While Wolfram|Alpha can do many sorts of computations, mathematical calculations are one of its particular specialties.In fact, using the power of Mathematica‘s computational capabilities under the hood, Wolfram|Alpha can do many things, ranging from the very simple to the fiendishly complicated, with mathematical functions.To highlight what we mean by “many” (and we do indeed mean it), let’s take the unassuming sine function as an example.Here is a list that started out as 93 possibilities of what Wolfram|Alpha can do with sine (one for each day of summer), but ended up including a number of bonus inputs above and beyond that which we couldn’t resist throwing in.Let’s start with the simple input sin(x) for the function itself: For those who know about such things, most of the results above apply to real values of the argument x. If on the other hand you use a different form of the argument such as sin(z) or sin(x + i y), Wolfram|Alpha will prefer an interpretation assuming complex variables, and so give a different set of plots and results: Of course, one can request each of the above computational results individually, for example, global maxima sin(x).

One can also generate a multitude of plots by specifically asking for them, including: One can also easily plot more complicated expressions of the sine function, such as: And even many functions using literal Mathematica syntax, such as: One can calculate special values of the sine function, such as sin(pi/88).Conversely, given the resulting value 0.0356923338389804557600… …one can recognize it as a value of the sine function at x = pi/88 (the first of the three possible closed forms returned as shown above).So motivated, one might consider asking for a list of all special values at once using the input special values of sin(x).A particularly beautiful example in this class is function expand sin(pi/17), which shows the classic result that Gauss found on his 17th birthday.One can ask for special forms of values and properties of values directly: And find minima and maxima of functions containing sine: One can draw 2D and 3D Lissajous figures: One can ask for inflection points: There are a multitude of mathematical formulas that can be requested.

For concrete cases, consider: As well as general formulas: Wolfram|Alpha also knows about many specific kinds of representations: One can ask for the relation of the sine functions with other functions: Including its relation with the inverse function and other functions expressed in terms of sine: One can solve equations containing the sine functions: Prove trigonometric identities involving the sine function: Find discontinuities and poles of expressions containing sine: And, of course, integrate functions of sine: One can get sample lists of sums, products, and integrals: One can expand the sine function in a series: Sum the Taylor series to finite order in closed form: Or get a Pade approximation of the sine function: One can simplify more complicated expressions involving sine: As well as expand a function into a Fourier sin series: The sine function has many relatives and generalizations, including but not limited to: One can use the sine function in probability calculations, for example, to find the expectation for sin(x) assuming x is distributed normally: One can solve differential equations involving the sine function: Calculate Wronskians involving the sine functions: And find the differential equation obeyed by an expression containing the sine function: One can use the sine law and the spherical law of sines: Express and approximate the value of pi through values of the sine function: And request an interactive version of the sine function: One can even do computations on URL-referenced objects on the internet, for example, providing analysis of an image of a sine function taken from The Wolfram Functions Site: Last, not least, one can “play” the sine function: We could go on (and on, and on), but that almost certainly suffices (“many” indeed).bitcoin mining software chrome

So as many of you are going back to school after what we hope was an enjoyable summer, please keep Wolfram|Alpha and its powerful collection of functionality in mind as you resume your studies.While we realize not everyone will enjoy getting back into the academic term as much as they did the allegedly halcyon days of summer, we hope there are at least a few of you out there who will have more fun learning with the help of Wolfram|Alpha, now that you know about it, than you did during your long, boring summer.litecoin pool gpuIn fault-tolerant computer systems, and in particular distributed computing systems, Byzantine fault tolerance is the characteristic of a system that tolerates the class of failures known as the Byzantine Generals' Problem, which is a generalized version of the Two Generals' Problem.ethereum lending

The phrases interactive consistency or source congruency have been used to refer to Byzantine fault tolerance, particularly among the members of some early implementation teams.The objective of Byzantine fault tolerance is to be able to defend against Byzantine failures, in which components of a system fail with symptoms that prevent some components of the system from reaching agreement among themselves, where such agreement is needed for the correct operation of the system.litecoin trend 2017Correctly functioning components of a Byzantine fault tolerant system will be able to immutably provide the system's service assuming there are not too many faulty components.situs bitcoin gratisThe following practical, concise definitions are helpful in understanding Byzantine fault tolerance: The terms fault and failure are used here according to the standard definitions originally created by a joint committee on "Fundamental Concepts and Terminology" formed by the IEEE Computer Society's Technical Committee on Dependable Computing and Fault-Tolerance and IFIP Working Group 10.4 on Dependable Computing and Fault Tolerance.bitcoin blockchain split

Note that the type of system services which Byzantine faults affect are agreement (a.k.a consensus) services.Byzantine refers to the Byzantine Generals' Problem, an agreement problem (described by Leslie Lamport, Robert Shostak and Marshall Pease in their 1982 paper, "The Byzantine Generals Problem") in which a group of generals, each commanding a portion of the Byzantine army, encircle a city.These generals wish to formulate a plan for attacking the city.In its simplest form, the generals must only decide whether to attack or retreat.Some generals may prefer to attack, while others prefer to retreat.The important thing is that every general agrees on a common decision, for a halfhearted attack by a few generals would become a rout and be worse than a coordinated attack or a coordinated retreat.The problem is complicated by the presence of traitorous generals who may not only cast a vote for a suboptimal strategy, they may do so selectively.For instance, if nine generals are voting, four of whom support attacking while four others are in favor of retreat, the ninth general may send a vote of retreat to those generals in favor of retreat, and a vote of attack to the rest.

Those who received a retreat vote from the ninth general will retreat, while the rest will attack (which may not go well for the attackers).The problem is complicated further by the generals being physically separated and must send their votes via messengers who may fail to deliver votes or may forge false votes.Byzantine fault tolerance can be achieved if the loyal (non-faulty) generals have a unanimous agreement on their strategy.Note that there can be a default vote value given to missing messages.For example, missing messages can be given the value .Further, if the agreement is that the votes are in the majority, a pre-assigned default strategy can be used (e.g., retreat).The typical mapping of this story on to computer systems is that the computers are the generals and their digital communication system links are the messengers.Known examples of Byzantine failures