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Created for Stanford Mathematics Camp 2016. A brief introduction to the Hopf Fibration for introductory topology students.

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as ”the rationals”, is usually denoted by a boldface Q (or blackboard bold , Unicode ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for ”quotient”. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal). A real number that is not rational is called irrational. Irrational numbers include √2, , e, and . The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost allreal numbers are irrational.

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Human life has developed in many aspects since the evolution of computer started. The main function of creating new technologies is to make human life easier. New technologies are being invented everyday which creates limitless opportunities for developers to make use of these technologies to serve a specific purpose or task. The Advertising used to be print paper or TV ads. Today mobile and online video creates new ways to contact with customers. Today Ads are kind of the most popular online advertising market in the world and trading ad spaces over the Internet. Currently the social online ads like Facebook and YouTube have some preferred set of users they wish to reach by showing their ads but some cases force the users to watch their Advertising. Ethan Zuckerman wad created the first pop-up advertising on the web was 15-Aug-2014. The pop-up advertising most using in the pc application and browser, and APPs for smartphone. The small internet windows that pop up on your screen can be useful, annoying or dangerous often used by advertisers to get your attention or by viruses to trick you into clicking on them. This guide gives them basic information about your identity. Some dangerous pop-up can leak your private information like your name, number, credit card member, and etc. In this paper we sum up, all these facts contribute to show how danger the pop-up advertising with developing new pop-up advertising to leak the user identity.

As of this writing, the algorithm employed for difficulty adjustment in the CryptoNote reference code is known by the Monero Research Lab to be flawed. We describe and illustrate the nature of the flaw and recommend a solution. By dishonestly reporting timestamps, attackers can gain disproportionate control over network difficulty. We verify this route of attack by auditing the CryptoNote reference difficulty adjustment code, which, we reimplement in the Python programming language. We use a stochastic model of blockchain growth to test the CryptoNote reference difficulty formula against the more traditional Bitcoin difficulty formula. This allows us to test our difficulty formula against various hash rate scenarios. This research bulletin has not undergone peer review, and reflects only the results of internal investigation.

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We present a geometric proof of the addition formulas for the hyperbolic sine and cosine functions, using elementary properties of linear transformations.

The accelerated rate of growth in the amount of web applications returns as a result an increase in the traffic that web servers must handle. This aggregated traffic, in addition to the demand of the clients to be served in a real time frame, leads to the requirement of a customized way to control web related resources. All processes of the web server are tied to the control that the OS scheduler has over them, and for the default settings, the scheduler is set to handle general purpose tasks instead of being optimized forweb serving purposes. To address this issue, the use of custom settings into the scheduler will allow the daemons needed to run a web page (such as Apache, PHP and a SQL DB) to be handled by the OS as efficiently as possible. The results of the test will be the comparison in performance of a web server for different settings on the Linux Scheduler.

In this paper we will study an algorithm designed by Madgwick which is commonly used to determine the orientation of a quadcopter. The algorithm uses a group of accelerometers, gyroscopes and magnetometers integrated in what is called an IMU as input. Some differences have been found between the results obtained by the original paper and the implementation done by the author. Therefore, a thorough study has been made, finding a miscalculation in the equations. The results show a relative average error in the orientation of 1,44 ppm.