## CMU Mathematics Summer Undergraduate Research Fellows, 2016

### Projects:

*On Optimal Assets Allocation of Retirement Portfolio*,**Jiaping Bian**,**Jingyi Chen****Advisor:**Willam Hrusa

**Abstract:**Optimal investment is a very important branch of mathematical finance. One of the most important applications concerns allocation of retirement assets between different types of funds. We consider a simple model in which there are two basic funds: a money market account and a stock fund. The stock fund has a higher expected return than the money market account, but this fund is riskier than the money market. We assume that the price (per share) of the stock fund evolves according to a binomial model. The investor's decisions concerning asset allocation should be based on his or her risk preferences. (Here we quantify risk preferences using utility functions.) We focus on the case of a young person who is entering the workforce and has no retirement savings yet. We assume that contributions of 10% annual salary will be made to the retirement account. So far, we have assumed that increases in salary are deterministic. Since there is a long time horizon until retirement, small difference in strategies can lead to substantial differences at retirement. We considered logarithmic and power-type utility functions. For these utility functions, there are explicit solutions to the utility maximization problem, but these solutions typically involve borrowing from the money market to buy stock. In practice, retirement accounts do not permit borrowing or short sales. Incorporating these assumptions into a model leads to an extremely complicated set of constraints when there are many investment periods. We have developed an algorithm that is very easy to implement and that we believe will create an optimal portfolio (or at least get very close). This algorithm shows that every investor, even if they are quite risk averse, should initially put all contributions into the stock account. At some point in time, money should start shifting to the money market account. The level of risk aversion can be used to quantify when the transition should occur. We have run simulations involving randomly generated strings of coin tosses and the results suggest that our algorithm is optimal. We are currently employing ideas from dynamic programming to develop a proof.*Online Purchasing of Trees with Uncertainty*,**Manuel Fernandez****Advisor:**Wesley Pegden

**Abstract:**Consider the complete graph $K_n$ and to each edge assign a random weight drawn from the uniform distribution $U(0,1)$, where each edge's weight is independent of all other edge weights. In 1985, Frieze proved that as $n \rightarrow \infty$, the expected weight of a Minimum Spanning Tree on $K_n$ with these edge weights equals $\gamma(3) \approx 1.20$. We consider a similar problem, but under an online purchasing setting (the purchaser views edges one at a time, the edges are shown to the purchaser in some order, and the cost is the sum of the weights of the purchased edges). The goal is to purchase a spanning tree of $K_n$ under the restriction that an edge must be examined before purchasing and must be immediately purchased or rejected upon examination (rejected edges cannot be purchased later.) Aldous, Angel and Berestycki proved that the expected cost of a tree that an optimal algorithm purchases under this restriction is at least 1.368 and at most 4. We proved that the expected cost is at most 2.1385. To do this, we construct an algorithm for purchasing a spanning tree from $K_n.$ The algorithm uses edge partition, multi-staged purchasing thresholds, and a final "cleanup" phase to purchase the rest of the spanning tree. We calculate the expected cost of the tree purchased in terms of its input parameters using results about sparse random graphs and basic probability theory. The upper bound is obtained by optimizing over the input parameters.*Existence, Uniqueness, and Asymptotic Behavior of Solutions to a Linear System of PDEs with Weak Viscoelastic Damping*,**Liuyu Jin**

**Advisor:**Willam Hrusa

**Abstract:**Consider the system of partial differential equations \begin{align*} \epsilon_{t}(x, t) &= v_{x}(x, t)\\ v_{t}(x, t) &= \sigma_{x}(x, t)\\ \sigma_{t}(x, t) &= 2v_{x}(x, t) + a(x)(\epsilon(x, t) - \sigma(x, t)),\\ \end{align*} together with the initial conditions \begin{align*} \epsilon(x, 0) &= \epsilon_{0}(x)\\ v(x, 0) &= v_{0}(x)\\ \sigma(x, 0) &= \sigma_{0}(x), \\ \end{align*} and the boundary condition \begin{align*} v(0, t) &= v(1, t) = 0,\\ \end{align*} where $a(x)$ is the damping function. This system models the motion of a one-dimensional body composed partly of elastic material and partly of viscoelastic. For $a(x) \equiv 0$, the body is elastic and there is no damping. (In particular, when $a\equiv 0$ the velocity $v$ does not have a large-time limit.) For $a(x) \equiv 1$, it would be the system for a viscoelastic damping material of Maxwell type and it is well-known that $v$ approaches 0 at $t\rightarrow\infty$. This research focuses on the asymptotic behavior of the solutions to the particular system with a smooth function $a(x)$ such that $0 \leq a(x) \leq 1$, and $a(x) > 0$, for $x$ in an interval $I \subset (0, 1)$. We use semigroup methods to establish existence and uniqueness of solutions to in an appropriate function space. The behavior of solutions as $t\rightarrow\infty$. Using the Invariance Principle (from the theory of dynamical systems), we show that if $\int_0^1a(x)\, dx>0$, then $v$ tends to zero as $t\rightarrow\infty$. We also identify the large-time limits of $\sigma$ and $\epsilon$.*$L_1$ Regularization for Compact Support*,**Timothy Li**

**Advisor:**Giovanni Leoni

**Abstract:**In this paper, we solve a conjecture of Osher and Yin that $L_1$ regularization of certain minimization problems ensures compact support of minimizers. In particular, we consider a class of minimization problems and show that adding an $L_1$ term forces compact support of solutions given certain sufficient conditions namely that the given forcing function sufficiently decays toward positive and negative infinity. It turns out that the condition found is sharp. Furthermore, we can give a bound on the measure of the support and a bound on the support in addition to the existence of minimizers. The main difficulty in analyzing this problem is the lack of a derivative of absolute value at 0. We can overcome this by using the convexity of the problem and computing the subdifferential instead of the derivative. We also use some tools from calculus of variations. It is hopeful that this approach can be generalized to other minimization problems as well as to higher dimension problems. The picture is courtesy of R. E. Caflisch, S. J. Osher, H. Schaffer, and G. Tran.*Continuous and Discrete Time Multi-Species Population Models*,**Hang Liao**,**Cameron Montag**,**Jacob Neumann**

**Advisor:**Willam Hrusa

**Abstract:**In this research project, we model various types of multi-species population relationships, using both differential equations and iterative discrete-time models. In general, we find that many effective population models have an equation with three components: a linear term for growth proportional to the species population, a term representing the benefit of the species' cooperation, and a damping term representing the effect of resource scarcity. The behavior of these systems typically depend on the choice of several constants; this approach affords our models a great degree of generality.

We focus on systems where the damping factor grows sufficiently quickly to ensure that the population will not diverge (e.g. polynomials of degree 2, 2.5, and 3). In both the discrete-time and continuous-time scenarios, this typically yields systems where we can demonstrate the existence of equilibria and/or periodic behavior. Often, such solutions are closed form expressions involving the values of the constants.

Ultimately, we arrive at a number of general results characterizing a variety of systems. This will allow us to describe the effects of different species' population counts and initial conditions on each other in both the short- and long-term.*A Combinatorial Interpretation for Super Catalan Numbers $T(3,n)$*,**Gidon Orelowitz**

**Advisor:**Irina Gheorghiciuc

**Abstract:**The super Catalan numbers $T(m,n)=(2m)!(2n)!/2m!n!(m+n)!$ are integers that generalize the Catalan numbers. Two different weighted combinatorial interpretations are known for $T(m,n)$, one by Allen and Gheorghiciuc, another one by Georgiadis, Munemasa and Tanaka. The weighted interpretation by Allen and Gheorghiciuc was converted into a conventional combinatorial interpretation in the case $m=2$. In this paper we expand their techniques to convert a weighted combinatorial interpretation into a conventional combinatorial interpretation in the case $m=3$.*Linearized Navier-Stokes with free surface and fractional surface tension*,**Samuel Zbarsky**

**Advisor:**Ian Tice

**Abstract:**We are interested in the problem of proving energy bounds on the linearized Navier-Stokes problem with gravity and free boundary on the surface. We investigate a toy problem and understand how the energy decays both with and without surface tension, as well as using a fractional power of the Laplacian to interpolate between these two possibilities, finding that the decay rate changes between algebraic and exponential at the halfway point. We use these energy bounds to prove local existence. We then derive energy bounds for the linearized Navier-Stokes problem (assuming local existence) and analyze how they depend on the power of the Laplacian and find similar decay to the toy model.