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Guide to Buy Industrial Steel Doors in Digah House Company

Guide to Buy Industrial Steel Doors in Digah House Company

2021-06-30
Digah Company
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On this page, you can find quality content focused on industrial steel doors. You can also get the latest products and articles that are related to industrial steel doors for free. If you have any questions or want to get more information on industrial steel doors, please feel free to contact us.

The reason why industrial steel doors is highly favored in the market can be summarized into two aspects, namely outstanding performance and unique design. The product is characterized by long-term life cycle, which can be attributed to the high-quality materials it adopts. Guangzhou House Empire Construction&Furnishing Co.,Ltd invests a lot to establish a professional design team, which is responsible for developing the stylish appearance for the product.Digah Company has been integrating our brand mission, that is, professionalism, into every aspect of the customer experience. The goal of our brand is to differentiate from the competition and to convince clients to choose to cooperate with us over other brands with our strong spirit of professionalism delivered in the Digah Company branded products and services.At Digah Company, the promotion of our service principle of integrity for our customers is greatly enhanced for getting industrial steel doors.
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Calculating the Out-the-Door Price for a New Car - CarGurus
Calculating the Out-the-Door Price for a New Car - CarGurus
Calculating the Out-the-Door Price for a New Car - CarGurusBiggest sale of the year! Thousands below invoice! New-car dealers commonly use phrases like these to draw prospective buyers into showrooms; yet, as with anything, the devil is in the details. More important than a vehicle's sale price is the actual out-the-door price, a figure that reflects all applicable taxes, fees, and other costs that may turn that great deal into something more overpriced. Some of these costs are unavoidable, but others are optional and possibly open to negotiation.The Cost of the Car All automakers, from Honda to Bentley, assign a Manufacturer's Suggested Retail Price (MSRP) to each new vehicle. Sometimes referred to as the "price of unit," it's nothing more than a recommendation made by the manufacturer to its dealerships, and it's typically the price that shoppers will see advertised at the dealership. Included in the MSRP are the base price of the vehicle plus any additional packages or accessories (added either at the factory or the dealership), such as a moonroof or a set of floor mats. While the MSRP is what the shopper is expected to pay, the invoice price (which you can find on CarGurus Research pages' Trims & Specs tabs) is the amount the dealer pays for the vehicle and any accessories from the manufacturer. Finally, the destination charge covers the cost of transporting the vehicle from the assembly plant to the dealer's lot. With the exception of the destination charge, shoppers can negotiate all of the figures above, as they do not reflect manufacturer-provided incentives made directly to the dealer. Unless the vehicle in question is rare or in particularly high demand, shoppers should use the invoice price as a starting point for negotiation, taking advantage of any rebates or promotions that work to lower the overall price. Since these promotions change frequently, it's worth spending a few minutes researching current offers before stepping foot inside a dealership. From military-personnel discounts to local credit-union-membership financing rates, there's usually some sort of discount for almost every buyer.The Value of Your Trade-In Purple Pintos and wheezing junkers being notable exceptions, choosing to trade in your car can have a significant impact on a new vehicle's out-the-door price. Depending on factors such as season, location, and current fuel prices, dealerships will have varying levels of interest in particular trade-in vehicles, and will make offers accordingly. If a buyer does not owe any additional money on their car loan, the amount the dealership pays for the trade will reduce the purchase price of the new vehicle, thereby allowing the buyer to pay less sales tax. If, however, the dealer's offer is less than what is still owed on the trade, the buyer will need to cover the difference. That amount can be a substantial sum, which will be added to the out-the-door price. This is called having negative equity, or being "under water," which is best avoided, even if that means delaying your new-car purchase until you've paid off your current ride.How Much You will Pay in Taxes and Fees Perhaps the most overlooked expense associated with a new-car purchase is that of taxes and fees. A lot of shoppers will spend time and effort hammering out the best price for a car or value for a trade, only to find themselves heading into round two with a finance manager outlining extra costs that are largely non-negotiable. Depending on where the buyer lives, these will include state sales tax, registration (license plates, or "tags" in some areas), inspection and/or smog fees, a prep fee (basically money to cover the dealer's overhead expenses), and the seemingly inescapable documentation fee. That last one can vary greatly, with some dealerships pre-printing a set price on all sales forms to indicate it's not up for discussion, while others make a point of noting that they do not ever charge a doc fee. Another expense to consider is excise tax, which is what is paid to your local DMV whenever the vehicle's registration is renewed (usually annually). The amount due typically correlates with the car's value, meaning it should decrease in subsequent years, until leveling off at a figure determined by the state. More often than not, however, excise tax is not paid while you are still at the dealership-typically a bill will arrive via mail directly from the DMV.Given the high prices of new cars, most buyers find themselves needing a loan-and that means interest. Though less common than it once was, some manufacturers still offer 0% interest rates for buyers who finance through the dealership, but in most cases buyers will still have to pay at least a small percentage in interest. Rates are determined based on factors such as credit score and the length of the loan's term. Buyers can do themselves a favor by checking their credit score and their bank's current auto loan rates before starting negotiations with a dealer. One way to keep costs down in this area is to pay any taxes and fees out of pocket, reducing the amount that needs to be borrowed and, therefore, the amount of interest you will pay.Those hard-to-believe low prices work to get people in the door, but the out-the-door prices determine whether shoppers can afford a a specific vehicle. With a solid understanding of what is involved and how the numbers get crunched, buyers can avoid the unexpected and make that next new-car purchase as stress-free as possible.
Summer Vacation Security Tips to Keep Your Home Safe | Directors Mortgage
Summer Vacation Security Tips to Keep Your Home Safe | Directors Mortgage
SUMMER VACATION SECURITY TIPS TO KEEP YOUR HOME SAFEPlanning on going somewhere fun for summer vacation? Do not forget to make sure your house is safe and secure before you leave. These summer vacation home security tips will make it easy to maintain peace of mind and ensure a happy homecoming.Open windows and doors are easy targets for would-be home intruders. Make sure to close and lock all of your home's doors and windows before you leave, and do not forget to set your home security alarm as well. Jog your memory for these small tasks with a to-do checklist or helpful phone alarm before you head out.If your pets are going to stay home while you are away, you might want to leave a window cracked so they can enjoy a cool breeze in-between visits from a pet sitter. While your pets will appreciate this, it's still important to keep your house secure. Luckily, you can invest in specialty locks for windows or sliding patio doors.Depending on the type of windows you have, you could install hinged wedge, keyed sash, folding and other window locks. These locks prevent would-be intruders from being able to open the window enough to fit through. You can also find locks that fit onto a sliding door's tracks to achieve a similar effect.If you are on a budget, you can use a thick piece of PVC pipe or wooden dowel to act as a makeshift lock. Open your window or sliding patio door just enough for some fresh air to come in, but not enough for a person to enter. Then measure the amount of track that's left exposed. Have a piece of pipe or dowel cut to that length, then put it in place. This makeshift lock will let you keep a window or sliding door cracked but secure.Before you go, test your lock installation to make sure it's sturdy and secure. If you live in a neighborhood where break-ins are common, it may be better to err on the side of caution and leave everything completely closed and locked.Home intruders generally seek easy opportunities to get in and out of a house unnoticed. That means they are always on the lookout for an obviously empty home. Luckily, it's not that hard to fool them.These are a few quick and easy things you can do to thwart potential break-ins by masking the symptoms of an empty house:• Ask neighbors to park in your driveway, especially overnight• Leave some lights on in your home, or put devices like TVs and lamps on a timer so your home shows regular signs of activity• Ask friends, relatives or trusted neighbors to check in on your home in the evenings a few times while you are gone. Invite them to stay and have dinner or watch a movie on your couch so it looks like someone's home• Stop mail and newspaper delivery so your mailbox does not overflow and papers do not sit untouched in your driveway• Ask a trusted neighbor to collect flyers, packages and other front-porch deliveries that might pile up while you are gone• Set your lawn sprinklers to run and have someone keep your garden plants watered. Wilting plants are a sign that garden maintenance is not happening, which could signal your absenceIf break-ins are common in your neighborhood or you are going on a really long vacation, you may want to hire a house sitter to keep up a seamless appearance of activity and take care of all the little things on this list. Make sure to get references if you need to hire a house sitter that you have not met before.Another important thing to think about: do not use social media to announce that you are going on vacation. It may seem harmless to make a fun Facebook post about how excited you are for your trip, but that can be an easy source of information for someone looking for empty homes. If your social media accounts are not set to private, it's best to wait until after your vacation is over to post about it. If you choose to post publicly about other topics while you are away, make sure to avoid using geotags that could let someone know you are not home.As part of their easy-opportunity strategy, home invaders often look through a home's windows to see if there are any obvious valuables they can quickly grab. This means you should avoid leaving valuables sitting out in plain sight whenever possible.Of course, that does not mean you need to lock up your 65" LED TV. Large, bulky objects are hard to hide, and that means they are also difficult to carry and conceal during a robbery. On the other hand, small things that can inconspicuously slip into a pocket or backpack are definitely worth hiding.A little common sense in reducing opportunity will likely do the trick. For example, do not leave your sleek new laptop sitting on the kitchen table in plain sight from the sidewalk. Put it inside your laptop bag in a closet. If you have valuable jewelry that you normally display on top of your bedroom dresser, store it in a drawer instead.Smart home security does not take a lot of effort. With just a few steps, you can make your home a lot less vulnerable while you are away. Taking care of these things before you depart will give you the freedom to fully unplug and unwind.adding new closet... how much space required to pass by comfortably - will use sliding doors to access?Easier to move the bed that do all this figuring
How to Set Up a Smartphone for a Child
How to Set Up a Smartphone for a Child
These days, it seems like younger and younger kids are getting smartphones. Even parents who would prefer to wait longer may be tempted, wanting to know that their child is safe during the day - or pressured, wanting their children to fit in. We can not forget, though, that smartphones are fun, portable touch-screen computers. Left to their own devices (no pun intended), children may spend hours playing games and watching YouTube videos. And kids will be kids, perhaps more likely than an adult to break or lose the smartphone. The good news is that we can reduce the risks. As with so many things, it will mainly take a bit of time and ongoing attention, but parental peace of mind is worth it. How to set up your child's smartphone right The settings I will go through are easy to tweak to your own preferences - you can make them as tough or easy as you like, once you know where they are. Let me start with a few basic assumptions, however: Kids play, and that's OK (with limits). The Internet is like the ocean - enticing but also dangerous. Everyone is entitled to a measure of privacy. The two most popular mobile platforms, Apple's iOS and Google's Android, both have built-in parental control tools, but Android lets users adjust more settings than iOS does. Android phones have another attractive quality: They are less expensive and therefore easier to replace. Some people prefer iOS, which may be better for households that have other Apple devices - and for parents who are not concerned about the price. You can read here about how to set up iOS for kids, although it's in the context of tablets and for younger children. Using an Android smartphone requires a Google account, but the terms and conditions stipulate that only a person who is 13 or older can create an account. If you do not already have a second account that you can use for this purpose, you will have to create one. The child does not need to know the password for their new account, and enabling two-factor authentication for it is advisable. Needless to say, the authentication should be linked to your smartphone, not to your child's. Note: You can create an account for your child on devices running Android 7.0 and above by using the Family Link feature, but it is currently available only in the United States. Now, power up the smartphone, add your existing Google account information in the setup wizard, enter your desired e-mail address and other data, enter your age and your phone number, and accept the terms and conditions. If it is not the first time you are starting up the smartphone, you set up the new account here: Settings Accounts Add accounts Google. In the new window, tap Or create a new account. At this point, skip entering payment details for the Google Play store. I will come back to that issue a bit later, but for now I will say that payment information should be linked to an adult's account. That said, we are talking about elementary-school-age children, who are unlikely to need e-mail on their phones. A Gmail account is more likely to collect spam than useful communications, so I recommend not enabling it on the smartphone: Open Settings, go to Google account settings, and uncheck the Sync Gmail option. If a parent is using an Android device too, then creating a family account may make sense. Using a family account lets family members share paid purchases across all of its devices as well as letting a child pay for purchases from their smartphone (using the parent's payment information - and only with the parent's explicit consent). You can create a family account in Google Play from your own smartphone. Launch the Google Play app, open up the menu, and tap on Settings Account Family Manage family members. You can invite your significant other and your children by entering their Gmail addresses on this screen. The invitation has to be accepted from the family member's smartphone. After that, take your smartphone, go to the family group's settings, open the child's profile, and select one of the following modes for purchase approvals: All content, Only paid content, Only in-app purchases, or No approval required. Anything that requires approval will now require either your password entered on the child's device or approval from your device. You can prevent your child from downloading adult games or songs with explicit lyrics using Google Play's parental controls. To do so, launch Google Play on your child's smartphone, open up the menu, and tap on Settings Parental Controls. Enter a PIN - something easy for you to remember but hard for your child to guess. The requirement to enter a PIN prevents your child from disabling the controls. After entering the PIN, you can select age restrictions for games, films, and music separately. To prevent unsupervised Internet use - and gain better control over mobile data usage - start by disabling mobile data use both in the settings of your child's smartphone and using your mobile service provider's self-service options. After that, use the smartphone's settings to join your home Wi-Fi network. Now it's up to you to monitor that use inside the home, of course. Disabling mobile data should help keep the child online only at home. True, free Wi-Fi can be had in a library or at a friend's home, but I will address that later in the post. By the way, when buying a SIM card for your child, ask the carrier about special plans and options for children - these plans and options usually include additional features for keeping children safe. Sure, the games installed on your child's smartphone are intended for children, but still, stay alert. Children can spend endless time mine crafting and launching angry birds. Limiting gaming duration with Android's built-in tools is problematic; therefore, look to dedicated apps such as Kaspersky Safe Kids. The Safe Kids app's many features include location and social-network controls, but here we are interested mainly in defining restrictions on launching apps and making the Internet safe to use. To use it, install Safe Kids on both a parent's and the child's smartphones. Set the modes, child mode for the child's smartphone and parent mode for yours. The child's app has almost no settings for you to worry about - just install the app according to the instructions and grant it administrator rights. You will have a bit more to do for the app on your own smartphone - you will need to set up a PIN for Safe Kids, check its settings, and enable options. The crucial ones are the ability to filter unwanted websites and app restriction by category and age, which are located on the Internet and Applications subsections respectively. Now, let's restrict the amount of time the child can spend at the display. You can allot, for example, 15 minutes, 1 hour, 2 hours, or more. After that, the smartphone will be able to make and receive calls, but most of the other apps will be blocked with a notification that time is up. If your child does not tend to spend too much time on pictures and SMS messages, then we can take the opposite tack and assign an amount of time allowed in specific apps, such as games. At the same time, we can completely block access to some apps, such as any browser other than Google Chrome and Android's built-in browser (which are protected by Safe Kids' content filter; other browsers may have access to unwanted websites). Safe Kids provides comprehensive parental control, including protection against its own deletion, and the recent update has brought one more useful feature - restricting access to smartphone's own settings. It's worth doing to make sure your child will not be able to connect to unknown Wi-Fi networks, reset the system time, or mess with settings in some other way you did not predict. I personally used to recommend an app called Smart Applock for that purpose, but now that Safe Kids has this feature I suggest using it and not installing additional software to execute this task. One last setting: the lock screen. Enable a lock screen to protect the smartphone from classmates and others who might find it. You can do that through Settings Security or Settings Lock screen. Work with your child to ensure he or she is comfortable entering the PIN or pattern that unlocks the phone. Software and settings are not a substitute for ongoing parental guidance. Get involved with your child's smartphone usage. Talk about surfing, gaming restrictions, and all that, but expect that your child will find ways around your "smartphone usage plan" - and adapt as needed.1. smartphone or phablet?What exactly is a phablet?2. Which is best smartphone in India 2017?S7 AND S7 EDGE WITH ALL FEATURES PACKED!!3. What smartphone apps work best for streaming to smart TVs?You have an android based TV like Sony? Can you install Kodi? That is not a streaming app as such , but it's a gateway for want of a better word to install add ins like "Cinema Box" .If you can install Kodi then it will pick give you a selection of streaming services suitable for TV Not phones
Modelling Lunar Motion in Python Closed
Modelling Lunar Motion in Python Closed
Well, just to be clear: I haven't looked at your code, I'm not responding to that. I'm submitting in the answer section because I have a general comment.The treatment by Richard Fitzpatrick that you link to is an exercise in how to go about obtaining an analytic solution for the case of the the Sun-Earth-Moon three body problem. In the general case of three bodies with mutual interactions there is no exhaustive solution, but in some cases some well chosen approximations can make it feasible to do the computations with pen and paper. As we know: for centuries this is what astronomers did in order to do astronomy: you find ways to bring the calculations within reach of being handled with pen and paper.These approaches are not used when electronic computers are used to calculate celestial motion. On an electronic computer you use Newton's law of gravity in its direct form: $$ F fracGm_1 m_2r^2 $$(Wich in the case of the Sun, Earth, Moon system of course means that you process three interactions: Sun-Earth, Sun-Moon, Earth-Moon.)To my knowledge that is how on electronic computers all simulations of celestial motion are done.My suggestion: use cartesian coordinates, and use a non-rotating coordinate system. If you also want to plot the motion with respect to some rotating coordinate system then you can apply that coordinate transformation to the result of the numerical integration. (Again, I suppose that in past centuries in order to bring the calculations within reach of pen and paper it was in some circumstances efficient to express the equation of motion and the intended coordinate transformation in a combined formula, and proceed from there.) Let me add some general considerations.Trade-offsWith todays computing power: for the purpose of processing a three body case your computational power is effectively limitless. Only when you want to to model, say, an entire galaxy you need a supercomputer.In the treatment by Fitzpatrick there are approximations, he mentions neglecting higher order terms. Those approximations introduce error. Negligable in the short run, but over time you get accumulation of error.If the only expression you evaluate is Newton's law in bare form then there is no approximation in the formula's themselves. Then only one source of error remains, the precision level of the numerical integration. You have a trade-off there. Higher precision level may limit you to, say, computing only a couple of thousand years of Moon motion. Lower precision level may allow you to compute millions of years, but probably by that time accumulation of error has made the end state meaningless.(Also, over time the distance between the Earth and the Moon is increasing. There are tidal effects that result in the Earth slowing down and the Moon being pulled to a higher orbit. So a long term model would have include that gravitational interaction.)So it's good to be aware of the trade-offs that apply in your case, and to try and make sensible decisions.One important trade-off, I think: the simpler the expressions that the numerical integration evaluates, the more debuggable the simulation is. Newton's law of gravity is a simple, transparent expression.Displaying the resultsAgain, since your computing power is effectively limitless you can display the results of the computation in any way you want. For instance, you can choose to set up a side by side display of the motion with respect to some particular inertial coordinate system, and next to it the motion with respect to some particular rotating coordinate system. The required coordinate transformation for that is trivial.That is, given your limitless computing power the decision on how to display the results is independent from the decision of how to execute the numerical integration itself.There is no such thing as: "I want to visualize the results as motion with respect to this rotating coordinate system so I must do all the computation in that rotating coordinate system." Obviously the two are independent.• Related QuestionsQuantum version of the Galton BoardYou might like this 110-page paper by me and Alex Arkhipov, which is all about a quantum bosonic analogue of Galton's board (we even use the same graphic you did -- see Section 1.1!). In particular, we gave strong evidence that such a board (with an arbitrary configuration of "pegs," and with multiple entry points for the "balls") is exponentially hard even to simulate using a classical computer. This suggests that such a quantum Galton's board (which is now called a "BosonSampler") could be used as a rudimentary, proof-of-principle quantum computer. And indeed, within the last year the first BosonSampling experiments were done in linear optics (see here), although so far only with 3 photons.To make our argument for computational hardness work, we needed two crucial assumptions:(1) The "balls" have to be indistinguishable particles. If they're distinguishable, then the distribution for any one individual ball could still exhibit interference fringes. But once you knew the probability distribution for one ball, the distribution for n balls would just be obtained by sampling from that distribution n times independently -- thereby producing "conventional, classical" Law of Large Numbers behavior. By contrast, identical quantum particles can famously become "correlated" even if they've never explicitly interacted, as seen for example in the Hong-Ou-Mandel dip.(2) The "balls" have to be bosons. In that case, their transition amplitudes are given by nxn matrix permanents, the calculation of which is a famous hard problem in computer science. By contrast, if the balls are fermions, then their transition amplitudes are given by nxn determinants, which are easy to calculate classically.Of course, there's also a "narrower" way to interpret your question, which might be closer to what you were actually asking about! Namely, rather than an "arbitrary" Galton-like board, we could consider the specific geometry from the figure: so let's say, a network of 50/50 interferometers arranged in a diamond pattern in the plane, with a single source of particles at the top. And we could then calculate (or, if we're lazier, numerically simulate...) the particular probability distribution over n-particle outcomes that that configuration leads to, under two different assumptions:(1) That the particles are distinguishable. (In this case, of course, the problem reduces to working out the distribution for a single particle.)(2) That the particles are indistinguishable bosons.(Note that a third case, that the particles are indistinguishable fermions, never arises -- for by the Pauli exclusion principle, n identical fermions couldn't even "fit" simultaneously through the single source at the top.)If I have some time later, I might work out the answers and post them here -- but in the meantime, anyone else should feel free to do so first.Addendum: OK, so let's consider the case of a single quantum particle passing through a "diamond-shaped" network of 50/50 interferometers. In that case, the probability distribution after n steps will be determined, not by the nth row of Pascal's triangle (as in the classical case), but by the nth row of what we might call the "interferometric Pascal's triangle." The latter is defined as follows: let A(i,j) be the jth entry in row i. Then:A(0,0)1A(0,j)0 for all j0For ij positive and odd: A(i,j)A(i-1,j)A(i-1,j1)For ij positive and even: A(i,j)A(i-1,j-1)-A(i-1,j)I'm almost certain that the result will asymptotically approximate the standard behavior for the "quantum random walk on the line": see here or here for good overviews. In particular, the distribution will not look anything like the Gaussian one: instead it should be nearly uniform, except with a bunch of oscillating peaks near the two edges, with the size of the peaks getting damped as you move closer to the center. (See the linked papers for example images.)A slight caveat is that usual analyses of quantum random walks assume there's a "coin" (i.e., a spin-1/2 internal degree of freedom), whereas I've used a staggered arrangement of interferometers to remove the need for the coin. I don't think that affects the walk's qualitative behavior, but I don't have a proof
Split
Split
Quilapayn's path has been marked by internal issues over the years, with some members leaving and others taking their place. In 1987 Willy Oddo, its most charismatic member, decided to leave France to be closer to Chile. He chose to live in Argentina. In October 1988, Pinochet was overthrown by a referendum. After 15 years of exile Quilapayun could return to Chile and performed tours in 1988, 1989, 1991 and 1992. In 1989 Eduardo Carrasco returned to Chile while most of the group continued to live in France and perform around the world. In November 1991 Willy Oddo was murdered by an offender in Santiago. After Rodolfo Parada registered the name "Quilapayn" without the authorization of the other members, other historic members refused to continue with Parada and Wang, resulting in the group splitting into two, both claiming the name and legacy of Quilapayn, and leading to subsequent litigation. The Chile-based historic faction is celebrating the group's 40 year anniversary performing concerts in Chile, Latin America and Europe, together with the "historic" version of Inti-Illimani, another important Chilean group. These joint concerts have been advertised and promoted as IntiQuila. The current "historic" lineup includes Eduardo Carrasco, Rubn Escudero, Ricardo Venegas, Guillermo Garca, Ismael Odd (son of Guillermo "Willy" Odd), Hugo Lagos, Hernn Gmez, Carlos Quezada and Sebastin Quezada (son of Carlos). On December 5, 2007, the Court of Appeal of Paris forbade Parada and Wang's group "from making use of the name QUILAPAYN, subject to a fine of 10 000 euros per infringement". This judgement was confirmed by the Highest Court of Appeal (Cour de Cassation de Paris) on June 11, 2009. In 2015 the band's career reached its fiftieth year and both factions celebrated this anniversary. Parada's group performed three big shows at the end August in Santiago, together with other well-known Chilean artists while the "historic" faction made a big concert in front of Palacio de la Moneda in Santiago and announced several other anniversary concerts in Chile, Colombia and Spain on its website. In November 2015, the Chilean trademark conflict ended, since the Instituto Nacional de Patentes Industriales (INAPI), after a thirteen years process gave the exclusive right for using the mark "Quilapayun" to the group headed by Carrasco.• Other Related Knowledge ofa split— — — — — —Non-split reductive groupsAs discussed above, the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. Some examples among the classical groups are: Every nondegenerate quadratic form q over a field k determines a reductive group G = SO(q). Here G is simple if q has dimension n at least 3, since G k displaystyle G_overline k is isomorphic to SO(n) over an algebraic closure k displaystyle overline k . The k-rank of G is equal to the Witt index of q (the maximum dimension of an isotropic subspace over k). So the simple group G is split over k if and only if q has the maximum possible Witt index, n / 2 displaystyle lfloor n/2floor . Every central simple algebra A over k determines a reductive group G = SL(1,A), the kernel of the reduced norm on the group of units A* (as an algebraic group over k). The degree of A means the square root of the dimension of A as a k-vector space. Here G is simple if A has degree n at least 2, since G k displaystyle G_overline k is isomorphic to SL(n) over k displaystyle overline k . If A has index r (meaning that A is isomorphic to the matrix algebra Mn/r(D) for a division algebra D of degree r over k), then the k-rank of G is (n/r) 1. So the simple group G is split over k if and only if A is a matrix algebra over k.As a result, the problem of classifying reductive groups over k essentially includes the problem of classifying all quadratic forms over k or all central simple algebras over k. These problems are easy for k algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions. A reductive group over a field k is called isotropic if it has k-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwise anisotropic. For a semisimple group G over a field k, the following conditions are equivalent: G is isotropic (that is, G contains a copy of the multiplicative group Gm over k); G contains a parabolic subgroup over k not equal to G; G contains a copy of the additive group Ga over k.For k perfect, it is also equivalent to say that G(k) contains a unipotent element other than 1. For a connected linear algebraic group G over a local field k of characteristic zero (such as the real numbers), the group G(k) is compact in the classical topology (based on the topology of k) if and only if G is reductive and anisotropic. Example: the orthogonal group SO(p,q) over R has real rank min(p,q), and so it is anisotropic if and only if p or q is zero. A reductive group G over a field k is called quasi-split if it contains a Borel subgroup over k. A split reductive group is quasi-split. If G is quasi-split over k, then any two Borel subgroups of G are conjugate by some element of G(k). Example: the orthogonal group SO(p,q) over R is split if and only if |pq| 1, and it is quasi-split if and only if |pq| 2.
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