How to of the Day

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• How to Strengthen Your Knees
• How to Integrate Gaussian Functions
• How to Make Mushroom Risotto

How to Strengthen Your Knees Posted: 09 Feb 2017 04:00 PM PST It's important to keep your knees strong and healthy so that your mobility doesn't deteriorate as you grow older. We often take the health of our knees for granted, not noticing there's a problem until everyday activities like lifting boxes or walking downhill become painful. Take the following measures to strengthen your knees and ensure you'll stay active for as long as possible. EditSteps EditUnderstanding Knee Health Get to know basic knee anatomy. The knee is the largest joint in the body, and is made up of the lower end of the thigh bone (femur), the upper end of the shin bone (tibia), and the kneecap (patella). These bones are connected by ligaments and cartilage, including the meniscus, which cushions the area where the femur and tibia meet. Be aware of common knee injuries. As one of the most heavily used joints in the body, the knee is subject to a range of injuries. The more you know, the better prepared you will be to avoid circumstances that lead to or exacerbate injuries.[1] The iliotibial band, or IT band, is the area of thicker tissue that runs from the outside of the pelvis to the outside of the knee. The IT band helps to stabilize the knee during physical activity. It can become inflamed and painful when it is overused, leading to iliotibial band syndrome (ITBS). Runners, hikers, and other active people often experience this injury. The anterior cruciate ligament (ACL) is commonly torn during activities like running, jumping, and landing from a jump. Other ligaments may be torn as well. The meniscus, which acts like a shock absorber to protect the knee joint from impact, can be easily torn during activities like twisting, pivoting, or decelerating. Understand how other parts of the leg affect the knees. The knees are supported by the other muscles in the legs, especially quadriceps, hamstrings, and glutes. Keeping these muscles strong is a crucial to having strong knees and preventing injury. EditExercising to Strengthen Your Knees Stretch your IT band. Spending some time stretching and warming up your IT band before diving into a strenuous activity is a good way to keep your knees strong.[2] Stand with your left foot crossed over your right and stretch your arms above your head. Lean your upper body as far as you can to the left without bending your knees. Repeat with your right foot crossed over your left, leaning your upper body to the right. Sit on the floor with your legs stretched in front of you. Cross one over the other and pull your knee as close as you can toward your chest, holding it in place for a few seconds. Repeat with your other leg. Take a brisk walk before launching into a more complicated exercise to give your IT band a chance to loosen up. Work out your quadriceps, hamstrings, and glutes. Do lunges to work on your quadriceps. Stand upright with your hands on your hips. Take a big step forward with your left foot, and lower your body toward the floor until your left leg is bent at a right angle. Your back knee will lower until it is nearly touching the floor. Repeat this exercise several times, then switch to the other side. Strengthen your hamstrings with step-ups. Stand in front of a raised surface and practice stepping up with one foot, then the other. Repeat on both sides. Do squats to get stronger glutes. Stand upright and simply squat to the ground, bending your knees and keeping your back upright. For a less strenuous version of this exercise, practice standing in front of a chair, sitting, and standing again. Learn to jump well. Jumping is great exercise, and done properly, it helps strengthen the knees. Try jumping rope in front of a mirror so you can study your form. Are you landing with your knees straight, or are they bent? Landing on straight knees puts too much pressure on your joint, and can eventually lead to injury. For stronger knees, practice landing in a half-squat position with your knees bent. Try recreational activities that build full-body muscle tone. If your leg muscles aren't strong, your knees won't be either. Yoga is a low-impact activity that tones your leg muscles. As a bonus, many yoga positions involve stretching and warming up your knees. Swimming is another great way to build leg and knee strength and flexibility. Walking and biking keep your legs and knees in shape for more strenuous activities. EditChanging Your Lifestyle to Strengthen Your Knees Add anti-inflammatory foods to your diet. Joints become weakened and painful when they are inflamed, so adding foods that are known to be anti-inflammatory to your diet may help keep your knees strong. [3] Fish, flax seeds, olive oil, avocados, and whole fruits and vegetables are said to have anti-inflammatory properties. Make sure you get enough vitamin E. Vitamin E is believed to hinder enzymes that break down the cartilage in the joints. Spinach, broccoli, peanuts, mango and kiwi are excellent sources of vitamin E. [4] Eat more calcium. Bone health is also important for knee strength, so take measures to prevent osteoporosis. Cow milk, yogurt, cheese and goat milk are good sources are calcium. Almonds and leafy greens are also excellent sources. Stop doing activities that hurt. If you experience a lot of knee pain doing certain activities, chances are you aren't strengthening your knees by continuing. Try lower impact exercises for awhile to give your knees a rest. After a few months of focusing on leg muscle strength and flexibility, you may find you're able to do your favorite activities without pain. EditVideo EditWarnings Running on a hard paved surface can destroy your knees over time. Always wear proper running shoes, and don't overdo the running. If you are experiencing any pain from the activities you do, stop right away. Don't twist your feet in such a way that your knees are wrenched sideways. You can permanently stretch or tear the ligaments that hold your knee together (ligaments, unlike muscles, are not supposed to stretch). EditRelated wikiHows Strengthen Ligaments Avoid Knee Injuries Keep Your Bones and Joints Healthy Get Your Legs and Butt in Shape Soothe Your Legs After Stress or Long Walk Avoid Feet and Leg Problems if Standing for Work Prevent Osteoporosis Tape a Knee Safely Strengthen Hamstrings After an Injury EditSources and Citations Cite error: <ref> tags exist, but no <references/> tag was found

How to Integrate Gaussian Functions Posted: 09 Feb 2017 08:00 AM PST The Gaussian function ${\displaystyle f(x)=e^{-x^{2}}}$ is one of the most important functions in mathematics and the sciences. Its characteristic bell-shaped graph comes up everywhere from the normal distribution in statistics to position wavepackets of a particle in quantum mechanics. Integrating this function over all of ${\displaystyle x}$ is an extremely common task, but it resists the techniques of elementary calculus. No amount of change of variables, integration by parts, trigonometric substitution, etc. will simplify the integral. In fact, the antiderivative of the Gaussian, the error function, cannot be written in terms of elementary functions. Nevertheless, there exists an exact solution for the definite integral, which we find in this article. We also generalize the Gaussian integral to obtain some more interesting results. EditSteps EditGaussian Integral Begin with the integral. ${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}{\mathrm {d} }x}$ Consider the square of the integral. We are expanding this integral into the ${\displaystyle xy}$ plane. The idea here is to turn this problem into a double integral for which we can easily solve, and then take the square root. ${\displaystyle \int _{-\infty }^{\infty }{\mathrm {d} }xe^{-x^{2}}\int _{-\infty }^{\infty }{\mathrm {d} }ye^{-y^{2}}}$ Convert to polar coordinates. Recall that the area integral of a polar rectangle is of the form ${\displaystyle r{\mathrm {d} }r{\mathrm {d} }\theta ,}$ with the extra ${\displaystyle r}$ there in order to scale the angle to units of length. This extra ${\displaystyle r}$ makes this problem trivial. ${\displaystyle \int _{-\infty }^{\infty }{\mathrm {d} }xe^{-x^{2}}\int _{-\infty }^{\infty }{\mathrm {d} }ye^{-y^{2}}=\int _{0}^{\infty }r{\mathrm {d} }r\int _{0}^{2\pi }{\mathrm {d} }\theta e^{-r^{2}}}$ Evaluate by means of a u-substitution. Let ${\displaystyle u=r^{2}.}$ Then the differential will cancel out the extra ${\displaystyle r}$ that we got from changing to polar. ${\displaystyle {\begin{aligned}\int _{0}^{\infty }r{\mathrm {d} }r\int _{0}^{2\pi }{\mathrm {d} }\theta e^{-r^{2}}&=\pi \int _{0}^{\infty }e^{-u}{\mathrm {d} }u\\&=\pi (-e^{-\infty }+e^{0})\\&=\pi \end{aligned}}}$ Arrive at the integral of a Gaussian. Since we were evaluating the square of the integral, we take the square root of our result. ${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}{\mathrm {d} }x={\sqrt {\pi }}}$ Importantly, the Gaussian function is even. ${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}{\mathrm {d} }x=2\int _{0}^{\infty }e^{-x^{2}}{\mathrm {d} }x=2\cdot {\frac {\sqrt {\pi }}{2}}}$ Consider the integral of the general Gaussian function. This function is determined by the parameters ${\displaystyle a}$ and ${\displaystyle \sigma ,}$ where ${\displaystyle a}$ is a (normalization) constant that determines the height of the bell curve, and ${\displaystyle \sigma }$ is the standard deviation, which determines the curve's width. ${\displaystyle f(x)=ae^{-{\frac {x^{2}}{2\sigma ^{2}}}}}$ Follow the steps shown above to verify this integral. ${\displaystyle \int _{-\infty }^{\infty }ae^{-{\frac {x^{2}}{2\sigma ^{2}}}}{\mathrm {d} }x=a\sigma {\sqrt {2\pi }}}$ Another way to formulate the problem is if we have a Gaussian in the form ${\displaystyle e^{-\alpha x^{2}}.}$ Verify this integral as well. ${\displaystyle \int _{-\infty }^{\infty }e^{-\alpha x^{2}}{\mathrm {d} }x={\sqrt {\frac {\pi }{\alpha }}}}$ (Optional) Normalize the area to find the normalization constant ${\displaystyle a}$. In many applications, it is desired that the area of the Gaussian be set to unity. In this case, we set ${\displaystyle a\sigma {\sqrt {2\pi }}=1}$ and solve for ${\displaystyle a.}$ ${\displaystyle a={\frac {1}{\sigma {\sqrt {2\pi }}}}}$ Here, we arrive at the normalized Gaussian, so desired in such applications as probability theory, quantum mechanics, and so forth. The diagram above lists a few normalized Gaussians with varying standard deviations (actually variances, since they are defined by ${\displaystyle \sigma ^{2}}$). ${\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}$ EditGeneralizations Consider the integral below. The Gaussian integral ${\displaystyle \int _{0}^{\infty }e^{-\alpha x^{2}}{\mathrm {d} }x={\frac {1}{2}}{\sqrt {\frac {\pi }{\alpha }}}}$ is a result that can be used to find numerous related integrals. The ones below are called moments of the Gaussian. Below, ${\displaystyle n}$ is a positive number. ${\displaystyle \int _{0}^{\infty }x^{n}e^{-x^{2}}{\mathrm {d} }x}$ If ${\displaystyle n}$ is even, consider the related integral (written below) and differentiate under the integral. The result from differentiating under the integral is that even powers of ${\displaystyle x}$ get brought down. Notice that as the integral gets negated, the result on the right also gets negated because of the negative power in ${\displaystyle \alpha ,}$ so the answers remain positive. Since differentiation is much easier than integration, we could do this all day, making sure to set ${\displaystyle \alpha =1}$ at a convenient time. We list some of these integrals below. Make sure to verify them for yourself. ${\displaystyle \int _{0}^{\infty }e^{-\alpha x^{2}}{\mathrm {d} }x={\frac {1}{2}}{\sqrt {\frac {\pi }{\alpha }}}}$ ${\displaystyle \int _{0}^{\infty }x^{2}e^{-x^{2}}{\mathrm {d} }x=-\int _{0}^{\infty }{\frac {\partial }{\partial \alpha }}e^{-\alpha x^{2}}{\mathrm {d} }x=-{\frac {\mathrm {d} }{{\mathrm {d} }\alpha }}\left({\frac {1}{2}}{\sqrt {\frac {\pi }{\alpha }}}\right)={\frac {\sqrt {\pi }}{4}}}$ ${\displaystyle \int _{0}^{\infty }x^{4}e^{-x^{2}}{\mathrm {d} }x={\frac {3{\sqrt {\pi }}}{8}}}$ ${\displaystyle \int _{0}^{\infty }x^{6}e^{-x^{2}}{\mathrm {d} }x={\frac {15{\sqrt {\pi }}}{16}}}$ If ${\displaystyle n}$ is not even, use the u-sub ${\displaystyle u=x^{2}}$. Then we can use the Gamma function to easily evaluate. Below, we choose ${\displaystyle n=9}$ and ${\displaystyle n=1/3}$ as examples. ${\displaystyle \int _{0}^{\infty }x^{9}e^{-x^{2}}{\mathrm {d} }x={\frac {1}{2}}\int _{0}^{\infty }u^{4}e^{-u}{\mathrm {d} }u={\frac {4!}{2}}=12}$ ${\displaystyle \int _{0}^{\infty }x^{1/3}e^{-x^{2}}{\mathrm {d} }x={\frac {1}{2}}\int _{0}^{\infty }u^{-1/3}e^{-u}{\mathrm {d} }u={\frac {\Gamma (2/3)}{2}}}$ It is interesting to note that we could've used the Gamma function for even ${\displaystyle n}$ as well. It is a more general method of evaluating these types of integrals that typically is no more involved than differentiating under the integral. Set ${\displaystyle \alpha =i}$ to obtain three integrals. The result is general enough such that ${\displaystyle \alpha }$ can even take on complex values. Recall Euler's formula relating the complex exponential function to the trigonometric functions. If we take the real and imaginary parts of our result, we obtain two integrals for free. Neither of the two real integrals have antiderivatives that can be written in closed form. ${\displaystyle e^{-ix^{2}}=\cos x^{2}-i\sin x^{2}}$ ${\displaystyle \int _{0}^{\infty }e^{-ix^{2}}{\mathrm {d} }x={\frac {1}{2}}{\sqrt {\frac {\pi }{i}}}={\frac {\sqrt {\pi }}{2}}e^{-i\pi /4}={\frac {\sqrt {\pi }}{2{\sqrt {2}}}}(1-i)}$ ${\displaystyle \int _{0}^{\infty }\cos x^{2}{\mathrm {d} }x=\int _{0}^{\infty }\sin x^{2}{\mathrm {d} }x={\frac {\sqrt {\pi }}{2{\sqrt {2}}}}}$ These two integrals are special cases of the Fresnel integrals, where they are important in the study of optics. If you are not very familiar with complex numbers, the number ${\displaystyle i}$ can be rewritten in polar form as ${\displaystyle e^{i\pi /2},}$ because imaginary exponents are rotations in the complex plane - in this case, by an angle of ${\displaystyle \pi /2.}$ Polar form simplifies almost everything associated with complex numbers, so we can easily take the square root. Calculate the Fourier transform of the Gaussian function by completing the square. Calculating the Fourier transform is computationally very simple, but it requires a slight modification. We opt to complete the square because we recognize the property that the integral is independent of the shift (see the tips). Since we have to add 0 in order to not change the integrand, we have to compensate by adding a ${\displaystyle -{\frac {\omega ^{2}}{4}}}$ term. Watch the signs - they can be tricky. ${\displaystyle {\begin{aligned}{\mathcal {F}}\{e^{-t^{2}}\}&=\int _{-\infty }^{\infty }e^{-t^{2}}e^{-i\omega t}{\mathrm {d} }t\\&=\int _{-\infty }^{\infty }e^{-(t^{2}+i\omega t-\omega ^{2}/4+\omega ^{2}/4)}{\mathrm {d} }t\\&=e^{-\omega ^{2}/4}\int _{-\infty }^{\infty }e^{-(t+i\omega /2)^{2}}{\mathrm {d} }t\\&={\sqrt {\pi }}e^{-\omega ^{2}/4}\end{aligned}}}$ Interestingly, the Fourier transform of a Gaussian is another (scaled) Gaussian, a property that few other functions have (the hyperbolic secant, whose function is also shaped like a bell curve, is also its own Fourier transform). For physicists and engineers, Gaussian integrals are incredibly useful. Proficiency in evaluating them is a must. EditVideo EditTips It turns out that the Gaussian as defined in step 6 of part 1 is not the most general form. As seen in the diagram, one can also shift the Gaussian some units ${\displaystyle \mu ,}$ so that the ${\displaystyle x^{2}}$ turns into a ${\displaystyle (x-\mu )^{2}}$ in the exponent. However, it is obvious that the translation does not matter when we are integrating over all ${\displaystyle x,}$ which is why completing the square while calculating the Fourier transform works. Nevertheless, the general form of the normalized Gaussian looks like this. ${\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}$

How to Make Mushroom Risotto Posted: 09 Feb 2017 12:00 AM PST Risotto is an Italian rice dish that is easily customized and tweaked to your liking once you get the technique down. Unlike white rice, which cooks without much fuss in a rice cooker or pot, risotto needs to be stirred and managed the entire time it is on the stove. The result, however, is a rich, creamy, and textured rice that can be served as an appetizer, main course, or side dish. EditIngredients EditClassic Mushroom Risotto 2 cups arborio, baldo, or carnaroli rice. Arborio is the "classic" option. 1 lb sliced portobello and/or crimini mushrooms 1 onion 2-3 cloves of garlic 1 tablespoon of butter 3 tablespoons of oil 6-8 cups of chicken, vegetable, or beef broth 1/2-1 cup dry white wine, like a Sauvignon Blanc Salt & Pepper, to taste 1-2 tablespoons of desired spices -- consider rosemary, thyme, a single bay leaf, and parsley. EditOptional Additional Ingredients 1/2 lb chicken or beef Cooked vegetables (carrots, brussel sprouts, green onions, etc.) 1 ounce porcini mushrooms, reconstituted 2-3 shallots 1 can cream of mushroom soup 1 ounce feta, goat, or asiago cheese Grated Parmesan cheese EditSteps Heat 3 tablespoons olive oil in a large skillet on medium heat. Always add your oil while the skillet is heating-- it can damage you pan to heat it up with nothing in it. Use the biggest pan you have-- this evenly distributes the heat and helps the rice cook better.[1] Dice 1 small onion and 2-3 cloves of garlic and add to the oil. You can cut these to whatever size you prefer, but they will generally cook best as small, diced chunks. Stir occasionally. Thinly slice your mushrooms while the onions and garlic cook. Remove the stems and cut the mushrooms to your desired thickness, usually around 1/2 a centimeter each. You want roughly 1-2 cups of mushroom, depending on your preferences. Add the mushrooms to the skillet once your onions are translucent. This usually takes 3-5 minutes. Translucent onions are almost, but not quite, see-through. Add salt and pepper and stir everything to coat the vegetables in oil. This does not need to be an exact science -- if you think the onions are translucent, go ahead and add the mushrooms. Heat up 6 cups of broth while the mushrooms cook. Using a medium sauce pot or the microwave, heat up all of your broth on medium high heat. It does not need to be boiling, just hot to the touch. If you add cold broth to your rice you greatly lower the temperature of the pan, effectively halting cooking and preventing the rice from absorbing the liquid.[2] Cook the mushrooms for 3-5 minutes, then add 1 tablespoon of butter and your herbs. The mushrooms should look slightly wilted, and they will have sucked up the liquid in the skillet. For a slightly healthier risotto you can skip the butter. Herbs to add include 1 teaspoon of the following (if using fresh, instead of dried, herbs, double the amounts). Thyme Sage Rosemary Basil Cracked red pepper. Add the 2 cups of arborio rice and stir regularly. Once the butter has melted, add the rice and stir so that it is coated with the oil and butter. Cook for 1 minute, until the edges of the rice begin to appear clear. This is also known as "toasting" the rice. You want to do this before adding any liquid. Cooking the rice without liquid hardens the starch on the outside, preventing it from sticking together later.[3] You can also use baldo or carnaroli rice for a creamier texture. Add 1/2 cup of white wine. Add the white wine and stir into the rice. Cook together until almost all of the wine is evaporated or absorbed into the rice. Don't worry, the final dish will not be alcoholic -- most of the alcohol will burn off. Wine "deglazes" the skillet, removing tasty bits of caramelized vegetables from the pan and adding them to the rice. Move on once all of the wine is absorbed. Add one ladle-full of hot broth. Once the wine is absorbed, use a ladle to add the first cup of broth to the rice mixture. Resist the urge to add more than one cup -- you need to take it slow to get the best risotto you can. Too much liquid, too quickly, will lead to mushy, unappetizing rice. Take it slow -- the patience will pay off.[4] Stir the broth and rice regularly. You cannot stir risotto enough. Though you can leave the pan briefly if you need, the more you stir the risotto the better the texture and flavor will be when you are done. Add another cup of broth once the first has been absorbed. Once most of the broth has been absorbed by the rice, add another cup. You'll know when it is absorbed, as there are no more puddles or visible liquid. Repeat until the broth is gone or the rice is your desired consistency. The rice should be slightly firm and creamy, but never mushy. Depending on the amount and quality of rice, you may need 1 cup more or less. After about half the broth is added, taste it regularly to get your desired consistency.[5] If your rice is still hard but you are out of broth, add a little water or milk and continue stirring. Remove from the heat and serve. Add any salt and pepper you desire and garnish on top with Parmesan cheese or truffle oil. To create a creamier risotto, substitute 1 cup cream of mushroom soup for 1 cup of broth. Make sure the cream of mushroom is hot before adding it, and add the cream of mushroom last to make sure the rice cooks properly. Use red wine instead of white for a pink, savory risotto. Red wine imparts a somewhat richer flavor on the risotto, and it often pairs well with the earthy, nutty flavor of mushrooms. Try to use a dry red wine like a Pinot noir or Chianti, or better yet use Sherry or Marsala -- two wines commonly used for cooking. Add meat or other vegetables to make risotto a main dish. Cook your meat and vegetables separately and add them to the risotto with your last cup of broth. You can use anything you would like, but some some combinations to try include: Beef/Steak cubes, red wine, and goat cheese. Brussel sprouts, carrots, and celery, all cooked until soft. Shallots, porcini mushroom, and green onions. EditTips Risotto needs constant attention and must be constantly stirred after the addition of the rice. Remember: stir, stir, stir Test your rice often to make sure it's at the correct firmness before you remove it from heat. EditWarnings If the heat is too high your rice will burn before becoming soft. The rice continuously absorbs the liquid and the rice increases drastically in size. If cooking for 5 or more, make sure you have an extra large pan. EditRelated wikiHows Make Risotto With Peas and Parmigiano Make Risotto Make Risotto Rice Make Four Vegetable Risotto Make Toasted Seed Risotto Balls EditSources and Citations Cite error: <ref> tags exist, but no <references/> tag was found

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