A Note on Confusion between Linear and Affine Functions and the Generalized Forms of Gradient

Background: Arguably the most frequently used term in science, particularly in mathematics and statistics, is linear. However, confusion arises from the various meanings of linearity instructed in different levels of mathematical courses. The definition of linearity taught in high school is less correct than the one learned in a linear algebra class. The correlation coefficient of two quantitative variables is a numerical measure of the affinity, not only linearity, of two variables. However, every statistics book loosely says it is a measure of linear relationship. This clearly show that there is some confusion between use of the terms the linear function and affine function. Objective: This article aims at clarifying the confusion between use of the terms linear function and affine function. It also provides more generalized forms of the gradient in different branches of mathematics and show their equivalency. Materials and Methods: We have used the pure analytical deductive methods to proof the statements. Results: We have clearly presented that gradient is the measure of affinity, not just linearity. It becomes a special case of the derivative in calculus, of the least-squares estimate of the regression coefficient in statistics and matrix theory. The gradient can ­­­­be seen in terms of the inverse of the informative matrix in the most general setting of the linear model estimation. Conclusion: The article has been clearly written to show the distinction between the linear and affine functions in a concise and unambiguous manner. We hope that readers will clearly see various generalizations of the gradient and article itself would be a simple exposition, enlightening, and fun to read.

Experiences in Multi-Platform Campaigns

This chapter discusses the relationship between engagement actions during the experiences in game advertising, with the aim to investigate the elements that support consumer action in a Virtual Reality (VR) game developed for Oculus Rift. Since VR can be understood as a potential investment in advertising platform, the proposal of this chapter is to comprehend the experience of engagement with brands inside the game, including aspects about VR gameplay and immersion. For that, the researchers developed a game with different levels of brand integration in a VR environment. In order to measure engagement levels in this experience, the researchers have invited 24 participants to play the game to observe their behaviors. As a result of this study, it is suggested that the VR can be a useful environment to insert advertising for multi-platforms experiences, since it can promote an emotional and cognitive experiences around the brand and not only a simple exposition of the advertising.

Free Agent Auctions and Revenue Sharing: A Simple Exposition

This article uses a simple approach to address the issue of how revenue sharing in professional sports leagues can affect the allocation of free agent players to teams. To affect the allocation of free agents, the imposition of revenue sharing must alter the ranking of bidding teams in terms of maximum salary offers. Two types of revenue sharing systems are considered: traditional gate revenue sharing and pooled revenue sharing. The article suggests that team rankings for ability to pay are not affected by pooled revenue sharing, however the distribution of player salaries will be affected asymmetrically. Traditional gate revenue sharing can alter the ability to pay rankings for teams, depending upon playing schedules and the closeness of revenues between closely ranked teams. Revenue data for two professional sports leagues provide evidence in favor of the model predictions.