#and if i go at it with angles and stable points and measuring proportions and comparing this bit to that bit | Explore Tumblr posts and blogs

gazegirls-blog1 · 6 years ago

Text

The Maximum Speed of Electric Scooter

Here are distinctive Electric Scooter models accessible in the market and every accompany diverse speed go. The Electric Scooters that are utilized for entertainment and fun exercises like kick bikes; the top speed of 5-10 mph can be frustrating. In electric scooters there are two sorts of characterized speeds, the open to cruising and greatest speed.

There are a few bikes that guarantee to convey the greatest speed of 20 mph, however who have attempted those bikes actually can't accomplish such top speed in any territory. It is conceivable to accomplish most noteworthy speed if the bike accompanies bigger air wheels or suspension. Likewise, there are a few systems how to cause an electric bike to speed up.

The Electric Scooter intended for children have an engine that can encourages you accomplish up to 10-12 miles for every hour, except the best models of grown-ups Electric Scooter can keep up the most extreme speed breaking point of 18-20 miles for every hour or considerably more. Much the same as any electrical gadget, on the off chance that you need higher speed and augment the presentation, you have to go for Electric Scooters that accompanies rock solid and incredible engine which can enable you to accomplish the most elevated speed.

Battery and Max Range

The vast majority of the models of Electric Scooters accompany lead-corrosive batteries that catalyst the engine. In any case, a portion of the extraordinarily structured and top of the line models of Electric Scooter significantly depend on the lithium particle battery innovation which is exceptionally advance and unrivaled in various manners. The lithium particle battery is lightweight and it takes around three to four hours to get completely energized. Then again, the lead-corrosive batteries are unreasonably high in weight and it additionally requires some investment period to get completely energized, settling on it the second rate decision for the majority of the riders. The battery limit of an Electric Scooter is essentially cited by the size and it is referenced in Amp or Ah. However, this data is very pointless until the ostensible voltage of the battery pack is known and after that it should be increased together to decide the limit watt-hours. In this way, it would be useful if the battery sizes are cited in Watt-hours.

Scope of an Electric Scooter is the critical component which should be considered while getting one. The significance of range should be underscored by the way that how the Electric Scooter is used will decide the exhibition. The bikes that convey substantial weight would cover lesser range contrasted with the bike that conveys lighter weight. For more subtleties, you can check this article about electric bike max extend or the area where the ability to weight proportion is referenced. It is evaluated that the grown-up Electric Scooters can cover scope of 7-12 miles on a solitary charge, while the bikes for children can just cover a scope of 5-8 miles for every charge.

There are numerous elements that impacts the scope of an Electric Scooter and this incorporates head wind, rider's weight, stature, geography or slopes, landscape, how much beginning and halting the rider does, every one of these elements are significant for computing and deciding the general range.

The Weight Range

Weight of the Electric Scooter is likewise a significant factor to consider while getting one. The heaviness of the e-bike impacts the general range that it can cover and henceforth you have to think of it as cautiously. The bigger measured bikes need additionally inclining while at the same time turning and it exceptionally hard for the riders to turn corners with their bikes. In addition, the heavier models accompany improved yapping framework which shows that it will take more time for you to stop the substantial models. In addition, the bigger size Electric Scooters are less affected by the outer elements like the windy breezes which you face while crossing the scaffold. Since it is overwhelming you will appreciate progressively stable ride.

The Style of an Electric Scooter

Today, the market is overflowed with a wide assortment of Electric Scooter plans and styles to look over. A portion of the Electric Scooter models accompany seats and deck, while others accompany no seats like the Electric great kick bikes. You are required to ride these bikes by remaining on it and moving the bike is conceivable with the additional high handle bars. There are a few models of Electric Scooter that are light in weight and it tends to be collapsed effectively for capacity and dealing with it in open vehicle. In the event that you need further developed models of Electric Scooters, at that point guarantee to search for some different highlights like the back and front suspension and the pneumatic tires.

The Safety Aspects of an Electric Scooter

Before obtaining an Electric Scooter or electric kick bike you should guarantee to check the security part of the bike, highlights and gadgets cautiously. It is in every case preferable to be protected over being heartbroken. You should not think little of the perils and dangers that are related with Electric Scooter riding. You should realize the conceivable electric bike risks and guarantee to check all the security angles which can avoid every one of the dangers. Besides, adhering to the directions and organizing the highlights over fun is the way to security.

The Comfort Level of an Electric Scooter

The Electric Scooters without expanded air tires and suspensions are especially awkward for riding on typical and smooth territories. Each pothole and knocks that you go over will be transmitted up to the handle bar and standing stages, raising a ruckus while riding. You likewise need to guarantee that you pick the Electric Scooter with negligible play/give in vertical handlebar. The best Electric Scooter models would make you feel firmer, steady and unsurprising while at the same time cruising at higher speed, paying little heed to the landscape.

The Motor of an Electric Scooter

Engine is the core of the Electric Scooter and it is the thing that drives the bike. Contingent on the planned use and quality, the Electric Scooter highlights shifting force engines. The child's Electric Scooters accompany engines of up to 350 watts and this engine offers nice lower normal rates and increasing speed. The ground-breaking engine introduced in Electric Scooter is 500 watts which offer uplifted increasing speed and speed and it gives your bike the expected capacity to climb delicate inclines. On the off chance that you are searching for the ground-breaking Electric Scooters which can accomplish higher speeds and can climb serenely any landscapes and soak inclines, at that point guarantee to pick Electric Scooter that accompanies 750 watts engine or 1000 watt engine for more power.

Breaking Systems of an Electric Scooter

When you are riding your Electric Scooter at its top speed of most extreme 40mph, it is constantly a smart thought to utilize both the brakes introduced at back and front of the bike. The engine forthright can likewise be utilized to stop the bike during crises as it accompanies worked in electric brakes which can productively hinder the speed of the bike. Having brakes both at front and back is demonstrated to powerful contrasted with the plate brake on back wheel. Sticking the brake at back can make the Electric Scooter slide and it might wind up crazy. In this way, guarantee to pick the bike that has both back and front brakes.

Cost of an Electric Scooter

The expense of the Electric Scooter differs relying on the highlights and power that it has. In this way, you are required to think about the paces of various models on the web and pick the one that suits your spending limit.

I trust you discovered this electric bike purchasing guide helpful. Remark beneath on the off chance that you need to include something different!

1 note · View note

nervousalpacahologram · 7 years ago

Text

A Shopping List, Lipstick, And A Taxi

For @swarmisazergling

O5-10 sat in her quarters, thoughtfully tapping her pen against the desk she sat at. A paper list of items sat in front of her, all things for her Factotum to buy and install into the Archive. It was more-or-less a glorified shopping list, but many of the words written down would mean nothing to the average man or woman.

Scranton Reality Anchors, automated special stabilizers, temporal sinks… A whole host of anomalous technologies being used to make her Archive a more stable anomaly.

As she continued to write things down, a faint knock on the door interrupted her. Glancing up, Ten took in the sight of Salt, her favorite Factotum and body double. They looked almost identical, toffee-dark skin and long, curly, brown hair. It was tied back in a loose ponytail and had begun to grey faintly at the edges. They had started dating about six months ago, and Ten didn’t know what that said about her own personality.

When she had mentioned it to Six, he had said something about never having expected her to be such a narcissist, a statement that she took a healthy amount of offense from.

“Ten, I’ve got a message for you. Witness gave it to me, to give to you.” Ten took the note she had been handed, Four’s elegant script running across the page. She thanked Salt, who smiled and left. When she was alone in her office, she carefully eyed the note. Whenever Four had something to say he would meet her in person, their friendship having extended back to when they had both joined in the aftermath of the Insurgency fiasco.

She picked up the note and read it, then re-read it again, and a third time for good measure:

“Ten, it’s rather simple. We need to talk. Meet me at the Intersection… You know which one. Make sure that you remain unfollowed, and that you do not forget the way home.”

She knew what the last part meant. It was code for bringing mnestics, as some form of memory-damaging effect was present. Twelve had coined it shortly after Ten had arrived and included it in a list of Factotum code. She stood up and walked to a small cabinet to the side of her office, removing a bottle of Lotus-class mnestics, slipping them into her pocket. She then swiftly exited and approached one of the several vehicles surrounding Site-158.

A factotum named Ash entered the driver’s seat and shot a questioning glance back at the grey-suited Overseer.

“Just an intersection.” Ash shrugged, putting the car into drive and speeding down the air-strip like roads, the fields and concrete dusted with bone-grey snow.

They soon arrived at the crossroads, dust rising into the air from the dirt road the car had stopped on. Ten stepped out and thanked Ash, who turned around and sped off without as much as a glance. Looking out over the intersection, she took a deep breath, stepped sideways, and found herself in a field of glimmering lights.

Crossroads were places with an above average concentration of Way intersections, usually giving access to the regular meeting place for Ten and Four. The only objects in the space were a single wooden table and two chairs. Four sat in the farther seat, a cigarette dangling from between his lips. He blew out a cloud of red smoke, which floated upwards and dispersed into the brisk air of the Ways. He was dressed in his formal wear, a red suit with a similar colored panama hat tilted precariously on his caramel-dark hair.

He looked up, and when he saw her, removed the cigarette and stood up, stretching his back. All was quiet for a moment, and then he spoke.

“Did you bring them?” His golden eyes glimmered in the light of the gossamer walls, webs of electrum light giving them the appearance of shattered glass.

“Yes.” Ten gestured to her purse, shaking it for good effect.

“Good shit Ten. Now, follow me. We have work to do.” With that, Four turned and began to walk away, and Ten trailed behind him. They travelled through the Ways for several minutes before ten spoke up, glancing over to the red-clad man.

“Now that I’m here, can you tell me what’s going on?” She cocked her head, watching as Four’s expression turned to one of thoughtfulness.

“Yeah, I suppose I can. But not right now. Follow me.” They continued to walk, and Four soon took a step sideways back into reality, followed closely by Ten. When the white-light had cleared from her eyes, she found herself at the edge of a cave, lit by blazing torchlight. Two men in blue robes studded with pink stars lay on the ground, bleeding out through stab wounds along their necks. It almost seemed black in the moonlight.

Another figure stood in the moonlight, and his Identity was immediately recognizable: Twelve. The Physician stood there, a curved dagger held calmly by his side.

“Told you you’d need the mnestics.” Four said, smirking slightly. Twelve turned back towards them, grinning. His slightly crooked teeth glinting.

“Aha! My friends, you’ve finally arrived! I trust you’ve informed Ten on our purpose?” His smile dropped when he was met with the silence of Four and Ten’s mild head-shake. “Really Ambassador? Must you always be so… Gah! What’s the word… 神秘的な、ない...不可解… Cryptic! You can at least tell our friend over here!”

“Fine, but why must you ruin my fun?” He turned to Ten as Twelve rolled his eyes behind his glasses. “Given the recent rise in anomalous activity, the Council and Foundation as a whole are leaning towards a less radical stance on the anomalous. You know that. For example, the increase in Thaumials, have you noticed that?” He cocked his head, leaning forward.

“Yes, of course. But which O5’s are adopting this stance?” She pulled a cigarette from her pocket and lit it, not caring if it smudged her lipstick. It was too late at night to deal with Four’s strangeness today.

“We’ve got Two, maybe Three, me, Five, maybe Six and Eight, Nine, you, and Twelve. I plan to introduce project NOSTOS soon, as I believe we definitely have enough support.”

“So, why are we here of all places with Twelve?”

“If our initiative works, we’ll need to have allies, and I want to make a good impression on the Keepers of The Seven-Pointed Star, and Twelve happens to be an expert on their beliefs. So since we’ll need to make a good impression I suggest that Twelve, you put away your knife and fix your lipstick Ten.” He gestured to the corpses on the ground. “Before more assassins show up at least. These guys aren’t exactly the most popular group among Fifthests.”

Ten rummaged through her bag, cursing when she couldn’t find her grey lipstick. Before her cussing could continue, Four tapped her on the shoulder, handing her a tube of grey lipstick, only a few shades off.

“Where did you get this Four? And more importantly, why do you even have this?”

“I have too many pockets Archivist, and you always complain about forgetting lipstick so…” She took it gratefully, turning as Twelve sheathed their knife, rolling his shoulders.

“Shall we begin?” Four said, adjusting his hat to sit at a slightly less crazy angle. With everyone prepared, thy strode inside the cave, greeted by several men and women. They all wore silver cloaks, large seven-pointed star masks obscuring their face save for the eyes. The apparent leader’s mask held an eye in the center. She and Four stepped forward, and the Ambassador greeted the woman.

“Hello, Ambassador. You wanted me to organize this meeting, and I pulled through. State your business.” She said, spreading her arms in a welcoming gesture.

“Well now, you see, rumor has it that you are in possession of a certain artifact, something called an Archon Gate?” Ten watched as a visible wave of unease spread through the assembled cultists, faint whispers spreading through their ranks.

“How did you learn about this?” She replied, taking an almost defensive stance.

“All I can say is: Insider knowledge. And don’t worry, your members are all loyal, but you might remember a man in your inner circle having disappeared a few months ago? Pity you never saw his face.” He grinned, silver tongue toying with his teeth. He walked forward and whispered something in her ear. He stepped back and motioned towards Ten. “Now please, Archivist.”

“We know what the Archon Gate can do, it can expand any room it is placed in to Non-Euclidian proportions, a massive area in a tiny space.” Four had told her once of this cults possession of the relic, in typical Four-fashion. “Our cause… requires it if we are to weather this oncoming storm.” She watched as the cultists murmured among each other before finally turning back to them.

“Tell us, why should we give this to you?”

“It is a necessary device, for as I’ve been shown, the anomalous is rising at an exponential curve, and if it cannot be contained, then a ΨK-Class “Crimson Skies” scenario is imminent. The anomalous will overrun the world, destroying everything in its path. Unless we can Contain.” She took a half-step back, standing up to her full height.

“How do we know you’re not all lying? How do we know that you won’t take the Gate and then send in your Task Forces and burn us out?” Her voice rose slowly as she spoke, indicating the fear she was feeling.

“I was afraid you’d say that. But be have just a bit of insurance for this. Twelve? Show our friends.” He gestured to the Chinese man, who grinned and strode forward, stopping in front of the leader. Despite his short stature, he seemingly towered over the cultists.

“I suggest you take the mnestics now Ten. Like, severely suggest.” Ten pulled the orange bottle from her purse and downed three pills, feeling the kick into effect when Twelve tore off his glasses. Her and everyone else in the room clutched at their heads as burning fractals and seven pointed stars filled their visions. She saw a vast void contained behind Twelve’s eyes and inside his heart, writing and laughing with every heartbeat.

She watched as the void became a seven-pronged star with a colorless green eye in the center, everything it put it’s baleful gaze upon burning in cold halogen fire. The images remained burned into her retinas when twelve returned the glasses to his face, and when the dark spots cleared, she saw the cultists on their knees, their hands clasped and mouths whispering dark prayers.

“I believe that they’re expecting the Gate my little ones.” Twelve said, turning back towards Ten and Four. They both stood in varying levels of awe, their eyes wide. Two cultists scrambled away, retrieving a black box etched with white lines, which they gave to Twelve in a supplicating manner. He thanked them in a language that Ten didn’t understand before he led Ten and Four from the cave.

As soon as they exited the cave, Four and Ten began to talk at the same time, both trying to make sense of what they had just witnessed. After several moments of interrupting each other, Ten finally spoke.

“What in One’s name was that Twelve!” The Physician laughed heartily, wiping a tear from the corner of his eye. As soon as his laughter died down, he replied.

“Ever wonder why there are all those rumors that I’m not human? All rumors have to come from somewhere after all.” With those cryptic words, Twelve stepped sideways into the Ways, disappearing from sight and leaving Four and Ten alone.

“What should we do now?” Four asked, “We’ve got the Archon Gate, Twelve is stranger than before, and we might have an ally in these guys.” They stood there for a second, simply enjoying the cool night air before Ten broke the silence.

“Fuck if I know… But Six does have an ample amount of expensive alcohol at his own Site which he never drinks…” She jokingly elbowed Four in the arm, prompting a bout of laughter from the short man.

“Didn’t think you drank Ten.”

“Not often, but when I do I tend to rival the Brights.” The smile that spread across Four’s face sealed the plans for the night.

=======================================================

“The… The point is… fucking-uh… I DON’T KNOW! There! I said it! Happy now!” Ten all but screamed, throwing her hands into the air. An empty bottle of whisky brought over from a parallel universe sat next to them, seal cracked open for the first time since it had been gifted to Six. Four grinned triumphantly as Ten conceded her argument and lowered her head into her hands.

“You don’t argue well when you’re drunk, do you Ten?” Four said, leaning forward with a shit-eating grin. Ten gave him the finger, and Four laughed raucously.

“How do you argue better when you’re hammered!?” Four was about to respond when they were interrupted by the sound of a door creaking open. They both turned and were greeted by the silhouette of O5-6: The Cowboy. His mouth moved, but no words came out. He stood there, trying to find his words, settling for a long list of angry expletives.

“You two fucking dumbasses! Why the fuck are you drinking my damned whisky!” He strode angrily into the room, and despite his angry countenance, the two drunk overseers burst into laughter. “Out! Get out! I swear to god!” He gripped them both by the collars of their shirts and began to pull. They were half-dragged and half-stumbled out of the Site, and Six loaded them into the taxi that sat conspicuously outside. Ash rolled their eyes at the two drunken overseers before speeding off into the night, not exchanging a single word with Six.

=====

They would wake up in their respective beds, memories of the previous night’s festivities foggy in their minds. They would both find a strongly-worded E-mail from Six, and they would both laugh, being close enough to the other O5 to know that his threats were hollow. Four would return to organizing Project NOSTOS, and Ten would return to her glorified shopping list. Six would fume about the whisky he would never drink, forgetting about it only a handful of days later.

One and Three would tell them that they were being reckless, not sending a double in their place, and Four would laugh, asking if they thought so low of him. Ten would smile as One would put his face in his hands and sigh.

And Life would go on.

#SCP Foundation #o5-10 #o5-4 #o5-12 #o5-6 #the archivist #the ambassador #the physician #mikell bright #smoking/#alchohol/#project NOSTOS #Crimson Skies #implied/referenced SCP-3125 #little gift for @swarmisazergling

23 notes · View notes

meditativeyoga · 6 years ago

Text

Yoga for Osteoporosis - An Interview with Loren Fishman, M.D. and Ellen Saltonstall

Osteoporosis is among the most extensive persistent conditions in the Western hemisphere. Exactly how does yoga aid counteract the impacts of bone loss and osteoporosis? In this interview, Dr. Loren Fishman as well as Ellen Saltonstall, co-authors of Yoga for Osteoporosis, discuss the outcomes of Dr. Fishman's research on yoga for weakening of bones and also a few of the factors to consider and also precautions for individuals with osteopenia or osteoporosis taking into consideration a yoga exercise technique.

Eva Norlyk Smith: You have written a book on Yoga for Osteoporosis and have conducted a major study on the bone-building benefits of yoga exercise on osteoporosis. What made you curious about this subject in the firsts place?

Loren Fishman: Weakening of bones is just one of the most widespread chronic conditions in the Western hemisphere, and it's tough to overemphasize its wellness results. Weakening of bones influences 44 million Americans. That's over half of everyone over the age of 50. It is HALF of all ladies of whatever age and also 25 percent of all males. It influences over 200 million individuals worldwide. So this is big time.

Osteoporosis creates a million fractures every year, many of which are vertebral fractures and also regarding 300,000 are hip cracks. We worry so much concerning bust cancer in females, however, in actuality the risk of a hip fracture is equivalent to the combined threat of bust, uterine and also ovarian cancer cells. And also it's not simply women who go to threat. For men over 50, although we listen to a great deal regarding prostate cancer, guys over 50 are in fact extra likely to have a hip crack than prostate cancer.

Fully 25% of the individuals that have hip fractures pass away. One more 25 percent go into an assisted living home never to leave, so fifty percent of people who acquire a hip crack have an extremely significant life change.

Eva Norlyk Smith: Exactly how does yoga aid counteract osteoporosis?

Loren Fishman: Well, let me count the methods. Most individuals have heard of Wolff's regulation: The architectonic, the architectural assistance of bone, follows the lines of pressure to which that bone is exposed. When bone cells get stimulated via being compressed or turned or extended, they generate more bone mass till that bone gets solid enough, to withstand the pressure. At that factor the pressure no more distorts the bone, and the bone-making cells quit making much more. What a remarkable feedback system. In weakening of bones, the bones flex much more, so pressure is a lot more efficient in boosting the cells to earn bone.

In short, like weightlifting, yoga works by worrying the bone. Yoga stimulates the bone with isometric tightening at almost every possible angle for long periods of time.

Ellen Saltonstall: Just what is usually overlooked in contemporary weight-training workout and absolutely when assessing the impacts of weakening of bones medications, is that there is a difference between framework as well as density. Dexa scans will obtain a measurement of thickness, yet they inform us absolutely nothing about the framework of the bone. Dense bone mass by itself doesn't always supply protection against cracks, unless the bone fibers are laid down in such a way to offer better toughness, the bone mass is not mosting likely to be very stable It resembles the distinction between a pile of steel light beams and also the George Washington Bridge. A bridge has been prepared by designers, so the light beams, when assembled, produce an efficient, completely incorporated framework, which could sustain significant quantities of weight-- because of the toughness developed by the structural interconnections.

In short, thickness as well as structure both issue for bone health. Sadly, we don't have hassle-free means to determine the structure of bones as of. We do have straightforward means to gauge the density. The weakening of bones medicines do function, they lower the danger of crack considerably, but the functional limitations of simply developing bone mass without correct structure and also stamina are completely ignored.

Loren Fishman: Yoga helps grow bone mass, however because yoga exercise positions draw and stretch the bones from every conceivable angle, yoga also might stimulate the development of a bone structure that has the ability to withstand better amounts of pressure, as well as several kinds of challenges.

As we talk about in our program, there countless various other crucial methods which yoga advantages individuals with weakening of bones, such as boosting equilibrium, muscle strength, series of motion as well as control, while reducing stress and anxiety. These are other vital advantages of yoga for people with weakening of bones since they each help in reducing the risk of falling.

Eva Norlyk Smith: Several yoga educators and yoga specialists run under the presumption that twisting postures must be stayed clear of by individuals with weakening of bones. Yet, you claim that spins are fine, also essential for people with weakening of bones. Might you clarify on that?

Ellen Saltonstall: It holds true that numerous people believe that twists are contraindicated. However, we have actually found that spins are extremely beneficial and also bring little bit, if any, danger. Spins provide such an in proportion stress to the vertebral body that it's not a crack risk. As well as though doctors caution their clients with weakening of bones never ever to do any type of turning, there seems to be no actual proof that spins are negative in and of themselves.

Of course, if you're turning and also lifting something at the exact same time, it's a different story. However, yoga exercise spins, as I stated, provide such a balanced stress, we feel that gentle, supported twists are risk-free to do as component of one's yoga exercise practice. And also they are crucial, because they give such a total stimulation to the vertebral body.

Loren Fishman: If you have certain various other back conditions, like a herniated disk, twists can be a problem. If all you have is weakening of bones, there's almost no other way of developing strength and promote bone development than revealing that vertebra to some stress. We do have to avoid forward flexes in individuals with weakening of bones, as well as the side bends they can do are limited. So benting the vertebra is just one of minority ways to obtain all the vertebrae under some compression and also promote bone growth in the vertebrae. This is extremely important, because a lot of weakening of bones fractures take place in the vertebrae.

Over the last Ten Years, we have given yoga techniques including twists to lots of, many individuals, normally over the age of 65, nearly all of which have actually had significant osteoporosis. This has been well over 30,000 hours of yoga. There hasn't been a single incidence of crack, Actually, nothing bad has actually taken place. We have not had one point go incorrect. This is something that both yoga teachers and yoga practitioners need to know.

Ellen Saltonstall: Obviously, the secret to getting the advantages of yoga is to make sure that the yoga exercise positions are performed with proper placement. Every joint in the body as well as every part of the body has an optimum placement as well as this is a clinical procedure. There are several yoga exercise designs that highlight this accuracy of alignment. If the joints are straightened effectively, not only do you obtain finest stimulation of the bones, however you have better equilibrium also. So one instance is the tops of the arms as well as the tops of the legs, i.e. the upper arm bone and also the thigh bone, both return in the outlets of the joints and also when they are rooted securely in the outlets of the joints, the congruency of the joints is better, it's stronger.

Another example of an alignment that we usually make use of is that the breast bones raises, and also this stops that stoop ahead pose that is generally seen in weakening of bones. So those are just 2 instances of placement, certainly, there are a great deal of alignment-based directions that we provide when instructing individuals with osteoporosis that are vital to obtaining the most effective, as well as best, results. For people with osteoporosis, just practicing yoga, without appropriate focus on alignment, could not be such a good idea.

Nothing in this meeting ought to be taken as clinical suggestions. Osteoporosis is a severe condition, always consult a doctor about any kind of medical concerns as well as before beginning any type of workout regimen.

Dr. Fishman is an M.D. as well as an Iyengar-trained yoga exercise teacher, along with the managing companion of Manhattan Physical Medicine and also Recovery. Ellen Saltonstall is an author, a yoga exercise specialist, a senior accredited Anusara yoga exercise instructor and also she holds a master's degree in the area of healing motion education.

#healing #meditation #Yoga #yoga works

2 notes · View notes

ggonecrane-ml4daprojects · 4 years ago

Text

Week 3 Assignment - Running a Lasso Regression Analysis

Introduction

In this assignment I am going to perform a Lasso Regression analysis using k-fold cross validation to identify a subset of predictors from 24 or so of predictor variables that best predicts my response variable. My data set is from Add Health Survey, Wave 1.

My response variable is students’ academic performance in the measurement of students grade point average, labelled as GPA1, in which 4.0 is the maximum value.

The predictor variables I have used for this analysis are as follows:

Data Management

Since my original raw dataset contains quite a few missing values in some columns and rows, I have created a new dataset by removing any rows with missing data via Pandas’ dropna() function. Also, I have managed BIO_SEX variable by re-coding its female value to 0 and male value to 1, and called it a new variable MALE.

Program

My Python source for this analysis is appended at the bottom of this report.

Analysis

In this analysis, I have programmed to randomly split my dataset into a training dataset consisting of 70% of the total observations and a test dataset consisting of the other 30% of the observations. In terms of model selection algorithm, I have used Least Angle Regression (LAR) algorithm.

First, I have run k-fold cross-validation with k =10, meaning 10 random folds from the training dataset to choose the final statistical model. The following is the list of coefficient values obtained:

-------------------------------------------------------------------------------------

Coefficient Table (Sorted by Absolute Value)

-------------------------------------------------------------------------------------

{'WHITE': 0.0,

'INHEVER1': 0.0002990082208926463,

'PARPRES': -0.002527493803234668,

'FAMCONCT': -0.009013886132829411,

'EXPEL1': -0.009800613179381865,

'NAMERICAN': -0.011971527818286374,

'DEP1': -0.013819592163450321,

'COCEVER1': -0.016004694549008488,

'ASIAN': 0.019825632747609248,

'ALCPROBS1': 0.026109998722006623,

'ALCEVR1': -0.030053393438026644,

'DEVIANT1': -0.03398238436410221,

'ESTEEM1': 0.03791720034719747,

'PASSIST': -0.041636965715495806,

'AGE': -0.04199151644577515,

'CIGAVAIL': -0.04530490276829347,

'HISPANIC': -0.04638996070701919,

'MAREVER1': -0.06950247134179253,

'VIOL1': -0.07079177062520403,

'BLACK': -0.08033742139890393,

'PARACTV': 0.08546097579217665,

'MALE': -0.108287101884112,

'SCHCONN1': 0.11836667293459634}

As the result shows, there was one predictor at the top, WHITE ethnicity, of that regression coefficient shrunk to zero after applying the LASSO regression penalty. Hence, WHITE predictor would not be made to the list of predictors in the final selection model. Among the predictors made to the list though, the last five predictors turned out to be the most influential ones: SCHCONN1, MALE, PARACTV, BLACK, and VIOL1.

SCHCONN1 (School connectedness) and PARACTV (Activities with parents family) show the largest regression coefficients, 0.118 and 0.085, respectively. They apparently are strongly positively associated with students GPA. MALE (Gender), BLACK (Ethnicity), and VIOL1 (Violence) however, coefficients of -0.108, -0.080, and -0.070, respectively, are the ones strongly negatively associated with students GPA.

The following Lasso Path plot depicts such observations graphically:

The plot above shows the relative importance of the predictor selected at any step of the selection process, how the regression coefficients changed with the addition of a new predictor at each step, as well as the steps at which each variable entered the model. As we already saw in the list of the regression coefficients table above, two of positive strongest predictors are paths, started from low x-axis, with green and blue color, SCHCONN1 and PARACTV, respectively. Three of negative strongest predictors are the ones, started from low x-axis, drawn downward as the alpha value on the x-axis increases, with brown (MALE), grey (BLACK), and cyan (VIOL1) colors, respectively.

The following plot shows mean square error on each fold:

We can see that there is variability across the individual cross-validation folds in the training data set, but the change in the mean square error as variables are added to the model follows the same pattern for each fold. Initially it decreases rapidly and then levels off to a point at which adding more predictors doesn't lead to much reduction in the mean square error.

The following is the average mean square error on the training and test dataset.

-------------------------------------------------------------------------------------

Training Data Mean Square Error

-------------------------------------------------------------------------------------

0.4785435409557714

-------------------------------------------------------------------------------------

Test Data Mean Square Error

-------------------------------------------------------------------------------------

0.44960217328334645

As expected, the selected model was less accurate in predicting students GPA in the test data, but the test mean square error was pretty close to the training mean square error. This suggests that prediction accuracy was pretty stable across the two data sets.

The following is the R square for the proportion of variance in students GPA:

-------------------------------------------------------------------------------------

Training Data R-Square

-------------------------------------------------------------------------------------

0.20331942870725228

-------------------------------------------------------------------------------------

Test Data R-Square

-------------------------------------------------------------------------------------

0.2183030945000226

The R-square values were 0.20 and 0.21, indicating that the selected model explained 20 and 21% of the variance in students GPA for the training and test sets, respectively.

======== Program Source Begin =======

#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ Created on Fri Feb 5 06:54:43 2021

@author: ggonecrane """

#from pandas import Series, DataFrame import pandas as pd import numpy as np import matplotlib.pylab as plt from sklearn.model_selection import train_test_split from sklearn.linear_model import LassoLarsCV import pprint

def printTableLabel(label): print('\n') print('-------------------------------------------------------------------------------------') print(f'\t\t\t\t{label}') print('-------------------------------------------------------------------------------------')

#Load the dataset data = pd.read_csv("tree_addhealth.csv")

#upper-case all DataFrame column names data.columns = map(str.upper, data.columns)

# Data Management recode1 = {1:1, 2:0} data['MALE']= data['BIO_SEX'].map(recode1) data_clean = data.dropna()

resp_var = 'GPA1' # 'SCHCONN1' # exp_vars = ['MALE','HISPANIC','WHITE','BLACK','NAMERICAN','ASIAN', 'AGE','ALCEVR1','ALCPROBS1','MAREVER1','COCEVER1','INHEVER1','CIGAVAIL','DEP1', 'ESTEEM1','VIOL1','PASSIST','DEVIANT1','GPA1','EXPEL1','FAMCONCT','PARACTV', 'PARPRES','SCHCONN1'] exp_vars.remove(resp_var)

#select predictor variables and target variable as separate data sets predvar= data_clean[exp_vars]

target = data_clean[resp_var]

# standardize predictors to have mean=0 and sd=1* predictors=predvar.copy() from sklearn import preprocessing

for key in exp_vars: predictors[key]=preprocessing.scale(predictors[key].astype('float64'))

# split data into train and test sets pred_train, pred_test, tar_train, tar_test = train_test_split(predictors, target, test_size=.3, random_state=123)

# specify the lasso regression model model=LassoLarsCV(cv=10, precompute=False).fit(pred_train,tar_train)

# print variable names and regression coefficients res_dict = dict(zip(predictors.columns, model.coef_)) pred_dict = dict(sorted(res_dict.items(), key=lambda x: abs(x[1]))) printTableLabel('Coefficient Table (Sorted by Absolute Value)') pprint.pp(pred_dict)

# plot coefficient progression m_log_alphas = -np.log10(model.alphas_) ax = plt.gca() plt.plot(m_log_alphas, model.coef_path_.T) plt.axvline(-np.log10(model.alpha_), linestyle='--', color='k', label='alpha CV') plt.ylabel('Regression Coefficients') plt.xlabel('-log(alpha)') plt.title('Regression Coefficients Progression for Lasso Paths')

# plot mean square error for each fold m_log_alphascv = -np.log10(model.cv_alphas_) plt.figure() plt.plot(m_log_alphascv, model.mse_path_, ':') plt.plot(m_log_alphascv, model.mse_path_.mean(axis=-1), 'k', label='Average across the folds', linewidth=2) plt.axvline(-np.log10(model.alpha_), linestyle='--', color='k', label='alpha CV') plt.legend() plt.xlabel('-log(alpha)') plt.ylabel('Mean squared error') plt.title('Mean squared error on each fold')

# MSE from training and test data from sklearn.metrics import mean_squared_error train_error = mean_squared_error(tar_train, model.predict(pred_train)) test_error = mean_squared_error(tar_test, model.predict(pred_test)) printTableLabel('Training Data Mean Square Error') print(train_error) printTableLabel('Test Data Mean Square Error') print(test_error)

# R-square from training and test data rsquared_train=model.score(pred_train,tar_train) rsquared_test=model.score(pred_test,tar_test) printTableLabel('Training Data R-Square ') print(rsquared_train) printTableLabel('Test Data R-Square ') print(rsquared_test)

======== Program Source End =======

0 notes

supplementmegamart-blog · 7 years ago

Photo

KetoFit is a dynamic new supplement to empower you To expend FAT FOR FUEL. Utilizing the freshest advancement in the supplement business, KetoFit will empower you to get more fit, increase essentialness, get more grounded and HEAL AT THE CELLULAR LEVEL. Exogenous ketones is the thing that makes KetoFit so extraordinary, and we have 14 grams of it in each serving.

KetoFit is a keto keep away from sustenance pill that empowers people to get more fit and get fit without losing mass.

This is done by energizing the body to make greater imperativeness by devouring fat and not starches.

Keto Fit is a standout amongst the most current weight decrease pills open in the UK.

This supplement contains Beta-hydroxybutyrate (BHB) as its guideline settling. It boasts of growing imperativeness levels by 225% by expending wealth fat.

This without anyone else is the essential clarification behind the feasible weight decrease results I saw distinctive customers posting on the web.

How Does KetoFit Work?

HCA is a well-look at normal manufactured that is assigned a trademark hunger suppressant. It infers that you are less excited and eat less with this settling.

This is possible in light of the way that this compound forms the making of serotonin in the brain. Serotonin is the neurotransmitter that is responsible for your mentality.

Bigger measures of this neurotransmitter improve the control of your eating and longing for wants.

Another way this pill empowers you get more fit is by smothering Citrate Lyase.

This impetus is a synergist concoction for the unsaturated fat biosynthesis process. This is a technique that produces and stores fat. Furthermore, without it, you won't put on additional fat weight.

What Are The Key Benefits To Expect?

Here are the potential points of interest you can expect with this pill:

Stays away from Fat Creation And Storage: The fundamental favored angle of this pill is its ability to limit fat age and limit. This keeps away from weight put on without limiting weight decrease. It's an exceptional technique to shed pounds and keep up your ideal weight whole deal.

Covers Appetite: The benefit of wanting disguise is possible because of the lift in serotonin levels. The upgraded serotonin levels empower you to control your excited eating for more trademark weight decrease.

Upgrades Metabolic Rate: The fat disaster pill upgrades your basal metabolic rate. This revives your body's processing of set away fat. Essentially review that this works best when you eat relentlessly and rehearse reliably.

Assembles Energy Production: Upgraded processing underpins the body's change of set away fat into imperativeness. This imperativeness diminishes weariness, improve recovery, help in weight decrease, and upgrade the general estimation of flourishing.

Upgrades Mood: The livening up is possible in view of the development in serotonin and diminishing in cortisollevels. Cortisol is the weight hormone that conservatives fat disaster while regularly extending fat storing.

Support Blood Glucose And Cholesterol Levels: KetoFit cuts down your LDL cholesterol and grows your HDL cholesterol levels. The body changes over set away fat as opposed to starches for imperativeness. This sponsorships all the more enduring glucose levels.

What Is Ketosis?

Our body cells anticipate that essentialness will perform required activities. This imperativeness is created when our fat and muscle cells are changed over to glucose.

This method is for the most part direct and it can take numerous months to lose stores of muscle to fat proportion.

In a territory of Ketosis, muscle to fat proportion is clearly changed over into ketones. These ketones are used as fuel for the cells.

Right when blood Ketones have a higher obsession than 0.5 mmol/L, the body is in the state of restorative ketosis.

Healthy Ketosis can be proficient by taking keto check calories pills like KetoFit.

The other option is to go on a ketogenic eat less that is high in fat, coordinate in protein and low in starches.

The benefits of ketosis join a lift in weight decrease, tendency, work out, mental execution, and ailment neutralizing activity.

The Active KetoFit Ingredient:

KetoFit Weight Loss Pills contain Beta-hydroxybutyrate.

This earth shattering settling is moreover named as BHB, 3-hydroxybutyric destructive and 3-hydroxybutanoic destructive (3HB).

BHB is one of three ketone bodies that furthermore joins Acetoacetate (AcAc) and Acetone.

BHB is most plenteous in the blood with 78% of total ketones being BHB.

Acetoacetate and Acetone are contained 20% and 2%, independently, of the total ketones in the body.

From the three ketone bodies, BHB is best at helping you get more fit.

This is by virtue of it is unlimited and adequately stable to be used over a broad stretch of time.

BHB salts in dietary supplements like KetoFit Pills help with giving various favorable circumstances to the prosperity.

Especially with respect to getting fit as a fiddle and upgrading exercise execution.

It improves fat devour without diminishing mass while upgrading recovery from depletion and quickly revives glycogen levels.

How To Buy KetoFit Weight Loss Pills?

I watched out for Amazon, GNC, and Wal-Mart with no say of KetoFit pills in their structures. Thusly, it creates the impression that the primary certified way we can get these tablets is through the official site.

A cautious and secure package is used to pass on KetoFit pills.

What's My Final Verdict On Buying This Fat Loss Pill?

Thusly, as it's plainly obvious, I found some extraordinary and some unpleasant things about this thing. The favorable circumstances are achievable, and the thing is of OK quality. In any case, the downside of being too much expensive and using shady business procedures are accessible.

Order Now - https://www.supplementmegamart.com/ketofit/

#KetoFit #KetoFit Trial #KetoFit Review #KetoFit Ingredients #KetoFit Free Trial #KetoFit Weight Loss #Where To Buy KetoFit #KetoFit Cons #KetoFit Pros #KetoFit SCAM

0 notes

displaysshopsrated-blog · 8 years ago

Text

Slatwall Displays For Shops

Try the bullet record of merchandise that Fashionable Plastics Designs offers -slatwall systems, slatwall hooks, slat wall hooks, peg board hooks, peg hooks, slatwall panels, pegboard accessories, gondola shelving, slatwall shelves, and slatwall fixtures. These are the kinds of questions you will need to ask your self as you are making the final choices about your new candy store's acrylic displays. For an improved exposure of your merchandise merchandise, we have now acrylic and chrome slatwall equipment which are really versatile and fast to put in and use in your shop. Use slatwall and slatwall panels to give texture to partitions round the focus. For top rated slatwall displays go to gogopanels.com website.

Brochure Holders are additionally very useful for exhibitions and commerce exhibits, being light-weight and very moveable. There is a large number of colourful PVC and aluminium slatwall inserts to choose from, enabling you to create a placing and distinctive show suited to your particular person type. Delivery of slatwall panels is costly resulting from their measurement and weight, it is important to understand how they will be delivered before buying. The slatboard can also be very best for retail enviroments and can accommodate small acrylic cabinets & product racks.

Slatwall Displays

Whereas new shelving may require only a fast wipe down and it is ready to go, cleansing used fixtures can add hours of additional time to the method as you make those used fixtures appear like new once more. If you happen to own or operate an automotive parts store, listed here are a few of the best decisions of retail display fixtures on your establishment. One thing completely different, one thing clean, one thing clear, the Round Cell Cellphone Show Group is a winner each time. Discovering the highest acrylic manufacturer is rarely an issue due to Modern Plastic Designs in Nyc. Slatwall is flexible so you should utilize it the place you wish to display many products.

Fixtures are objects which used by retailer owners to serve objective corresponding to for show, to accentuate, and to maximise space. Retailer Fixtures or showroom displays are used for visual merchandising, show merchandise and in addition create a center of attention among the shopper. And you never know once you'll need to do something completely different together with your clear acrylic trays, so it is a good suggestion to keep several of every design on hand. You should definitely get some additional equipment like hooks, locks and lighting along with quality retail shop shows and counters sold in our UK based mostly store. We also have hundreds of slatwall hooks, fittings and cabinets in metal and acrylic / perspex.

Many alternative equipment are available for a Slatwall Panel for hanging bagged items or goods on a peg kind hook, a novel shelf system can be configured on Slatwall creating a large part of wall shelving that will not take loads of store space, hangers for every type of Clothes, T-Shirts and Ball Caps.

Slatwall shows are useful retail show racks as a result of fact these merchandise are a certain assurance to make your clients fascinated in the products you showcase. Promote Your self : - A mannequin may also be used to promote the shop by capturing the passing customer's attention and attractive them to enter and look at the objects for sale. Mannequins Shows: - It's wonderful approach to be a focus for customers towards business by displaying merchandise by mannequins.

As the retailer you could have the right retail display fixtures to show the different merchandise in your store. After getting these tools, you'll be able to situate the display rack somewhere on your retailer's flooring that may provide the best visibility and convenience after which position your acrylic bins on the rack. Trendy retailer fixtures introduce slatwall equipment all styles and sizes to accommodate the many show needs that retailers have for varied products.

Our Maple Slatwall, which has a melamine finish, is an outstanding and efficient way of displaying a variety of merchandise by using a wide range of our slatwall accessories. Slatwall display fixtures are one of many ways that you can use to showcase your products in a beautiful manner.

Even when you've employed a high-notch retail retailer designer, he or she should be buying your fixtures and other materials at costs even higher than wholesale. Slatwall Towers and Gondolas Displays: - Slatwall Tower and Gondolas with advance features, interior design, cost-effective and extra over our display merchandisers are visually interesting and appeal to buyers with their size. Then measure one foot and one quarter inch from the bottom point of the ceiling and utilizing the level and chalk line make one other line where you plane to put up the slatwall panel.

Typically finished with any combination of Low or High Stress Laminates and obtainable with extras akin to casters, groove inserts, logos and more, these fixtures haven't any bother displaying practically any sort of merchandise effectively and effectively for slatwall displays.

The three most typical places to create candy displays are on countertops or tables (acrylic round face bins, acrylic mini bins, acrylic gravity bins, and acrylic towers work nicely in these display areas), on your store's ground area (where you need to use any of the previous acrylic bulk dispensers paired with other display fixtures like display racks), and alongside your store's wall (by which case you will want wall shelves or slatwall fixtures).

As an acrylic fabrication firm, we are able to carry your thoughts about an acrylic wall display from concept into bodily product shortly and simply. Whether or not it's clothes, accessories, or some other product, Slatwall can be organized to successfully display anything. Ensure your retail fixtures will suit your store's theme and are properly positioned. I like to recommend altering shows month-to-month, so a topic like Hispanic Heritage Month works great for that. Love books - cannot afford all I want, so haunt two beautiful guide trade outlets that additionally make delightful shows - and the store owner and I DO love to chat about books and 'good reads'. The intelligent Acrylic Countertop Mini Cell Cellphone Hand will take your cellphone from its hand, and put it into theirs. The plastic strips hook up with the edges of the slatwall to make sure angles.

With the rise in choices for finishes and laminates used to customize slatwall displays over time, you should have ample choices to find a display that fits your retailer's or business' goal, branding theme and budget. Coloured plastic containers are nice ways to boost your merchandise shows or organize the merchandise you're display.

For mannequin options over USA, Fashionable Store Fixtures among the finest retail retailer design and custom retailer fixtures manufacturing firm in New York and New Jersey (U.S.A). We serve with broad experience and capabilities to design your retail retailer that help to draw prospects in the direction of respective businesses.

For more modular show wants like commerce reveals, job gala's, gross sales conferences, and out of doors occasions, GOGO Panels rugged ABS shows are the way in which to go. Fixtures are most reasonable and dynamic approach for success of any retail enterprise or showroom. Commonplace slatwall finish: With a sturdy stable shade or patterend Melamine Low Pressure Laminate (LPL) finish.

The same site has loads of good info on set up, and so on, and primarily based on the data I've been studying, it appears to be like like I will in all probability up the number of screws I've used in my set up. It's vital for purchasers to find a reliable supply for signal holders, acrylic bins, acrylic sign holders, and acrylic show stands and Slatwall Bin related merchandise in Miami, FL. Finding this will likely have been difficult prior to now. We supply a big selection of wire and slat grid panel store fixtures, plus grid hardware and equipment that will help you create elegant and practical retail displays. Retail shows are actually not an overhead price - they are nearly essentially the most crucial parts of your sales team.

You're most likely in no way surprised to see display racks listed as one of the high three display fixtures for pet shops - or any type of store, for that matter. So, if you are planning to buy a number of the automobile equipment then, go for them and make one of the best out of them. They're usually spinning shows which provide a large amount of area in proportion to their dimension. We are the one of the leading suppliers for advertising retailer display collections, dwelling enchancment merchandise and likewise retail retailer fixtures.

There is no such thing as a doubt that the thought for acrylic retail displays begins in your thoughts, and with the help of an acrylic fabrication or plastic fabrication company, you can convey that concept from thought into reality. Our slatwall panels are incredibly versatile and you may simply connect shelving accessories to your slatwall panels together with hooks and acrylic shelves. Business homeowners get the engaging approach to publicity to their retailer products and customer with the assistance of fixtures. Few shows like these can have each an extremely-light-weight design, that minimizes delivery cost and gives a solution that every one businesses can benefit from. Slatwall Show Cases can be utilized with priceless items that you'll want to hold out of consumers arms. Acrylic mini bins are just like spherical face bins with two variations: Their hinged lids aren't round and, true to their name, they are much smaller.

Slatwall shows can really showcase your objects so choose superior high quality ones that occur to last for years. When you're fascinated about utilizing slatwall display fixtures in your retailer however aren't quite certain tips on how to get started, take a look at the five ideas beneath. Sumetall slatwall panel is product of Medium Density Fiberboard (MDF), which is a high quality engineered timber product made with hardwood/softwood fibers. Thanks for submitting your assessment of Kin-Lai Retailer Fixtures Ltd.. We'll e mail you as quickly as it's published, (typically within forty eight hours). Your article exhibits how fascinating and inventive their work is. Thanks for an inspiring lens! Palay Show shares one of the strongest slatwall products produced in the business at present with slatwall displays.

youtube

#slatwall displays

0 notes

spottypotatostuff · 8 years ago

Text

How to Build Beautiful 3-D Fractals Out of the Simplest Equations

Ordinary equations can be transformed into complex 3-D figures that offer new questions to explore.Olena Shmahalo/Quanta Magazine; original figure by Laurent Bartholdi and Laura DeMarco

If you came across an animal in the wild and wanted to learn more about it, there are a few things you might do: You might watch what it eats, poke it to see how it reacts, and even dissect it if you got the chance.

Mathematicians are not so different from naturalists. Rather than studying organisms, they study equations and shapes using their own techniques. They twist and stretch mathematical objects, translate them into new mathematical languages, and apply them to new problems. As they find new ways to look at familiar things, the possibilities for insight multiply.

That’s the promise of a new idea from two mathematicians: Laura DeMarco, a professor at Northwestern University, and Kathryn Lindsey, a postdoctoral fellow at the University of Chicago. They begin with a plain old polynomial equation, the kind grudgingly familiar to any high school math student: f(x) = x2 – 1. Instead of graphing it or finding its roots, they take the unprecedented step of transforming it into a 3-D object.

With polynomials, “everything is defined in the two-dimensional plane,” Lindsey said. “There isn’t a natural place a third dimension would come into it until you start thinking about these shapes Laura and I are building.”

The 3-D shapes that they build look strange, with broad plains, subtle bends and a zigzag seam that hints at how the objects were formed. DeMarco and Lindsey introduce the shapes in a forthcoming paper in the Arnold Mathematical Journal, a new publication from the Institute for Mathematical Sciences at Stony Brook University. The paper presents what little is known about the objects, such as how they’re constructed and the measurements of their curvature. DeMarco and Lindsey also explain what they believe is a promising new method of inquiry: Using the shapes built from polynomial equations, they hope to come to understand more about the underlying equations—which is what mathematicians really care about.

Breaking Out of Two Dimensions

In mathematics, several motivating factors can spur new research. One is the quest to solve an open problem, such as the Riemann hypothesis. Another is the desire to build mathematical tools that can be used to do something else. A third—the one behind DeMarco and Lindsey’s work—is the equivalent of finding an unidentified species in the wild: One just wants to understand what it is. “These are fascinating and beautiful things that arise very naturally in our subject and should be understood!” DeMarco said by email, referring to the shapes.

Laura DeMarco, a professor at Northwestern University.Courtesy of Laura DeMarco

“It’s sort of been in the air for a couple of decades, but they’re the first people to try to do something with it,” said Curtis McMullen, a mathematician at Harvard University who won the Fields Medal, math’s highest honor, in 1988. McMullen and DeMarco started talking about these shapes in the early 2000s, while she was doing graduate work with him at Harvard. DeMarco then went off to do pioneering work applying techniques from dynamical systems to questions in number theory, for which she will receive the Satter Prize—awarded to a leading female researcher—from the American Mathematical Society on January 5.

Meanwhile, in 2010 William Thurston, the late Cornell University mathematician and Fields Medal winner, heard about the shapes from McMullen. Thurston suspected that it might be possible to take flat shapes computed from polynomials and bend them to create 3-D objects. To explore this idea, he and Lindsey, who was then a graduate student at Cornell, constructed the 3-D objects from construction paper, tape and a precision cutting device that Thurston had on hand from an earlier project. The result wouldn’t have been out of place at an elementary school arts and crafts fair, and Lindsey admits she was kind of mystified by the whole thing.

“I never understood why we were doing this, what the point was and what was going on in his mind that made him think this was really important,” said Lindsey. “Then unfortunately when he died, I couldn’t ask him anymore. There was this brilliant guy who suggested something and said he thought it was an important, neat thing, so it’s natural to wonder ‘What is it? What’s going on here?’”

In 2014 DeMarco and Lindsey decided to see if they could unwind the mathematical significance of the shapes.

A Fractal Link to Entropy

To get a 3-D shape from an ordinary polynomial takes a little doing. The first step is to run the polynomial dynamically—that is, to iterate it by feeding each output back into the polynomial as the next input. One of two things will happen: either the values will grow infinitely in size, or they’ll settle into a stable, bounded pattern. To keep track of which starting values lead to which of those two outcomes, mathematicians construct the Julia set of a polynomial. The Julia set is the boundary between starting values that go off to infinity and values that remain bounded below a given value. This boundary line—which differs for every polynomial—can be plotted on the complex plane, where it assumes all manner of highly intricate, swirling, symmetric fractal designs.

Lucy Reading-Ikkanda/Quanta Magazine

If you shade the region bounded by the Julia set, you get the filled Julia set. If you use scissors and cut out the filled Julia set, you get the first piece of the surface of the eventual 3-D shape. To get the second, DeMarco and Lindsey wrote an algorithm. That algorithm analyzes features of the original polynomial, like its degree (the highest number that appears as an exponent) and its coefficients, and outputs another fractal shape that DeMarco and Lindsey call the “planar cap.”

“The Julia set is the base, like the southern hemisphere, and the cap is like the top half,” DeMarco said. “If you glue them together you get a shape that’s polyhedral.”

The algorithm was Thurston’s idea. When he suggested it to Lindsey in 2010, she wrote a rough version of the program. She and DeMarco improved on the algorithm in their work together and “proved it does what we think it does,” Lindsey said. That is, for every filled Julia set, the algorithm generates the correct complementary piece.

The filled Julia set and the planar cap are the raw material for constructing a 3-D shape, but by themselves they don’t give a sense of what the completed shape will look like. This creates a challenge. When presented with the six faces of a cube laid flat, one could intuitively know how to fold them to make the correct 3-D shape. But, with a less familiar two-dimensional surface, you’d be hard-pressed to anticipate the shape of the resulting 3-D object.

“There’s no general mathematical theory that tells you what the shape will be if you start with different types of polygons,” Lindsey said.

Mathematicians have precise ways of defining what makes a shape a shape. One is to know its curvature. Any 3-D object without holes has a total curvature of exactly 4π; it’s a fixed value in the same way any circular object has exactly 360 degrees of angle. The shape—or geometry—of a 3-D object is completely determined by the way that fixed amount of curvature is distributed, combined with information about distances between points. In a sphere, the curvature is distributed evenly over the entire surface; in a cube, it’s concentrated in equal amounts at the eight evenly spaced vertices.

A unique attribute of Julia sets allows DeMarco and Lindsey to know the curvature of the shapes they’re building. All Julia sets have what’s known as a “measure of maximal entropy,” or MME. The MME is a complicated concept, but there is an intuitive (if slightly incomplete) way to think about it. First, picture a two-dimensional filled Julia set on the plane. Then picture a point on the same plane but very far outside the Julia set’s boundary (infinitely far, in fact). From that distant location the point is going to take a random walk across two-dimensional space, meandering until it strikes the Julia set. Wherever it first strikes the Julia set is where it comes to rest.

The MME is a way of quantifying the fact that the meandering point is more likely to strike certain parts of the Julia set than others. For example, the meandering point is more likely to strike a spike in the Julia set that juts out into the plane than it is to intersect with a crevice tucked into a region of the set. The more likely the meandering point is to hit a point on the Julia set, the higher the MME is at that point.

In their paper, DeMarco and Lindsey demonstrated that the 3-D objects they build from Julia sets have a curvature distribution that’s exactly proportional to the MME. That is, if there’s a 25 percent chance the meandering point will hit a particular place on the Julia set first, then 25 percent of the curvature should also be concentrated at that point when the Julia set is joined with the planar cap and folded into a 3-D shape.

“If it was really easy for the meandering point to hit some area on our Julia set we’d want to have a lot of curvature at the corresponding point on the 3-D object,” Lindsey said. “And if it was harder to hit some area on our Julia set, we’d want the corresponding area in the 3-D object to be kind of flat.”

This is useful information, but it doesn’t get you as far as you’d think. If given a two-dimensional polygon, and told exactly how its curvature should be distributed, there’s still no mathematical way to identify exactly where you need to fold the polygon to end up with the right 3-D shape. Because of this, there’s no way to completely anticipate what that 3-D shape will look like.

“We know how sharp and pointy the shape has to be, in an abstract, theoretical sense, and we know how far apart the crinkly regions are, again in an abstract, theoretical sense, but we have no idea how to visualize it in three dimensions,” DeMarco explained in an email.

She and Lindsey have evidence of the existence of a 3-D shape, and evidence of some of that shape’s properties, but no ability yet to see the shape. They are in a position similar to that of astronomers who detect an unexplained stellar wobble that hints at the existence of an exoplanet: The astronomers know there has to be something else out there and they can estimate its mass. Yet the object itself remains just out of view.

A Folding Strategy

Thus far, DeMarco and Lindsey have established basic details of the 3-D shape: They know that one 3-D object exists for every polynomial (by way of its Julia set), and they know the object has a curvature exactly given by the measure of maximal entropy. Everything else has yet to be figured out.

In particular, they’d like to develop a mathematical understanding of the “bending laminations,” or lines along which a flat surface can be folded to create a 3-D object. The question occurred early on to Thurston, too, who wrote to McMullen in 2010, “I wonder how hard it is to compute or characterize the pair of bending laminations, for the inside and the outside, and what they might tell us about the geometry of the Julia set.”

Kathryn Lindsey, a mathematician at the University of Chicago.Courtesy of Kathryn Lindsey

In this, DeMarco and Lindsey’s work is heavily influenced by the mid 20th-century mathematician Aleksandr Aleksandrov. Aleksandrov established that there is only one unique way of folding a given polygon to get a 3-D object. He lamented that it seemed impossible to mathematically calculate the correct folding lines. Today, the best strategy is often to make a best guess about where to fold the polygon—and then to get out scissors and tape to see if the estimate is right.

“Kathryn and I spent hours cutting out examples and gluing them ourselves,” DeMarco said.

DeMarco and Lindsey are currently trying to describe the folding lines on their particular class of 3-D objects, and they think they have a promising strategy. “Our working conjecture is that the folding lines, the bending laminations, can be completely described in terms of certain dynamical properties,” DeMarco said. Put another way, they hope that by iterating the underlying polynomial in the right way, they’ll be able to identify the set of points along which the folding line occurs.

From there, possibilities for exploration are numerous. If you know the folding lines associated to the polynomial f(x) = x2– 1, you might then ask what happens to the folding lines if you change the coefficients and consider f(x) = x2 – 1.1. Do the folding lines of the two polynomials differ a little, a lot or not at all?

“Certain polynomials might have similar bending laminations, and that would tell us all these polynomials have something in common, even if on the surface they don’t look like they have anything in common,” Lindsey said.

It’s a bit early to think about all of this, however. DeMarco and Lindsey have found a systematic way to think about polynomials in 3-D terms, but whether that perspective will answer important questions about those polynomials is unclear.

“I would even characterize it as being sort of playful at this stage,” McMullen said, adding, “In a way that’s how some of the best mathematical research proceeds—you don’t know what something is going to be good for, but it seems to be a feature of the mathematical landscape.”

Source: How to Build Beautiful 3-D Fractals Out of the Simplest Equations

The post How to Build Beautiful 3-D Fractals Out of the Simplest Equations appeared on Spotty Potato.

0 notes

d33-alex · 8 years ago

Text

The Trouble with Quantum Mechanics

The physicist Eric J. Heller’s Transport XIII, inspired by electron flow experiments conducted at Harvard. According to Heller, the image ‘shows two kinds of chaos: a random quantum wave on the surface of a sphere, and chaotic classical electron paths in a semiconductor launched over a range of angles from a particular point. Even though one is quantum mechanical and the other classical, they are related: the chaotic classical paths cause random quantum waves to appear when the classical system is solved quantum mechanically.’

The development of quantum mechanics in the first decades of the twentieth century came as a shock to many physicists. Today, despite the great successes of quantum mechanics, arguments continue about its meaning, and its future.

The first shock came as a challenge to the clear categories to which physicists by 1900 had become accustomed. There were particles—atoms, and then electrons and atomic nuclei—and there were fields—conditions of space that pervade regions in which electric, magnetic, and gravitational forces are exerted. Light waves were clearly recognized as self-sustaining oscillations of electric and magnetic fields. But in order to understand the light emitted by heated bodies, Albert Einstein in 1905 found it necessary to describe light waves as streams of massless particles, later called photons.

Then in the 1920s, according to theories of Louis de Broglie and Erwin Schrödinger, it appeared that electrons, which had always been recognized as particles, under some circumstances behaved as waves. In order to account for the energies of the stable states of atoms, physicists had to give up the notion that electrons in atoms are little Newtonian planets in orbit around the atomic nucleus. Electrons in atoms are better described as waves, fitting around the nucleus like sound waves fitting into an organ pipe.1 The world’s categories had become all muddled.

Worse yet, the electron waves are not waves of electronic matter, in the way that ocean waves are waves of water. Rather, as Max Born came to realize, the electron waves are waves of probability. That is, when a free electron collides with an atom, we cannot in principle say in what direction it will bounce off. The electron wave, after encountering the atom, spreads out in all directions, like an ocean wave after striking a reef. As Born recognized, this does not mean that the electron itself spreads out. Instead, the undivided electron goes in some one direction, but not a precisely predictable direction. It is more likely to go in a direction where the wave is more intense, but any direction is possible.

Probability was not unfamiliar to the physicists of the 1920s, but it had generally been thought to reflect an imperfect knowledge of whatever was under study, not an indeterminism in the underlying physical laws. Newton’s theories of motion and gravitation had set the standard of deterministic laws. When we have reasonably precise knowledge of the location and velocity of each body in the solar system at a given moment, Newton’s laws tell us with good accuracy where they will all be for a long time in the future. Probability enters Newtonian physics only when our knowledge is imperfect, as for example when we do not have precise knowledge of how a pair of dice is thrown. But with the new quantum mechanics, the moment-to-moment determinism of the laws of physics themselves seemed to be lost.

All very strange. In a 1926 letter to Born, Einstein complained:

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.2

As late as 1964, in his Messenger lectures at Cornell, Richard Feynman lamented, “I think I can safely say that no one understands quantum mechanics.”3 With quantum mechanics, the break with the past was so sharp that all earlier physical theories became known as “classical.”

The weirdness of quantum mechanics did not matter for most purposes. Physicists learned how to use it to do increasingly precise calculations of the energy levels of atoms, and of the probabilities that particles will scatter in one direction or another when they collide. Lawrence Krauss has labeled the quantum mechanical calculation of one effect in the spectrum of hydrogen “the best, most accurate prediction in all of science.”4 Beyond atomic physics, early applications of quantum mechanics listed by the physicist Gino Segrè included the binding of atoms in molecules, the radioactive decay of atomic nuclei, electrical conduction, magnetism, and electromagnetic radiation.5 Later applications spanned theories of semiconductivity and superconductivity, white dwarf stars and neutron stars, nuclear forces, and elementary particles. Even the most adventurous modern speculations, such as string theory, are based on the principles of quantum mechanics.

Many physicists came to think that the reaction of Einstein and Feynman and others to the unfamiliar aspects of quantum mechanics had been overblown. This used to be my view. After all, Newton’s theories too had been unpalatable to many of his contemporaries. Newton had introduced what his critics saw as an occult force, gravity, which was unrelated to any sort of tangible pushing and pulling, and which could not be explained on the basis of philosophy or pure mathematics. Also, his theories had renounced a chief aim of Ptolemy and Kepler, to calculate the sizes of planetary orbits from first principles. But in the end the opposition to Newtonianism faded away. Newton and his followers succeeded in accounting not only for the motions of planets and falling apples, but also for the movements of comets and moons and the shape of the earth and the change in direction of its axis of rotation. By the end of the eighteenth century this success had established Newton’s theories of motion and gravitation as correct, or at least as a marvelously accurate approximation. Evidently it is a mistake to demand too strictly that new physical theories should fit some preconceived philosophical standard.

In quantum mechanics the state of a system is not described by giving the position and velocity of every particle and the values and rates of change of various fields, as in classical physics. Instead, the state of any system at any moment is described by a wave function, essentially a list of numbers, one number for every possible configuration of the system.6 If the system is a single particle, then there is a number for every possible position in space that the particle may occupy. This is something like the description of a sound wave in classical physics, except that for a sound wave a number for each position in space gives the pressure of the air at that point, while for a particle in quantum mechanics the wave function’s number for a given position reflects the probability that the particle is at that position. What is so terrible about that? Certainly, it was a tragic mistake for Einstein and Schrödinger to step away from using quantum mechanics, isolating themselves in their later lives from the exciting progress made by others.

Even so, I’m not as sure as I once was about the future of quantum mechanics. It is a bad sign that those physicists today who are most comfortable with quantum mechanics do not agree with one another about what it all means. The dispute arises chiefly regarding the nature of measurement in quantum mechanics. This issue can be illustrated by considering a simple example, measurement of the spin of an electron. (A particle’s spin in any direction is a measure of the amount of rotation of matter around a line pointing in that direction.)

All theories agree, and experiment confirms, that when one measures the amount of spin of an electron in any arbitrarily chosen direction there are only two possible results. One possible result will be equal to a positive number, a universal constant of nature. (This is the constant that Max Planck originally introduced in his 1900 theory of heat radiation, denoted h, divided by 4π.) The other possible result is its opposite, the negative of the first. These positive or negative values of the spin correspond to an electron that is spinning either clockwise or counter-clockwise in the chosen direction.

But it is only when a measurement is made that these are the sole two possibilities. An electron spin that has not been measured is like a musical chord, formed from a superposition of two notes that correspond to positive or negative spins, each note with its own amplitude. Just as a chord creates a sound distinct from each of its constituent notes, the state of an electron spin that has not yet been measured is a superposition of the two possible states of definite spin, the superposition differing qualitatively from either state. In this musical analogy, the act of measuring the spin somehow shifts all the intensity of the chord to one of the notes, which we then hear on its own.

This can be put in terms of the wave function. If we disregard everything about an electron but its spin, there is not much that is wavelike about its wave function. It is just a pair of numbers, one number for each sign of the spin in some chosen direction, analogous to the amplitudes of each of the two notes in a chord.7 The wave function of an electron whose spin has not been measured generally has nonzero values for spins of both signs.

There is a rule of quantum mechanics, known as the Born rule, that tells us how to use the wave function to calculate the probabilities of getting various possible results in experiments. For example, the Born rule tells us that the probabilities of finding either a positive or a negative result when the spin in some chosen direction is measured are proportional to the squares of the numbers in the wave function for those two states of the spin.8

The introduction of probability into the principles of physics was disturbing to past physicists, but the trouble with quantum mechanics is not that it involves probabilities. We can live with that. The trouble is that in quantum mechanics the way that wave functions change with time is governed by an equation, the Schrödinger equation, that does not involve probabilities. It is just as deterministic as Newton’s equations of motion and gravitation. That is, given the wave function at any moment, the Schrödinger equation will tell you precisely what the wave function will be at any future time. There is not even the possibility of chaos, the extreme sensitivity to initial conditions that is possible in Newtonian mechanics. So if we regard the whole process of measurement as being governed by the equations of quantum mechanics, and these equations are perfectly deterministic, how do probabilities get into quantum mechanics?

One common answer is that, in a measurement, the spin (or whatever else is measured) is put in an interaction with a macroscopic environment that jitters in an unpredictable way. For example, the environment might be the shower of photons in a beam of light that is used to observe the system, as unpredictable in practice as a shower of raindrops. Such an environment causes the superposition of different states in the wave function to break down, leading to an unpredictable result of the measurement. (This is called decoherence.) It is as if a noisy background somehow unpredictably left only one of the notes of a chord audible. But this begs the question. If the deterministic Schrödinger equation governs the changes through time not only of the spin but also of the measuring apparatus and the physicist using it, then the results of measurement should not in principle be unpredictable. So we still have to ask, how do probabilities get into quantum mechanics?

One response to this puzzle was given in the 1920s by Niels Bohr, in what came to be called the Copenhagen interpretation of quantum mechanics. According to Bohr, in a measurement the state of a system such as a spin collapses to one result or another in a way that cannot itself be described by quantum mechanics, and is truly unpredictable. This answer is now widely felt to be unacceptable. There seems no way to locate the boundary between the realms in which, according to Bohr, quantum mechanics does or does not apply. As it happens, I was a graduate student at Bohr’s institute in Copenhagen, but he was very great and I was very young, and I never had a chance to ask him about this.

Today there are two widely followed approaches to quantum mechanics, the “realist” and “instrumentalist” approaches, which view the origin of probability in measurement in two very different ways.9 For reasons I will explain, neither approach seems to me quite satisfactory.10

The instrumentalist approach is a descendant of the Copenhagen interpretation, but instead of imagining a boundary beyond which reality is not described by quantum mechanics, it rejects quantum mechanics altogether as a description of reality. There is still a wave function, but it is not real like a particle or a field. Instead it is merely an instrument that provides predictions of the probabilities of various outcomes when measurements are made.

It seems to me that the trouble with this approach is not only that it gives up on an ancient aim of science: to say what is really going on out there. It is a surrender of a particularly unfortunate kind. In the instrumentalist approach, we have to assume, as fundamental laws of nature, the rules (such as the Born rule I mentioned earlier) for using the wave function to calculate the probabilities of various results when humans make measurements. Thus humans are brought into the laws of nature at the most fundamental level. According to Eugene Wigner, a pioneer of quantum mechanics, “it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.”11

Thus the instrumentalist approach turns its back on a vision that became possible after Darwin, of a world governed by impersonal physical laws that control human behavior along with everything else. It is not that we object to thinking about humans. Rather, we want to understand the relation of humans to nature, not just assuming the character of this relation by incorporating it in what we suppose are nature’s fundamental laws, but rather by deduction from laws that make no explicit reference to humans. We may in the end have to give up this goal, but I think not yet.

Some physicists who adopt an instrumentalist approach argue that the probabilities we infer from the wave function are objective probabilities, independent of whether humans are making a measurement. I don’t find this tenable. In quantum mechanics these probabilities do not exist until people choose what to measure, such as the spin in one or another direction. Unlike the case of classical physics, a choice must be made, because in quantum mechanics not everything can be simultaneously measured. As Werner Heisenberg realized, a particle cannot have, at the same time, both a definite position and a definite velocity. The measuring of one precludes the measuring of the other. Likewise, if we know the wave function that describes the spin of an electron we can calculate the probability that the electron would have a positive spin in the north direction if that were measured, or the probability that the electron would have a positive spin in the east direction if that were measured, but we cannot ask about the probability of the spins being found positive in both directions because there is no state in which an electron has a definite spin in two different directions.

These problems are partly avoided in the realist—as opposed to the instrumentalist—approach to quantum mechanics. Here one takes the wave function and its deterministic evolution seriously as a description of reality. But this raises other problems.

Erwin Schrödinger; drawing by David Levine

The realist approach has a very strange implication, first worked out in the 1957 Princeton Ph.D. thesis of the late Hugh Everett. When a physicist measures the spin of an electron, say in the north direction, the wave function of the electron and the measuring apparatus and the physicist are supposed, in the realist approach, to evolve deterministically, as dictated by the Schrödinger equation; but in consequence of their interaction during the measurement, the wave function becomes a superposition of two terms, in one of which the electron spin is positive and everyone in the world who looks into it thinks it is positive, and in the other the spin is negative and everyone thinks it is negative. Since in each term of the wave function everyone shares a belief that the spin has one definite sign, the existence of the superposition is undetectable. In effect the history of the world has split into two streams, uncorrelated with each other.

This is strange enough, but the fission of history would not only occur when someone measures a spin. In the realist approach the history of the world is endlessly splitting; it does so every time a macroscopic body becomes tied in with a choice of quantum states. This inconceivably huge variety of histories has provided material for science fiction,12 and it offers a rationale for a multiverse, in which the particular cosmic history in which we find ourselves is constrained by the requirement that it must be one of the histories in which conditions are sufficiently benign to allow conscious beings to exist. But the vista of all these parallel histories is deeply unsettling, and like many other physicists I would prefer a single history.

There is another thing that is unsatisfactory about the realist approach, beyond our parochial preferences. In this approach the wave function of the multiverse evolves deterministically. We can still talk of probabilities as the fractions of the time that various possible results are found when measurements are performed many times in any one history; but the rules that govern what probabilities are observed would have to follow from the deterministic evolution of the whole multiverse. If this were not the case, to predict probabilities we would need to make some additional assumption about what happens when humans make measurements, and we would be back with the shortcomings of the instrumentalist approach. Several attempts following the realist approach have come close to deducing rules like the Born rule that we know work well experimentally, but I think without final success.

The realist approach to quantum mechanics had already run into a different sort of trouble long before Everett wrote about multiple histories. It was emphasized in a 1935 paper by Einstein with his coworkers Boris Podolsky and Nathan Rosen, and arises in connection with the phenomenon of “entanglement.”13

We naturally tend to think that reality can be described locally. I can say what is happening in my laboratory, and you can say what is happening in yours, but we don’t have to talk about both at the same time. But in quantum mechanics it is possible for a system to be in an entangled state that involves correlations between parts of the system that are arbitrarily far apart, like the two ends of a very long rigid stick.

For instance, suppose we have a pair of electrons whose total spin in any direction is zero. In such a state, the wave function (ignoring everything but spin) is a sum of two terms: in one term, electron A has positive spin and electron B has negative spin in, say, the north direction, while in the other term in the wave function the positive and negative signs are reversed. The electron spins are said to be entangled. If nothing is done to interfere with these spins, this entangled state will persist even if the electrons fly apart to a great distance. However far apart they are, we can only talk about the wave function of the two electrons, not of each separately. Entanglement contributed to Einstein’s distrust of quantum mechanics as much or more than the appearance of probabilities.

Strange as it is, the entanglement entailed by quantum mechanics is actually observed experimentally. But how can something so nonlocal represent reality?

What then must be done about the shortcomings of quantum mechanics? One reasonable response is contained in the legendary advice to inquiring students: “Shut up and calculate!” There is no argument about how to use quantum mechanics, only how to describe what it means, so perhaps the problem is merely one of words.

On the other hand, the problems of understanding measurement in the present form of quantum mechanics may be warning us that the theory needs modification. Quantum mechanics works so well for atoms that any new theory would have to be nearly indistinguishable from quantum mechanics when applied to such small things. But a new theory might be designed so that the superpositions of states of large things like physicists and their apparatus even in isolation suffer an actual rapid spontaneous collapse, in which probabilities evolve to give the results expected in quantum mechanics. The many histories of Everett would naturally collapse to a single history. The goal in inventing a new theory is to make this happen not by giving measurement any special status in the laws of physics, but as part of what in the post-quantum theory would be the ordinary processes of physics.

One difficulty in developing such a new theory is that we get no direction from experiment—all data so far agree with ordinary quantum mechanics. We do get some help, however, from some general principles, which turn out to provide surprisingly strict constraints on any new theory.

Obviously, probabilities must all be positive numbers, and add up to 100 percent. There is another requirement, satisfied in ordinary quantum mechanics, that in entangled states the evolution of probabilities during measurements cannot be used to send instantaneous signals, which would violate the theory of relativity. Special relativity requires that no signal can travel faster than the speed of light. When these requirements are put together, it turns out that the most general evolution of probabilities satisfies an equation of a class known as Lindblad equations.14 The class of Lindblad equations contains the Schrödinger equation of ordinary quantum mechanics as a special case, but in general these equations involve a variety of new quantities that represent a departure from quantum mechanics. These are quantities whose details of course we now don’t know. Though it has been scarcely noticed outside the theoretical community, there already is a line of interesting papers, going back to an influential 1986 article by Gian Carlo Ghirardi, Alberto Rimini, and Tullio Weber at Trieste, that use the Lindblad equations to generalize quantum mechanics in various ways.

Lately I have been thinking about a possible experimental search for signs of departure from ordinary quantum mechanics in atomic clocks. At the heart of any atomic clock is a device invented by the late Norman Ramsey for tuning the frequency of microwave or visible radiation to the known natural frequency at which the wave function of an atom oscillates when it is in a superposition of two states of different energy. This natural frequency equals the difference in the energies of the two atomic states used in the clock, divided by Planck’s constant. It is the same under all external conditions, and therefore serves as a fixed reference for frequency, in the way that a platinum-iridium cylinder at Sèvres serves as a fixed reference for mass.

Tuning the frequency of an electromagnetic wave to this reference frequency works a little like tuning the frequency of a metronome to match another metronome. If you start the two metronomes together and the beats still match after a thousand beats, you know that their frequencies are equal at least to about one part in a thousand. Quantum mechanical calculations show that in some atomic clocks the tuning should be precise to one part in a hundred million billion, and this precision is indeed realized. But if the corrections to quantum mechanics represented by the new terms in the Lindblad equations (expressed as energies) were as large as one part in a hundred million billion of the energy difference of the atomic states used in the clock, this precision would have been quite lost. The new terms must therefore be even smaller than this.

How significant is this limit? Unfortunately, these ideas about modifications of quantum mechanics are not only speculative but also vague, and we have no idea how big we should expect the corrections to quantum mechanics to be. Regarding not only this issue, but more generally the future of quantum mechanics, I have to echo Viola in Twelfth Night: “O time, thou must untangle this, not I.”

by Steven Weinberg, January 19, 2017 issue of The New York Review of Books – https://goo.gl/wWnS3Q

1. Conditions on sound waves at the closed or open ends of an organ pipe require that either an odd number of quarter wave lengths or an even or an odd number of half wave lengths must just fit into the pipe, which limits the possible notes that can be produced by the pipe. In an atom the wave function must satisfy conditions of continuity and finiteness close to and far from the nucleus, which similarly limit the possible energies of atomic states.

2. Quoted by Abraham Pais in ‘Subtle Is the Lord’: The Science and the Life of Albert Einstein (Oxford University Press, 1982), p. 443.

3. Richard Feynman, The Character of Physical Law (MIT Press, 1967), p. 129.

4. Lawrence M. Krauss, A Universe from Nothing (Free Press, 2012), p. 138.

5. Gino Segrè, Ordinary Geniuses (Viking, 2011).

6. These are complex numbers, that is, quantities of the general form a+ib, where a and b are ordinary real numbers and i is the square root of minus one.

7. Simple as it is, such a wave function incorporates much more information than just a choice between positive and negative spin. It is this extra information that makes quantum computers, which store information in this sort of wave function, so much more powerful than ordinary digital computers.

8. To be precise, these “squares” are squares of the absolute values of the complex numbers in the wave function. For a complex number of the form a+ib, the square of the absolute value is the square of a plus the square of b.

9. The opposition between these two approaches is nicely described by Sean Carroll in The Big Picture (Dutton, 2016).

10. I go into this in mathematical detail in Section 3.7 of Lectures on Quantum Mechanics, second edition (Cambridge University Press, 2015).

11. Quoted by Marcelo Gleiser, The Island of Knowledge (Basic Books, 2014), p. 222.

12. For instance, Northern Lights by Philip Pullman (Scholastic, 1995), and the early “Mirror, Mirror” episode of Star Trek.

13. Entanglement was recently discussed by Jim Holt in these pages, November 10, 2016.

14. This equation is named for Göran Lindblad, but it was also independently discovered by Vittorio Gorini, Andrzej Kossakowski, and George Sudarshan.

#science #quantum mechanics #quantum physics

0 notes