What I really want to say:why are negative numbers considered real they don't actually exist except if you use the example of debt but debt is a mere human construct it doesn't actually exist in reality it's an illusion created by man and fuelled by capitalism so back to negative numbers why don't they fit into complex numbers it is a true conundrum
The first day back from winter break can be restless.
Many children are still coming down from the excitement of the holidays. Two unstructured weeks away from school — with strange food, rituals and relatives — can be overwhelming for many children, especially when it grinds to a halt after the new year and normality resumes.
But for students whose families are struggling in poverty, time away from school isn’t an exciting blip on an otherwise calm school year. For them, it can be a crippling time of insecurity when it comes to food and shelter.
When shallow water (like runoff from melting snow) flows across pavement, it creates small repeated wave-like ripples. What creates that texture and why isn’t it just a steady flow?
This is a great question that’s probably crossed the mind of anyone who’s seen water running down the gutter of a street after a storm. The short answer is that this gravity-driven flow is becoming unstable.
Fluid dynamicists often like to characterize flows into two main types: laminar and turbulent. Most flows in nature are turbulent, like the wild swirls you see behind cars driving in the rain. But there are laminar flows in nature as well. Often flows that begin as laminar will become turbulent. This happens because those laminar flows are unstable to disturbances.
The classic example of stability is a ball on a hill. If the ball is at the top of the hill and you disturb it, it will roll down the hill because its original position was unstable. If, on the other hand, the ball is in a depression, then you can prod the ball and it will eventually settle back down into its original place because that position was stable. Another way of looking at it is that, in the unstable case, the disturbance–how far the ball is from its original position–grows uncontrollably. In the stable case, on the other hand, the disturbance can be initially large but eventually decays away to nothing.
There are many ways to disturb a laminar flow–surface roughness, vibrations, curvature, noise, etc., etc. These disturbances enter the flow and they can either grow (and become unstable) or decay (because the flow is stable to the disturbance). Just as one can look at the stability of a pendulum, one can mathematically examine the stability of a fluid flow. When one does this for water flowing down an incline, one finds that the flow is quite unstable, even in the ideal case of a pure, inviscid fluid flowing down a smooth wall.
The reason that one sees distinctive waves with a particular wavelength (assuming that they aren’t caused by local obstructions) is directly related to this idea of instability. Essentially, the waves are the disturbance, having grown large enough to see. One could imagine that any wavelength disturbance is possible in a flow, but mathematically, what one finds, is that different wavelengths have different growth rates associated with them. The wavelength we observe is the most unstable wavelength in the flow. This is the wavelength that grows so much quicker than the others that it just overwhelms them and trips the flow to turbulence. This is very common. For example, you can see distinctive waves showing up before the flow goes turbulent in both this mixing layer simulation and this boundary layer flow.
(Image credits: anataman, mo_cosmo; also special thanks to Garth G. who originally asked a similar question via email)
I only train my core isometrically. Without movement. No sit-ups, no side bends, no toes-to-bars, no Russian twists, none of that. I do this for two reasons: safety and functionality.
Firstly let’s look at the spine. It is made up of a series of bony vertebrae with vertebral disks in between them. Vertebral disks are made of two parts, the outer, solid layer called the Annulus Fibrosis, and the inner, viscous layer called the Nucleus pulposus. A herniation occurs when the inner layer pushes its way out through the outer layer, often (very painfully) impinging a nerve coming off the spinal cord. I don’t need to tell you that as an athlete, you don’t want this. No one does. If you do herniate a disk and it hits a nerve root, you’re looking at a lot of pain, rehab, and potentially months off of your sport.
Core exercises that involve movement of the spine increase your chances of suffering a disk herniation. Research shows that repeated spinal flexion (bending forward) is needed to cause disk herniations . If a researcher wants to herniate a spine specimen, they will put it through thousands of cycles of flexion and extension with moderate compression. That is exactly what you are putting your spine through every time you do a sit-up or a poorly executed back extension on a GHD (compression in this case is caused by your own core muscles, think of each vertebrae as a book stacked on top of another, your muscles squeeze down on the stack of books to keep them from falling over). Why would you put your spine through that? There are several safer, more effective alternatives to train these muscles.
Research also shows that repeated twisting also makes you more vulnerable to a herniation by slowly wearing away at layers of the annulus fibrosis, making it easier for the nucleus pulposus to herniate . Sure, Russian twists are working your obliques, but at what cost?
The human spine is very good at absorbing compressive forces, the vertebrae–disks and vertebral curves all allow for this. The spine is not, however, nearly as good at handling forces like shear. For example NIOSH, a health and safety board recommends a spinal compression force of no higher than 3400N during work tasks, while the limit for shear forces is only 1000N . The exact numbers are not important, but safety experts agree that our spines are about 3.5 times better at handling compression than they are shear. Excessive shear becomes a problem when the spine is fully flexed forward (think sit-ups, toes-to-bars, etc). These exercises definitely do a good job of working your rectus abdominus, but not without introducing potentially dangerous and unnecessary shear forces on your lumbar spine. Exercises that keep the spine in neutral or near neutral are safer because they put the spine in a position to handle forces compressively instead of introducing shear forces.
At this point is when someone would usually say something along the lines of “Well I do exercises X and my back is fine! This can’t be true.” Your back may be fine if you’re doing these exercises now, but it may not be in the future. You may be fine in the future too even if you continue to do these exercises, but you are definitely increasing your risk by continuing to do so. To succeed in any sport, you need to stay healthy. There is no reason to put your spine through potentially dangerous exercises when safer alternatives exist (more on these alternatives later). Also, you should keep in mind that the absence of pain does not mean the absence of injury. Only the outer layers of the annulus fibrosis contain sensory nerve fibers , so during the early stages of a herniation, pain would not be an issue.
Almost every sport I can think of requires a core that is neutral or near neutral (don’t think of neutral as a perfect position, think of it as a certain small range of motion around that position), and a core that is braced isometrically. The only example I can think of where this isn’t true is gymnastics (but gymnasts are freaks of nature, so let’s ignore them) and maybe swimming or rowing. I’m sure there are more, but that doesn’t really matter. Most sports require a neutral, isometrically braced spine.
This is especially true of strength sports. The squat, Snatch, and Clean & Jerk all require a neutral spine that is braced isometrically. Those who bench with an arch won’t have a neutral spine and some deadilfters prefer pulling with their spine in a little bit of flexion, but both of these exercises definitely require the spine to be braced isometrically. If strength athletes always need isometric core strength, the majority of their core training should be isometric as well. Since the spine is capable of moving in three different planes, the core should be trained isometrically resisting motion in all three of these planes. You should select exercises that resist flexion and extension (bending forward and backwards), lateral flexion (bending to either side) and rotation (twisting). Below is a list of exercises that I have had good success with implementing in my training, that challenge the core in all three planes of motion. Assuming you have the basic stability needed to do them, these exercises are a solid foundation.
This list if by no means exhaustive. Get creative and find what works for you.
At the end of the day, you can train your core muscles in whatever way you want to train them. Just be aware of the risks and rewards that come with your choice of exercises and take this information into consideration before your next core workout; it’s probably not worth it. Stay safe!