Degree Complexity of a Family of Birational Maps
We compute the degree complexity of a family of birational mappings of the plane with high order singularities.
math.DS↗
arXiv subjects
Kyounghee Kim
Publications and source records attributed to Kyounghee Kim.
Linear Recurrences in the Degree Sequences of Monomial Mappings
Let $A$ be an integer matrix, and let $f_A$ be the associated monomial map. We give a connection between the eigenvalues of $A$ and existence of a linear recurrence relation in the sequence of degrees.
math.DS↗
The Number of Periodic Orbits of a Rational Difference Equation
We show how a variant of the Lefschetz Fixed Point Theorem may be used to count the number of periodic orbits for certain rational difference equations.
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Exponential Martingales and Time integrals of Brownian Motion
We find a simple expression for the probability density of $\int \exp (B_s - s/2) ds$ in terms of its distribution function and the distribution function for the time integral of $\exp (B_s + s/2)$. The relation is obtained with a change of measure argument where expectations over events determined by the time integral are replaced by expectations over the entire probability space. We develop precise information concerning the lower tail probabilities for these random variables as well as for time integrals of geometric Brownian motion with arbitrary constant drift. In particular, $E[ \exp\big(θ/ \int \exp (B_s)ds\big) ]$ is finite iff $θ< 2$. We present a new formula for the price of an Asian call option.
math.PR↗
One-Factor Term Structure without Forward Rates
We construct a no-arbitrage model of bond prices where the long bond is used as a numeraire. We develop bond prices and their dynamics without developing any model for the spot rate or forward rates. The model is arbitrage free and all nominal interest rates remain positive in the model. We give examples where our model does not have a spot rate; other examples include both spot and forward rates.
math.PR↗
Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices
We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set ${\cal SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to ${\cal SC}_q$.
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Periodicities in Linear Fractional Recurrences: Degree growth of birational surface maps
We consider the set of all 2-step recurrences (difference equations) that are given by linear fractional maps. These give birational maps of the plane. We determine the degree growth of these birational maps. We find the all the maps in this family that are periodic. This also leads to new surface automorphisms with positive entropy.
math.DS↗
On the Degree Growth of Birational Mappings in Higher Dimension
Let $f$ be a birational map of ${\bf C}^d$, and consider the degree complexity, or asymptotic degree growth rate $δ(f)=\lim_{n\to\infty}({\rm deg}(f^n))^{1/n}$. We introduce a family of elementary maps, which have the form $f=L\circ J$, where $L$ is (invertible) linear, and $J(x_1,...,x_d)=(x_1^{-1},...,x_d^{-1})$. We develop a method of regularization and show how it can be used to compute $δ$ for an elementary map.
math.DS↗