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on 19-Dec-2018 (Wed)

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The observation that the universe is expanding has led to the current view that about 14 billion years ago the currently visible universe was concentrated into a point-like region that exploded in an event called the Big Bang .

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the electromagnetic force , a relatively weak but long-range force between electric charges, bound electrons to nuclei to form atoms, and the universe acquired the potential for complex chemistry and the existence of life

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Therefore, your data stream must include a process called error checking. Error checking is accomplished by adding some kind of standard information to each byte. In a simple computer data network, the handshaking informa- tion is called the parity bit, which tells the device receiving each byte whether the sum of the ones and zeroes inside the byte is odd or even. This value is called a checksum. If the receiving device discovers that the parity bit is not correct, it instructs the transmitter to send the same byte again.

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More complex networks, including wireless systems, include additional error-checking hand- shaking data with each string of data.

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First it has to warn the device at the other end that it is ready to send and make sure the intended recipient is ready to accept data. To accomplish this, a series of “handshaking” requests and answers must surround the actual data.

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it’s important to understand that not every bit that moves through a computer data network is part of the original block of information that arrived at the input computer. In a complex net- work such as a wireless data channel, as much as 40 percent or more of the transmitted data is handshaking and other overhead. It’s all essential, but every one of those bits increases the amount of time that the message needs to move through the network.

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Ethernet was introduced in the 1970s as a method for connecting multiple computers and related equipment in the same building. Ethernet offers several advantages: It’s fast, it’s extremely flexible, it’s relatively easy to install and use, and it’s inexpensive. It has become an industry standard supported by dozens of manufacturers, so you can use different brands of equipment in the same network. Today, more than 85 percent of all local area networks (LANs), including just about every modern home and office network, use some form of Ethernet to provide the physical connection between computers through twisted-pair cables, coaxial cables, or fiber optic cables

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One of Ethernet’s most important features is the method it uses to prevent conflicts among nodes, called Carrier Sense Multiple Access with Collision Detection (CSMA/CD). Every time a network node is ready to transmit a frame, it checks if another frame is already using the network; if the network is clear, the node sends the frame. But if the node detects that another frame is using the net- work (a condition called a collision), it waits a random period of time before it tries again. CSMA/CD is important because it allows a relatively large number of computers and other devices to operate through the same network without interference.

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There are many Ethernet specifications that cover different data transfer speeds and different kinds of cables and connectors. The ones you’re most likely to see in a small LAN include the following: 10Base-T: 10 Mbps through twisted-pair cables 100Base-T or Fast Ethernet: 100 Mbps through twisted-pair cables 1000Base-T or Gigabit Ethernet: 1000 Mbps through twisted-pair or fiber optic cables Wireless or Wi-Fi: any of several systems that use radio signals instead of wires—the latest 802.11n Wi-Fi networks can operate at speeds up to 70 Mbps

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A 10Base-T network is adequate for a small home network. It’s faster than most broadband Internet services, so it’s sufficient for handling the inbound and outbound data (including audio and video) that you exchange with the Internet. However, most new network ports, hubs, and switches can handle both 10Base-T and 100Base-T, so there’s very little point to limiting the network to the slower speed. 100Base-T will also allow you to move pictures, music, and videos and play multiplayer games within your own network much faster than a 10Base-T network, and it will not limit the speed of an 802.11n link. Considering the insignificant difference in cost, today’s 100Base-T networks are always a better choice than the older 10Base-T versions.

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If a 100Base-T network can’t handle 100 Mbps because of interference or some other problem, it will automatically drop down to 10Base-T.

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A 10Base-T device can work on a 100Base-T network, but it will force the whole network to drop down to 10 Mbps.

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You might also see the word Ethernet used to identify the connector on a computer, printer, or other network device that mates with an Ethernet cable to connect the device to a network. The instruction manual or the label on every piece of Ethernet-compatible equipment should tell you which type of connection it uses.

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Twisted-pair cables are bundles of wires in which each pair of wires is twisted together, as shown in Figure 2-3. Because data normally moves in only one direction through each pair of wires, a 10Base-T or 100Base-T network connection uses two pairs—one for each direction. The most common Ethernet cables include a total of eight wires in four color-coded wire pairs, so you can use the remaining wires as spares.

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Another name for Wi-Fi is wireless Ethernet, because Wi-Fi uses many of the same data-handling rules and specifications as a wired Ethernet network. However, every Wi-Fi packet must include additional handshaking data, so the overall data transfer speed is often slower than a conventional Ethernet link.

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In a powerline network, computer data moves through a building’s existing electric wiring. Each computer connects through a parallel port, a USB port, or an Ethernet port to a data adapter that plugs directly into an AC wall outlet. The same power transformer that feeds your house wiring also isolates your data network from your neighbors

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The most widely used standard for powerline networks is called HomePlug. The greatest advantage of HomePlug and other powerline networks is that the wires are already in place. Every AC wall socket in the house can double as a network connection point. It’s also more secure than Wi-Fi, and it can reach greater distances than a Wi-Fi network with just one base station. Wi-Fi signals are often blocked by thick walls and other obstacles that make no difference to a powerline system.

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You must plug all your powerline adapters directly into wall outlets. Surge protectors and powerline conditioners often absorb powerline network data, because they see the data as “noise” on the AC power voltage. Conversely, if you’re using a powerline net- work, you will want to connect your stereo or home theater system to power conditioners to filter out the noise produced by the network.

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Two more home networking methods are possible, but they’re almost always provided as supplements to other services. These systems use the internal telephone wiring that connects extension telephones in several rooms or the coaxial cable (coax) that provides cable TV signals.

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There’s one more concept that every network planner should understand: the difference between data terminal equipment (DTE) and data communications equipment or data circuit-terminating equipment (DCE)

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Data can move through a wire in only one direction. When a data link sends and receives signals at the same time, it must use separate wires to send data from the DTE to the DCE, and from the DCE to the DTE. Therefore, a network device uses separate inputs and outputs on the same multipin connector. The specific pin assignment is different in different connection types, but the inputs and outputs are always different pins or sockets.

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Therefore, when you connect two pieces of equipment, the outputs at each end must go to inputs at the other end. If Pin 2 on one device is an output, Pin 2 on the other device must be an input. Most standard data cables connect each connector pin to the same numbered pin at the other end, so connecting two devices through a cable is exactly the same as plugging one device directly into another.

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When you connect a terminal to a control device, the output pins on the DTE device connect to the input pins on the DCE device.

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Direct computer-to-computer communication requires a special cable because you can’t connect a DTE device directly to another DTE device. When you connect two DTE devices with serial data ports, you connect the output on one computer to the output on the other computer, and the input to the input, so neither computer will actually receive any data. Therefore, you must flip the connections, so each output connects to an input. A cable or adapter that connects output pins to input pins is called a null modem.

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Most of the time, we think of a computer network as a structure that can link one computer to any other computer connected to the same network. But sometimes all you need is a direct connection between two computers. This kind of connection is called a point-to-point network.

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For example, if you’re in a meeting where some- body asks for a copy of a report or drawing, you could use the built-in infrared network tools built into many laptop computers to shoot the file across the table from your computer to your colleague’s. Or if you want to copy a file from a friend’s computer, you could plug a transfer cable into both machines or set up a point-to-point Wi-Fi link

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Most Wi-Fi networks connect wireless nodes to a LAN through a wireless access point, but Wi-Fi network adapters can also support wireless links directly from one computer to another. This kind of connection is called an ad hoc network, because it’s usually set up as a temporary link rather than as part of a permanent network infrastructure (wireless networks with one or more central access points are called infrastructure networks)

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Infrared connections use invisible flashing light (it’s invisible because it uses frequencies outside the range of human sight) to exchange data between computers, mobile telephones, digital cameras, and other devices. Most of the wireless remote control units that you use with your television, DVD player, and home stereo system also use infrared light signals. Infrared channels are often called IrDA connections, because the Infrared Data Association (IrDA) has set the standards for infrared communication.

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Many laptop computers have built-in IrDA ports, usually in an incon- spicuous location along the edge of the case. The IrDA port is usually an infrared lens under a transparent plastic cover, like the one shown in Figure 2-8. The camera captured the flashing infrared light, even though it’s not normally visible to the human eye.

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As you have probably noticed with your TV’s remote control, infrared signals can bounce off walls and other objects, so it’s not absolutely necessary to point a pair of IrDA ports directly at each other, especially when they’re both indoors. When two computers with active IrDA ports are in the same room, they will usually detect each other automatically.

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The infrared port on a laptop computer can detect an IrDA signal from another computer in the same room and automatically set up a network link between the two devices. It’s best, then, to disable the infrared port anytime you’re not planning to use it. To disable or enable infrared communications in Windows, open the Device Manager (Control Panel System Hardware Device Manager), expand the list of infrared devices, right-click the name of the infrared port, and choose Disable or Enable from the pop-up menu

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This has several didactic advantages worth mentioning: both posets and monoids are themselves special kinds of categories, which in a certain sense represent the two “dimen- sions” (objects and arrows) that a general category has.

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Many phenomena occurring in categories can best be understood as generalizations from posets or monoids. On the other hand, the categories of posets (and monotone maps) and monoids (and homomorphisms) provide two further, quite different examp- les of categories in which to consider various concepts. The notion of a limit, for instance, can be considered both in a given poset and in the category of posets.

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The selection of material was easy. There is a standard core that must be included: categories, functors, natural transformations, equi- valence, limits and colimits, functor categories, representables, Yoneda’s lemma, adjoints, and monads. That nearly fills a course. The only “optional” topic inclu- ded here is cartesian closed categories and the λ-calculus, which is a must for computer scientists, logicians, and linguists. Several other obvious further topics were purposely not included: 2-categories, topoi (in any depth), and monoidal categories. These topics are treated in Mac Lane, which the student should be able to read after having completed the course.

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What is category theory? As a first approximation, one could say that category theory is the mathematical study of (abstract) algebras of functions.

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group theory is the abstraction of the idea of a system of permutations of a set or symmetries of a geometric object

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category theory arises from the idea of a system of functions among some objects

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We think of the composition g ◦ f as a sort of “product” of the functions f and g, and consider abstract “algebras” of the sort arising from collections of functions. A category is just such an “algebra,” consisting of objects A,B,C,... and arrows f : A → B, g : B → C, ..., that are closed under composition and satisfy certain conditions typical of the composition of functions.

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category theory was invented in the tradition of Felix Klein’s Erlanger Programm, as a way of studying and characterizing different kinds of mathematical structures in terms of their “admissible trans- formations.” The general notion of a category provides a characterization of the notion of a “structure-preserving transformation,” and thereby of a species of structures admitting such transformations

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1945 Eilenberg and Mac Lane’s “General theory of natural equivalences” was the original paper, in which the theory was first formulated.

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Late 1940s The main applications were originally in the fields of algebraic topology, particularly homology theory, and abstract algebra

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1950s A. Grothendieck et al. began using category theory with great success in algebraic geometry.

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1960s F.W. Lawvere and others began applying categories to logic, revealing some deep and surprising connections.

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1970s Applications were already appearing in computer science, linguistics, cognitive science, philosophy, and many other areas.

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For example, the important notion of an adjoint functor occurs in logic as the existential quantifier and in topology as the image operation along a continuous function. From a categorical point of view, these turn out to be essentially the same operation.

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The concept of adjoint functor is in fact one of the main things that the reader should take away from the study of this book. It is a strictly category-theoretical notion that has turned out to be a conceptual tool of the first magnitude—on par with the idea of a continuous function.

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In fact, just as the idea of a topological space arose in connection with con- tinuous functions

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the notion of a category arose in order to define that of a functor, at least according to one of the inventors. The notion of a functor arose—so the story goes on—in order to define natural transformati- ons.

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everywhere that functions are, there are categories. Indeed, the subject might better have been called abstract function theory,

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Let f be a function from a set A to another set B, we write f : A → B. To be explicit, this means that f is defined on all of A and all the values of f are in B. In set theoretic terms, range(f) ⊆ B.

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Now suppose we also have a function g : B C, then there is a composite function g ◦ f : A → C, given by (g ◦ f)(a)=g(f(a)) a ∈ A.

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Now this operation ” of composition of functions is associative, as follows. If we have a further function h : C D and form h ◦ g and g ◦ f, then we can compare (h ◦ g) ◦ f and h ◦ (g ◦ f) as indicated in the diagram given above. It turns out that these two functions are always identical, (h ◦ g) ◦ f = h ◦ (g ◦ f ) since for any a ∈ A,wehave ((h ◦ g) ◦ f)(a)=h(g(f(a))) = (h ◦ (g ◦ f))(a)

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By the way, this is, of course, what it means for two functions to be equal: for every argument, they have the same value.

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Finally, note that every set A has an identity function 1 A : A → A given by 1 A (a)=a. These identity functions act as “units” for the operation ◦ of composition, in the sense of abstract algebra. That is to say, f ◦ 1 A = f =1 B ◦ f for any f : A → B.

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Definition 1.1. A category consists of the following data:
• Objects: A,B,C,...
• Arrows : f,g,h,...
• For each arrow f, there are given objects dom(f), cod(f) called the domain and codomain of f. We write f : A → B to indicate that A =dom(f)andB =cod(f).
• Given arrows f : A → B and g : B → C, that is, with cod(f)=dom(g)

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there is given an arrow g ◦ f : A → C called the composite of f and g.
• For each object A, there is given an arrow 1 A : A → A called the identity arrow of A.
These data are required to satisfy the following laws:
• Associativity: h ◦ (g ◦ f)=(h ◦ g) ◦ f for all f : A → B, g : B → C, h : C → D.
• Unit: f ◦ 1 A = f =1 B ◦ f for all f : A → B

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A category is anything that satisfies this definition—and we will have plenty of examples very soon. For now I want to emphasize that, unlike in Section 1.2, the objects do not have to be sets and the arrows need not be functions. In this sense, a category is an abstract algebra of functions, or “arrows” (sometimes also called “morphisms”), with the composition operation “◦” as primitive. If you are familiar with groups, you may think of a category as a sort of generalized group.

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A “homomorphism of categories” is called a functor.

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