There are no magic formulas or secret sales tips for boosting your sales metrics and performance. Sales are dependent on a myriad of factors lining up in your favor, and many of them are out of the numbers on your sales dashboard. The best you can do is do your best putting in the work.

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The customer acquisition process is a long journey. As any salesperson knows, there are many phases to this journey, and while the close might get all the glory, the most important part of the process may be all the way at the beginning: prospecting.

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Chances are good, few habits need more of an overhaul right now than your sleep routine. You probably picked up a few bad sleep habits during the cold, dark winter and any additional stress you're facing is helping the situation, says Brown University associate professor and medical doctor Katherine Sharkey.

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It is hard to believe that six weeks ago we were all living life as we previously knew it and making plans for the life we thought was ahead.

It has been an exhausting six weeks. It has been a frustrating and angering six weeks.  It has been a scary and anxiety-inducing six weeks. It has been a life- and business-altering six weeks.

And yet, every day — over and over and over again — I have seen small-business owners around the world step up, persist and display the type of character that I am inspired by.

I saw an amazing quote last week. It reminded that character is not built during crisis, but that character is revealed during crisis. How true is that? I look around and I see people who have been revealed as power hungry and self-centered, but I have also seen people who are kind, generous and committed to the greater good. And so many small business owners fit into that category.

Email marketing continues to be one of the most powerful tools for growing your business. (Need proof? Check out stat #1 below.)

And although email marketing has been around for decades, it’s everchanging.

New email marketing tools — like AMP for Email — pop up all of the time. Marketers are taking innovative approaches to email marketing. And email marketing benchmarks change every year.

Want to see how your own email marketing strategy compares and what other businesses are doing to innovate?

Check out this ultimate list of email marketing statistics to find out how you stack up and see email marketing benchmarks to compare your results.

General email marketing statistics and usage
79% of small businesses say email marketing is important to their business strategy. [AWeber]

60% of small businesses say their email marketing strategy is effective or very effective. [AWeber]

64% of B2B marketers use a dedicated email marketing platform. [Content Marketing Institute]

40% of B2B marketers claim that email newsletters are the most important tactic in their content marketing strategy. [Content Marketing Institute]

43% of small businesses have 500 or less email subscribers. [AWeber]

42% of small businesses with over 500 subscribers have effective or very effective email marketing strategies. [AWeber]

46% of emails are opened on mobile devices. [Litmus]

61% of consumers prefer to be contacted by brands through email. [Statista]

On average, consumers spend 2.5 hours checking email on a typical weekday. [Adobe]

Make communication a priority
In response to the restrictions temporarily in place, companies large and small have made changes to the way they do business. Restaurants are offering curbside pickup. Many retailers have closed their brick-and-mortar stores but are ramping up e-commerce with free delivery and 24-hour customer support. Grocery stores have introduced new cleaning protocols and special senior shopping hours.

No matter your industry, be proactive in sharing this information with customers and keeping them updated. In this digital-first era, all types of businesses are much better equipped to reach customers, both existing and prospective. By using multiple platforms — posting on social media, sending mobile messages, and updating your website and directory listings such as Google My Business — your business has the ability to connect with customers quickly and easily.  

Be transparent
We are living through a period of uncertainty in which nearly every American is affected by this pandemic in some way. It’s important to acknowledge that publicly. Practicing sensitivity and transparency in light of our current economic climate is not only appropriate — it’s necessary. Soften the tone in your messaging and infuse empathy in recognition of what’s happening all around us.

For small businesses, resources can be tight. Especially right now.
Limited budgets can present challenges for how businesses raise awareness, acquire customers, and generate revenue.

In fact, 39% of small business owners agree that a limited budget is a major roadblock in growing their businesses.

However, there are plenty of low-cost, high-yielding marketing hacks that can help you reach your customers.
Get featured in press by responding to journalists and podcasters
Personalize cold outreach campaigns
Correct Existing Mentions of Your Brand
Add videos to your landing page
Repurpose old blog content

For small business owners, tidying up your digital space means conducting a virtual sweep of your website and organizations security practices. I recommend that small business owners use the following cybersecurity spring cleaning checklist to ensure they don't miss any important items.

Spring Clean Your Website
The first step of cybersecurity spring cleaning is to deep clean your business website. By clearing out what you do not need, you can improve the overall health of your site.

Start with decluttering plugins and software. You should only keep the ones you're using and delete the rest to reduce your sites cybersecurity risk.

After you remove any plugins and software you are not using, make sure the ones you keep are updated. In addition, continue to proactively monitor your plugins and software on a regular basis and check for security updates.

It's also the perfect time to take a closer look at the data you're gathering from customers. Ask yourself if you are collecting information that truly benefits your business, such as information that drives value for marketing, sales, and services. If the type of information you collect has changed over time, delete any data and applications you're no longer using such as analytics code, remarketing snippets, affiliate tracking, and CRM tracking.

Social distancing. Telemedicine. Self-quarantine. These are all words that at the start of 2020 weren't part of our vocabulary, but several months into the new decade we are all hearing and using them daily. There is no denying that the coronavirus outbreak has dramatically changed just about every facet of just about every person’s life around the world.

From a business perspective, the stock market saw its largest one day loss and largest one day gain in history. The U.S. saw the largest job-loss report ever. We are in uncharted waters, and how long we will remain in them remains uncertain. However, there is one thing that we all know, and that is that this outbreak will change the lives of everyone for years or decades to come. Nearly 20 years after 9-11, enhanced airport security, no-fly lists and counterterrorism efforts are still the norm. The same will be true of the COVID-19 aftermath. Is your business ready for the five largest macro trends we are about to see?

1. The rise of enhanced websites and digital tools
2. Cybersecurity concerns take center stage
3. An increase in virtual meetings
4. Increased control in expenses
5. Even more remote employees

The opportunities to answer questions for potential publication can come from various sources. Help a Reporter, known as HARO, matches journalists with sources for their stories. Quora allows users to answer any question posed by their community. And professional membership organizations offer business leaders the opportunity to share their insights in industry publications.

The opportunities to add your own insights are out there. But why would you, as a business owner, devote your time to answering these questions? What are the benefits?

Here are five ways that answering questions online can help promote your business.

Connections
Search engine optimization (SEO)
Brand
Website traffic
PR opportunities

In the current environment, cybersecurity is essential for businesses of all sizes. Many small or medium-sized businesses find themselves without adequate cybersecurity, either as a result of believing that they don’t need it or simply overlooking it among the many demands that come with running a business.

1. Use protection against ransomware
2. Invest in employee security training
3. Adopt multifactor authentication
4. Use a Security Information and Event Management system
5. Implement effective systems for protecting and monitoring data
6. Have a plan for mobile device security

1. Develop a professional mobile-friendly business website.

Your website must be professional and should provide a rich experience and the feel of your brand to users. Here are a few tips to make your website professional and SEO-friendly:

• Use your business logo and branding on your website.

• Make sure your website is mobile-friendly (more than half of local searches are mobile searches)

• Your website should load in less than three seconds.

• Use a clean, minimal design, and avoid fancy styles.

• Your website must be well structured and easy to use.

Many business owners waste too much money creating a professional small business website. But there is no need to waste too much money on a business website. You can create a professional website for less than $500. So do not waste more money on website design, but invest that extra money in SEO.

2. Identify profitable keywords.

The success of an SEO campaign depends on the targeted keywords. Thats why you need to choose the right and profitable keywords for your small business. All keywords are not equal; some keywords could have high search volume but not profitability, while some could be profitable but have small volumes.

Also, focus on long-tail keywords because they are easy to rank and more profitable than short keywords. The profitability of a keyword depends on the nature of the keyword. For that, you need to understand the intent behind that keyword — why a user is searching that.

For example, when a person searches ice cream on Google, they want to know about ice cream, which means it has informational intent. But when a person searches for best choco-milk ice cream near me, the person wants to eat ice cream. This will be a profitable keyword for you to target.

3. Create a separate page for each product.

I found that many small businesses list all their products on one page like the homepage, but this is not a good practice. If you want to get more profit from your local SEO efforts, then you should create specific pages for each product. That way you can rank higher for each product page for multiple keywords.

For example, if you have a clothing store website, then you need to create separate pages for each product, such as one page for jeans and another for shirts. Also, you can create further subpages, such as jeans> men jeans> blue jeans. This can boost your rankings and revenue and reduce your efforts.

4. Use schema.

Structured data is helping Google better understand webpages. That’s why it can help you rank higher and get the advantage of other SERP features like featured snippets, knowledge graphs, etc. Product schema, local business, FAQ and others are must-use schema types for every business.

Unless Congress extends FFCRA past its December 31 deadline or increases the number of weeks past 12, many parents have used up this time anyway. You can not run your business without employees, and employees will find it very difficult to work without schools. And if they extend FFCRA leave, it may help your employees, but your business needs people to run.

On March 16, Lisa Eskenazi Boyer was working up a sweat with her students one last time at her bustling Queens, New York, fitness studio. Covid-19 lockdown orders were about to take effect, and Simply Fit Astoria — along with all other local gyms — would have to close its doors later that night. She never expected it to be for good.....

March 11, 2020 is a day destined for the history books: WHO Declares Coronavirus Outbreak a Pandemic. It was that day that, all around the world, leaders began scrambling, ripping through the pages of their crisis playbooks (or quickly creating them), searching for their pandemic play-by-play. Shortly after came the day the markets crashed on March 16, turning the crisis to both a health and economic calamity.

Though etched in our minds with great infamy, it’s days like these that I believe make true leaders. Abraham Lincoln, Franklin Delano Roosevelt and Martin Luther King Jr. wouldn’t be the leaders we remember if it were not for the fiery trials that forged their legacies. This is true not only for politicians and activists, but also for business leaders. As president of The UPS Store, a business deemed essential throughout the pandemic, I’ve seen what works (and what does not) when leading through a crisis, and how leaders can turn even a global pandemic into an opportunity.

Take a step back
The lightning pace and innovation of technology in todays world has trained us to think that speedy decisions are good decisions.

The food and beverage industry is a tough game. Sixty percent of restaurants don’t make it past their first year, and 80 percent go out of business within five years. Those are hard odds.

Franchising takes some of the risks out of the equation by giving you a proven model to work with. But being a franchisor with a proven model under your arm does not mean you’re suddenly bulletproof or immune to the laws of economics. If you start making unforced errors, you are going to fail.

Here are the five reasons most people fail as the owner of a franchise. Avoid these deadly sins at all costs:

Sin 1: Financial complacency
Sin 2: Operational obtuseness
Sin 3: Poor hiring choices
Sin 4: Myopic risk management
Sin 5: Mediocre offerings

1. Focus on the positive
2. Hit the reset button
3. Never give up
4. Network

Digital marketers often identify social media as one of the best forms of content marketing, but it can often feel like we’re just going through the motions. If the social media content isn’t attracting leads, what good is it? It’s likely you just need a quick boost in strategy to make sure your content is appealing to your target audience and getting inbound requests and messages.

In fact, 90 percent of social media users have used the platform to communicate directly with a business before. So if none of your customers or followers are reaching out to you, it’s a telltale sign that something should be changed. Ideally, you’ll post a picture or video with a robust caption that offers value and the floodgates will open: direct messages, likes, comments and queries should start coming (or even just trickling at first) in, proving that your content struck a chord and inspired action. Not there yet? Here are four ways to bolster your strategy to attract those leads.

1. Focus your content on interesting stories
2. Do a poll asking what type of content people want most
3. Host a Q & A on Facebook or Instagram Live
4. Make sure you have a call to action in every post

Traditional tax planning is transactional and, honestly, not nearly as beneficial as one might think. You ask your taxes preparer questions and figure out what to do in the spur of the moment. Creating a long-term plan of action for your taxes is how to create real savings, but it takes months to create an effective plan. Now is the time for business owners and investors to be planning to reap the rewards for the rest of 2020 and into 2021.

Analyze income
Many accountants suggest pushing income to a later year. There are a few different factors to consider when deciding whether to do this. First, is your income so low you lose deductions? Many personal deductions don’t carry over to the next year. Rather than taking deductions now, you may want to accelerate your income to make use of all your deductions. Another factor to consider is the next year’s tax rates. There’s a real chance that income tax rates could increase in 2021, so the best plan would be to accelerate your income into 2020 to avoid paying at a higher rate.

While major corporations furloughing workers and declaring bankruptcy tends to get the biggest headlines, our culture's dramatic shift to working from home is the true breakout business story from this pandemic. The transition has certainly had its share of ups and downs, but rapidly growing acceptance indicates this is a trend that is almost certainly going to shape the future of work.

The transition began before 2020
While Covid-19 restrictions caused an abrupt shift, working from home was already accelerating. Research from FlexJobs found that the number of people in the United States who worked from home grew by an astounding 159 percent between 2005 and 2017.

Much of this growth can be attributed to freelancing. Upworks Freelancing in America 2019 survey found that the number of Americans who did freelance work grew from 53 million to 57 million between 2014 and 2019. Younger generations were especially likely to participate, with 40 percent of millennials and 53 percent of Generation Z contributing to the gig economy.

The former vice president won the top job, but undoing Trumps discordant legacy will not be easy.

Many of the elements of Trumps biggest legislative achievement--the Tax Cuts and Jobs Act (TCJA) of 2017--are set to expire in the next few years. To name two: the research and development tax credit, which in 2022 requires businesses to amortize expenses over five years rather than doing so immediately, as is the current case. And full expensing for short-term business investments will begin phasing out in 2023.

News of the vaccine also raises questions for business owners: Can you require employees to be vaccinated, and if so, how do you go about it?

Employment lawyers and HR professionals say that policies regarding the flu vaccine are a good place to start. Many states mandate that hospital workers and other health care professionals, as well as school children and preschoolers in daycare, get flu shots and other vaccines. But it's not required for most professions. Generally, employers can require a flu vaccination, but an employee may be entitled to an exemption if he or she has a particular disability that needs to be accommodated, or a sincerely-held religious objection to taking the vaccine, says Michael Schmidt, a New York-based employment lawyer for Cozen O'Connor.

In both cases, the employer may have to pay for the vaccine or reasonable accommodation. If you refuse to make accommodations for an anti-vaxxer, it's possible to face a claim for violating the Americans with Disabilities Act (ADA), the Civil Rights Act, or the Occupational Safety and Health Administration's whistleblower protection program. Most of the time, Schmidt says, the advice is for employers to encourage employees to get a flu shot rather than try to create a policy that demands it. However, he notes, many would argue that the Covid-19 situation is far more threatening than the flu is at this point, meaning a vaccine may be more crucial to the overall health of a workplace.

Whether your cell phone has a broken screen, is no longer the latest technology, or you are simply itching for the latest cell phone model, it is important that you take appropriate care in disposing of your old cell phone.

Retain and Protect Your Data
Cell phones are a treasure trove of information; think of the panic that strikes each time you mistakenly misplace your cell phone. The information contained in our cell phones is meaningful and it is important that you backup any data on the phone that you wish to retain. The data you backup would include: any music, photos, messages etc...  If you are moving to a new cell phone you can often transfer the data you wish to retain to the new cell device.

Protect Your Personal Information
Cell phones often contain sensitive personal data, including passwords, bank information, payment methods, messages, contacts, videos, account numbers and much more. After transferring the data to a new device, information is still retained on your original cell phone.  Providing this type of sensitive information to strangers could put you at risk for identity theft, other other types of fraud, or simply leave you very uncomfortable.

How to Dispose of Old Cell Phones; Protect Your Data and Privacy

1. Be clear in your direction.
2. Know who your customer is.
3. Focus on the channels that matter.
4. Only talk about what matters.
5. Remember what makes you special.
6. Find your niche.
7. Learn something new.
8. Plan well to save time.
9. Engage with your customers more.

Arguably the most critical aspect of digital transformation, digital payments ensure that small merchants or kiranas can continue to accept or send money and preserve cash flow in an increasingly contactless world.

For small businesses,traditional marketing can be expensive and difficult to maintain. Inbound marketing can level the playing field and give even the smallest business a chance to stand out and grow.

Why use inbound marketing?

1. It is cost effective
2. It helps build customers trust.
3. It increases brand awareness and boosts your online presence.
4. It can improve your marketing decision making
5. You can craft customer-focused content.
6. Inbound marketing provides two-way communication.
7. It helps bring in organic traffic to your website

The Covid-19 pandemic caused an unprecedented financial ripple effect in nearly every industry but hit the small businesses that define America the hardest. Throughout the first six months of the pandemic, more than 60 million Americans filed for unemployment insurance. That’s 23 million more than the 37 million who filed claims during the 18-month Great Recession.

By now, many small-business owners who made the difficult decision to shrink or temporarily pause are rebuilding. As they have already learned, though, rebuilding your business is not as easy as flipping a switch and watching your business rebound to its pre-Covid-19 state. As a small-business owner, your plan to rebuild should focus on rehiring employees who can fulfill your immediate needs while simultaneously paving the groundwork for growth in the new norm.

The accelerated churn rate of the S&P 500 indicates that at least half of todays top U.S. companies will get replaced by someone new over the next decade. That is a mind-boggling market value of $13.5 trillion up for grabs. And the craziest part is who replaces the old market leaders: It is often companies that, just a few years before, were considered scrappy little startups.

To unseat a champion, a smaller company has to play by a completely different set of rules.

1. Change the basis of competition.
2. Exploit taboos.
3. Optimize for power.
4. Dramatic simplification.

With the tax filing deadline approaching, make sure your company is getting all the assistance available from government programs. For instance, that means checking that you've fully utilized the Employee Retention Credit (ERC), the refundable tax credit designed to make it easier for businesses to keep employees on the payroll.

The credit is getting extended as part of the American Rescue Plan Act, the $1.9 trillion relief package just signed by President Biden. Originally scheduled to end on June 30, ERC will continue through year end, giving business owners access to as much as $33,000 per employee in incentives.

How the credit works, depending on the time frame
First half of 2021:

Eligible employers can claim a refundable credit against the employer share of Social Security tax equal to 70 percent of a full-time employee's qualified wages paid--including certain health plan expenses--from January 1 through June 30, 2021. The maximum ERC amount available is $7,000 per employee per quarter or $14,000 for eligible wages paid in the first half of 2021.

Title: Anxiety, Stress, Worry, and Your Body
Category: Slideshows
Created: 8/22/2018 12:00:00 AM
Last Editorial Review: 1/12/2022 12:00:00 AM

Title: Can You Go Back to Normal After Psychosis?
Category: Diseases and Conditions
Created: 6/28/2022 12:00:00 AM
Last Editorial Review: 6/28/2022 12:00:00 AM

Title: Can You Get Yourself Back to Normal After Psychosis?
Category: Diseases and Conditions
Created: 7/5/2022 12:00:00 AM
Last Editorial Review: 7/5/2022 12:00:00 AM

The rumours are true. Well, the good ones are, anyway. Netflix is delighted and thrilled that so many of you, all over the world, have been watching and loving Sandman, which means that the thing we were all hoping would happen...?

It's happened.

And that's not all! You dared to Dream (and, y'know, kept asking me when and whether they were ever going to show up). And it's happening! The Sandman profile icons are coming to Netflix! Let joy be unconfined!

(I'm going to be Goldie. No, Matthew. No, Goldie.)

Title: See How Your Life Affects Your Skin
Category: Slideshows
Created: 5/30/2012 12:00:00 AM
Last Editorial Review: 7/13/2022 12:00:00 AM

About Prime Air. For over 25 years, Prime Air has been a leading worldwide supplier of quality aircraft components and services for all civil aircraft and engine platforms. Prime Air was acquired by HEICO Aerospace in 2006 and functions as the Asset Management Specialist within the HEICO Repair Group family. Loading... Certifications. ASA-100 and FAA 00-56B. ISO9001:2015 and AS9120B. CONTACT ...

The Great War is a channel on YouTube that covers the 1914-18 conflict that engulfed a large part of the planet.  A unique documentary with a massive scale that is filled with video clips and still images taken at the...

Once McDonald’s fries were something that plenty of people enjoyed, even if they were a bit dubious about the hamburgers, to say nothing of the McRib (if there ever was an item on the franchise’s menu that the term “mystery meat” could apply to, that’s the one), which apparently will soon be coming back to an Arches outlet near you.

But last week Lamb Weston, the largest french fry producer in the U.S., announced it has closed a plant in Washington state because people are opting for small fries, not medium or large.

According to McDonald’s:

“Everyone wants to know why McDonald’s French Fries taste so good—it’s a simple answer. McDonald’s World Famous Fries® are made with premium potatoes such as the Russet Burbank and the Shepody. With 0g of trans fat per labeled serving, these epic fries are crispy and golden on the outside and fluffy on the inside.”

Evidently the notation about the lack of trans fats isn’t enough to convince people that the 480 calories of a large order is something they want to eat, premium potatoes notwithstanding. (A bag of small fries is 230 calories.)

Time change, and with it things that were once familiar.

Read more at Glorious Noise...

Inclined Bed Therapy or IBT is the brain child of Andrew Fletcher, who discovered in the 1990's that gravity actually helps to drive circulation of the sap in trees. From there, it was a short step to ask whether this was also true for animals and humans. This posed the question "why on Earth do people sleep flat?" So Andrew suggested that people slightly raise the head end of their bed and see if any changes in their health are noticeable. This is one of the numerous anecdotes ... stories of personal success people have reported after raising the head end of their bed by just five or six inches. "Over two years ago I sat in the armchair reading a small advert which asked people to raise their bed by six inches at the head and to reply and tell what benefits had been noted. (No explanation was given) At the time I could not move my neck to my left or right side and it ached continuously. I was unable to sleep at night as i could not get comfortable. I was only able to turn by gently easing myself. It took about three to four turns. Getting out of bed was a major obstacle. I needed help to dress and undress. I spent most of my nights in the chair with the result that I was always tired and had no energy. My problem is osteoporosis of the upper and lower spine. I had tried hormone replacement therapy and wasted a small fortune with bone specialists and osteopaths. I was resigned to living my days out as best I could, having been told that there was nothing more that could be done for me. I expected nothing but had nothing to lose, so Harry raised the bed by six inches. We did not take it very seriously but were happy to try anything. On the fourth night I had the first full nights sleep since I don't remember when. By the end of the week I was sleeping naturally and turning over with ease. My dressing was a problem no longer, each day it became easier. I was able to turn my head without pain, right or left, to see the clock without getting up from my chair. There have been many other benefits too. I have worn glasses from the age of seven years and I am now sixty eight years. Last year was the first time I was told that there was a small improvement. My hair appears thicker, my hair brush needs cleaning less often. Harry had a large suppurating scar since he was six years old. He has had to continually dress it all of his life. But now it has healed up. His ear which constantly gave him trouble with a discharge has now cleared up completely. We both feel that the clock has been put back for us!" Ruby, 2nd April 1998 Other such stories can be found on the Inclined Bed Therapy website and on the facebook page with the same name...

Note to Curt:

Just because the state claims the authority to apprehend and punish rapists doesn’t mean that apprehending and punishing rapists is a form of state coercion. Nor is the notion that rape is bad an example of state coercion. Depending on your perspective, this is either a moral truth derived from God/reason/whatever or a widely-accepted social convention. Similarly, the notion that one can own property is (again, depending on your perspective) either morally necessary or a widely-accepted social convention that seems to work pretty well (here I’m dispensing with Communists and other fools who have nothing intelligent to say on the matter). Either way, the fact that the state claims ultimate authority to adjudicate property disputes does not make private property a form of state coercion. (Further reading)

“Young man, in mathematics you don’t understand things, you just get used to them.” —John von Neumann¹

This, in a sense, is at the heart of why mathematics is so hard. Math is all about abstraction, about generalizing the stuff you can get a sense of to apply to crazy situations about which you otherwise have no insight whatsoever. Take, for example, one way of understanding the manifold structure on SO(3), the special orthogonal group on 3-space. In order to explain what I’m talking about, I’ll have to give several definitions and explanations and each, to a greater or lesser extent, illustrates both my point about abstraction and von Neumann’s point about getting used to things.

First off, SO(3) has a purely algebraic definition as the set of all real (that is to say, the entries are real numbers) 3 × 3 matrices A with the property A^TA = I and the determinant of A is 1. That is, if you take A and flip rows and columns, you get the transpose of A, denoted A^T; if you then multiply this transpose by A, you get the identity matrix I. The determinant has its own complicated algebraic definition (the unique alternating, multilinear functional…), but it’s easy to compute for small matrices and can be intuitively understood as a measure of how much the matrix “stretches” vectors. Now, as with all algebraic definitions, this is a bit abstruse; also, as is unfortunately all too common in mathematics, I’ve presented all the material slightly backwards.

This is natural, because it seems obvious that the first thing to do in any explication is to define what you’re talking about, but, in reality, the best thing to do in almost every case is to first explain what the things you’re talking about (in this case, special orthogonal matrices) really are and why we should care about them, and only then give the technical definition. In this case, special orthogonal matrices are “really” the set of all rotations of plain ol’ 3 dimensional space that leave the origin fixed (another way to think of this is as the set of linear transformations that preserve length and orientation; if I apply a special orthogonal transformation to you, you’ll still be the same height and width and you won’t have been flipped into a “mirror image”). Obviously, this is a handy thing to have a grasp on and this is why we care about special orthogonal matrices. In order to deal with such things rigorously it’s important to have the algebraic definition, but as far as understanding goes, you need to have the picture of rotations of 3 space in your head.

Okay, so I’ve explained part of the sentence in the first paragraph where I started throwing around arcane terminology, but there’s a bit more to clear up; specifically, what the hell is a “manifold”, anyway? Well, in this case I’m talking about differentiable (as opposed to topological) manifolds, but I don’t imagine that explanation helps. In order to understand what a manifold is, it’s very important to have the right picture in your head, because the technical definition is about ten times worse than the special orthogonal definition, but the basic idea is probably even simpler. The intuitive picture is that of a smooth surface. For example, the surface of a sphere is a nice 2-dimensional manifold. So is the surface of a donut, or a saddle, or an idealized version of the rolling hills of your favorite pastoral scene. Slightly more abstractly, think of a rubber sheet stretched and twisted into any configuration you like so long as there are no holes, tears, creases, black holes or sharp corners.

In order to rigorize this idea, the important thing to notice about all these surfaces is that, if you’re a small enough ant living on one of these surfaces, it looks indistinguishable from a flat plane. This is something we can all immediately understand, given that we live on an oblate spheroid that, because it’s so much bigger than we are, looks flat to us. In fact, this is very nearly the precise definition of a manifold, which basically says that a manifold is a topological space (read: set of points with some important, but largely technical, properties) where, at any point in the space, there is some neighborhood that looks identical to “flat” euclidean space; a 2-dimensional manifold is one that looks locally like a plane, a 3-dimensional manifold is one that looks locally like normal 3-dimensional space, a 4-dimensional manifold is one that looks locally like normal 4-dimensional space, and so on.

In fact, these spaces look so much like normal space that we can do calculus on them, which is why the subject concerned with manifolds is called “differential geometry”. Again, the reason why we would want to do calculus on spaces that look a lot like normal space but aren’t is obvious: if we live on a sphere (as we basically do), we’d like to be able to figure out how to, e.g., minimize our distance travelled (and, thereby, fuel consumed and time spent in transit) when flying from Denver to London, which is the sort of thing for which calculus is an excellent tool that gives good answers; unfortunately, since the Earth isn’t flat, we can’t use regular old freshman calculus.² As it turns out, there are all kinds of applications of this stuff, from relatively simple engineering to theoretical physics.

So, anyway, the point is that manifolds look, at least locally, like plain vanilla euclidean space. Of course, even the notion of “plain vanilla euclidean space” is an abstraction beyond what we can really visualize for dimensions higher than three, but this is exactly the sort of thing von Neumann was talking about: you can’t really visualize 10 dimensional space, but you “know” that it looks pretty much like regular 3 dimensional space with 7 more axes thrown in at, to quote Douglas Adams, “right angles to reality”.

Okay, so the claim is that SO(3), our set of special orthogonal matrices, is a 3-dimensional manifold. On the face of it, it might be surprising that the set of rotations of three space should itself look anything like three space. On the other hand, this sort of makes sense: consider a single vector (say of unit length, though it doesn’t really matter) based at the origin and then apply every possible rotation to it. This will give us a set of vectors based at the origin, all of length 1 and pointing any which way you please. In fact, if you look just at the heads of all the vectors, you’re just talking about a sphere of radius 1 centered at the origin. So, in a sense, the special orthognal matrices look like a sphere. This is both right and wrong; the special orthogonal matrices do look a lot like a sphere, but like a 3-sphere (that is, a sphere living in four dimensions), not a 2-sphere (i.e., what we usually call a “sphere”).

In fact, locally SO(3) looks almost exactly like a 3-sphere; globally, however, it’s a different story. In fact, SO(3) looks globally like , which requires one more excursion into the realm of abstraction. , or real projective 3-space, is an abstract space where we’ve taken regular 3-space and added a “plane at infinity”. This sounds slightly wacky, but it’s a generalization of what’s called the projective plane, which is basically the same thing but in a lower dimension. To get the projective plane, we add a “line at infinity” rather than a plane, and the space has this funny property that if you walk through the line at infinity, you get flipped into your mirror image; if you were right-handed, you come out the other side left-handed (and on the “other end” of the plane). But not to worry, if you walk across the infinity line again, you get flipped back to normal.

Okay, sounds interesting, but how do we visualize such a thing? Well, the “line at infinity” thing is good, but infinity is pretty hard to visualize, too. Instead we think about twisting the sphere in a funny way:

You can construct the projective plane as follows: take a sphere. Imagine taking a point on the sphere, and its antipodal point, and pulling them together to meet somewhere inside the sphere. Now do it with another pair of points, but make sure they meet somewhere else. Do this with every single point on the sphere, each point and its antipodal point meeting each other but meeting no other points. It’s a weird, collapsed sphere that can’t properly live in three dimensions, but I imagine it as looking a bit like a seashell, all curled up on itself. And pink.

This gives you the real projective plane, . If you do the same thing, but with a 3-sphere (again, remember that this is the sphere living in four dimensions), you get . Of course, you can’t even really visualize or, for that matter, a 3-sphere, so really visualizing is going to be out of the question, but we have a pretty good idea, at least by analogy, of what it is. This is, as von Neumann indicates, one of those things you “just get used to”.

Now, as it turns out, if you do the math, SO(3) and look the same in a very precise sense (specifically, they’re diffeomorphic). On the face of it, of course, this is patently absurd, but if you have the right picture in mind, this is the sort of thing you might have guessed. The basic idea behind the proof linked above is that we can visualize 3-space as living inside 4-space (where it makes sense to talk about multiplication); here, a rotation (remember, that’s all the special orthogonal matrices/transformations really are) is just like conjugating by a point on the sphere. And certainly conjugating by a point is the same as conjugating by its antipodal point, since the minus signs will cancel eachother in the latter case. But this is exactly how we visualized , as the points on the sphere with antipodal points identified!

I’m guessing that most of the above doesn’t make a whole lot of sense, but I would urge you to heed von Neumann’s advice: don’t necessarily try to “understand” it so much as just to “get used to it”; the understanding can only come after you’ve gotten used to the concepts and, most importantly, the pictures. Which was really, I suspect, von Neumann’s point, anyway: of course we can understand things in mathematics, but we can only understand them after we suspend our disbelief and allow ourselves to get used to them. And, of course, make good pictures.

² As a side note, calculus itself is a prime example of mathematical abstraction. The problem with the world is that most of the stuff in it isn’t straight. If it were, we could have basically stopped after the Greeks figured out a fair amount of geometry. And, even worse, not only is non-straight stuff (like, for example, a graph of the position of a falling rock plotted against time) all over the place, but it’s hard to get a handle on. So, instead of just giving up and going home, we approximate the curvy stuff in the world with straight lines, which we have a good grasp of. As long as we’re dealing with stuff that’s curvy (rather than, say, broken into pieces) this actually works out pretty well and, once you get used to it all, it’s easy to forget what the whole point was, anyway (this, I suspect, is the main reason calculus instruction is so uniformly bad; approximating curvy stuff with straight lines works so well that those who who are supposed to teach the process lose sight of what’s really going on).

Over at Catallarchy, Micha Ghertner discusses “How To Tell You’ve Won An Argument;” namely, when your opponent concedes that his position is less coherent than your own, you’ve won. Now, I don’t want to dispute his point, but rather to question how relevant it is. I’ve touched on this before, but I’m a bit dubious of the notion that the “correct” position is the one that wins arguments between advocates of two different positions.

Obviously, in the first place, there’s nothing to prevent both arguers from being wrong; the relative lack of coherence of one of their positions means, at best, that the other’s position is “less” wrong (assuming that even makes sense and assuming that coherence is a measure of correctness).¹ But this is somewhat superficial (and besides, already mentioned and acknowledged in the comments to Ghertner’s post); more importantly, I want to cast doubts upon the parenthetical assumption I made above, that coherence is some sort of infallible metric for measuring correctness/validity.

In fact, Ghertner (perhaps unconsciously) alludes to this very issue when he quotes Wittgenstein’s famous seventh proposition from the Tractatus: “Whereof one cannot speak, thereof one must be silent.” Within the context of the Tractatus (as an attempt to construct or at least describe a perfect language), this supports the notion that being right and being coherent are synonymous, but Wittgenstein himself later rejects this perspective and, to me, the more apropos quotation is: “Explanations come to an end somewhere” (Philosophical Investigations, I§1). That is, no argument (and certainly none about abstract principles) is completely coherent; we always run up against that whereof we cannot speak and therefore must be silent. The question is simply at what stage in the investigation we enter the realm of unsupported assertion.

And even if we scale back our expectations and choose to embrace the position that manages to maintain coherence as far back as possible, there’s still no guarantee that we’re on the right track. Although much of the world can apparently be explained without the need to stipulate a deity, this doesn’t really make it any less likely that theism is right. In the words of Chuck Klosterman:

Math [or, perhaps more fittingly in this context, logic] is the antireligion, because it splinters the gravity of life’s only imperative equation: Either something is true, or it isn’t.

In fact, if we really want to get all Wittgensteinian about this (not that we necessarily should), we might even begin to question those positions which do appear to be coherent:

In the actual use of expressions we make detours, we go by side roads. We see the straight highway before us, but of course we cannot use it, because it is permanently closed. (PI, I§426)

Anyway, getting back to whatever semblance of a point I was trying to make, when someone admits that their position is incoherent, that does indeed mean that they’ve lost the argument, but I just wonder how important that really is. Giving up your high-paying job and live-in girlfriend to go back home and take care of your sick mother isn’t going to win a lot of arguments if we’re taking logical coherence as the criterion of victory (seriously, think about it), but that doesn’t mean it’s not the right thing to do. That doesn’t mean that coherence is totally irrelevant to what is right/correct, either (and, I should point out, in the above example helping your sick mom isn’t necessarily the right thing to do; as is almost always true, it depends on the circumstances), but let’s not give argument-winning more importance than it merits. Or, as some smarmy new-age intellectual might put it, in the pursuit of knowledge, our goal shouldn’t be to win arguments, but, rather, to discover truth.

The law of the excluded middle says here: It must either look like this, or like that. So it really—and this is a truism—says nothing at all, but gives us a picture. And the problem ought now to be: does reality accord with the picture or not? And this picture seems to determine what we have to do, what to look for, and how—but it does not do so, just because we do not know how it is to be applied. Here saying “There is no third possibility” or “But there can’t be a third possibility!”—expresses our inability to turn our eyes away from this picture: a picture which looks as if it must already contain both the problem and its solution, while all the time we feel that it is not so. (PI I§352)

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The Movie:

I think both in fictional movies and documentaries, when airplane trouble happens when a musical act is one of the passengers on the plane, the comedy to diffuse the tension is palpable, whether it is a tribute to "The Night the Music Died" when the Big Bopper, Richie Valens and Buddy Holly passed, or when members of Lynyrd Skynrd died in a 1977 plane crash. Neverthless, I'll Never Forget You attempts to put some emotion into the tragic event.

The film is based on the novel by Gene Odom, friend of singer Ronnie Van Zandt and who wrote the book that serves as the foundation for the film. In it, he, along with others that survived the crash (backup singer Leslie Hawkins and guitar roadie Craig Reed) as they share their thoughts on the crash and offer some thoughts on the days and hours leading up to the tragic events.

So when it comes to the film itself, the sto...Read the entire review

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