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nothing to do either with each other or with Fermat’s theorem.

7. The problem with elliptic curves is to find if they have integer solutions, and if so, how many.

8. The problem with elliptic curves is to find if they have integer solutions, and if so, how many. For example the equation y^2=x^3-2 with a = b = 0 and c = -2 has only one set of integer solutions, namely x = 3, y = 5, but proving that there are no other solutions is extremely difficult.

9. Modular forms come in various shapes and sizes, but each one is built from the same basic ingredients. What differentiates each modular form is the amount of each ingredient it contains.

10. At this point mathematicians Yutaka Taniyama and Goro Shimura found a strange affinity between some elliptic curves and some modular forms. This led to the Taniyama-Shimura conjecture that: all elliptic curves over Q (the field of rational numbers) are modular.

Albert Einstein, who fancied himself as a violinist, was rehearsing a Haydn string quartet. When he failed for the fourth time to get his entry in the second movement, the cellist looked up and said, «The problem with you, Albert, is that you simply can' t count.»

IX. Fostering Critical Thinking Skills

Read the text. Find additional material to expand the topic and write a commented essay in Russian on The Great Theorems of Mathematics:

Fermat’s Last Theorem

The 17th-century mathematician Pierre de Fermat wrote in 1637 in his copy of the famous Arithmetica of Diophantus: “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”. However, for the next 357 years mathemati-

cians tried in vain to find a proof.

Long after all the other statements made by Fermat had been either proved or disproved, this remained; hence it is called Fermat›s *Last* Theorem (actually, Conjecture would be more accurate than Theorem). This conjecture was worked on by many famous mathematicians. Fermat himself proved this theorem for n = 4, and Leonhard Euler did n = 3. Special cases were dispatched one after another, new theories were de-

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veloped to attack this problem, but all attempts at a general proof failed. Indeed, some mathematicians devoted much of their life’s work to the pursuit of that goal, and the search for a proof led to the development of whole new branches of mathematics, but it was not until this decade that the English mathematician Andrew Wiles, from Princeton University, finally completed the task.

Fermat‘s Last Theorem states that

xn + yn = zn

has no non-zero integer solutions for x, y and z when n > 2. This is a generalization of the Pythagorean theorem stating that in a right triangle (where one angle equals 90°), the sum of the squares of two sides equals the square of the hypotenuse.

Ancient Greeks and Babylonians knew that this equation had integer solutions, such as (3,4,5) (32 + 42 = 52) or (5,12,13). These solutions are known as Pythagorean triples, and there exist an infinite number of them (even excluding trivial solutions for which a, b and c have a common divisor). According to Fermat’s last theorem, no such solution exists when the exponent 2 is replaced by a larger integer number and a, b, c > 0.

The difficulty in proving is that the case revolves around the fact that there is an infinite number of equations, and an infinite number of possible values for a, b, and c. The proof has to prove that no solutions exist within this infinity of infinities. The actual proof is very indirect, and involves sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the branches of mathematics, which at face value appear to have nothing to do either with each other or with Fermat’s theorem.

Elliptic curves are of the form: y2 = x3 + ax2 + bx + c, where a, b, c are integers. The problem with elliptic curves is to find if they have integer solutions, and if so, how many. For example the equation

y2 = x3 - 2 with a = b = 0 and c = -2

has only one set of integer solutions, namely x = 3, y = 5, but proving that there are no other solutions is extremely difficult. The problem is simplified by making the possible numbers finite, i.e. working in ‘clock’

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arithmetic: the arithmetic of hours on the clock face. So 5-clock arithmetic uses only 0, 1, 2, 3, 4 then 5 = 0 again. If we begin at 7 o’clock and add 8 hours, then rather than ending at 15 o’clock (as in usual addition), we are at 3 o’clock. Essentially, when we reach 12, we start over. It was then possible to make progress with determining the number of integer solutions of the elliptic curves. For a particular elliptic curve, the number of integer solutions in each clock arithmetic forms an L-series for that curve. Because we cannot say how many solutions there are in normal number space, extending to infinity as it does, the L-series gives a great deal of information about the elliptic curve it describes. The idea is that studying the L-series you can learn all you want to know about its elliptic curve.

A modular form is defined by two axes, x and y, but each axis has a real and imaginary part. In effect it is four dimensional (xr, xi, yr, yi) where xr means real part of x, xi means imaginary part of x, and similarly with yr and yi. The four-dimensional space is called hyperbolic space.

The interesting thing about modular forms is that they exhibit infinite symmetry under transformations of the type:

az+b f(z) -> f[------] cz+d

These are functions that remain unchanged when the complex variable z is changed according to the above transformation. Here the elements a, b, c, d, arranged as a matrix, form an algebraic group. There are infinitely many possible variations. They all commute with each other and the function f is invariant under the group of transformations. Modular forms come in various shapes and sizes, but each one is built from the same basic ingredients. What differentiates each modular form is the amount of each ingredient it contains.

At this point mathematicians Yutaka Taniyama and Goro Shimura found a strange affinity between some elliptic curves and some modular forms. This led to the Taniyama-Shimura conjecture that: all elliptic curves over Q (the field of rational numbers) are modular. It was in proving this conjecture that Andrew Wiles established the proof of Fermat’s Last theorem. The reason elliptic curves and modular forms are connected is as follows. In 1986, Ken Ribet had proved Gerhard Frey’s epsilon conjecture that every counterexample an + bn = cn to Fermat’s last theorem would yield an elliptic curve defined as:

y2 = x(x - an)(x + bn),

which would provide a counterexample to the Taniyama-Shimura con-

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jecture. So now we have the following chain of reasoning:

(1)If the Taniyama-Shimura conjecture can be proved, then every elliptic curve is modular.

(2)If every elliptic curve must be modular, then the Frey elliptic curve is forbidden to exist.

(3)If the Frey elliptic curve does not exist, then there can be no solutions to the Fermat equation.

(4)Therefore Fermat’s Last Theorem is true.

The greatest difficulty was in proving that the Taniyama-Shimura con-

jecture was true. This is the contribution made by Andrew Wiles, and the final stage in establishing Fermat’s Last theorem.

X. Organizing Ideas

Make a concept map on Mathematics and fill it with basic notions, associated words and phrases you’ve learned in this unit.

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Computer Science

Unit I

Hardware

I. Getting Started

Read the text “Anatomy of Laptop Computer“. Divide it into several key parts and compose 3-5 questions to the each part. Put your questions to class.

II. Working With Vocabulary

Place the words and phrases below into the “Word“ column and complete the table:

computing power, PDA, palmtop, mobile use, microprocessor, operating system, memory, disk drives, display, input/output ports: serial ports, parallel ports, USB ports, touch pad, trackball, hard drive, floppy drive, CD/DVD drive, kernel, user-interface management, ROM, RAM, computer (primary) storage, sequential memory, built-in keyboard, compact flash drive, integrated peripherals, liquid crystal active matrix display, external AC adapter, lithium-ion battery, light-duty use.

III. Practising Translation Techniques

Make a written translation of the following text:

Anatomy of Laptop Computer

Alaptop computer (also known as notebook computer) is a small mobile personal computer, usually weighing from 1 to 3 kg (2 to 7 pounds) and having just as much computing power as desktops, without taking up as much space. Powerful laptops (often heavy) designed to compete with the computing power offered by a typical desktop are often known as desktop replacements. Computers larger than PDAs (Personal Digital Assistant—handheld computers with small keyboards and/or pen-based input systems providing calendar, web-browser, me-

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diaplayer and other applications) but smaller than notebooks are also sometimes called palmtops.

Laptops are capable of many of the same tasks that desktop computers perform, contain components that are similar to those in their desktop counterparts but are miniaturized and optimized for mobile use and efficient power consumption. The most essential parts of laptops are: microprocessor, operating system, memory, disk drive, display, input/output ports, sound cards and speakers.

The microprocessor is the brain of the laptop and coordinates all of the computer’s functions according to a set of internal programmed instructions (that is, the operating system software) stored in memory. A typical laptop processor can receive instructions or data from the user through a keyboard in combination with another device (mouse, touch pad, trackball), can receive and store data through several data storage devices (hard drive, floppy drive, CD/DVD drive), can display data on CRT or LCD computer monitors, can send data to printers, modems, networks and wireless networks through various input/output ports, is powered by AC power and/or batteries.

There are a wide range of notebook processors available from Intel (Pentium M, Celeron, Centrino), AMD (Athlon, Opteron) which develop and manufacture for the different Microsoft operating systems. Motorola and IBM develop and manufacture the PowerPC chips for Apple notebooks. Generally, notebook processors are less powerful than their desktop counterparts, owing to the need to conserve electricity and reduce heat output.

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The operating system is the master control program that is loaded first when the computer is turned on. Its main part, the “kernel”, resides in memory permanently. The operating system sets the standards for all application programs that run in the computer during all user-inter face and file management operations. The primary operating systems in use are the many versions of Windows (95, 98, NT, ME, 2000, XP), Macintosh OS X, Linux, many versions of Unix and different OS for IBM mainframes. For some applications DOS is still used.

Memory used in laptops includes both ROM and RAM. An acronym for Read Only Memory, ROM is computer memory on which programming codes and/or data has been pre-recorded at the factory. It can be read, but it cannot be erased or removed. ROM retains its content even when the power is switched off, unlike a computer’s Random Access Memory, or RAM, which needs a constant charge of electricity to keep its information. For this reason, ROM is considered to be ‘non-volatile’ and RAM is ‘volatile’. RAM is a type of computer storage whose contents can be accessed in any (i.e. random) order in contrast to sequential memory devices such as magnetic tapes and discs in which the mechanical movement of the storage medium forces the computer to access data in a fixed order. RAM is typically used for primary storage (main memory) in computers to hold actively used and actively changing information.

Various disk drive storage devices in laptops include the hard disk drive (HDD), with a usual disk size of 20–100 Gb, used for storing OS, application programs and data files, and, in addition, some type of removable disk storage systems: floppy disks, compact flash drives and optical disks such as compact discs (CD), digital video discs or digital versatile discs (DVD) and others.

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Computer nay-sayers

«I think there is a world market for maybe five com- puters.»–Thomas Watson, chairman of IBM, 1943

«There is no reason anyone would want a computer in their home.»–Ken Olson, president, chairman and founder of Digital Equipment Corp., 1977

«640K ought to be enough for anybody.»–Bill Gates, 1981

Input/Output ports can include serial ports, parallel ports and Universal Serial Bus (USB) ports and are used for transferring data to or from a computer and to or from peripheral devices – internal (such as a CD-ROM drive or internal modem) or external (such as a mouse, keyboard, printer, monitor) parts of a computer other than the CPU or working memory. Internal peripheral devices are often referred to as integrated peripherals.

Some parts for a modern laptop have no corresponding part in a desktop computer. Laptops usually have integrated liquid crystal active matrix displays with resolutions of 1024 by 768 pixels (XGA) and above. In addition to a built-in keyboard, Laptops may utilize one of three input devices to move the cursor on the LCD screen: trackball (rotating scroll ball), trackpoint (pushable point set among typing keys) or touchpad. All of these devices have buttons that act like the right and left buttons on a mouse. Also laptops can be powered or recharged from an external AC adapter using mains electricity, but also run on batteries. Current models use lithium-ion batteries, which have largely replaced the older nickel metal-hydride technology. Typical battery life for most laptops is two to five hours with light-duty use, but may drop to as little as one hour with intensive use. Batteries gradually degrade over time and eventually need to be replaced, depending largely on the charging and discharging pattern, from one to five years.

IV. Knowing Ins And Outs

Language reflects social, cultural and scientific renewal of modern society by coining neologisms. Morphological neologisms are created by compounding (Webmaster, cyberspace), derivation (digital—digi- tize, digitization), conversion (“In English you can verb almost everything!”), blending (infomercial=information+commercial), acronymy and initialization (TCP/IP, SCSI).

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Semantic neologisms can result from adapting or extending the meaning of existing words, i.e. in computing, the word “toaster” is used to describe a product that joins various components in one easy- to-use package, referring to the ease of using a kitchen toaster. New computing terms come mainly from the English language, and it is a translator who introduces new words into another language.

Below is the list of computing-related terms with definitions. Study the list, explain the coinage patterns for the terms and translate them into Russian. Keep in mind the golden rule for terminology translation: Brevity is the soul of wit!

New Computing Terms

Anticipointment—The feeling you get when a product or event does not live up to its own hype, i.e. “Windows Me was said to be a huge anticipointment for users who upgraded from 2000.”

Blog (a.k.a. weblog or Web log)—A Web site (or section of a Web site) where users can post a chronological, up-to-date e-journal entry of their thoughts. Each post usually contains a Web link. Basically, it is an open forum communication tool that, depending on the Web site, is either very individualistic or performs a crucial function for a company.

Clickable graphic or imagemap—An image or graphic that has been coded to contain interactive areas. When it’s clicked on, it launches another Web page or program.

Creeping featurism—The tendency for programmers to add more and more features to a software product in an attempt to “keep up with the Joneses” or in this case, to keep up with stiff competition from other companies. It generally produces a slow, clunky program.

Digiterati—The literati of the digital world, it includes people in the industry who are considered knowledgeable, hip, or otherwise in-the- know with regard to the online revolution.

Dot-com (dot-commers)—Based on the suffix .com, this refers to a company in the industry (with a .com Web site) whose primary focus is on the financial aspects of taking the company public (versus addressing any real market need or establishing a successful, long-term business model). An example of its usage goes like this: “He met with plenty of young American dot-commers overseas, and they simply weren’t prepared to do international business.”

E-commerce—Put simply, it means conducting business online. Selling goods, in the traditional sense, is possible to do electronically because of certain software programs that run the main functions of an e-commerce Web site, including product display, online ordering, and inventory management.

Ego-surfing—To perform an Internet search on one’s own name.

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Emoticon aka :-) or smiley—An emoticon is a sequence of typed characters that creates a rough picture of something, such as a facial expression. If you don’t see the picture in the emoticon shown above, try tilting your head to the left—the colon represents the eyes, the dash represents the nose, and the right parenthesis represents the mouth. The term “emoticon” literally means “an icon that represents emotion.” Emoticons grew out of the need to display feeling in the two-dimension- al, online, written world.

Googlewhacking—The name of a game for search-obsessed fans of Google.com whereby a user types two words into the Google search line with the intent of trying to retrieve a single search result. With more than three billion Web pages indexed by Google, if you see “Results 1-1 of 1” appear under your Google search, you’re a winner.

Hard copy—A printed copy of some kind of information (as opposed to an electronic version). You’ll hear someone ask, “Do you have a hard copy of that?” The opposite is known as a soft copy.

Key pal—The online equivalent of a pen pal. A key pal is a person you correspond with using a keyboard and e-mail (versus using a pen to write handwritten letters).

Link farm—This is a Web site that has no meaningful content of its own, just a bunch of links.

Mouselexia—The inability to use a mouse correctly. Some people are naturally “mouselexic,” while others are struck with mouselexia only when an IT guru is looking over their shoulder.

Netiquette—The code of conduct and unofficial rules that govern interaction and behavior in the net, i.e. Do not spam.

Ohnosecond—The fraction of time it takes to realize you’ve just goofed; for example, right after you hit the send button on an e-mail and realize you forgot to include the attachment. Another great example is that moment of horror when you see the key in the ignition switch just as you’re slamming the car door shut.

Privacation—A cross between “private” and “publication” in which information owners distribute products but retain controls on large-scale copyright infringement. An example of this is Stephen King’s honor-system serialization in which he asks each of his readers to snail mail him a one dollar bill each time they download one of his e-books.

Shopping cart technology—This is a software program designed to make products available for online ordering (kind of like an electronic catalogue). The behind-the-screens technology that enables this e- commerce component to work involves cookies and SSL.

Signal-to-noise ratio—A measure of the amount of useful information found in a given Usenet newsgroup. This phrase is often used derogatorily; for example, “The signal-to-noise ratio in this newsgroup

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