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.

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.

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

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.

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.

The effect of consumer risk perceptions and information search on willingness to buy GM food products: A cross cultural analysis

This paper seeks to address the widespread call in the literature for the cross-cultural examination ( and validation) of accepted concepts within consumer behaviour, such as consumer risk perceptions and information search. The findings of the study provide support for a number of accepted relationships, whilst identifying distinct cross cultural differences in external information search and willingness to buy genetically modified (GM) food products by consumers.

Cooperation using a robotic ad hoc network made from Bluetooth, JXTA, OSGi and other commercial off the shelf (COTS) products

Abstract - Mobile devices in the near future will need to collaborate to fulfill their function. Collaboration will be done by communication. We use a real world example of robotic soccer to come up with the necessary structures required for robotic communication. A review of related work is done and it is found no examples come close to providing a RANET. The robotic ad hoc network (RANET) we suggest uses existing structures pulled from the areas of wireless networks, peer to peer and software life-cycle management. Gaps are found in the existing structures so we describe how to extend some structures to satisfy the design. The RANET design supports robot cooperation by exchanging messages, discovering needed skills that other robots on the network may possess and the transfer of these skills. The network is built on top of a Bluetooth wireless network and uses JXTA to communicate and transfer skills. OSGi bundles form the skills that can be transferred. To test the nal design a reference implementation is done. Deficiencies in some third party software is found, specifically JXTA and JamVM and GNU Classpath. Lastly we look at how to fix the deciencies by porting the JXTA C implementation to the target robotic platform and potentially eliminating the TCP/IP layer, using UDP instead of TCP or using an adaptive TCP/IP stack. We also propose a future areas of investigation; how to seed the configuration for the Personal area network (PAN) Bluetooth protocol extension so a Bluetooth TCP/IP link is more quickly formed and using the STP to allow multi-hop messaging and transfer of skills.

The S.T.A.B. Trial - Standardised Testing of Artificial Blood. A comparative study of various products that may be used as artificial blood for high fidelity simulation training in the critical care setting

Aim: In the current climate of medical education, there is an ever-increasing demand for and emphasis on simulation as both a teaching and training tool. The objective of our study was to compare the realism and practicality of a number of artificial blood products that could be used for high-fidelity simulation. Method: A literature and internet search was performed and 15 artificial blood products were identified from a variety of sources. One product was excluded due to its potential toxicity risks. Five observers, blinded to the products, performed two assessments on each product using an evaluation tool with 14 predefined criteria including color, consistency, clotting, and staining potential to manikin skin and clothing. Each criterion was rated using a five-point Likert scale. The products were left for 24 hours, both refrigerated and at room temperature, and then reassessed. Statistical analysis was performed to identify the most suitable products, and both inter- and intra-rater variability were examined. Results: Three products scored consistently well with all five assessors, with one product in particular scoring well in almost every criterion. This highest-rated product had a mean rating of 3.6 of 5.0 (95% posterior Interval 3.4-3.7). Inter-rater variability was minor with average ratings varying from 3.0 to 3.4 between the highest and lowest scorer. Intrarater variability was negligible with good agreement between first and second rating as per weighted kappa scores (K = 0.67). Conclusion: The most realistic and practical form of artificial blood identified was a commercial product called KD151 Flowing Blood Syrup. It was found to be not only realistic in appearance but practical in terms of storage and stain removal.

Drivers of technology substitution : successive generations of high tech products

Principal Topic High technology consumer products such as notebooks, digital cameras and DVD players are not introduced into a vacuum. Consumer experience with related earlier generation technologies, such as PCs, film cameras and VCRs, and the installed base of these products strongly impacts the market diffusion of the new generation products. Yet technology substitution has received only sparse attention in the diffusion of innovation literature. Research for consumer durables has been dominated by studies of (first purchase) adoption (c.f. Bass 1969) which do not explicitly consider the presence of an existing product/technology. More recently, considerable attention has also been given to replacement purchases (c.f. Kamakura and Balasubramanian 1987). Only a handful of papers explicitly deal with the diffusion of technology/product substitutes (e.g. Norton and Bass, 1987: Bass and Bass, 2004). They propose diffusion-type aggregate-level sales models that are used to forecast the overall sales for successive generations. Lacking household data, these aggregate models are unable to give insights into the decisions by individual households - whether to adopt generation II, and if so, when and why. This paper makes two contributions. It is the first large-scale empirical study that collects household data for successive generations of technologies in an effort to understand the drivers of adoption. Second, in comparision to traditional analysis that evaluates technology substitution as an ''adoption of innovation'' type process, we propose that from a consumer's perspective, technology substitution combines elements of both adoption (adopting the new generation technology) and replacement (replacing the generation I product with generation II). Based on this proposition, we develop and test a number of hypotheses. Methodology/Key Propositions In some cases, successive generations are clear ''substitutes'' for the earlier generation, in that they have almost identical functionality. For example, successive generations of PCs Pentium I to II to III or flat screen TV substituting for colour TV. More commonly, however, the new technology (generation II) is a ''partial substitute'' for existing technology (generation I). For example, digital cameras substitute for film-based cameras in the sense that they perform the same core function of taking photographs. They have some additional attributes of easier copying and sharing of images. However, the attribute of image quality is inferior. In cases of partial substitution, some consumers will purchase generation II products as substitutes for their generation I product, while other consumers will purchase generation II products as additional products to be used as well as their generation I product. We propose that substitute generation II purchases combine elements of both adoption and replacement, but additional generation II purchases are solely adoption-driven process. Extensive research on innovation adoption has consistently shown consumer innovativeness is the most important consumer characteristic that drives adoption timing (Goldsmith et al. 1995; Gielens and Steenkamp 2007). Hence, we expect consumer innovativeness also to influence both additional and substitute generation II purchases. Hypothesis 1a) More innovative households will make additional generation II purchases earlier. 1 b) More innovative households will make substitute generation II purchases earlier. 1 c) Consumer innovativeness will have a stronger impact on additional generation II purchases than on substitute generation II purchases. As outlined above, substitute generation II purchases act, in part like a replacement purchase for the generation I product. Prior research (Bayus 1991; Grewal et al 2004) identified product age as the most dominant factor influencing replacements. Hence, we hypothesise that: Hypothesis 2: Households with older generation I products will make substitute generation II purchases earlier. Our survey of 8,077 households investigates their adoption of two new generation products: notebooks as a technology change to PCs, and DVD players as a technology shift from VCRs. We employ Cox hazard modelling to study factors influencing the timing of a household's adoption of generation II products. We determine whether this is an additional or substitute purchase by asking whether the generation I product is still used. A separate hazard model is conducted for additional and substitute purchases. Consumer Innovativeness is measured as domain innovativeness adapted from the scales of Goldsmith and Hofacker (1991) and Flynn et al. (1996). The age of the generation I product is calculated based on the most recent household purchase of that product. Control variables include age, size and income of household, and age and education of primary decision-maker. Results and Implications Our preliminary results confirm both our hypotheses. Consumer innovativeness has a strong influence on both additional purchases (exp = 1.11) and substitute purchases (exp = 1.09). Exp is interpreted as the increased probability of purchase for an increase of 1.0 on a 7-point innovativeness scale. Also consistent with our hypotheses, the age of the generation I product has a dramatic influence for substitute purchases of VCR/DVD (exp = 2.92) and a strong influence for PCs/notebooks (exp = 1.30). Exp is interpreted as the increased probability of purchase for an increase of 10 years in the age of the generation I product. Yet, also as hypothesised, there was no influence on additional purchases. The results lead to two key implications. First, there is a clear distinction between additional and substitute purchases of generation II products, each with different drivers. Treating these as a single process will mask the true drivers of adoption. For substitute purchases, product age is a key driver. Hence, implications for marketers of high technology products can utilise data on generation I product age (e.g. from warranty or loyalty programs) to target customers who are more likely to make a purchase.

A national building products inventory

Australia has no nationally accepted building products life cycle inventory (LCI) database for use in building Ecologically Sustainable Development (ESD) assessment (BEA) tools. More information about the sustainability of the supply chain is limited by industry’s lack of real capacity to deliver objective information on process and product environmental impact. Recognition of these deficits emerged during compilation of a National LCI database to inform LCADesign, a prototype 3 dimensional object oriented computer aided design (3-D CAD) commercial building design tool. Development of this Australian LCI represents 24 staff years of effort here since 1995. Further development of LCADesign extensions is proposed as being essential to support key applications demanded from a more holistic theoretical framework calling for modules of new building and construction industry tools. A proposed tool, conceptually called LCADetails, is to serve the building product industries own needs as well as that of commercial building design amongst other industries’ prospective needs. In this paper, a proposition is examined that the existing national LCI database should be further expanded to serve Australian building product industries’ needs as well as to provide details for its client-base from a web based portal containing a module of practical supply and procurement applications. Along with improved supply chain assessment services, this proposed portal is envisaged to facilitate industry environmental life cycle improvement assessment and support decision-making to provide accredited data for operational reporting capabilities, load-based reasoning as well as BEA applications. This paper provides an overview of developments to date, including a novel 3-D CAD information and communications technology (ICT) platform for more holistic integration of existing tools for true cost assessment. Further conceptualisation of future prospects, based on a new holistic life cycle assessment framework LCADevelop, considering stakeholder relationships and their need for a range of complementary tools leveraging automated function off such ICT platforms to inform dimensionally defined operations for such as automotive, civil, transport and industrial applications are also explored.

Drivers of adoption for successive generation of high-tech products

To understand the diffusion of high technology products such as PCs, digital cameras and DVD players it is necessary to consider the dynamics of successive generations of technology. From the consumer’s perspective, these technology changes may manifest themselves as either a new generation product substituting for the old (for instance digital cameras) or as multiple generations of a single product (for example PCs). To date, research has been confined to aggregate level sales models. These models consider the demand relationship between one generation of a product and a successor generation. However, they do not give insights into the disaggregate-level decisions by individual households – whether to adopt the newer generation, and if so, when. This paper makes two contributions. It is the first large scale empirical study to collect household data for successive generations of technologies in an effort to understand the drivers of adoption. Second, in contrast to traditional analysis in diffusion research that conceptualizes technology substitution as an “adoption of innovation” type process, we propose that from a consumer’s perspective, technology substitution combines elements of both adoption (adopting the new generation technology) and replacement (replacing generation I product with generation II). Key Propositions In some cases, successive generations are clear “substitutes” for the earlier generation (e.g. PCs Pentium I to II to III ). More commonly the new generation II technology is a “partial substitute” for existing generation I technology (e.g. DVD players and VCRs). Some consumers will purchase generation II products as substitutes for their generation I product, while other consumers will purchase generation II products as additional products to be used as well as their generation I product. We propose that substitute generation II purchases combine elements of both adoption and replacement, but additional generation II purchases are solely adoption-driven process. Moreover, drawing on adoption theory consumer innovativeness is the most important consumer characteristic for adoption timing of new products. Hence, we hypothesize consumer innovativeness to influence the timing of both additional and substitute generation II purchases but to have a stronger impact on additional generation II purchases. We further propose that substitute generation II purchases act partially as a replacement purchase for the generation I product. Thus, we hypothesize that households with older generation I products will make substitute generation II purchases earlier. Methods We employ Cox hazard modeling to study factors influencing the timing of a household’s adoption of generation II products. A separate hazard model is conducted for additional and substitute purchases. The age of the generation I product is calculated based on the most recent household purchase of that product. Control variables include size and income of household, age and education of decision-maker. Results and Implications Our preliminary results confirm both our hypotheses. Consumer innovativeness has a strong influence on both additional purchases and substitute purchases. Also consistent with our hypotheses, the age of the generation I product has a dramatic influence for substitute purchases of VCR/DVD players and a strong influence for PCs/notebooks. Yet, also as hypothesized, there was no influence on additional purchases. This implies that there is a clear distinction between additional and substitute purchases of generation II products, each with different drivers. For substitute purchases, product age is a key driver. Therefore marketers of high technology products can utilize data on generation I product age (e.g. from warranty or loyalty programs) to target customers who are more likely to make a purchase.