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Three Questions To Ask Your Lawyer Before Starting A Small Business

Gaining customers. Producing a quality product. Finding new markets. These are the things that small business owners focus on when starting their latest venture. But while entrepreneurs are busy planning for the future, many fail to prepare for the worst.

Businesses large and small are loaded with risk. Liability is—or should be—a chief concern. Too many entrepreneurs, however, overlook the need to plan for and manage risk when launching their businesses.

complete article




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10 Business Development Techniques to Grow Sales Fast

Time to crank up sales. Thankfully, the economy is starting to heat up, and as such, it may be an ideal time to learn how to crank up your sales.

complete article




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Quick Guide to Small Business Health Insurance in 2015

A major trend this year among small employers is to drop group health insurance coverage to help their employees pay for individual health insurance policies. By switching their employees to the individual health insurance marketplace, the small employers are ensuring that their employees get to choose the coverage that is best for their families. In addition, by not offering a group health insurance policy, the employers are making sure their employees have access to premium tax credits to assist them with the cost of their premiums.

complete article




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How To Quickly Master Social Media Marketing For Small Business

According to a new study, one in six marriages that began in the last three years has resulted from an interaction that started online. At the same time, Nielsen also reports that four out of every five corporations in America are now leveraging social media to help expand their client base and build relationships with customers.

complete article




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9 Things Managers Do That Make Good Employees Quit

Managers tend to blame their turnover problems on everything under the sun, while ignoring the crux of the matter: people do not leave jobs; they leave managers.

complete article




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7 Questions to Ask Before Designing a Logo

Attempting to be creative when it will not come to you naturally can be an excercise in futility. Traditional means of conjuring those creative impulses, like taking a walk in the park, can often spur creativity, but getting a better idea of what a business needs can be an even more effective way to divine logo inspiration. When working with a business, it is important to gain a sense of what the business is about and what role the logo will play in the clients business and marketing plans. To make this process simpler, we have compiled 7 questions to consider asking a client before commencing logo design:

7 Questions to Ask Before Designing a Logo




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15 Qualities You Need to Develop Mental Toughness

We all reach critical points in our lives where our mental toughness is tested. It might be because of a toxic friend or colleague, a dead-end job, or a struggling relationship.

Whatever the challenge, you have to be strong, see things through a new lens, and take decisive action if you want to move through it successfully.

It sounds easy. We all want good friends, good jobs, and good relationships.

complete article




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9 Things Managers Do That Make Good Employees Quit

It is pretty incredible how often you hear managers complaining about their best employees leaving, and they really do have something to complain about—few things are as costly and disruptive as good people walking out the door.

Managers tend to blame their turnover problems on everything under the sun, while ignoring the crux of the matter: people don’t leave jobs; they leave managers.

complete article




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4 Quick End of Year Legal Tips for Small Businesses

To make lives easier, Moore offers the following legal tips to help small and medium-sized business owners.

complete article




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11 Cyber Security Questions Every Small Business Should Ask

The big question therefore is: are you prepared for a cyber attack? Small business credit provider Headway Capital, has compiled data to help you check yourself.

complete article




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Square Expands Lending To Reach Small Business

Last summer, Square began offering loans to non-Square merchants with the help of a restaurant software partner called Upserve. Now, the payments and financial services company has created a formal partnership program with the long-term goal of reaching millions more.
Square Capital is launching the partnership platform with BigCommerce, an e-commerce software company that is letting Square make loan offers to its tens of thousands of small business clients.

Square is also extending loan offers to restaurants that work with its own delivery service, Caviar, but which do not already process payments through Square.

Taken together, these new partnerships give Jack Dorseys company a way to jumpstart further expansion of its lending program, which has already extended $1.8 billion in funds to more than 140,000 businesses since its launch three-plus years ago.

complete article




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Feeling Stuck? Here Are 34 Quotes on Creativity from the Worlds Most Inspirational Leaders.

Every successful leader or entrepreneur has used creativity to help them get to where they are today. In fact, creativity is a core component to anyones success. Take a look at J.K. Rowling -- using creativity and imagination, she created a successful franchise full of witches and wizards. Or how about Steve Jobs? Without creatively thinking about the simplest, sleekest way to create products for consumers, Apple would have never been born.

complete article




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10 Pro Techniques to Improve Your Business Blogging

Use a Content Scorecard to Publish Better Blog Posts
When it comes to creating blog content for your business, quality is just as important as quantity, if not more so. To make sure all of your posts are up to professional standards, use the content scorecard included in this Social Media Examiner post by Garrett Moon.

Choose the Right Keywords for Your Blog Niche
If you want readers to be able to organically find our blog content, you need to include some relevant keywords. Choosing those keywords can be a challenge, so check out the recent Abtech Blog post by Abasifreke Etop for some useful tips.

complete article




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Employees Are 32 Percent Less Likely to Quit if They Get This 1 Thing From Their Boss

There are a lot of reasons to hate a job: annoying co-workers; bad working conditions; long hours for too little pay; the stress level; and more.

But one of the worst things -- arguably the worst -- is having a bad boss.

When you don't feel seen, appreciated, or listened to by your direct manager, it can have a dramatic impact on not only your work performance, but also your sense of self-esteem. Over time, you can begin to feel anxious, worthless, and demoralized.

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17 Powerfully Inspiring (and Funny) Quotes for Every Leader

Being a leader can be a difficult, and sometimes thankless, job. You work long hours, make tough decisions, and try your best to make your employees and your customers happy. But things don't always work out the way you hope.

Every leader can use a bit of inspiration and humor every once in a while. Adding some fun to your day can make it all come together and, at the same time, perhaps even put a smile on your face.

The next time you feel down, here are 17 funny leadership quotes to perk you up.

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Small Business Is Suffering A Squeeze

A Small Business can be identified as a being a privately-owned corporation, partnership, or sole proprietor that have fewer staff and less annual revenue than a regular-sized corporation. The European Union (EU) defines a small business as one with under 50 employees.

Small business or as they are sometimes known, small and medium-sized enterprises (SMEs) contribute 47% of revenue to the U.K. economy. They have a key role in boosting productivity but need support to expand activities and increase their economic impact.

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7 Signs It is Time to Quit Your Business

If time is the only resource that we cannot acquire, then spending your days trying to run a business you no longer love -- or that does not serve your needs -- is one of the biggest ways to squander our lives. With that in mind, here are the seven signals and truths that indicate it might be time for you to find something else to do with your precious life span:

1. Your dreams have stopped.
Remember back when you were so excited about your business? You went to bed thinking about it. Brainstorms popped into your head at the dinner table. You told everyone who would hold still to listen. Maybe you dreamed you would take over or revolutionize your industry or imagined the lifestyle this little baby company of yours would grow up to afford you. If now it is  just about making it through the day -- or even more stressful, making it through payroll -- it could be time to consider doing something else.

complete article




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50 Motivational Quotes From Inspiring Women Leaders

The climb to the top can be tough going for anyone. But as women, there are particular hurdles that can make the journey upward toward positions of power and influence -- in myriad spheres -- especially challenging.

Here, in 50 inspiring quotes, businesswomen, role models, activists, entertainers, authors and politicians share their thoughts on leadership and success -- and what exactly those words mean to them.

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If You Say Yes to Any of These 6 Questions, Your Emotional Intelligence May Be Higher Than Most People

The World Economic Forum recently uncovered something that would not have registered on the radar screen of most HR leaders even ten years ago.

One of the top 10 job skills required for workers to thrive--a skill projected to trend in the year 2022--is emotional intelligence.

Emotional intelligence (EQ) has embedded itself in the business lexicon as a force to be reckoned with; it is by far one of the most desired qualities for personal and professional development.

Managers are hiring workers with more right-brained skills like EQ because they know these people contribute to the workplace on a relational and interpersonal level that is unmatched.

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How answering questions helps promote your business

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




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5 Interview Questions for Any Prospective Employee

1. A question that delves into past experiences within working cultures and extrapolates where future environments may/may not align. For example: Creating the right culture is important for our company. Can you share details on a time you were in a toxic work environment, and how did management handle the situation? Describe when you were in a wonderful work environment and what management did correctly.

2. Deciphering how the candidates prefer to be managed as a key to ensuring management-style alignment. For example: Taking guidance from leaders is a critical part of any working arrangement. Describe how you like to be managed and how would you describe the ideal leader for your personality style?

3. Understanding motivations and expectations for long-term engagement. For example: We believe in long-term retention, and so we are cautious on the type of candidate we onboard. If you were going to commit to helping build this company long-term, what would it take for you to be happy and motivated?

4. Perspectives on their self-awareness of personal growth objectives and needs. For example: We all have challenges we are working to overcome. If you had to assess your challenges, where would be areas that you would focus to develop and look for support to improve?


5. Communication competence and tolerance for different interaction styles. For example: We all have different communication and coordination styles. What is the method you most effectively consolidate new information to inform your work, and what is your preferred communication style to align with your leadership and stakeholders?




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A Covid-19 Vaccine Looks Promising. Can You Require Employees to Take It?

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.




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An Entrepreneurs Quick Guide to Invoice Financing for Small Businesses

Invoice financing is a type of business funding wherein the business sells its outstanding invoices or account receivables (A/R) to financing companies to get an immediate cash flow boost. The financing company takes over the invoices, and sometimes be in charge of collecting customer payments (as in invoice factoring).

Invoice financing is a popular financing option for businesses that have to wait 30,60, or 90 days to get their clients payments.




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quinine

Title: quinine
Category: Medications
Created: 5/14/2022 12:00:00 AM
Last Editorial Review: 5/14/2022 12:00:00 AM




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Strings for the Deaf, The String Quartet Tribute to Queens of the Stone Age

Compulsive purchases are so often wrong that I nearly put this one back on the shelf. I'm glad I didn't....




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Europe Wireless Headphones Market Expands as Demand for High-Quality Audio and Mobility Grows, as per Maximize Market Research.

(EMAILWIRE.COM, November 01, 2024 ) The Europe Wireless Headphones market is experiencing substantial growth, driven by rising consumer preference for high-quality audio and seamless mobility. With advancements in noise-canceling technology, battery life, and Bluetooth connectivity, wireless headphones...




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Quantum Communication Market Projected to Reach $5.54 Billion by 2030

(EMAILWIRE.COM, November 08, 2024 ) The global quantum communication market size is projected to grow from USD 0.74 billion in 2024 to USD 5.54 billion by 2030 at a Compound Annual Growth Rate (CAGR) of 39.6% during the forecast period. Download PDF Brochure@ https://www.marketsandmarkets.com/pdfdownloadNew.asp?id=143942501&utm_source=emailwire.com&utm_medium=paidpr&utm_campaign=quantum-communication-market Increasing...




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***** Quotation - Prime Aviation (rank 11)

Prime Aviation. Charter of your choice. Login; empty; About Us; VIP; empty; Request a Quote; Air Ambulance




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***** SMBC Aviation Capital :: SMBC Aviation Capital to acquire Goshawk ... (rank 13)

Goshawk Aviation Limited is a top ten global aircraft lessor with an owned, managed and committed fleet of 216 aircraft. Formed in 2013, the business has quickly grown to become one of the premier global aircraft leasing companies building a high-quality portfolio of young new technology aircraft placed on long leases to airlines globally.




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***** Carlyle Aviation Completes Acquisition of Fly Leasing (rank 15)

Carlyle Aviation is the commercial aviation investment and servicing arm within Carlyle’s $61 billion Global Credit platform. The closing of the transaction follows the receipt of regulatory approval from all government authorities required by the merger agreement and approval by FLY’s shareholders. Carlyle Aviation used funds from its ...




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J.M. DeMatteis, writer of "Constantine: City of Demons" interview

New York Comic Con 2018: Constantine: City of Demons At the 2018 New York Comic Con, DVDTalk’s Francis Rizzo III sat down with the cast and crew of the new feature-length animated film Constantine: City if Demons, due out...




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The Creators and Actors of "Batman: The Animated Series"

New York Comic Con 2018: Batman: The Animated Series At the 2018 New York Comic Con, DVDTalk’s Francis Rizzo III sat down with Batman: The Animated Series creators Bruce Time and Eric Radomski, and actors Kevin Conroy, Loren Lester,...




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Einladung an die Medien: Stolpersteine in Lübeck in die App "Stolpersteine Digital" aufgenommen




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Landesbeauftragter für politische Bildung: Alle Stolpersteine in Schleswig-Holstein jetzt in der App "Stolpersteine Digital" verfügbar




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Schleswig-Holsteinischer Bürgerpreis 2024: Neun Projekte in den Kategorien "Alltagshelden" und "U27" nominiert!




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Menschenrechte sind unteilbar. Requiem zum Gedenken der Toten an den Grenzen Europas am 20. November, 18 Uhr in Lübeck




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New Hard Quartet: Rio’s Song

Video: The Hard Quartet – “Rio’s Song”

Directed by Jared Sherbert. From The Hard Quartet, out October 4 on Matador.

The whole idea of an indie rock supergroup is kind of ridiculous but here we are. The Hard Quartet is Stephen Malkmus from Pavement and Matt Sweeney from Chavez, both on Matador Records, plus Drag City’s Emmett Kelly and Jim White. Put ’em together and what have you got? Skibidi-Bobbidi-Boo!

The song’s good (we love the 90s!) but the video’s great. Especially if you’re familiar with the Stones’ 1981 “Waiting On A Friend” promo. They went to great lengths to capture the vibe. Director Jared Sherbert told Pitchfork, “The original staircase is now surrounded by businesses, so we shot at the nearly identical staircase next door. The St. Marks Bar & Grill is gone, and, although there was another bar in the neighborhood with almost the exact same layout, International Bar was incredibly accommodating and encouraged us to shoot there, which just felt right for this. The apartment window with the guy daydreaming has changed, but a neighbor a few doors down let us use hers.”

Sweeney captures Mick’s aura without stooping to an impression while Malkmus totally nails Keith’s elegantly wasted swagger.

Read more at Glorious Noise...




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Quackbusters, Skeptics and the Web of Trust

What are 'quackbusters', you might ask. Well, Tim Bolen has the answer to that question. On his site (quackpotwatch.org) he explains: The "quackbuster" operation is a conspiracy. It is a propaganda enterprise, one part crackpot, two parts evil. It's sole purpose is to discredit, and suppress, in an "anything goes" attack mode, what is wrongfully named "Alternative Medicine." It has declared war on reality. The conspirators are acting in the interests of, and are being paid, directly and indirectly, by the "conventional" medical-industrial complex. These so-called quackbusters seem to be a branch of a larger movement, the "skeptics". Their website at www.skeptic.com/ shows who they are. Skeptics think of themselves as having opinions based on scientific 'truth'. They are very outspoken and very much "out there" to disabuse the rest of us of any idea that does not fit into their version of the scientific world view. While real scientific procedure requires there to be observation and experiment, formation and testing of hypotheses, and open discussion of both experiment and theories, the skeptics have firmly made up their mind on a number of issues. And they don't hesitate to tell us where we are going wrong... Mercury and fluoride for instance are not poisons for skeptics, and anyone who thinks they are must clearly be a conspiracy nut. Vaccination is good for you, as are chemotherapy and radiation cancer treatments offered by conventional medicine. If you oppose either of them you are simply a 'quack' or at the least you are an easy target for those who take advantage of your stupidity. The practices of alternative medicine, including "chiropractic, the placebo effect, homeopathy, acupuncture, and the questionable benefits of organic food, detoxification, and ‘natural’ remedies" are a favorite subject of the skeptics. They know that only mainstream medicine should be relied on and everyone who is into those practices really needs to have their head examined....




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Vaccine damage in Great Britain: The consequences of Dr Wakefield’s trials

More and more evidence is coming to light that Dr. Wakefield was on the right track when he researched the connection between the MMR vaccine and intestinal inflammation in the vaccinated children. Was Dr. Andrew Wakefield Right After All? Wakefield’s Lancet Paper Vindicated New Published Study Verifies Andrew Wakefield’s Research on Autism But how did Dr. Wakefield first get into the sights of the UK vaccine industry and how was the campaign against him mounted? Martin Walker, the author of "Dirty Medicine" and a number of other books on health, closely followed the case that eventually resulted in Dr. Wakefield's exile from the UK. He describes how it all happened and how the vaccine manufacturers were able to bring down the full weight of government and the courts against both Wakefield and the many parents who were suing for recognition of the damage vaccines had done to their children. "As a campaigner of 40 years, I think that what surprises me most about Dr Wakefield’s case, is how easily and how completely we were defeated by the pharmaceutical companies, how over a thousand parents and children were written out of history together with their adverse drug reactions. Part of this defeat for the parents, the children and the doctors concerned was grounded in an unfortunate understanding that pharmaceutical company executives were decent people and humanitarians. In fact the pharmaceutical companies, their corporate structure and their relentless pursuit of profit, their fraudulent practices represent one of the last remaining shibboleths, in our society which need to be completely reformed, democratised, divested of vested interests and made public from top to bottom." We do learn from experience. That is why we should pay attention to how this case went so wrong and why the campaign to ruin those researchers and to leave the damaged children by the wayside was mounted in the first place. So it won't happen again. Here is Martin Walker's essay....




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But what about the antiquarians?

"Those who do not learn from the mistakes of the past are doomed to repeat them." This is good advice, especially for democracies, which tend (as even Plato and Thucydides noted) to have short-term attention spans. But we shouldn't forget an equally important lesson, articulated most forcefully by Nietzsche: The health of a person and a people also depends vitally on the capacity to forget. Forgetting is necessary to free ourselves from imperfectly understood "lessons of history," so that we can see the challenges ahead clearly, without preconceptions or prejudice. Forgetting is also the better part of forgiving, and there are whole domains of political controversy -- indeed, whole regions of the world -- where a little less history could be of service in this respect.
--Eliot Noyes, Getting Past the Past




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"...you just get used to them"

“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 ATA = 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 AT; 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.


1 This, by the way, is my second-favorite math quote of the year, behind my complex analysis professor’s imprecation, right before discussing poles vs. essential singularities, to “distinguish problems that are real but not serious from those that are really serious.”

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).




q

Ist die "Schilderwald"-Novelle der StVO nichtig?

In einer




q

Please help with your critique

Stop what you are doing.

Click on this link to the Personal Bee home page:

http://www.personalbee.com


Then email or comment on this message with your thoughts about what we are doing right and wrong. Tell me, from looking at the home page, what business you think we are in. Tell me how you would use this and how you would get others to use it...

thanks!

Ted