arXiv subjects

Corey Manack

Publications and source records attributed to Corey Manack.

7 recordsLinked to original sources

arXiv2015

Taking the risk out of RISK: Conquer Odds in the Board Game RISK

Dice odds in the board game RISK were first investigated by Tan, fixed by Osbourne, and extended by Blatt. We generalized dice odds further, varying the number of sides and the number of dice used in a single battle. We show that the attacker needs two more than 86% of the defending armies to have an over 50% chance to conquer an enemy territory. By normal approximation, we show that the conquer odds transition rapidly from low chance to high chance of conquering around the 86%+2 threshold.

math.HO↗

arXiv2015

Digit sequences of rational billiards on tables which tile $\mathbb{R}^2$

We classify the periodic digit strings which arise from periodic billiard orbits on the four convex $n$-gons $\Delta$ which tile $\mathbb{R}^2$ under reflection, answering problem a posed by Baxter and Umble. $\Delta$ is either an equilateral triangle, a $45-45-90$ triangle a $30-60-90$, triangle or a rectangle.

math.DS↗

arXiv2015

Leading Digit Laws on Linear Lie Groups

We determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\mathbb{R}^+$ is $dx/x$ and that the scale invariance of $dx/x$ implies the distribution of the digits follow Benford's law, which is the probability of observing a significand base $B$ of at most $s$ is $\log_B(s)$; thus the first digit is $d$ with probability $\log_B(1 + 1/d)$). Viewing this scale invariance as left invariance of Haar measure, we determine the power laws in significands from one matrix component of various such $G$. We also determine the leading digit distribution of a fixed number of components of a unit sphere, and find periodic behavior when the dimension of the sphere tends to infinity in a certain progression.

math.NT↗

arXiv2013

Linear Triangle Dynamics: The Pedal Map and Beyond

We present a moduli space for similar triangles, then classify triangle maps $f$ that arise from linear maps on this space, with the well-studied pedal map as a special case. Each linear triangle map admits a Markov partition, showing that $f$ is mixing, hence ergodic.

math.DS↗

arXiv2013

Character Estimates of Adjoint Simple Lie Groups

Call a compact, connected, simple Lie group $G$ {\emph{adjoint simple}} if it has trivial center. Let $C\subset G$ be a nontrivial conjugacy class, $e\in G$ the identity element of $G$. We prove the existence of an $N\in\mathbb{N}$, depending on $G$ but not $C$, such that $e$ lies in the interior of $C^n$ for all $n\geq N$. We then prove that a disk $D\subset\mathbb{C}$ of radius less than 1, contained in the unit disk $D_1$ and tangent to $D_1$ at $z=1$, contains the image of every normalized character $\chi(e)^{-1}\chi$ of $G$.

math.RT↗

arXiv2012

Enumeration of k-Exceedance Lattice Paths with an Application to Comparing Chains of Order Statistics

We enumerate the number of monotonic lattice paths starting at $(0,0)$ and terminating at $(m,n)$ in which $l$ of the first $k$ steps lie below the line $y=x\ (0\leq k\leq m\leq n)$. These closed formulas consist of terms which are a product Catalan numbers, ballot numbers and binomial coefficients. We then apply the combinatorial formulas to failure analysis by deriving a probability distribution that compares the performance of a $k$-out-of-$m$ system to a $k$-out-of-$n$ system of continuous, independent, and identically distributed random variables. Lastly, we provide asymptotics in a few special cases of $k,m,n$ and leave others as conjecture.

math.CO↗

arXiv2010

Low Degree Representations of Simple Lie Groups

We give upper bounds for the number of irreducible representations of dimension at most n for a compact semisimple Lie group. In particular, we prove that there are at most n irreducible representations of dimension at most n for a simple compact Lie group. We use this prove that the number of conjugacy classes of maximal subgroups of a compact simple Lie group is O(r) where r is the Lie rank. We also give a short proof that the dimensions of the weight spaces of a maximal torus are small relative to the dimension of an irreducible module.

math.RT↗