Zeros of Weakly Holomorphic Modular Forms of Levels 2 and 3

Let M-k(#)(N) be the space of weakly holomorphic modular forms for Gamma(0)(N) that are holomorphic at all cusps except possibly at infinity. We study a canonical basis for M-k(#)(2) and M-k(#)(3) and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.

Duration discrimination in a series of rhythmic events

Duration discrimination of the last of a series of four clicks was investigated. Examination of psychophysical functions from eight subjects revealed evidence for a Weber’s law model relating discrimination to base interclick interval. Also, the point of subjective equality was seen to change reliably as a function of base rate.

Biomechanics on the Half Shell: Functional Performance Influences Patterns of Morphological Variation in the Emydid Turtle carapace

This study uses the carapace of emydid turtles to address hypothesized differences between terrestrial and aquatic species. Geometric morphometrics are used to quantify shell shape, and performance is estimated for two shell functions: shell strength and hydrodynamics. Aquatic turtle shells differ in shape from terrestrial turtle shells and are characterized by lower frontal areas and presumably lower drag. Terrestrial turtle shells are stronger than those of aquatic turtles; many-to-one mapping of morphology to function does not entirely mitigate a functional trade-off between mechanical strength and hydrodynamic performance. Furthermore, areas of morphospace characterized by exceptionally poor performance in either of the functions are not occupied by any emydid species. Though aquatic and terrestrial species show no significant differences in the rate of morphological evolution, aquatic species show a higher lineage density, indicative of a greater amount of convergence in their evolutionary history. The techniques employed in this study, including the modeling of theoretical shapes to assess performance in unoccupied areas of morphospace, suggest a framework for future studies of morphological variation.

Prime and Semiprime Inner Functions

This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided.

On the composition of Frostman Blaschke products

We construct an infinite uniform Frostman Blaschke product B such that B composed with itself is also a uniform Frostman Blaschke product. We also show that the set of uniform Frostman Blaschke products is open in the set of inner functions with the uniform norm.

Prime and semiprime inner functions

Abstract This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided.

Decomposing Blaschke Products And Polynomials

The goal of this paper is to contribute to the understanding of complex polynomials and Blaschke products, two very important function classes in mathematics. For a polynomial, $f,$ of degree $n,$ we study when it is possible to write $f$ as a composition $f=g\circ h$, where $g$ and $h$ are polynomials, each of degree less than $n.$ A polynomial is defined to be \emph{decomposable }if such an $h$ and $g$ exist, and a polynomial is said to be \emph{indecomposable} if no such $h$ and $g$ exist. We apply the results of Rickards in \cite{key-2}. We show that $$C_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,(z-z_{1})(z-z_{2})...(z-z_{n})\,\mbox{is decomposable}\},$$ has measure $0$ when considered a subset of $\mathbb{R}^{2n}.$ Using this we prove the stronger result that $$D_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,\mbox{There exists\,}a\in\mathbb{C}\,\,\mbox{with}\,\,(z-z_{1})(z-z_{2})...(z-z_{n})(z-a)\,\mbox{decomposable}\},$$ also has measure zero when considered a subset of $\mathbb{R}^{2n}.$ We show that for any polynomial $p$, there exists an $a\in\mathbb{C}$ such that $p(z)(z-a)$ is indecomposable, and we also examine the case of $D_{5}$ in detail. The main work of this paper studies finite Blaschke products, analytic functions on $\overline{\mathbb{D}}$ that map $\partial\mathbb{D}$ to $\partial\mathbb{D}.$ In analogy with polynomials, we discuss when a degree $n$ Blaschke product, $B,$ can be written as a composition $C\circ D$, where $C$ and $D$ are finite Blaschke products, each of degree less than $n.$ Decomposable and indecomposable are defined analogously. Our main results are divided into two sections. First, we equate a condition on the zeros of the Blaschke product with the existence of a decomposition where the right-hand factor, $D,$ has degree $2.$ We also equate decomposability of a Blaschke product, $B,$ with the existence of a Poncelet curve, whose foci are a subset of the zeros of $B,$ such that the Poncelet curve satisfies certain tangency conditions. This result is hard to apply in general, but has a very nice geometric interpretation when we desire a composition where the right-hand factor is degree 2 or 3. Our second section of finite Blaschke product results builds off of the work of Cowen in \cite{key-3}. For a finite Blaschke product $B,$ Cowen defines the so-called monodromy group, $G_{B},$ of the finite Blaschke product. He then equates the decomposability of a finite Blaschke product, $B,$ with the existence of a nontrivial partition, $\mathcal{P},$ of the branches of $B^{-1}(z),$ such that $G_{B}$ respects $\mathcal{P}$. We present an in-depth analysis of how to calculate $G_{B}$, extending Cowen's description. These methods allow us to equate the existence of a decomposition where the left-hand factor has degree 2, with a simple condition on the critical points of the Blaschke product. In addition we are able to put a condition of the structure of $G_{B}$ for any decomposable Blaschke product satisfying certain normalization conditions. The final section of this paper discusses how one can put the results of the paper into practice to determine, if a particular Blaschke product is decomposable. We compare three major algorithms. The first is a brute force technique where one searches through the zero set of $B$ for subsets which could be the zero set of $D$, exhaustively searching for a successful decomposition $B(z)=C(D(z)).$ The second algorithm involves simply examining the cardinality of the image, under $B,$ of the set of critical points of $B.$ For a degree $n$ Blaschke product, $B,$ if this cardinality is greater than $\frac{n}{2}$, the Blaschke product is indecomposable. The final algorithm attempts to apply the geometric interpretation of decomposability given by our theorem concerning the existence of a particular Poncelet curve. The final two algorithms can be implemented easily with the use of an HTML

The Behavior Rating Inventory Of Executive Function - Self-Report Version (Brief-Sr) And The Delis-Kaplan Executive Function Sys

The assessment of executive functions is an area of study that has seen considerable development in recent years. Despite much research examining the validity of various measures of executive functions from both a direct and indirect format, little evidence exists in the extant literature evaluating the correspondence between these types of measures. The current study examined the extent of correspondence, comprising concurrent validity, between the Delis-Kaplan Executive Function System (D-KEFS) and the Behavior Rating Inventory of Executive Function ¿ Self-Report Version (BRIEF-SR). Participants included 30 undergraduate and high school students 18 years of age. Results indicated mixed evidence of concurrent validity between the two measures of executive functions. The findings obtained suggest both expected significant, negative correlation as well as lack of expected correlation between the measures. Suggestions for future research in the assessment of executive functions are discussed.