Do you know the root cause of your dog's ear infections?
If you have a dog that's prone to ear infections, you know they're no fun.
When I first rescued my Lab, Honey, it seemed just as I'd resolved one infection the next one was brewing. That's when I realized I was only treating the problem—I wasn't getting to the root of it.
I dreaded chasing her through the house to clean and treat those ears. Let's just say, she wasn't the easiest dog.
Some dogs are more prone to ear infections, particularly floppy-eared dogs. And Labs are one of those dogs. Breeds with ears that stand up are less likely to get them.
Plus Honey loves to swim. So I assumed her problem was that I wasn't doing a good job drying her ears when she got out of the pool. And that may have been part of it.
But interestingly, her food made the difference. When we got her, she was eating a cheap box store food. We transitioned her to a super premium food, and it made all the difference once she was on it for a little while.
The bacteria and/or yeast that cause infection make their home in dark, moist, alkaline places such as the ear. These organisms may bring about an infection but it's often secondary to something else going on in your fur baby's ear.
Many things can change the environment in the ear, turning it into a great place for bacteria and/or yeast to take hold.
So what are those things?
Allergies – You can read my post, 5 signs your dog may have a food allergy, to find out how a food allergy can bring about an ear infection. It's often the first sign that your dog is allergic to its food.
Parasites – Ear mites are more common in cats than dogs. But dogs that get them are often super sensitive to them. Extreme scratching can cause trauma to the ear that results in an infection.
Foreign bodies – Any kind of dirt or debris, i.e. a leaf, a piece of grass, etc., can cause an irritation that leads to infection. A dog's ear canal is "L" shaped. A foreign object can settle in the bottom where the canal turns and cause a problem.
Trauma – An injury can irritate the ear which leads to incessant scratching, and ultimately an infection.
Hormone imbalance – Hormones affect the skin and ears. An imbalance can cause skin and ear trouble. Irritated ears will make your dog scratch and that can lead to infection.
Excess moisture – Yeast loves moisture. If your dog is a swimmer like mine is, be sure to dry their ears every time they swim. And the same for bath time. If those ears are damp, you're giving infection an opportunity to take hold. Some breeds with heavy floppy ears like spaniels have moist ears making them more susceptible to infection.
Ear anatomy – Sometimes a dog's ears are just ripe for infection because of their structure. Creases, excessive hair, and other areas for dirt, bacteria and yeast to build up may make your dog more prone.
Heredity – There are hereditary diseases like dermatomyositis in Collies and primary seborrhea in Shar Peis that can affect the ears and lead to infection.
Tumors – A tumor or polyp in your dog's ear can lead to scratching and ultimately an infection.
To rid your dog of ear infections, and not just treat them, you must get to the cause and fix it. If your dog has a tumor in their ear, antibiotics won't fix it.
Take the signs of an infection seriously. If you notice a strong odor, discharge from the ear, head shaking or tilting, pawing at the ear, or even wobbliness see the vet.
How your vet treats the infection will depend on the cause. Your vet should do a thorough exam to see if there's anything in the ear that's causing the problem. And then treat the problem In addition to the infection.
If the vet thinks it's just bacterial, they'll prescribe an antibiotic—or an antifungal for yeast. If there's a lot of inflammation, you'll probably get a steroid too.
For an environmental allergy, the treatment may include antihistamines, steroids and regular cleaning with an ear wash.
The point I want to stress is that all these treatments are great if they fix the problem. But if the infection recurs, talk to your vet about digging deeper to find a permanent solution.
The most common cause of ear infections is food allergies. If your vet doesn't bring this up and you are fighting one infection after the other, it may be time to consider changing your dog's food.
My post, 5 signs your dog may have a food allergy, can help you through the food allergy dilemma.
I know a food change made all the difference for my Honey.
Husse's Lamm & Ris (Lamb & Rice) or Lax & Ris (Salmon & Rice) are ideal for dogs with food allergies. Rice is highly digestible and gluten-free. Gluten can be a problem for a dog with food allergies. All Husse foods are balanced to provide your dog with the nutrition they need.
If your dog's infection is truly just an infection brought on by bacteria or yeast, then the antibiotic or antifungal should do the trick. But be sure to keep the ears dry and clean.
Clean your dog's ears regularly with an ear wash. If you prefer something more holistic, brewed green tea at room temperature is commonly used as a natural ear cleaner.
Ask your vet to show you how to clean your dog's ears safely. And ask how often to clean them. You can cause more harm than good with over-cleaning.
My advice is to listen to your inner voice. If it's telling you there's something more than an infection going on because your dog is battling one after the next, you need to look into other possible causes for your dog's trouble. Don't be afraid to push your vet to explore the possible causes I've mentioned.
We all want the same thing for our pets… a happy, healthy and quality life with no pain or discomfort. Getting rid of those ear infections once and for all is essential to achieving that.
Does your dog suffer from frequent ear infections? What was the underlying cause? Share in the comment section at the top of the page.
If you have a cat, you know they can suffer from dental problems. But were you aware that close to 75% of cats over 5 years old suffer from a painful problem called tooth resorption?
It's the most common cause of tooth loss in cats.
Purebred cats over the age of 4 are most susceptible, particularly Siamese and Abyssinians.
To understand what resorption is, you need to understand the structure of the feline tooth. Their teeth are like human teeth. Here's an illustration.
Within the tooth, there's a chamber called the pulp chamber or root canal. That chamber is filled with blood vessels, lymphatic vessels and nerves.
This tissue is surrounded by a bony substance called dentin. Dentin creates most of the structure of the tooth. Enamel overlays the dentin.
Resorption starts when a lesion forms on the enamel. This lesion is not a cavity. But in time, it will spread to the dentin and then on to the pulp chamber.
Sometimes you can see the lesion on the tooth, but sometimes it lies below the gumline or is hidden by plaque.
If you can't see the lesion then how do you detect a problem?
Here are 10 signs to look out for to keep your kitty from suffering too long with this nasty condition.
Often, owners don't realize how much pain their cat was in from this condition until after it's treated. All of a sudden they have a happier more playful kitty.
Good dental hygiene, which includes brushing your cat's teeth daily and annual dental exams, will help you spot a problem. But it's not likely to prevent it because no one knows for sure what causes tooth resorption.
As with most other health conditions, early detection can help avoid serious suffering.
Resorption progresses through 5 stages. The goal is to catch it before it's gotten beyond the first stage.
Depending on the stage, your cat's tooth may need to be extracted. Unfortunately, that's the only treatment.
If your vet recommends removal, it's tricky. The tooth is fragile and will often break. The vet will take an X-ray during the procedure to help find any fractured root pieces. The whole tooth must be removed to avoid infection.
Because resorption is one problem that can't be prevented, the best course of action is vigilance. Check your kitty's teeth every day while brushing and talk to your vet if you see signs that your cat's mouth is hurting them.
Once your cat develops this problem, your vet is likely to suggest a dental exam every 6 months.
Has your cat suffered from tooth resorption? How did you detect the problem? Share your experience in the comment section at the top of the page.
In my last few posts, I've talked about food… Decoding the Dog Food Label and Digestibility and Dog Food.
But what's really in those ingredients on the label? The terms are either vague or incomprehensible.
Well, this week I will give you the Association of American Feed Control Officials' (AAFCO) guidelines for what's allowed to be included in each of the ingredients on your pet's food label.
I have to warn you that the permissible ingredients will horrify you. It's another argument for feeding your pet the highest quality food you can afford.
Premium pet foods have ingredients you can understand.
Here's how AAFCO defines 9 of the most common ingredients found in your pet's food.
Just like the meat we eat, the term meat on the label means mostly muscle tissue. It can also include fat and gristle just like when you buy meat for yourself.
But meat in pet food can also include some very unappealing things like heart muscle or the muscle that separates the heart and lungs from the other organs.
It doesn't include bone. And because the muscle is mechanically separated from the bone, it has the consistency of paste.
The manufacturer can also name the species the meat comes from in the ingredient list like beef or pork. But if they use the term meat, it must come from cattle, pigs, sheep or goats. Any other mammal must be identified by name.
Poultry and fish wouldn't fall in this category either. They have to be identified separately on the label.
Rendering is the process of extracting proteins and minerals from the animal carcass using heat and pressure to remove water and fat. By-products are not rendered.
These parts have to come from cattle, pigs, sheep and goats just like meat.
By-products are scary. You want to avoid them in your pet's food because they're a very poor quality source of protein. They're almost everything except the muscle tissue… including organs and bones.
Some of those organs may be organs we would eat, but many are not fit for human consumption. Like udders for instance. The USDA considers these parts to be safe for animals though. But does that mean you want to feed it to your beloved pet?
And remember too that lower quality ingredients may mean lower digestibility. So your pet's body isn't able to use the nutrients in those ingredients.
It's basically the parts of the bird you'd find at the grocery store if you bought a chicken or turkey. Often it's the parts that most people don't want like backs and necks.
Poultry can also include bone, unlike meat which cannot include bone. If the processor removed the bone, it would say "deboned poultry".
Pet food makers will often be more specific and list the poultry ingredient as chicken or turkey.
Similar to meat by-products, you don't want your pet's protein source to come from poultry by-products.
The next 5 ingredients are rendered ingredients. Rendered ingredients are cooked to destroy harmful bacteria. They're made up of the extracted proteins and minerals from animals and are called meals because they're ground to a uniform sized particle.
If the manufacturer doesn't specify what mammal the meal came from, like beef meal for instance, it can come from any mammal. The maker is not required to specify the mammal. And it doesn't have to come from cattle, pigs, sheep, or goats.
This is similar to meat meal but it includes added bone, not just the amount of bone normally found on the whole carcass of the animal.
Who the heck knows what this means. Seems like anything and everything that can't be specifically identified. Yuck!
This is the same as "poultry by-products," but it's rendered. Most of the water and fat has been removed to make a concentrated protein/mineral ingredient.
Again, poultry meal is like "poultry" but in rendered form to make a concentrated protein/mineral ingredient.
If you're interested in the details of these other ingredients, the AAFCO website is a great resource.
Next time you're in the pet store, take a look at pet food labels and compare them to a Husse label. Now that you understand the specifics behind the vague ingredient names, you'll see that Husse ingredients are exactly what you think they are.
Husse premium pet food is made with only the highest quality non-GMO human grade ingredients.
Are you surprised what's allowed in pet food? Share in the comment section at the top of the page.
After reading about digestibility last week, you're probably realizing that not all dog food is created equal. And the ambiguity of the labels on the bags and cans doesn't make it easy to decipher the differences.
But digestibility isn't the only factor to consider when choosing a high quality food for your best friend. The food label provides lots of information about the quality of the food inside. But you need to know how to decode it.
This week I will help you understand that label. From Guaranteed Analysis to Ingredients, from the AAFCO pledge to Dry Matter Basis, understanding these terms will make you an educated dog food consumer.
And a better pet parent.
Reading and understanding the food label is really the only way to know what you're feeding your dog. Here's what to look for.
Dog food, whether canned or kibble, has water in it. Dry food less so than wet, obviously. Canned food can be up to 80% water and dry food can be as low as 6% water. This is a huge disparity so keep that in mind if you are ever comparing a dry food to a wet food.
To figure out just how much protein, fat and fiber your dog is getting, you need to calculate the percentages of these nutrients on a dry matter basis. Basically, take the water out of the food when doing your analysis.
When calculating dry matter basis, start with the percentages of protein and fat. You'll find this info on the Guaranteed Analysis part of the label.
We'll use Husse Optimal as an example. Optimal has a 12% moisture content and 23% crude protein. The protein number is on an "as fed" basis meaning as it's fed from the bag with the water content. We need to convert to a dry matter basis to see how much protein your dog is really getting.
If the dry food is 12% moisture, then 88% is dry matter. If that dry food has 23% crude protein, we divide 23% by 88% to find out how much protein it has on a dry matter basis. It's 26% protein on a dry matter basis.
Let's analyze an actual canned food that shall remain nameless. This food is 79% moisture—so it's 21% dry matter—and has 8% crude protein. We divide 8% protein by 21% dry matter and we get the protein on a dry matter basis—38%.
If we were comparing these two foods and hadn't converted to a dry matter basis, we would have assumed that the canned food had much less protein than the kibble. When in fact, canned food is typically higher in protein and lower in fiber than dry food.
This section of the label is equivalent to the Nutrition Facts on people food. It tells us the nutrient content of the food.
The Association of American Feed Control Officials (AAFCO) requires pet food manufacturers to disclose the minimum amount of protein and fat, and the maximum amount of fiber and moisture in the food.
Any other nutrient guarantees in the Guaranteed Analysis are voluntary unless the manufacturer makes a claim on the label like "high in vitamin E" or "high in calcium". Then they must support their claims by including those nutrients in the Guaranteed Analysis.
Husse Optimal's Guaranteed Analysis includes calcium, phosphorus, zinc, selenium and omega 3 and 6, in addition to protein, fat, fiber and moisture.
The percentages in the Guaranteed Analysis are minimums and maximums but not exact amounts. AAFCO expects that there will be batch-to-batch variations.
But manufacturers must be careful to account for those fluctuations by being conservative in their estimates. If state authorities test the food, they expect those percentages to be accurate or there will be penalties.
Protein and fat are listed as crude sources, not as digestible sources. This is where digestibility percentages come in. The body will only use some of those nutrients. The digestibility of the food will affect how much the body absorbs and uses.
Just like people food, dog food must be labeled with a list of the ingredients in the food in order of weight.
High quality foods will have a protein source as their first ingredient, not corn, wheat or rice. That protein source might be chicken, beef, salmon, or a chicken, beef or salmon meal. Whatever it is… be sure it's protein.
Be careful here! This is where manufacturers can be deceitful.
If the maker breaks the ingredients down into smaller components, each one will weigh less and can be added towards the end of the list. But when grouped together, these ingredients could weigh more than the protein source.
For instance, ground corn, corn gluten and corn bran can be listed separately. If they were listed together on the ingredient list as corn, which is what they are, they would be higher on the list… possibly before the protein source.
If a manufacturer uses a protein meal, it may be further down the ingredient list if it's considered in its dehydrated form (without moisture).
For this reason, you must read through the whole list—right to the end— to understand what's in the food. Also, if they've used preservatives and artificial colorings, they'll be listed at the end.
AAFCO establishes standards for the production, labeling and sale of animal foods. There are two different adequacy statements that can be used on the label.
This means the food maker tested the food in the lab and it had the recommended levels of protein, fat, etc. But… it doesn't say what those proteins and fats are. And they could be anything.
I've read that manufacturers have submitted mock products to AAFCO for guaranteed analysis testing that consisted of shoe leather, motor oil and coal. They met the 10% protein, 6.5% fat, and 2.4% fiber on the guaranteed analysis but certainly weren't nutritious… or safe.
To carry this label, the manufacturer must have tested the food on animals and the food provided proper nutrition. That would avoid the shoe leather and motor oil concerns, but this statement has its own problems.
With this label, a manufacturer allowed to use it on one food—a food that's been tested on animals—can use it on any of their foods with equal or greater nutritional value, even if that food was never tested on dogs.
Neither certification gives any guarantees of nutritional quality. But at least the food has met some standards.
The last part of the label to pay attention to is the section that tells you how much to feed your dog. These recommendations are only a starting point. Every dog is different.
Read my article How do I know how much to feed my dog? If you aren't sure what's right for your dog.
Or, if you'd prefer to just use the guidelines on the label, start with the middle of the suggested range and see if your dog is gaining or losing weight. Or, if they're hungry all the time.
If in doubt, always talk with your vet. They'll tell you how much to feed your dog… and they can help you pick a high quality food too.
Deciphering pet food labels, that are often misleading, can seem daunting. But a little knowledge can go a long way in choosing the food that will keep your dog happy and healthy.
What confuses you most about dog food labels? You're probably not alone. Share in the comment section at the top.


text:
Why Aren't There Pussy Pics?
Laura McGuire | Published: November 30, 2018
Takeaway: Why don't people with vulvas (consensually) share photos of them? It may be because we don't live in a culture that celebrates them enough.
Dick pics. You know them, you love or hate them, and there is a significant chance you have received or sent one in your lifetime. Since the advent of digital cameras being added to our phones, pictures of penises have been sent at an ever-increasing rate. According to YouGov's survey of 1,156 women 53% of millennial women reported receiving a dick pic. While this is a largely millennial phenomenon, 8% of the women who have received these pictures are over 55, meaning this is now an inter-generational experience.
As a sexologist and consent educator I am fascinated by how popular dick pics are. There's even a Tumblr site, "Critique My Dick Pic," where users can get tips on taking better genital shots. Taking a picture of your penis has gone from purely sketchy to relatively culturally accepted and normalized. And fortunately, there has been an increase in the conversation about assuring that these shots are shared consensually, and that the rights of all parties to opt in or out of giving, receiving, and sharing such sensitive information is respected and discussed.
Recently, I was asked a very interesting question about dick pics from a gay, gender-queer friend. They asked me, "Do cis-women, gay or straight, share pictures of their genitals?"
I have asked this question myself and was interested to see someone else bring it up. I said the honest answer was no. As someone who dates people of all genders, I have never asked or received a picture of another woman's vulva. Even with men asking for 'nudes' this rarely, if ever, includes up close shots of the female partner's genitals. But why? I asked some fellow sexologists their thoughts, and this is what they said.
Paula Leech, a certified sex therapist working primarily around issues pertaining to relationships, intimacy, and sexuality said:
"Speaking in very heteronormative terms, cis-men tend to be very visual in how they engage with and activate their arousal. Women tend to like a narrative, thus explaining the popularity of romance novels and books such as 'Fifty Shades of Grey.' Often what we find is that women will put a great deal of effort into creating an erotic scene, where the lingerie tells a story, and she puts thought into how she poses and what she chooses to show as a result of that story. It may not be obvious, nor overly complicated, but it is often part of the excitement when taking "sexy" pics to send to a partner or partners.
Part of the larger, terribly misguided and harmful story about what it means to have a penis, is that your worth or masculinity is defined by how large or girthy your penis is. It has been held up as the ultimate pleasure source in partnered sex with a penis owner, which is why I see so many people incredibly distressed by erectile dysfunction or rapid ejaculation. We treat the penis as if it is the main route to pleasure in partnered sex; therefore, if something goes awry and the penis doesn't 'perform' as expected, it often causes a great deal of distress for the individual, and frequently, the couple.
The penis has also been symbolic in our culture throughout history. Power, domination, sexual prowess and manhood are a few of the common associations. Dick pics, therefore, not only reflect the more visual nature of men's sexuality but can also be a form of peacocking – proudly displaying one's manhood in an attempt to impress the object of one's desire. "
Ashley N. Grinonneau-Denton, another certified sex therapist, added:
"To me, this phenomenon has much to do with how negatively stigmatized the vulva has been. Historically, up until relatively recently, the discourse regarding sexuality has been based on a very masculine model of sex, which is obscenely far from representative of a one-size-fits-all approach. While men, from a very young age, tend to marvel at their erections, women are not afforded the same visual convenience of looking at their genitals, and often end up feeling like it should remain tucked away, even from themselves. The grand majority of women I work with come into therapy never having had the experience of looking at their own vulva. From that perspective, if a woman is apprehensive about looking at her own body, sharing a quick pic of her vulva to a [potential or current] partner runs odd somewhere in the range of slim-to-none."
Neither touched on why this holds true for queer women couples as well, though. Perhaps it's because both have internalized this form of misogyny and do not feel it would be appropriate or rational to request this from each other?
Vulvas Are Amazing!
While men may marvel at their erections from early on, there is no reason why the vulva shouldn't get just as much admiration. Our clitorises do get erect and engorge; they are made of the same erectile tissue as the penis. Our bodies produce the most delicious fluids that bubble and drip in ways that can be beautifully captured on film. And, yes, media and culture have long taught us shame for the ways our vulvas look, but we live in an era where we do not need to stand for this any longer.
In 2003, photographer Nick Karras created the photo-book Petals, which depicted a wide variety of sepia toned vulvas from a myriad of ages, races and experiences. The pictures are beautiful and have been used in many of my sexual health classes. Appreciating the beauty of the vulva, vagina, clitoris and entire pubic region is important to our health and well-being. Vulvas are like snowflakes; no two are alike. Why don't we take the time to appreciate this a little more?
If this discussion has got you thinking, I encourage you to take the time to appreciate the vulvas in your life. Look at your partner's intimate body intently, slowly and with admiration. Then do this for yourself as well. Look at books of un-airbrushed pussies, like in the Petals book. Too often in porn, the images of vulvas are surgically and digitally altered, and thus unrealistic. If you feel comfortable with this take some pictures just for yourself. See what you can capture especially when you are turned on. Who knows, maybe pussy pics will catch on and the world will come to appreciate the power of the pussy better? Viva la vulva!
LELO Antibacterial (Toy) Cleaning Spray 2 fl oz
LELO Luna Beads
Sliquid H2O Lubricant - 4.2OZ
Written by Laura McGuire
Dr. Laura McGuire (they/them or she/her) is a nationally recognized sexuality educator, trauma-informed specialist, and inclusion consultant at The National Center for Equity and Agency.
Dr. McGuire earned their bachelor's degree in social sciences from Thomas Edison State University and graduate degrees in Educational Leadership for Change from Fielding Graduate University.
THeir experience includes both public and private sectors, middle schools, high schools, and university settings. In 2015, she served as the first Sexual Violence Prevention and Education Program Manager at the University of Houston, and in 2017, she became the first Victim Advocate/Prevention Educator at the US Merchant Marine Academy.
Your Vulva Isn't Ugly, It Just Looks That Way
Sexting for Men: How to Keep It Classy and Still Get Your Rocks Off
A Delicate Flower: The Vulva and Why We Should Appreciate It
My Pussy, Personified
Dick Pics: This Is Not Your Grandfather's Exhibitionism
Mirror Mirror: Why We Should All Look at Our Vulvas More
Pictophilia
Clitoral Complex
Circumcised Penis
When I'm having sex, my inner critic seems to take over. How can I learn to say more positive things about myself and feel sexier?
What causes the birth control pill to fail?
What causes a condom to tear?
Sex Therapist

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Model for circular statistics I am looking for advice on circular statistics. In particular, I'd like to know if any one had any advice/ references that deal with regression models for circular variables and whether it is possible to include random effects as well.
At the moment I can fit very simple models in WinBUGS using wrapped cauchy distributions, but I don't know how to go on to the next step and add either fixed or random effects. Below is the WinBUGS code I have been using so far. I have tested it with simulated data and so far it has performed well, but trying to add in fixed/ random effects has, so far, not worked.
model{
 for (t in 1:N) {

 # likelihood for angles. We use the "ones" trick to sample from the
 # Wrapped Cauchy distribution (see WinBUGS manual)

 ones[t]<- 1
 ones[t] ~ dbern(wc[t])
 wc[t] <- (1/(2*Pi)*(1-rho[t]*rho[t])/(1+rho[t]*rho[t]-2*rho[t]*cos(theta[t]-
 mu.t[t])))/ 300 # Density function for Wrapped Cauchy distribution

 rho[t] <- lambda.t    # mean cosine for the circular distribution
 mu.t[t]<- nu.t# mean direction for turns
 }

 ###### priors for mean direction of angles
 nu.t ~ dunif(-3.14159265359, 3.14159265359)
lambda.t ~ dunif(0,1) # prior for mean cosine of circular distribution
 Pi <- 3.14159265359 # define Pi
 }

## Simulated Data ##
list(theta=c(1.57086666107637,0.624281203067249,4.83586153543422,5.52517105399153,0.250167755691792,5.24413183188724,0.175711907822086,0.503670499719972,0.00587906094477884,0.290131613934322,0.759047889069672,0.57973291007534,3.03128168541491,0.497790655905849,6.24730873150114,2.61159637947433,6.19811892339656,2.21476872674273,0.163464826891718,5.79300356573004,5.65352466175931,-0.0100726021401003,0.00574503925995024,0.260777171784755,5.8545805891331,6.09628602098184,6.07018161953988,5.90921466125829,0.0387070377090986,5.96019978900552,0.270388591408335,0.539775794451919,6.16303548945592,5.54317029065067,1.09867887761604,0.546155012914554,5.73154203573232,6.04837644493341,.242217723020124,0.201937287826239,6.19111529531002,0.602897213838987,5.53590129760264,0.304328180646957,6.12364810518025,0.0781317192586082,2.12148311222615,5.41742779164167,0.109722984863423,0.546244633029087,1.72483899231817,5.81142848191977,5.77431670621736,5.94852063016486,1.21880980868771,0.761391412364464,6.13385885651117,2.3278212791841,-0.00886837423371834,0.0509442654103693,0.919346146608449,0.22243489212092,0.0605109486858312,6.26215798187548,3.35930515203348,4.49262316826849,0.393662386151002,0.408276217352091,5.48604197934124,1.2319358669625,0.290890698266516,0.0356807866706245,5.01603150661483,2.13110569190685,5.58637984768018,0.705401496640296,0.474940761772081,5.58728776070886,6.12311166642116,0.00848809261322299,3.35074107197193,5.82089972193407,0.0531213061461832,5.97904289602246,4.31610462188531,5.61206825679503,0.184081838885041,0.288450927211418,0.594322121025956,1.07062485671203,0.400068367390392,5.08834932305335,4.35542895067301,6.08614182924595,6.14530696852739,5.25070254271081,5.91716602109256,1.78589020077607,6.23955405139402,6.09356179129423), N =100)

A: This book by Nick Fisher is very popular.  It covers many circular distributions including Cardioid, wrapped Cauchy, wrapped Normal and von MIses distribution.  I don't know if it tells you how to generate observations from these distributions, but it gives you a lot of distributions, theory and methods.
Statistical Analysis of Circular Data

redpajama_set_name: RedPajamaStackExchange


What better place than Hawaii is there when you envision a romantic getaway? Visit Honolulu, Hawaii's capitol city to experience all the romance of a tropical getaway for yourselves. This island has plenty of fun in the sun and romance is just around the bend.
For couples that are interested in history, Pearl Harbor is just a short way away from Honolulu, HI. Here you can experience a place in history that will never be forgotten. Take a tour of the site of the sunken USS Arizona and the Pacific Aviation Museum while you are at it.
The best way to get out and see the island while experiencing romantic nostalgia with your sweetheart is to take the Waikiki Trolley. The Trolley is a charming replica of a San Francisco cable car and will take you all over the island to see the sights and soak in the history.
Beyond cruising the island by trolley, Honolulu has many historical tours to take and even has a cruise line that offers sunset cruises with dinner and a show.
Honolulu, HI being the largest city in Hawaii is known as a tourist city. It has many attractions to visit including a zoo, many museums, an arboretum, an aquarium, many beaches, and a few botanical gardens. Some of the attractions to be sure to visit include the Honolulu Museum of Art, the Aloha Tower, the Lyon Arboretum, and the Kapiolani Park.
Honolulu, HI has no professional sports teams, however if you don't mind rooting for the home team, you can catch a game with your love and cheer for the University team. Honolulu also hosts three major road races throughout the year; The great Aloha Run, The Honolulu Triathalon, and the Honolulu marathon. If you and your sweetheart are into jogging then you are on the right island.


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Q: Properly outer automorphisms on type II$_1$ von Neumann algebras Let $M$ be a von Neumann algebra with separable predual. Let us assume that $M$ is of type II$_1$, meaning that it is finite but has no type I part. Let $\tau$ be a faithful normal tracial state on $M$.
First, I would like to choose some free ultrafilter $\omega$ on $\mathbb N$ and consider the tracial ultrapower
$$
M^\omega = \ell^\infty(\mathbb N, M)/\{ (x_n)_n \mid \tau(x_n^*x_n)\to 0 \}.
$$
Next, let $\alpha$ be an automorphism on $M$. It is called properly outer, if for every non-zero projection $p\in M^\alpha$, the induced automorphism on $pMp$ is not inner.
Note that each automorphism gives rise to $\alpha^\omega\in\operatorname{Aut}(M^\omega)$ via pointwise application, and restricts to an automorphism on the relative commutant $M^\omega\cap M'$.
Theorem: (Connes) If $M$ is the hyperfinite II$_1$-factor, then $\alpha$ is (properly) outer if and only if $\alpha^\omega$ is (properly) outer on $M^\omega\cap M'$.

My question: Is the above also true if $M$ is a hyperfinite type II$_1$ von Neumann algebra, i.e., not assumed to be a factor? So is an automorphism $\alpha$ properly outer if and only if $\alpha^\omega$ is properly outer on $M^\omega\cap M'$?

I am sure that somebody must have worked this out somewhere, but I cannot seem to find it. I would be grateful either for the argument showing/disproving this statement or a reference with the answer.
EDIT: I realized that it is not necessary for $\alpha$ to assume that it preserves the trace $\tau$. The question makes perfect without it.

A: The answer to your question is yes, although I haven't found a precise reference for this fact.
Let $M$ be a hyperfinite, type II$_1$ von Neumann algebra with separable predual. Later on I will use that it is possible to decompose $M$ as $Z(M) \overline{\otimes} \mathcal{R}$, where $Z(M)$ is the centre of $M$ and $\mathcal{R}$ is the hyperfinite II$_1$ factor (this is Theorem 1.5 in Chapter XVI of Takesaki's third book on operator algebras)
Definition 1. An automorphism $\alpha \in \text{Aut}(M)$ is centrally (non-)trivial if $\alpha^\omega$ is (non-)trivial on $M^\omega \cap M'$.
It is properly centrally non-trivial if none of its restriction to $pM$, for some $\alpha$-invariant, non-zero, central projection $p \in M$, is centrally trivial.
The proof is composed of two parts:

*

*Every properly outer automorphism on $M$ is properly centrally non-trivial.

*If $\alpha \in \text{Aut}(M)$ is properly centrally non-trivial, then $\alpha^\omega$ is properly outer on $M^\omega \cap M'$.

Part 1 is addressed in the following claim.
Claim 1. If $\alpha$ is an outer automorphism of $M$, then $\alpha$ is centrally non-trivial.
Proof. If the restriction of $\alpha$ to $Z(M)$ is non-trivial, then the conclusion is obvious, as $Z(M) \subset M^\omega \cap M'$. Suppose that $\alpha$ is trivial on $Z(M) \cong L^\infty(X)$, for some measure space $X$. Then $\alpha$ can be decomposed as
\begin{equation}
\alpha = \int_X^\oplus \alpha^x,
\end{equation}
where $\alpha^x \in \text{Aut}(\mathcal{R})$ for every $x \in X$ (a good reference for disintegration of von Neumann algebras and of their automorphisms is Chapter IV of Takesaki's first book on operator algerbas; see in particular Theorem 8.23 for the passage above). The claim follows by contradiction using the following three results:

*

*$\alpha$ is centrally trivial if and only if $\alpha^x$ is centrally trivial for almost every $x \in X$ (I'm not sure who originally proved this, but it is proved as Theorem 9.14 here).

*$\beta \in \text{Aut}(\mathcal{R})$ is centrally trivial if and only if it is inner (Connes).

*$\alpha$ is inner if and only if $\alpha^x$ is inner for almost every $x \in X$ (this is due to Lance, Theorem 3.4).

The second part of the proof, namely that if $\alpha$ is properly centrally non-trivial then $\alpha^\omega$ is properly outer on $M^\omega \cap M'$ was proved by  Ocneanu (see Lemma 5.6). One has to backtrack a bit in the paper to make sense of the proof, because of all the notation and definition appearing in the lemma (the setting there is much more general than what we need), but the argument is similar to the one given by Connes to prove the analogous result for $\mathcal{R}$ (Proposition 2.1.2 in Outer conjugacy classes of automorphisms of factors).