Let (V, ||·||) be a normed space, and W a d^Norm_V,||·|| -closed vector subspace of V.

(a) Prove that a function |||·||| : V /W → R_≥0 can be consistently defined by ∀v ∈ V : |||v + W||| df= inf({||v + w|| : R_≥0 | w ∈ W}).

(b) Prove that |||·||| is a norm on V /W.

(c) Prove that if (V, ||·||) is a Banach space, then so is (V /W, |||·|||)

Case Study 9—Social Media Is Gateway to eBook Sales Success: Amazon Kindle eBooks Thomas S. Mueller, Appalachian State University The Amazon Kindle eBook brand is one of the most popular forms of digital publishing in today’s social community. As of early 2016, the direct-to-consumer online retailer was selling 1.06 million eBooks (paid downloads) per day. Consumers spent $5.75 million per day during a recent month, with $1.76 million of those sales directed to author royalties. Though it is somewhat undocumented regarding distribution, authors received an additional $140 million in funds from Kindle Unlimited, Amazon’s subscription reading service.32 The industry as a whole is generating over $2.1 billion in sales annually. It is interesting to note that as established publisher sales decreased, eBooks written by independent authors increased.33 During an assessment in 2016, 56 of Amazon’s 100 top-selling eBooks were self-published titles. Topically, the bestselling independent authors write about paranormal activities, romance, thrillers, urban fiction, suspense, and science fiction. With the ascent of digital titles on Amazon and other sources, industry experts posit that printed versions of books will become a niche market, similar to the effect noted in magazine and newsprint.34 Digital publishing strategist Ben Thompson applies aggregation theory to the process, suggesting that sales portals like Amazon eliminate intermediaries, which allows independent publishers to avoid publishers and go direct to consumers.35 With a substantial faction of the social publishing market operating independently, advertising and promotion become the responsibility of the author. One portion of an author’s time is allocated to writing the novel, while the other half of the time is invested in social commerce, which includes marketing to expand the author’s fan base. The challenge is to differentiate an eBook from millions of competitive titles, then convince readers to download the digital offering onto a Kindle reader or other digital device. Pre-selling is essential to assure return on investment for the self-published author. One viable and affordable option is to build a social media marketing plan. Social media platforms provide opportunities to develop leads, generate a follower list, leverage other prominent users, access large topical groups, and promote content “teasers” that showcase your upcoming eBook.36 Most eBook authors develop a social media plan through content generation. These are most often short, concise articles of 250 to 500 words. Most successful social media plans offer shorter posts, with frequent placement. Social media content is usually housed in a blog, such as Google’s Blogger platform, or the highly customizable WordPress.37 It’s important to claim a URL that is unique and specific to the author and her or his work. A strong URL, along with key search words included in blog post titles, can help the author’s social media content climb in ranking on search engines. Social media portals such as LinkedIn and Google+ offer each user a personal landing page, but much more is available through groups and communities, where individuals share common interests. For example, a fashion community in Google+ hosts 557,000 followers.38 A marketing communication group in LinkedIn has collected 621,000 members.39 And, the LinkedIn Book Marketing group has 28,000 members.40 It is essential to identify a topical group to understand the relevance of posts, how interaction and replies work, and who posts the top viewed submissions. Authors can integrate into these platforms by reading posts, replying to posts, and finally crafting content that resonates with potential readers. Some authors incorporate introductory promotional comments on their upcoming eBook. It is important for authors to remember that blatant selling is not admissible; posted content must provide value and prove interesting to other group members.41 Facebook has been the most influential platform for many eBook authors. It has grown to over 1.4 billion users worldwide, with 70% checking their feed multiple times each day. eBook authors create a “landing page” for their profile as an author, or sometimes for a specific title. The challenge is to direct friends from the page to a personal website, and to target specific Facebook users who are profiled as sharing an interest in the eBook topic. Some authors host events or make promotional offers to engage users and increase traffic, which can be measured through social media data management. Mark Dawson initially started writing for a traditional publishing firm. His sales were weak and he transitioned to self-publishing. After learning how to leverage Facebook, Mark invested time and energy and now earns over $450,000 per year.42 Video platforms such as Vimeo and YouTube have been integrated into the social media promotion of eBooks. YouTube, the predominant player, was purchased in 2006 for $1.65 billion by Google. YouTube currently reports that it generates over 1 billion views per month.43 Successful eBook authors have integrated YouTube segments into their social media plans, including interviews or “teasers” that offer special introductory offers at eBook launch time. It’s beneficial to cross-promote YouTube posts across Facebook pages and embedded content in visual sharing sites such as Pinterest and Instagram. Studies indicate that the human brain assimilates images 60,000 times more quickly than text. eBook authors are intentional about visual sharing; Instagram is predominantly 18 to 24 female (56%) and allows space to share images. Pinterest represents a female and male audience, with two thirds between the ages of 18 and 24. One half are located outside the United States. It allows users to aggregate content that appeals to their interests and sensibilities. The author’s challenge is to entice user engagement, create awareness for the latest published eBook, and generate reader feedback, which can channel into digital book reviews. Many authors use the visual advantage to promote their eBook through storytelling in multiple posts.44 A social media platform that serves as the “messenger” for all social media activity is Twitter. Each message is limited to 140 characters, which can in turn be reposted and shared. Twitter now has 320 million monthly users, with over 1 billion monthly visits.45 Savvy eBook authors use Twitter to include “@” user handles to attract other key authors, or hashtags to create topical searches. Twitter can also steer followers to other platforms, where blog or web content has been placed. Some have also leveraged Twitter’s new live video broadcast program Periscope to promote special programs and activities. Some of the most highly successful eBook authors are also prominent podcast hosts. Technology entrepreneur James Altucher has published 11 books, including the Wall Street Journal bestseller Choose Yourself! Altucher’s unique position in the marketplace is that he lost his fortune, earned it back, and then lost it again. He now publishes eBooks such as Reinvent Yourself and promotes his projects on the popular “James Altucher Show” podcast, which is accessible on podbay.fm and iTunes.46 One social media opportunity that is sometimes overlooked is within the Amazon portal itself. An Amazon author page can showcase the author, include a biography, feature a compilation of the author’s Amazon Kindle publications, and link to other social media content such as Twitter or personal blogs. Most anyone can become a digital author, but not everyone can sell books. The holistic author understands how to write, publish, network, create, associate, and entertain. Social media presents the gateway where authors and customers connect in the open and unregulated marketplace.

Choose a topic, then create a book title for that topic. What order of social media would you utilize to make the announcement to potential readers?

Broussard Skateboard's sales are expected to increase by 20% from $7.6 million in 2016 to $9.12 million in 2017. Its assets totaled $3 million at the end of 2016. Broussard is already at full capacity, so its assets must grow at the same rate as projected sales. At the end of 2016, current liabilities were $1.4 million, consisting of $450,000 of accounts payable, $500,000 of notes payable, and $450,000 of accruals. The after-tax profit margin is forecasted to be 4%. Assume that the company pays no dividends. Under these assumptions, what would be the additional funds needed for the coming year? Do not round intermediate calculations. Round your answer to the nearest dollar.

Why is this AFN different from the one when the company pays dividends?
I. Under this scenario the company would have a lower level of retained earnings but this would have no effect on the amount of additional funds needed.
II. Under this scenario the company would have a higher level of retained earnings which would reduce the amount of additional funds needed.
III. Under this scenario the company would have a higher level of retained earnings which would increase the amount of additional funds needed.
IV. Under this scenario the company would have a higher level of retained earnings but this would have no effect on the amount of additional funds needed.
V. Under this scenario the company would have a lower level of retained earnings which would reduce the amount of additional funds needed.

Broussard Skateboard's sales are expected to increase by 15% from $7.6 million in 2016 to $8.74 million in 2017. Its assets totaled $5 million at the end of 2016. Broussard is already at full capacity, so its assets must grow at the same rate as projected sales. At the end of 2016, current liabilities were $1.4 million, consisting of $450,000 of accounts payable, $500,000 of notes payable, and $450,000 of accruals. The after-tax profit margin is forecasted to be 4%. Assume that the company pays no dividends. Under these assumptions, what would be the additional funds needed for the coming year? Do not round intermediate calculations. Round your answer to the nearest dollar.

$

Why is this AFN different from the one when the company pays dividends?

I. Under this scenario the company would have a lower level of retained earnings but this would have no effect on the amount of additional funds needed.

II. Under this scenario the company would have a higher level of retained earnings which would reduce the amount of additional funds needed.

III. Under this scenario the company would have a higher level of retained earnings which would increase the amount of additional funds needed

IV. Under this scenario the company would have a higher level of retained earnings but this would have no effect on the amount of additional funds needed.

V. Under this scenario the company would have a lower level of retained earnings which would reduce the amount of additional funds needed.

AFN Equation

Broussard Skateboard's sales are expected to increase by 15% from $7.4 million in 2016 to $8.51 million in 2017. Its assets totaled $5 million at the end of 2016. Broussard is already at full capacity, so its assets must grow at the same rate as projected sales. At the end of 2016, current liabilities were $1.4 million, consisting of $450,000 of accounts payable, $500,000 of notes payable, and $450,000 of accruals. The after-tax profit margin is forecasted to be 5%. Assume that the company pays no dividends. Under these assumptions, what would be the additional funds needed for the coming year? Do not round intermediate calculations. Round your answer to the nearest dollar.
$

Why is this AFN different from the one when the company pays dividends?
I. Under this scenario the company would have a higher level of retained earnings but this would have no effect on the amount of additional funds needed.
II. Under this scenario the company would have a lower level of retained earnings which would reduce the amount of additional funds needed.
III. Under this scenario the company would have a lower level of retained earnings but this would have no effect on the amount of additional funds needed.
IV. Under this scenario the company would have a higher level of retained earnings which would reduce the amount of additional funds needed.
V. Under this scenario the company would have a higher level of retained earnings which would increase the amount of additional funds needed.
-Select-IIIIIIIVV

Problem 12-03
AFN Equation

Broussard Skateboard's sales are expected to increase by 20% from $7.0 million in 2016 to $8.40 million in 2017. Its assets totaled $5 million at the end of 2016. Broussard is already at full capacity, so its assets must grow at the same rate as projected sales. At the end of 2016, current liabilities were $1.4 million, consisting of $450,000 of accounts payable, $500,000 of notes payable, and $450,000 of accruals. The after-tax profit margin is forecasted to be 4%. Assume that the company pays no dividends. Under these assumptions, what would be the additional funds needed for the coming year? Do not round intermediate calculations. Round your answer to the nearest dollar.
$

Why is this AFN different from the one when the company pays dividends?
I. Under this scenario the company would have a higher level of retained earnings which would increase the amount of additional funds needed.
II. Under this scenario the company would have a higher level of retained earnings but this would have no effect on the amount of additional funds needed.
III. Under this scenario the company would have a lower level of retained earnings which would reduce the amount of additional funds needed.
IV. Under this scenario the company would have a lower level of retained earnings but this would have no effect on the amount of additional funds needed.
V. Under this scenario the company would have a higher level of retained earnings which would reduce the amount of additional funds needed.
-Select-IIIIIIIVV

Broussard Skateboard's sales are expected to increase by 15% from $7.6 million in 2016 to $8.74 million in 2017. Its assets totaled $5 million at the end of 2016. Broussard is already at full capacity, so its assets must grow at the same rate as projected sales. At the end of 2016, current liabilities were $1.4 million, consisting of $450,000 of accounts payable, $500,000 of notes payable, and $450,000 of accruals. The after-tax profit margin is forecasted to be 7%. Assume that the company pays no dividends. Under these assumptions, what would be the additional funds needed for the coming year? Do not round intermediate calculations. Round your answer to the nearest dollar.

_________________$

Why is this AFN different from the one when the company pays dividends?

I. Under this scenario the company would have a higher level of retained earnings which would reduce the amount of additional funds needed.
II. Under this scenario the company would have a higher level of retained earnings which would increase the amount of additional funds needed.
III. Under this scenario the company would have a higher level of retained earnings but this would have no effect on the amount of additional funds needed.
IV. Under this scenario the company would have a lower level of retained earnings which would reduce the amount of additional funds needed.
V. Under this scenario the company would have a lower level of retained earnings but this would have no effect on the amount of additional funds needed.

Carlsbad Corporation's sales are expected to increase from $5 million in 2016 to $6 million in 2017, or by 20%. Its assets totaled $2 million at the end of 2016. Carlsbad is at full capacity, so its assets must grow in proportion to projected sales. At the end of 2016, current liabilities are $1 million, consisting of $250,000 of accounts payable, $500,000 of notes payable, and $250,000 of accrued liabilities. Its profit margin is forecasted to be 4%

a. Assume that the company pays no dividends. Under these assumptions, what would be the additional funds needed for the coming year? Write out your answer completely. For example, 5 million should be entered as 5,000,000. Round your answer to the nearest cent. $

b

Why is this AFN different from the one when the company pays dividends?

I. Under this scenario the company would have a lower level of retained earnings, which would increase the amount of additional funds needed.

II Under this scenario the company would have a lower level of retained earnings, which would decrease the amount of additional funds needed.

III Under this scenario the company would have a higher level of retained earnings, which would reduce the amount of additional funds needed

IV Under this scenario the company would have a higher level of retained earnings, which would reduce the amount of assets needed

v Under this scenario the company would have a higher level of spontaneous liabilities, which would reduce the amount of additional funds needed.

Broussard Skateboard's sales are expected to increase by 20% from $8.0 million in 2016 to $9.60 million in 2017. Its assets totaled $4 million at the end of 2016. Broussard is already at full capacity, so its assets must grow at the same rate as projected sales. At the end of 2016, current liabilities were $1.4 million, consisting of $450,000 of accounts payable, $500,000 of notes payable, and $450,000 of accruals. The after-tax profit margin is forecasted to be 6%. Assume that the company pays no dividends. Under these assumptions, what would be the additional funds needed for the coming year? Do not round intermediate calculations. Round your answer to the nearest dollar.
$

Why is this AFN different from the one when the company pays dividends?
I. Under this scenario the company would have a higher level of retained earnings which would increase the amount of additional funds needed.
II. Under this scenario the company would have a higher level of retained earnings but this would have no effect on the amount of additional funds needed.
III. Under this scenario the company would have a lower level of retained earnings which would reduce the amount of additional funds needed.
IV. Under this scenario the company would have a lower level of retained earnings but this would have no effect on the amount of additional funds needed.
V. Under this scenario the company would have a higher level of retained earnings which would reduce the amount of additional funds needed.

4.) Determine whether the following probability experiment is a binomial experiment. If the probability experiment is not a binomial experiment, state why?

a.) In a town with 400 citizens, 100 randomly selected citizens are asked to identify their religion. The number who identify with a Christian religion is recorded.

b.) An experiment is conducted in which a single die is cast until a 3 comes up. The number of throws required is recorded.