, which would be a sad end to an otherwise interesting problem.

    A related question is the behavior of for really large .  It seems intuitively likely that if the street is very long, the complicated factors that affect the precise expected value will stay relatively small, and will look like a nice function of like multiplication by some number .  This number would be a measure of the macroscopic efficiency of this random parking method.  It would be a fun number to know, if it exists, but I can’t see how to find it.

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Tags: math.PR

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PRIMES: Research Papers

2017 Research Papers

132) Richard Xu, Algebraicity regarding Graphs and Tilings (27 Jan 2018)

Given a planar graph G, we prove that there exists a tiling of a rectangle by squares such that each square corresponds to a face of the graph and the side lengths of the squares solve an extremal problem on the graph. Furthermore, we provide a practical algorithm for calculating the side lengths. Finally, we strengthen our theorem by restricting the centers and side lengths of the squares to algebraic numbers and explore the application of our technique in proving algebraicity in packing problems.


131) Anlin Zhang (PRIMES) and Laura P. Schaposnik (University of Illinois at Chicago), Modelling epidemics on d-cliqued graphs (published in Letters in Biomathematics 5:1 (Jan 16, 2018)

Since social interactions have been shown to lead to symmetric clusters, we propose here that symmetries play a key role in epidemic modelling. Mathematical models on d-ary tree graphs were recently shown to be particularly effective for modelling epidemics in simple networks. To account for symmetric relations, we generalize this to a new type of networks modelled on d-cliqued tree graphs, which are obtained by adding edges to regular d-trees to form d-cliques. This setting gives a more realistic model for epidemic outbreaks originating within a family or classroom and which could reach a population by transmission via children in schools. Specifically, we quantify how an infection starting in a clique (e.g. family) can reach other cliques through the body of the graph (e.g. public places). Moreover, we propose and study the notion of a safe zone, a subset that has a negligible probability of infection.

130) Andy Xu and Wendy Wu, Higher Gonalities of Erdös-Rényi Random Graphs (22 Dec 2017)

We consider the asymptotic behavior of the second and higher gonalities of an Erdös-Rényi random graph and provide upper bounds for both via the probabilistic method. Our results suggest that for sufficiently large $n$, the second gonality of an Erdös-Rényi random Graph $G(n,p)$ is strictly less than and asymptotically equal to the number of vertices under a suitable restriction of the probability $p$. We also prove an asymptotic upper bound for all higher gonalities of large Erdös-Rényi random graphs that adapts and generalizes a similar result on complete graphs. We suggest another approach towards finding both upper and lower bounds for the second and higher gonalities for small $p=\frac{c}{n}$, using a special case of the Riemann-Roch Theorem, and fully determine the asymptotic behavior of arbitrary gonalities when $c\leq 1$.

129) Dylan Pentland, Coefficients of Gaussian polynomials modulo N (16 Dec 2017)

The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to permutations and in representation theory.
Stanley conjectured that the function $f_{k,R}(n) = \#\left\{i : [q^{i}] \left[{n \atop k}\right]_q \equiv R \pmod{N}\right\}$ is quasipolynomial for $N=2$. We generalize, showing that this is in fact true for any integer $N\in \mathbb{N}$ and determine a quasi-period $\pi'_N(k)$ derived from the minimal period $\pi_N(k)$ of partitions with at most $k$ parts modulo $N$.

128) Espen Slettnes, Carl Joshua Quines, Shen-Fu Tsai, and Jesse Geneson (CrowdMath-2017), Variations of the cop and robber game on graphs (, 31 Oct 2017)

We prove new theoretical results about several variations of the cop and robber game on graphs. First, we consider a variation of the cop and robber game which is more symmetric called the cop and killer game. We prove for all $c < 1$ that almost all random graphs are stalemate for the cop and killer game, where each edge occurs with probability $p$ such that $\frac{1}{n^{c}} \le p \le 1-\frac{1}{n^{c}}$. We prove that a graph can be killer-win if and only if it has exactly $k\ge 3$ triangles or none at all. We prove that graphs with multiple cycles longer than triangles permit cop-win and killer-win graphs. For $\left(m,n\right)\neq\left(1,5\right)$ and $n\geq4$, we show that there are cop-win and killer-win graphs with $m$ $C_n$s. In addition, we identify game outcomes on specific graph products.
Next, we find a generalized version of Dijkstra's algorithm that can be applied to find the minimal expected capture time and the minimal evasion probability for the cop and gambler game and other variations of graph pursuit.
Finally, we consider a randomized version of the killer that is similar to the gambler. We use the generalization of Dijkstra's algorithm to find optimal strategies for pursuing the random killer. We prove that if $G$ is a connected graph with maximum degree $d$, then the cop can win with probability at least $\frac{\sqrt d}{1+\sqrt d}$ after learning the killer's distribution. In addition, we prove that this bound is tight only on the $\left(d+1\right)$-vertex star, where the killer takes the center with probability $\frac1{1+\sqrt d}$ and each of the other vertices with equal probabilities.

127) Ayush Agarwal (PRIMES) and Christian Gaetz (MIT), Differential posets and restriction in critical groups (, 23 Oct 2017)

In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when $G$ is an element in a differential tower of groups, critical groups of certain representations are closely related to words of up-down maps in the associated differential poset. We use this to generalize an explicit formula for the critical group of the permutation representation of the symmetric group given by the second author, and to enumerate the factors in such critical groups.

126) Louis Golowich (PRIMES) and Chiheon Kim (MIT), New Classes of Set-Sequential Tree (, 14 Oct 2017)

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the label for either a single vertex or a single edge. We resolve certain cases of a conjecture of Balister, Gyori, and Schelp in order to show many new classes of trees to be set-sequential. We show that all caterpillars $T$ of diameter $k$ such that $k \leq 18$ or $|V(T)| \geq 2^{k-1}$ are set-sequential, where $T$ has only odd-degree vertices and $|T| = 2^{n-1}$ for some positive integer $n$. We also present a new method of recursively constructing set-sequential trees.

2016 Research Papers

125) Piotr Suwara (MIT) and Albert Yue (PRIMES), An Index-Type Invariant of Knot Diagrams Giving Bounds for Unknotting Framed Unknots (, 7 Jul 2017)

We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves.
We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under $\Omega$3 moves depends only on the forward/backward character of the move, similar to how the change of St or cowrithe depends only on the positive/negative quality of the move.

124) P.A. CrowdMath, Results on Pattern Avoidance Games (, 18 Apr 2017)

A zero-one matrix $A$ contains another zero-one matrix $P$ if some submatrix of $A$ can be transformed to $P$ by changing some ones to zeros. $A$ avoids $P$ if $A$ does not contain $P$. The Pattern Avoidance Game is played by two players. Starting with an all-zero matrix, two players take turns changing zeros to ones while keeping $A$ avoiding $P$. We study the strategies of this game for some patterns $P$. We also study some generalizations of this game.

123) P.A. CrowdMath, Algorithms for Pattern Containment in 0-1 Matrices (, 18 Apr 2017)

We say a zero-one matrix $A$ avoids another zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeros. A fundamental problem is to study the extremal function $ex(n,P)$, the maximum number of nonzero entries in an $n \times n$ zero-one matrix $A$ which avoids $P$. To calculate exact values of $ex(n,P)$ for specific values of $n$, we need containment algorithms which tell us whether a given $n \times n$ matrix $A$ contains a given pattern matrix $P$. In this paper, we present optimal algorithms to determine when an $n \times n$ matrix $A$ contains a given pattern $P$ when $P$ is a column of all ones, an identity matrix, a tuple identity matrix, an $L$-shaped pattern, or a cross pattern. These algorithms run in $\Theta(n^2)$ time, which is the lowest possible order a containment algorithm can achieve. When $P$ is a rectangular all-ones matrix, we also obtain an improved running time algorithm, albeit with a higher order.

122) Alec Leng, Independence of the Miller-Rabin and Lucas Probable Prime Tests (30 Mar 2017)

In the modern age, public-key cryptography has become a vital component for secure online communication. To implement these cryptosystems, rapid primality testing is necessary in order to generate keys. In particular, probabilistic tests are used for their speed, despite the potential for pseudoprimes. So, we examine the commonly used Miller-Rabin and Lucas tests, showing that numbers with many nonwitnesses are usually Carmichael or Lucas-Carmichael numbers in a specific form. We then use these categorizations, through a generalization of Korselt’s criterion, to prove that there are no numbers with many nonwitnesses for both tests, affirming the two tests’ relative independence. As Carmichael and Lucas-Carmichael numbers are in general more difficult for the two tests to deal with, we next search for numbers which are both Carmichael and Lucas-Carmichael numbers, experimentally finding none less than $10^{16}$. We thus conjecture that there are no such composites and, using multivariate calculus with symmetric polynomials, begin developing techniques to prove this.

121) Andrew Gritsevskiy and Adithya Vellal, Development and Biological Analysis of a Neural Network Based Genomic Compression System (3 Mar 2017)

The advent of Next Generation Sequencing (NGS) technologies has resulted in a barrage of genomic data that is now available to the scientific community. This data contains information that is driving fields such as precision medicine and pharmacogenomics, where clinicians use a patient’s genetics in order to develop custom treatments. However, genomic data is immense in size, which makes it extremely costly to store, transport and process. A genomic compression system which takes advantage of intrinsic biological patterns can help reduce the costs associated with this data while also identifying important biological patterns. In this project, we aim to create a compression system which uses unsupervised neural networks to compress genomic data. The complete compression suite, GenComp, is compared to existing genomic data compression methods. The results are then analyzed to discover new biological features of genomic data. Testing showed that GenComp achieves at least 40 times more compression than existing variant compression solutions, while providing comparable decoding times in most applications. GenComp also provides some insight into genetic patterns, which has significant potential to aid in the fields of pharmacogenomics and precision medicine. Our results demonstrate that neural networks can be used to significantly compress genomic data while also assisting in better understanding genetic biology.

120) Vivek Bhupatiraju, John Kuszmaul, and Vinjai Vale, On the Viability of Distributed Consensus by Proof of Space (3 Mar 2017)

In this paper, we present our implementation of Proof of Space (PoS) and our study of its viability in distributed consensus. PoS is a new alternative to the commonly used Proof of Work, which is a protocol at the heart of distributed consensus systems such as Bitcoin. PoS resolves the two major drawbacks of Proof of Work: high energy cost and bias towards individuals with specialized hardware. In PoS, users must store large “hard-to-pebble” PTC graphs, which are recursively generated using subgraphs called superconcentrators. We implemented two types of superconcentrators to examine their differences in performance. Linear superconcentrators are about 1:8 times slower than butterfly superconcentrators, but provide a better lower bound on space consumption. Finally, we discuss our simulation of using PoS to reach consensus in a peer-to-peer network. We conclude that Proof of Space is indeed viable for distributed consensus. To the best of our knowledge, we are the first to implement linear superconcentrators and to simulate the use of PoS to reach consensus on a decentralized network.

119) Albert Yue, An Index-Type Invariant of Knot Diagrams and Bounds for Unknotting Framed Knots (3 Mar 2017)

We introduce a new knot diagram invariant called self-crossing index, or $\mathrm{SCI}$. We found that $\mathrm{SCI}$ changes by at most $\pm 1$ under framed Reidemeister moves, and specifically provides a lower bound for the number of 3 moves. We also found that $\mathrm{SCI}$ is additive under connected sums, and is a Vassiliev invariant of order 1. We also conduct similar calculations with Hass and Nowik's diagram invariant and cowrithe, and present a relationship between forward/backward, ascending/descending, and positive/negative 3 moves.


118) Valerie Zhang, Computer-Based Visualizations and Manipulations of Matching Paths (2 Mar 2017)

Given n points in the 2-D plane, a matching path is a path that starts at one of these n points and ends at a different one without going through any of the other n - 2 points. Matching paths, as well as an important operation called the Hurwitz move, come up naturally in the study of complex algebraic varieties. At the heart of the Hurwitz move is the twist operation, which “twists” one matching path along another to produce a new (third) matching path. Performing the twist operation by hand, however, is not only tedious but also prone to errors and unnecessary complications. Therefore, using computer-based methods to represent matching paths and perform the twist operation makes sense. In this project, which was coded in Java, computer-based methods are developed to perform the twist operation efficiently and accurately, providing a framework for visualizing and manipulating matching paths with computers. The computer program performs fast computations and represents matching paths as simply as possible in a simple visual interface. This program could be utilized when solving open problems in symplectic geometry: potential applications include characterizing the overtwistedness of contact manifolds, as well as better understanding braid group actions.

117) Harshal Sheth, Nihar Sheth, and Aashish Welling, Read-Copy Update in a Garbage Collected Environment (1 Mar 2017)

Read-copy update (RCU) is a synchronization mechanism that allows efficient parallelism when there are a high number of readers compared to writers. The primary use of RCU is in Linux, a highly popular operating system kernel. The Linux kernel is written in C, a language that is not garbage collected, and yet the functionality that RCU provides is effectively that of a “poor man’s garbage collector” (P. E. McKenney). RCU in C is also complicated to use, and this can lead to bugs. The purpose of this paper is to investigate whether RCU implemented in a garbage collected language (Go) is easier to use while delivering comparable performance to RCU in C. This is tested through the implementation and benchmarking of 4 linked lists, 2 using RCU and 2 using mutexes. One RCU linked list and one mutex linked list are implemented in each language. This paper finds that RCU in a garbage collected language is indeed significantly easier to use, has similar overall performance to, and on very high read loads, outperforms, RCU in C.

116) Xiangyao Yu (MIT), Siye Zhu (PRIMES), Justin Kaashoek (PRIMES), Andrew Pavlo (Carnegie Mellon University), and Srinivas Devadas (MIT), Taurus: A Parallel Transaction Recovery Method Based on Fine-Granularity Dependency Tracking (28 Feb 2017)

Logging is crucial to performance in modern multicore main-memory database management systems (DBMSs). Traditional data logging (ARIES) and command logging algorithms enforce a sequential order among log records using a global log sequence number (LSN). Log flushing and recovery after a crash are both performed in the LSN order. This serialization of transaction logging and recovery can limit the system performance at high core count. In this paper, we propose Taurus to break the LSN abstraction and enable parallel logging and recovery by tracking fine-grained dependencies among transactions. The dependency tracking lends Taurus three salient features. (1) Taurus decouples the transaction logging order with commit order and allows transactions to be flushed to persistent storage in parallel independently. Transactions that are persistent before commit can be discovered and ignored by the recovery algorithm using the logged dependency information. (2) Taurus can leverage multiple persistent devices for logging. (3) Taurus can leverage multiple devices and multiple worker threads for parallel recovery. Taurus improves logging and recovery parallelism for both data and command logging. .

115) Louis Golowich (PRIMES), Chiheon Kim (MIT), and Richard Zhou (PRIMES), Maximum Size of a Family of Pairwise Graph-Different Permutations (, 27 Feb 2017), published in The Electronic Journal of Combinatorics 24:4 (2017)

Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise $G$-different permutations for various graphs $G$. We show that for all balanced bipartite graphs $G$ of order $n$ with minimum degree $n/2 - o(n)$, the maximum number of pairwise $G$-different permutations of the vertices of $G$ is $2^{(1-o(1))n}$. We also present examples of bipartite graphs $G$ with maximum degree $O(\log n)$ that have this property. We explore the problem of bounding the maximum size of a family of pairwise graph-different permutations when an unlimited number of disjoint vertices is added to a given graph. We determine this exact value for the graph of 2 disjoint edges, and present some asymptotic bounds relating to this value for graphs consisting of the union of $n/2$ disjoint edges.

114) Sathwik Karnik, On the Classification and Algorithmic Analysis of Carmichael Numbers (, 26 Feb 2017)

In this paper, we study the properties of Carmichael numbers, false positives to several primality tests. We provide a classification for Carmichael numbers with a proportion of Fermat witnesses of less than 50%, based on if the smallest prime factor is greater than a determined lower bound. In addition, we conduct a Monte Carlo simulation as part of a probabilistic algorithm to detect if a given composite number is Carmichael. We modify this highly accurate algorithm with a deterministic primality test to create a novel, more efficient algorithm that differentiates between Carmichael numbers and prime numbers.

113) Felix Wang, Functional equations in Complex Analysis and Number Theory (26 Feb 2017)

We study the following questions:
(1) What are all solutions to $f\circ \hat{f} = g\circ \hat{g}$ with $f,g,\hat{f},\hat{g}\in\mathbb{C}(X)$ being complex rational functions?
(2) For which rational functions $f(X)$ and $g(X)$ with rational coefficients does the equation $f(a)=g(b)$ have infinitely many solutions with $a,b\in$ $Q$?
We utilize various algebraic, geometric and analytic results in order to resolve both (1) and a variant of (2) in case the numerator of $f(X)-g(Y)$ is an irreducible polynomial in $\mathbb{C}[X,Y]$. Our results have applications in various mathematical fields, such as complex analysis, number theory, and dynamical systems. Our work resolves a 1973 question of Fried, and makes significant progress on a 1924 question of Ritt and a 1997 question of Lyubich and Minsky. In addition, we prove a quantitative refinement of a 2015 conjecture of Cahn, Jones and Spear.

112) Albert Gerovitch, Automatically Improving 3D Neuron Segmentations for Expansion Microscopy Connectomics (25 Feb 2017)

Understanding the geometry of neurons and their connections is key to comprehending brain function. This is the goal of a new optical approach to brain mapping using expansion microscopy (ExM), developed in the Boyden Lab at MIT to replace the traditional approach of electron microscopy. A challenge here is to perform image segmentation to delineate the boundaries of individual neurons. Currently, however, there is no method implemented for assessing a segmentation algorithm’s accuracy in ExM. The aim of this project is to create automated assessment of neuronal segmentation algorithms, enabling their iterative improvement. By automating the process, I aim to devise powerful segmentation algorithms that reveal the “connectome” of a neural circuit. I created software, called SEV-3D, which uses the pixel error and warping error metrics to assess 3D segmentations of single neurons. To allow better assessment beyond a simple numerical score, I visualized the results as a multilayered image. My program runs in a closed loop with a segmentation algorithm, modifying its parameters until the algorithm yields an optimal segmentation. I am further developing my application to enable evaluation of multi-cell segmentations. In the future, I aim to further implement the principles of machine learning to automatically improve the algorithms, yielding even better accuracy.

111) Laura Pierson, Signatures of Stable Multiplicity Spaces in Restrictions of Representations of Symmetric Groups (25 Feb 2017)

Representation theory is a way of studying complex mathematical structures such as groups and algebras by mapping them to linear actions on vector spaces. Recently, Deligne proposed a new way to study the representation theory of finite groups by generalizing the collection of representations of a sequence of groups indexed by positive integer rank to an arbitrary complex rank, creating an abelian tensor category. In this project, we focused on the case of the symmetric groups $S_n,$ the groups of permutations of $n$ objects. Elements of the Deligne category Rep $S_t$ can be constructed by taking a stable sequence of $S_n$ representations for increasing $n$ and interpolating the associated formulas to an arbitrary complex number $t.$ In this project, we studied the case of restriction multiplicity spaces $V_{\lambda,\rho}$, counting the number of copies of an irreducible representation $V_{\rho}$ of $S_{n-k}$ in the restriction $\text{Res}_{S_{n-k}}^{S_n} V_{\lambda}$ of an irreducible representation of $S_n.$ We found formulas for norms of orthogonal basis vectors in these spaces, and ultimately for signatures (the number of basis vectors with positive norm minus the number with negative norm), an invariant that multiplies over tensor products and has important combinatorial connections.

110) Kevin Chang, Upper Bounds for Ordered Ramsey Numbers of Small 1-Orderings (, 7 Feb 2017)

A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_{<} (H)$ is the minimum $n$ such that every edge-2-coloring of the complete graph on the vertex set $\{1, \ldots, n\}$ contains a copy of $H$, the $i$th smallest vertex of which either has order-label $i$ in $H$ or no order-label in $H$.
This paper conducts the first systematic study of ordered Ramsey numbers for $1$-orderings of small graphs. We provide upper bounds for $R_{<} (H)$ for each connected $1$-ordering $H$ on $4$ vertices. Additionally, for every $1$-ordering $H$ of the $n$-vertex path $P_n$, we prove that $R_{<} (H) \in O(n)$. Finally, we provide an upper bound for the generalized ordered Ramsey number $R_{<} (K_n, H)$ which can be applied to any $k$-ordering $H$ containing some vertex with order-label $1$.

109) Nikhil Marda, On Equal Point Separation by Planar Cell Decompositions (, 17 Jan 2017)

In this paper, we investigate the problem of separating a set $X$ of points in $\mathbb{R}^{2}$ with an arrangement of $K$ lines such that each cell contains an asymptotically equal number of points (up to a constant ratio). We consider a property of curves called the stabbing number, defined to be the maximum countable number of intersections possible between the curve and a line in the plane. We show that large subsets of $X$ lying on Jordan curves of low stabbing number are an obstacle to equal separation. We further discuss Jordan curves of minimal stabbing number containing $X$. Our results generalize recent bounds on the Erd\H{o}s-Szekeres Conjecture, showing that for fixed $d$ and sufficiently large $n$, if $|X| \ge 2^{c_dn/d + o(n)}$ with $c_d = 1 + O(\frac{1}{\sqrt{d}})$, then there exists a subset of $n$ points lying on a Jordan curve with stabbing number at most $d$.

108) Samuel Cohen and Peter Rowley, Results of Triangles Under Discrete Curve Shortening Flow (7 Jan 2017)

In this paper, we analyze the results of triangles under discrete curve shortening flow, specifically isosceles triangles with top angles greater than $\frac{\pi}{3}$, and scalene triangles. By considering the location of the three vertices of the triangle after some small time $\epsilon$, we use the definition of the derivative to calculate a system of differential equations involving parameters that can describe the triangle. Constructing phase plane diagrams and then analyzing them, we find that the singular behavior of discrete curve shorting flow on isosceles triangles with top angles greater than $\frac{\pi}{3}$ is a point, and for scalene triangles is a line segment.

107) Matthew Hase-Liu (PRIMES) and Nicholas Triantafillou (MIT), Efficient Point-Counting Algorithms for Superelliptic Curves (7 Jan 2017;, 7 Sep 2017)

In this paper, we present efficient algorithms for computing the number of points and the order of the Jacobian group of a superelliptic curve over finite fields of prime order p. Our method employs the Hasse-Weil bounds in conjunction with the Hasse-Witt matrix for superelliptic curves, whose entries we express in terms of multinomial coefficients. We present a fast algorithm for counting points on specific trinomial superelliptic curves and a slower, more general method for all superelliptic curves. For the first case, we reduce the problem of simplifying the entries of the Hasse-Witt matrix modulo p to a problem of solving quadratic Diophantine equations. For the second case, we extend Bostan et al.'s method for hyperelliptic curves to general superelliptic curves. We believe the methods we describe are asymptotically the most efficient known point-counting algorithms for certain families of trinomial superelliptic curves.

106) P.A. CrowdMath, Bounds on parameters of minimally non-linear patterns (, 31 Dec 2016), published in theElectronic Journal of Combinatorics 25:1 (2018)

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$ of $P$. We prove that the ratio between the length and width of any minimally non-linear 0-1 matrix is at most $4$, and that a minimally non-linear 0-1 matrix with $k$ rows has at most $5k-3$ ones. We also obtain an upper bound on the number of minimally non-linear 0-1 matrices with $k$ rows.
In addition, we prove corresponding bounds for minimally non-linear ordered graphs. The minimal non-linearity that we investigate for ordered graphs is for the extremal function $ex_{<}(n, G)$, which is the maximum possible number of edges in any ordered graph on $n$ vertices with no ordered subgraph isomorphic to $G$.

105) Seth Shelley-Abrahamson (MIT) and Alec Sun (PRIMES), Towards a Classification of Finite-Dimensional Representations of Rational Cherednik Algebras of Type D (, 15 Dec 2016)

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations $L_c(\lambda^\pm)$ of the rational Cherednik algebra $H_c(D_n, \mathbb{C}^n)$ of type $D$ for symmetric bipartitions $\lambda$ are infinite dimensional for all parameters $c$. In particular, all finite-dimensional irreducible representations of rational Cherednik algebras of type $D$ arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type $B$.

104) Nicholas Guo (PRIMES) and Guangyi Yue (MIT), An Application of Rational Hyperplane Arrangements in Counting Independent Sets of Graphs (, 13 Dec 2016)

We develop a method in counting independent sets of disjoint union of certain types of graphs. This method is based upon the $n$ to 1 covering from the points in the finite field $\mathbb{F}_q^n$ which the characteristic polynomial of a rational hyperplane arrangement in $\mathbb{R}^n$ are counting to the independent sets of the corresponding graph. We show that for graphs $G(k)$ with $k$ vertices corresponding to a class of rational hyperplane arrangements, the number of $n$-element independent sets of any disjoint union $G = G(k_1)+ G(k_2)+\cdots + G(k_s)$ depends only on the total number of vertices in the entire disjoint union, $\sum_i{k_i}$. We also give some results with broader conditions. This new technique has importance in simplifing the complexity of counting independent sets in a disjoint union of graphs, and provides closed-form solutions for certain cases.

103) Yatharth Agarwal (PRIMES), Vishnu Murale (PRIMES), Jason Hennessey (Boston University), Kyle Hogan (Boston University), and Mayank Varia (Boston University), Moving in Next Door: Network Flooding as a Side Channel in Cloud Environments (14-16 Nov 2016), published in Sara Foresti and Giuseppe Persiano, eds., Cryptology and Network Security: 15th International Conference Proceedings, CANS 2016, Milan, Italy, November 14–16, 2016, pp. 755-760.

Co-locating multiple tenants’ virtual machines (VMs) on the same host underpins public clouds’ affordability, but sharing physical hardware also exposes consumer VMs to side channel attacks from adversarial co-residents. We demonstrate passive bandwidth measurement to perform traffic analysis attacks on co-located VMs. Our attacks do not assume a privileged position in the network or require any communication between adversarial and victim VMs. Using a single feature in the observed bandwidth data, our algorithm can identify which of 3 potential YouTube videos a co-resident VM streamed with 66 % accuracy. We discuss defense from both a cloud provider’s and a consumer’s perspective, showing that effective defense is difficult to achieve without costly under-utilization on the part of the cloud provider or over-utilization on the part of the consumer.

102) Dhruv Rohatgi, A Connection Between Vector Bundles over Smooth Projective Curves and Representations of Quivers (31 Oct 2016)

We create a partition bijection that yields a partial result on a recent conjecture by Schiffmann relating the problems of counting over a finite field (1) vector bundles over smooth projective curves, and (2) representations of quivers.




101) Aaron Yeiser, A Next Generation Partial Differential Equation Solver (30 Oct 2016)

When solving differential equations in multiple dimensions, mesh generation is required to discretize the geometry of the domain. Current numerical techniques, such as finite element methods, are numerically unstable on meshes containing skinny triangles. Here, we develop a novel numerical method that is numerically stable even on meshes with skinny elements. Our method is spectrally accurate on each element, and the discretization size can be adapted to suit a wide variety of applications. Our algorithm alleviates the current burden on mesh generation algorithms of avoiding skinny triangles, allowing meshes to instead be optimized for the efficient solution of time-dependent partial differential equations. We can also simulate the Navier-Stokes equations at moderate Reynolds numbers with our method.


100) Tanya Khovanova (MIT) and Rafael Saavedra (PRIMES), Discreet Coin Weighings and the Sorting Strategy (, 23 Sep 2016)

In 2007, Alexander Shapovalov posed an old twist on the classical coin weighing problem by asking for strategies that manage to conceal the identities of specific coins while providing general information on the number of fake coins. In 2015, Diaco and Khovanova studied various cases of these "discreet strategies" and introduced the revealing factor, a measure of the information that is revealed.
In this paper we discuss a natural coin weighing strategy which we call the sorting strategy: divide the coins into equal piles and sort them by weight. We study the instances when the strategy is discreet, and given an outcome of the sorting strategy, the possible number of fake coins. We prove that in many cases, the number of fake coins can be any value in an arithmetic progression whose length depends linearly on the number of coins in each pile. We also show the strategy can be discreet when the number of fake coins is any value within an arithmetic subsequence whose length also depends linearly on the number of coins in each pile. We arrive at these results by connecting our work to the classic Frobenius coin problem. In addition, we calculate the revealing factor for the sorting strategy.

99) Kai-Siang Ang (PRIMES) and Laura P. Schaposnik (University of Illinois at Chicago), On the geometry of regular icosahedral capsids containing disymmetrons (, 29 Aug 2016), published in Journal of Structural Biology (19 Jan 2017)

Icosahedral virus capsids are composed of symmetrons, organized arrangements of capsomers. There are three types of symmetrons: disymmetrons, trisymmetrons, and pentasymmetrons, which have different shapes and are centered on the icosahedral 2-fold, 3-fold and 5-fold axes of symmetry, respectively. In 2010 [Sinkovits & Baker] gave a classification of all possible ways of building an icosahedral structure solely from trisymmetrons and pentasymmetrons, which requires the triangulation number T to be odd. In the present paper we incorporate disymmetrons to obtain a geometric classification of icosahedral viruses formed by regular penta-, tri-, and disymmetrons. For every class of solutions, we further provide formulas for symmetron sizes and parity restrictions on h, k, and T numbers. We also present several methods in which invariants may be used to classify a given configuration.

98) Tanya Khovanova (MIT) and Shuheng Niu (PRIMES), m-Modular Wythoff (, 2 Aug 2016)

We discuss a variant of Wythoff's Game, $m$-Modular Wythoff's Game, and identify the winning and losing positions for this game.




2015 Research Papers

97) Caleb Ji, Robin Park, and Angela Song, Combinatorial Games of No Strategy (20 Aug 2016)

In this paper, we study a particular class of combinatorial game motivated by previous research conducted by Professor James Propp, called Games of No Strategy, or games whose winners are predetermined. Finding the number of ways to play such games often leads to new combinatorial sequences and involves methods from analysis, number theory, and other fields. For the game Planted Brussel Sprouts, a variation on the well-known game Sprouts, we find a new proof that the number of ways to play is equal to the number of spanning trees on n vertices, and for Mozes’ Game of Numbers, a game studied for its interesting connections with other fields, we use prior work by Alon to calculate the number of ways to play the game for a certain case. Finally, in the game Binary Fusion, we show through both algebraic and combinatorial proofs that the number of ways to play generates Catalan’s triangle.

96) Meena Jagadeesan, The Exchange Graphs of Weakly Separated Collections (, 19 Aug 2016)

Weakly separated collections arise in the cluster algebra derived from the Pl\"ucker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace $\mathcal{I}$ and proved the connectivity of every exchange graph. Oh and Speyer later introduced a generalization of exchange graphs that we call $\mathcal{C}$-constant graphs. They characterized these graphs in the smallest two cases. We prove an isomorphism between exchange graphs and a certain class of $\mathcal{C}$-constant graphs. We use this to extend Oh and Speyer's characterization of these graphs to the smallest four cases, and we present a conjecture on a bound on the maximal order of these graphs. In addition, we fully characterize certain classes of these graphs in the special cases of cycles and trees.

95) Nicholas Diaco, Counting Counterfeit Coins: A New Coin Weighing Problem (, 13 Jun 2016)

In 2007, a new variety of the well-known problem of identifying a counterfeit coin using a balance scale was introduced in the sixth International Kolmogorov Math Tournament. This paper offers a comprehensive overview of this new problem by presenting it in the context of the traditional coin weighing puzzle and then explaining what makes the new problem mathematically unique. Two weighing strategies described previously are used to derive lower bounds for the optimal number of admissible situations for given parameters. Additionally, a new weighing procedure is described that can be adapted to provide a solution for a broad spectrum of initial parameters by representing the number of counterfeit coins as a linear combination of positive integers. In closing, we offer a new form of the traditional counterfeit coin problem and provide a lower bound for the number of weighings necessary to solve it.

94) Jesse Geneson (MIT) and Meghal Gupta (PRIMES), Bounding extremal functions of forbidden 0-1 matrices using (r,s)-formations (19 Mar 2016)

First, we prove tight bounds of $n 2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ on the extremal function of the forbidden pair of ordered sequences $(1 2 3 \ldots k)^t$ and $(k \ldots 3 2 1)^t$ using bounds on a class of sequences called $(r,s)$-formations. Then, we show how an analogous method can be used to derive similar bounds on the extremal functions of forbidden pairs of $0-1$ matrices consisting of horizontal concatenations of identical identity matrices and their horizontal reflections.

93) Varun Jain, Novel Relationships Between Circular Planar Graphs and Electrical Networks (20 Feb 2016)

Circular planar graphs are used to model electrical networks, which arise in classical physics. Associated with such a network is a network response matrix, which carries information about how the network behaves in response to certain potential differences. Circular planar graphs can be organized into equivalence classes based upon these response matrices. In each equivalence class, certain fundamental elements are called critical. Additionally, it is known that equivalent graphs are related by certain local transformations. Using wiring diagrams, we first investigate the number of Y-∆ transformations required to transform one critical graph in an equivalence class into another, proving a quartic bound in the order of the graph. Next, we consider positivity phenomena, studying how testing the signs of certain circular minors can be used to determine if a given network response matrix is associated with a particular equivalence class. In particular, we prove a conjecture by Kenyon and Wilson for some cases.

92) Arthur Azvolinsky, Explicit Computations of the Frozen Boundaries of Rhombus Tilings of Polygonal Domains (12 Feb 2016)

Consider a polygonal domain $\Omega$ drawn on a regular triangular lattice. A \textit{rhombus tiling} of $\Omega$ is defined as a complete covering of the domain with $60^{\textrm{o}}$-rhombi, where each one is obtained by gluing two neighboring triangles together. We consider a uniform measure on the set of all tilings of $\Omega$. As the mesh size of the lattice approaches zero while the polygon remains fixed, a random tiling approaches a deterministic limit shape. An important phenomenon that occurs with the convergence towards a limit shape is the formation of \textit{frozen facets}; that is, areas where there are asymptotically tiles of only one particular type. The sharp boundary between these ordered facet formations and the disordered region is a curve inscribed in $\Omega$. This inscribed curve is defined as the \textit{frozen boundary}. The goal of this project was to understand the purely algebraic approach, elaborated on in a paper by Kenyon and Okounkov, to the problem of explicitly computing the frozen boundary. We will present our results for a number of special cases we considered.

91) David Amirault, Better Bounds on the Rate of Non-Witnesses of Lucas Pseudoprimes (3 Feb 2016)

Efficient primality testing is fundamental to modern cryptography for the purpose of key generation. Different primality tests may be compared using their runtimes and rates of non-witnesses. With the Lucas primality test, we analyze the frequency of Lucas pseudoprimes using MATLAB. We prove that a composite integer n can be a strong Lucas pseudoprime to at most 16 of parameters P, Q unless n belongs to a short list of exception cases, thus improving the bound from the previous result of 415: We also explore the properties obeyed by such exceptions and how these cases may be handled by an extended version of the Lucas primality test.

90) Daniel Guo, An Infection Spreading Model on Binary Trees (26 Jan 2016)

An important and ongoing topic of research is the study of infectious diseases and the speed at which these diseases spread. Modeling the spread and growth of such diseases leads to a more precise understanding of the phenomenon and accurate predictions of spread in real life. We consider a long-range infection model on an infinite regular binary tree. Given a spreading coefficient $\alpha>1$, the time it takes for the infection to travel from one node to another node below it is exponentially distributed with specific rate functions such as $2^{-k}k^{-\alpha}$ or $\frac{1}{\alpha^k}$, where $k$ is the difference in layer number between the two nodes. We simulate and analyze the time needed for the infection to reach layer $m$ or below starting from the root node. The resulting time is recorded and graphed for different values of $\alpha$ and $m$. Finally, we prove rigorous lower and upper bounds for the infection time, both of which are approximately logarithmic with respect to $m$. The same techniques and results are valid for other regular $d$-ary trees, in which each node has exactly $d$ children where $d>2$.

89) Jacob Klegar, Bounded Tiling-Harmonic Functions on the Integer Lattice (25 Jan 2016)

Tiling-harmonic functions are a class of functions on square tilings that minimize a specific energy. These functions may provide a useful tool in studying square Sierpinski carpets. In this paper we show two new Maximum Modulus Principles for these functions, prove Harnack's Inequality, and give a proof that the set of tiling-harmonic functions is closed. One of these Maximum Modulus Principles is used to show that bounded infinite tiling-harmonic functions must have arbitrarily long constant lines. Additionally, we give three sufficient conditions for tiling-harmonic functions to be constant. Finally, we explore comparisons between tiling and graph-harmonic functions, especially in regards to oscillating boundary values.

88) Richard Yi, A Probability-Based Model of Traffic Flow (22 Jan 2016)

Describing the behavior of traffic via mathematical modeling and computer simulation has been a challenge confronted by mathematicians in various ways throughout the last century. In this project, we introduce various existing traffic flow models and present a new, probability-based model that is a hybrid of the microscopic and macroscopic views, drawing upon current ideas in traffic flow theory. We examine the correlations found in the data of our computer simulation. We hope that our results could help civil engineers implement efficient road systems that fit their needs, as well as contribute toward the design of safely operating unmanned vehicles.

87) Kenz Kallal, Matthew Lipman, and Felix Wang, Equal Compositions of Rational Functions (21 Jan 2016)

We study the following questions:
(1) What are all solutions to $f\circ \hat{f} = g\circ \hat{g}$ in complex rational functions $f,g\in\mathbb{C}(X)$ and meromorphic functions $\hat{f}, \hat{g}$ on the complex plane?
(2) For which rational functions $f(X)$ and $g(X)$ with coefficients in an algebraic number field $K$ does the equation $f(a)=g(b)$ have infinitely many solutions with $a,b\in K$?
We utilize various algebraic, geometric and analytic results in order to resolve both questions in the case that the numerator of $f(X)-g(Y)$ is an irreducible polynomial in $\mathbb{C}[X,Y]$ of sufficiently large degree. Our work answers a 1973 question of Fried in all but finitely many cases, and makes significant progress towards answering a 1924 question of Ritt and a 1997 question of Lyubich and Minsky.

86) Dhruv Medarametla, Bounding Norms of Locally Random Matrices (21 Jan 2016)

Recently, several papers proving lower bounds for the performance of the Sum Of Squares Hierarchy on the planted clique problem have been published. A crucial part of all four papers is probabilistically bounding the norms of certain \locally random" matrices. In these matrices, the entries are not completely independent of each other, but rather depend upon a few edges of the input graph. In this paper, we study the norms of these locally random matrices. We start by bounding the norms of simple locally random matrices, whose entries depend on a bipartite graph H and a random graph G; we then generalize this result by bounding the norms of complex locally random matrices, matrices based o of a much more general graph H and a random graph G. For both cases, we prove almost-tight probabilistic bounds on the asymptotic behavior of the norms of these matrices.

85) Rachel Zhang, Statistics of Intersections of Curves on Surfaces (19 Jan 2016)

Each orientable surface with nonempty boundary can be associated with a planar model, whose edges can then be labeled with letters that read out a surface word. Then, the curve word of a free homotopy class of closed curves on a surface is the minimal sequence of edges of the planar model through which a curve in the class passes. The length of a class of curves is defined to be the number of letters in its curve word. We fix a surface and its corresponding planar model.
Fix a free homotopy class of curves ω on the surface. For another class of curves c, let i(ω; c) be the minimal number of intersections of curves in ω and c. In this paper, we show that the mean of the distribution of i(ω; c), for random curve c of length n, grows proportionally with n and approaches μ(ω) ⋅ n for a constant μ(ω). We also give an algorithm to compute μ(ω) and have written a program that calculates μ(ω) for any curve ω on any surface. In addition, we prove that i(ω; c) approahces a Gaussian distribution as n → ∞ by viewing the generation of a random curve as a Markov Chain.

84) Cristian Gutu and Fengyao Ding, SecretRoom: An Anonymous Chat Client (16 Jan 2016)

While many people would like to be able to communicate anonymously, the few existing anonymous communication systems sacrifice anonymity for performance, or vice­versa. The most popular such app is Tor, which relies on a series of relays to protect anonymity. Though proven to be efficient, Tor does not guarantee anonymity in the presence of strong adversaries like ISPs and government agencies who can conduct in­depth traffic analysis. In contrast, our messaging application, SecretRoom, implements an improved version of a secure messaging protocol called Dining Cryptographers Networks (DC­Nets) to guarantee true anonymity in moderately sized groups. However, unlike traditional DC­Nets, SecretRoom does not require direct communication between all participants and does not depend on the presence of honest clients for anonymity. By introducing an untrusted server that performs the DC­Net protocol on behalf of the clients, SecretRoom manages to reduce the O(n2) communication associated with traditional DC­Nets to O(n) for n clients. Moreover, by introducing artificially intelligent clients, SecretRoom makes the anonymity set size independent of the number of “real” clients. Ultimately SecretRoom reduces the communication to O(n) and allows the DC­Net protocol to scale to hundreds of clients compared to a few tens of clients in traditional DC­Nets.

83) Girishvar Venkat, Signatures of the Contravariant Form on Representations of the Hecke Algebra and Rational Cherednik Algebra associated to G (r,1,n) (15 Jan 2016)

The Hecke algebra and rational Cherednik algebra of the group G (r,1,n) are non-commutative algebras that are deformations of certain classical algebras associated to the group. These algebras have numerous applications in representation theory, number theory, algebraic geometry and integrable systems in quantum physics. Consequently, understanding their irreducible representations is important. If the deformation parameters are generic, then these irreducible representations, called Specht modules in the case of the Hecke algebra and Verma modules in the case of the Cherednik algebra, are in bijection with the irreducible representations of G (r,1,n). However, while every irreducible representation of G (r,1,n) is unitary, the Hermitian contravariant form on the Specht modules and Verma modules may only be non-degenerate. Thus, the signature of this form provides a great deal of information about the representations of the algebras that cannot be seen by looking at the group representations. In this paper, we compute the signature of arbitrary Specht modules of the Hecke algebra and use them to give explicit formulas of the parameter values for which these modules are unitary. We also compute asymptotic limits of existing formulas for the signature character of the polynomial representations of the Cherednik algebra which are vastly simpler than the full signature characters and show that these limits are rational functions in t. In addition, we show that for half of the parameter values, for each k, the degree k portion of the polynomial representation is unitary for large enough n.

82) Mehtaab Sawhney (PRIMES) and Jonathan Weed (MIT), Further results on arc and bar k-visibility graphs (, 6 Jan 2016)

We consider visibility graphs involving bars and arcs in which lines of sight can pass through up to k objects. We prove a new edge bound for arc k-visibility graphs, provide maximal constructions for arc and semi-arc k-visibility graphs, and give a complete characterization of semi-arc visibility graphs. We show that the family of arc i-visibility graphs is never contained in the family of bar j-visibility graphs for any i and j, and that the family of bar i-visibility graphs is not contained in the family of bar j-visibility graphs for $i \neq j$. We also give the first thickness bounds for arc and semi-arc k-visibility graphs. Finally, we introduce a model for random semi-bar and semi-arc k-visibility graphs and analyze its properties.

81) Harshal Sheth and Aashish Welling, An Implementation and Analysis of a Kernel Network Stack in Go with the CSP Style (30 Dec 2015)

Modern operating system kernels are written in lower level languages such as C. Although the low level functionalities of C are often useful within kernels, they also give rise to several classes of bugs. Kernels written in higher level languages avoid many of these potential problems, at the possible cost of decreased performance. This research evaluates the advantages and disadvantages of a kernel written in a higher level language. To do this, the network stack subsystem of the kernel was implemented in Go with the Communicating Sequential Processes (CSP) style. Go is a high level programming language that supports the CSP style, which recommends splitting large tasks into several smaller ones running in independent \threads". Modules for the major networking protocols, including Ethernet, ARP, IPv4, ICMP, UDP, and TCP, were implemented. In this study, the implemented Go network stack, called GoNet, was compared to a representative network stack written in C. The GoNet code is more readable and generally performs better than that of its C stack counterparts. From this, it can be concluded that Go with CSP style is a viable alternative to C for the language of kernel implementations.

80) Xiangyao Yu (MIT), Hongzhe Liu (PRIMES), Ethan Zou (PRIMES), and Srini Devadas (MIT), Tardis 2.0: An Optimized Time Traveling Coherence Protocol (, 27 Nov 2015)

The scalability of cache coherence protocols is a significant challenge in multicore and other distributed shared memory systems. Traditional snoopy and directory-based coherence protocols are difficult to scale up to many-core systems because of the overhead of broadcasting and storing sharers for each cacheline. Tardis, a recently proposed coherence protocol, shows potential in solving the scalability problem, since it only requires O(logN) storage per cacheline for an N-core system and needs no broadcasting support. The original Tardis protocol, however, only supports the sequential consistency memory model. This limits its applicability in real systems since most processors today implement relaxed consistency models like Total Store Order (TSO). Tardis also incurs large network traffic overhead on some benchmarks due to an excessive number of renew messages. Furthermore, the original Tardis protocol has suboptimal performance when the program uses spinning to communicate between threads. In this paper, we address these downsides of Tardis protocol and make it significantly more practical. Specifically, we discuss the architectural, memory system and protocol changes required in order to implement TSO consistency model on Tardis, and prove that the modified protocol satisfies TSO. We also propose optimizations for better leasing policies and to handle program spinning. Evaluated on 20 benchmarks, optimized Tardis at 64 (256) cores can achieve average performance improvement of 15.8% (8.4%) compared to the baseline Tardis and 1% (3.4%) compared to the baseline directory protocol. Our optimizations also reduce the average network traffic by 4.3% (6.1%) compared to the baseline directory protocol. On this set of benchmarks, optimized Tardis improves on a fullmap directory protocol in the metrics of energy, performance and storage, while being simpler to implement.

79) Allison Paul, Spectral Inference of a Directed Acyclic Graph Using Pairwise Similarities (11 Nov 2015)

A gene ontology graph is a directed acyclic graph (DAG) which represents relationships among biological processes. Inferring such a graph using a gene similarity matrix is NP-hard in general. Here, we propose an approximate algorithm to solve this problem efficiently by reducing the dimensionality of the problem using spectral clustering. We show that the original problem can be simplified to the inference problem of overlapping clusters in a network. We then solve the simplified problem in two steps: first we infer clusters using a spectral clustering technique. Then, we identify possible overlaps among the inferred clusters by identifying maximal cliques over the cluster similarity graph. We illustrate the effectiveness of our method over various synthetic networks in terms of both the performance and computational complexity compared to existing methods.


78) Niket Gowravaram, A Variation of nil-Temperley-Lieb Algebras of type A (26 Sep 2015)

We investigate a variation on the nil-Temperley-Lieb algebras of type A. This variation is formed by removing one of the relations and, in some sense, can be considered as a type B of the algebras. We give a general description of the structure of monomials formed by generators in the algebras. We also show that the dimension of these algebras is the sequence ${2n \choose n}$, by showing that the dimension is the Catalan transform of the sequence $2^n$.


77) Caleb Ji, Tanya Khovanova (MIT), Robin Park, and Angela Song, Chocolate Numbers (, 21 Sep 2015), published in Journal of Integer Sequences, vol. 19 (2016)

In this paper, we consider a game played on a rectangular $m \times n$ gridded chocolate bar. Each move, a player breaks the bar along a grid line. Each move after that consists of taking any piece of chocolate and breaking it again along existing grid lines, until just $mn$ individual squares remain.
This paper enumerates the number of ways to break an $m \times n$ bar, which we call chocolate numbers, and introduces four new sequences related to these numbers. Using various techniques, we prove interesting divisibility results regarding these sequences.

76) Niket Gowravaram and Tanya Khovanova (MIT), On the Structure of nil-Temperley-Lieb Algebras of type A (, 1 Sep 2015)

We investigate nil-Temperley-Lieb algebras of type A. We give a general description of the structure of monomials formed by the generators. We also show that the dimensions of these algebras are the famous Catalan numbers by providing a bijection between the monomials and Dyck paths. We show that the distribution of these monomials by degree is the same as the distribution of Dyck paths by the sum of the heights of the peaks minus the number of peaks.


75) Tanya Khovanova (MIT) and Karan Sarkar, P-positions in Modular Extensions to Nim (, 27 Aug 2015), published in International Journal of Game Theory, vol. 46 (2017)

In this paper, we consider a modular extension to the game of Nim, which we call $m$-Modular Nim, and explore its optimal strategy. In $m$-Modular Nim, a player can either make a standard Nim move or remove a multiple of $m$ tokens in total. We develop a winning strategy for all $m$ with $2$ heaps and for odd $m$ with any number of heaps.



74) Nicholas Diaco and Tanya Khovanova (MIT), Weighing Coins and Keeping Secrets (, 20 Aug 2015), published in Mathematical Intelligencer (September 2016)

In this expository paper we discuss a relatively new counterfeit coin problem with an unusual goal: maintaining the privacy of, rather than revealing, counterfeit coins in a set of both fake and real coins. We introduce two classes of solutions to this problem --- one that respects the privacy of all the coins and one that respects the privacy of only the fake coins --- and give several results regarding each. We describe and generalize 6 unique strategies that fall into these two categories. Furthermore, we explain conditions for the existence of a solution, as well as showing proof of a solution's optimality in select cases. In order to quantify exactly how much information is revealed by a given solution, we also define the revealing factor and revealing coefficient; these two values additionally act as a means of comparing the relative effectiveness of different solutions. Most importantly, by introducing an array of new concepts, we lay the foundation for future analysis of this very interesting problem, as well as many other problems related to privacy and the transfer of information.


73) Luke Sciarappa, Simple commutative algebras in Deligne's categories Rep($S_t$) (, 24 Jun 2015)

We show that in the Deligne categories $\mathrm{Rep}(S_t)$ for $t$ a transcendental number, the only simple algebra objects are images of simple algebras in the category of representations of a symmetric group under a canonical induction functor. They come in families which interpolate the families of algebras of functions on the cosets of $H\times S_{n-k}$ in $S_n$, for a fixed subgroup $H$ of $S_k$.


2014 Research Papers


72) Geoffrey Fudenberg (Harvard), Maxim Imakaev (MIT), Carolyn Lu (PRIMES), Anton Goloborodko (MIT), Nezar Abdennur (MIT), and Leonid Mirny (MIT), Formation of Chromosomal Domains by Loop Extrusion (bioRxiv, 14 Aug 2015), published in Cell Reports 15:9 (31 May 2016): 2038–2049.

Characterizing how the three-dimensional organization of eukaryotic interphase chromosomes modulates regulatory interactions is an important contemporary challenge. Here we propose an active process underlying the formation of chromosomal domains observed in Hi-C experiments. In this process, cis-acting factors extrude progressively larger loops, but stall at domain boundaries; this dynamically forms loops of various sizes within but not between domains. We studied this mechanism using a polymer model of the chromatin fiber subject to loop extrusion dynamics. We find that systems of dynamically extruded loops can produce domains as observed in Hi-C experiments. Our results demonstrate the plausibility of the loop extrusion mechanism, and posit potential roles of cohesin complexes as a loop-extruding factor, and CTCF as an impediment to loop extrusion at domain boundaries.

71) Kavish Gandhi, Maximal Monochromatic Geodesics in an Antipodal Coloring of Hypercube (4 April 2015)

A geodesic in the hypercube is the shortest possible path between two vertices. Leader and Long (2013) conjectured that, in every antipodal $2$-coloring of the edges of the hypercube, there exists a monochromatic geodesic between antipodal vertices. For this and an equivalent conjecture, we prove the cases $n = 2, 3, 4, 5$. We also examine the \emph{maximum} number of monochromatic geodesics of length $k$ in an antipodal $2$-coloring and find it to be $2^{n-1}(n-k+1)\binom{n-1}{k-1}(k-1)!$. In this case, we classify all colorings in which this maximum occurs. Furthermore, we explore the maximum number of antipodal geodesics in a subgraph of the hypercube with a fixed proportion of edges, providing a conjectured optimal configuration as a lower bound, which, interestingly, contains a constant proportion of geodesics with respect to $n$. Finally, we present a series of smaller results that could be of use in finding an upper bound on the maximum number of antipodal geodesics in such a subgraph of the hypercube.


70) Jesse Geneson (MIT) and Peter M. Tian (PRIMES), Sequences of formation width $4$ and alternation length $5$ (, 13 Feb 2015)

Sequence pattern avoidance is a central topic in combinatorics. A sequence $s$ contains a sequence $u$ if some subsequence of $s$ can be changed into $u$ by a one-to-one renaming of its letters. If $s$ does not contain $u$, then $s$ avoids $u$. A widely studied extremal function related to pattern avoidance is $Ex(u, n)$, the maximum length of an $n$-letter sequence that avoids $u$ and has every $r$ consecutive letters pairwise distinct, where $r$ is the number of distinct letters in $u$.
We bound $Ex(u, n)$ using the formation width function, $fw(u)$, which is the minimum $s$ for which there exists $r$ such that any concatenation of $s$ permutations, each on the same $r$ letters, contains $u$. In particular, we identify every sequence $u$ such that $fw(u)=4$ and $u$ contains $ababa$. The significance of this result lies in its implication that, for every such sequence $u$, we have $Ex(u, n) = \Theta(n \alpha(n))$, where $\alpha(n)$ denotes the incredibly slow-growing inverse Ackermann function. We have thus identified the extremal function of many infinite classes of previously unidentified sequences.


69) William Wu (PRIMES), Nicolaas Kaashoek (PRIMES), Matthew Weinberg (MIT), Christos Tzamos (MIT), and Costis Daskalakis (MIT), Game Theory based Peer Grading Mechanisms for MOOCs, paper for the Learning at Scale 2015 conference, March 14-18, 2015, Vancouver, BC, Canada (4 February 2015)

An efficient peer grading mechanism is proposed for grading the multitude of assignments in online courses. This novel approach is based on game theory and mechanism design. A set of assumptions and a mathematical model is ratified to simulate the dominant strategy behavior of students in a given mechanism. A benchmark function accounting for grade accuracy and workload is established to quantitatively compare e ectiveness and scalability of various mechanisms. After multiple iterations of mechanisms under increasingly realistic assumptions, three are proposed: Calibration, Improved Calibration, and Deduction. The Calibration mechanism performs as predicted by game theory when tested in an online crowd-sourced experiment, but fails when students are assumed to communicate. The Improved Calibration mechanism addresses this assumption, but at the cost of more e ort spent grading. The Deduction mechanism performs relatively well in the benchmark, outperforming the Calibration, Improved Calibration, traditional automated, and traditional peer grading systems. The mathematical model and benchmark opens the way for future derivative works to be performed and compared.


68) Alexandria Yu, Towards the classification of unital 7-dimensional commutative algebras (19 Jan 2015)

An algebra is a vector space with a compatible product operation. An algebra is called commutative if the product of any two elements is independent of the order in which they are multiplied. A basic problem is to determine how many unital commutative algebras exist in a given dimension and to find all of these algebras. This classification problem has its origin in number theory and algebraic geometry. For dimension less than or equal to 6, Poonen has completely classified all unital commutative algebras up to isomorphism. For dimension greater than or equal to 7, the situation is much more complicated due to the fact that there are infinitely many algebras up to isomorphism. The purpose of this work is to develop new techniques to classify unital 7-dimensional commutative algebras up to isomorphism. An algebra is called local if there exists a unique maximal ideal m. Local algebras are basic building blocks for general algebras as any finite dimensional unital commutative algebra is isomorphic to a direct sum of finite dimensional unital commutative local algebras. Hence, in order to classify all finite dimensional unital commutative algebras, it suffices to classify all finite dimensional unital commutative local algebras. In this article, we classify all unital 7-dimensional commutative local algebras up to isomorphism with the exception of the special case k1 = 3 and k2 = 3, where, for each positive integer i, mi is the subalgebra generated by products of i elements in the maximal ideal m and ki is the dimension of the quotient algebra mi/mi+1. When k2 = 1, we classify all finite dimensional unital commutative local algebras up to isomorphism. As a byproduct of our classification theorems, we discover several new classes of unital finite dimensional commutative algebras.


67) Niket Gowravaram and Uma Roy, Diagrammatic Calculus of Coxeter and Braid Groups (, 15 Mar 2015)

We investigate a novel diagrammatic approach to examining strict actions of a Coxeter group or a braid group on a category. This diagrammatic language, which was developed in a series of papers by Elias, Khovanov and Williamson, provides new tools and methods to attack many problems of current interest in representation theory. In our research we considered a particular problem which arises in this context. To a Coxeter group $W$ one can associate a real hyperplane arrangement, and can consider the complement of these hyperplanes in the complexification $Y_W$. The celebrated $K(\pi,1)$ conjecture states that $Y_W$ should be a classifying space for the pure braid group, and thus a natural quotient ${Y_W}/{W}$ should be a classifying space for the braid group. Salvetti provided a cell complex realization of the quotient, which we refer to as the Salvetti complex. In this paper we investigate a part of the $K(\pi,1)$ conjecture, which we call the $K(\pi,1)$ conjecturette, that states that the second homotopy group of the Salvetti complex is trivial. In this paper we present a diagrammatic proof of the $K(\pi,1)$ conjecturette for a family of braid groups as well as an analogous result for several families of Coxeter groups.


66) Arjun Khandelwal, Compact dot representations in permutation avoidance (3 Mar 2015)

A paper published by a Eriksson et. al in 2001 introduced a new form of representing a permutation, referred to as the compact dot representation, with the goal of constructing a smaller superpattern. We study this representation and give bounds on its size. We also consider a variant of the problem, where limitations on the alphabet size are imposed, and obtain lower bounds. Lastly, we consider the Mobius function of the poset of permutations ordered by containment.



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