Bundling Strategies When Products Are Vertically Differentiated and Capacities Are Limited

We consider a seller who owns two capacity-constrained resources and markets two products (components) corresponding to these resources as well as a bundle comprising the two components. In an environment where all customers agree that one of the two components is of higher quality than the other and that the bundle is of the highest quality, we derive the seller's optimal bundling strategy. We demonstrate that the optimal solution depends on the absolute and relative availabilities of the two resources as well as upon the extent of subadditivity of the quality of the products. The possible strategies that can arise as equilibrium behavior include a pure components strategy, a partial- or full-spectrum mixed bundling strategy, and a pure bundling strategy, where the latter strategy is optimal when capacities are unconstrained. These conclusions are contrary to findings in the prior literature on bundling that demonstrated the unambiguous dominance of the full-spectrum mixed bundling strategy. Thus, our work expands the frontier of bundling to an environment with vertically differentiated components and limited resources. We also explore how the bundling strategies change as we introduce an element of horizontal differentiation wherein different types of customers value the available components differently.

INTERPOLATING BLASCHKE PRODUCTS AND ANGULAR DERIVATIVES

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra H-infinity[(b) over bar : b has finite angular derivative everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

Interpolating Blaschke Products and Angular Derivatives

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra generated by the algebra of bounded analytic functions and the conjugates of Blaschke products with angular derivative finite everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

Groups Acting on Tensor Products

Groups preserving a distributive product are encountered often in algebra. Examples include automorphism groups of associative and nonassociative rings, classical groups, and automorphism groups of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving tensor products over semisimple and semiprimary rings, and present effective algorithms to construct generators for these groups. We also discuss applications of our methods to algorithmic problems for which all currently known methods require an exponential amount of work. (C) 2013 Elsevier B.V. All rights reserved.

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.

Understanding Electrophoretic Stacking Of In-Capillary Generated Reaction Products

The separation of small molecules by capillary electrophoresis is governed by a complex interplay among several physical effects. Until recently, a systematic understanding of how the influence of all of these effects is observed experimentally has remained unclear. The work presented in this thesis involves the use of transient isotachophoretic stacking (tITP) and computer simulation to improve and better understand an in-capillary chemical assay for creatinine. This assay involves the use of electrophoretically mediated micro-analysis (EMMA) to carry out the Jaffé reaction inside a capillary tube. The primary contribution of this work is the elucidation of the role of the length and concentration of the hydroxide plug used to achieve tITP stacking of the product formed by the in-capillary EMMA/Jaffé method. Computer simulation using SIMUL 5.0 predicts that a 3-4 fold gain in sensitivity can be recognized by timing the tITP stacking event such that the Jaffé product peak is at its maximum height as that peak is electrophoresing past the detection window. Overall, the length of the hydroxide plug alters the timing of the stacking event and lower concentration plugs of hydroxide lead to more rapidly occurring tITP stacking events. Also, the inclusion of intentional tITP stacking in the EMMA/Jaffé method improves the sensitivity of the assay, including creatinine concentrations within the normal biological range. Ultimately, improvement in assay sensitivity can be rationally designed by using the length and concentration of the hydroxide plug to engineer the timing of the tITP stacking event such that stacking occurs as the Jaffé product is passing the detection window.

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