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78-530: A result (also called upshot ) is the outcome or consequence of a sequence of actions or events. Possible results include gain , injury , value , and victory . Some types of results include the outcome of an action , the final value of a calculation , and the outcome of a vote . A result is the final consequence of a sequence of actions or events expressed qualitatively or quantitatively . Possible results include advantage , disadvantage , gain , injury , loss , value , and victory . There may be

156-438: A n ) n = − ∞ ∞ {\textstyle {(a_{n})}_{n=-\infty }^{\infty }} is a bi-infinite sequence , and can also be written as ( … , a − 1 , a 0 , a 1 , a 2 , … ) {\textstyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )} . In cases where

234-464: A n ) . {\textstyle (a_{n}).} Here A is the domain, or index set, of the sequence. Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets . A net is a function from a (possibly uncountable ) directed set to a topological space. The notational conventions for sequences normally apply to nets as well. The length of

312-663: A distance from L {\displaystyle L} less than d {\displaystyle d} . For example, the sequence a n = n + 1 2 n 2 {\textstyle a_{n}={\frac {n+1}{2n^{2}}}} shown to the right converges to the value 0. On the other hand, the sequences b n = n 3 {\textstyle b_{n}=n^{3}} (which begins 1, 8, 27, ...) and c n = ( − 1 ) n {\displaystyle c_{n}=(-1)^{n}} (which begins −1, 1, −1, 1, ...) are both divergent. If

390-599: A function of n . Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions. Not all sequences can be specified by a recurrence relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...). There are many different notions of sequences in mathematics, some of which ( e.g. , exact sequence ) are not covered by

468-415: A high speed compared to mass storage which is slower but less expensive per bit and higher in capacity. Besides storing opened programs and data being actively processed, computer memory serves as a mass storage cache and write buffer to improve both reading and writing performance. Operating systems borrow RAM capacity for caching so long as it is not needed by running software. If needed, contents of

546-416: A limit if the elements of the sequence become closer and closer to some value L {\displaystyle L} (called the limit of the sequence), and they become and remain arbitrarily close to L {\displaystyle L} , meaning that given a real number d {\displaystyle d} greater than zero, all but a finite number of the elements of the sequence have

624-483: A natural number N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} we have If ( a n ) {\displaystyle (a_{n})} is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that | ⋅ | {\displaystyle |\cdot |} denotes

702-413: A range of possible outcomes associated with an event depending on the point of view , historical distance or relevance. Reaching no result can mean that actions are inefficient, ineffective, meaningless or flawed. Some types of result are as follows: Sequence For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also,

780-415: A recurrence relation is Recamán's sequence , defined by the recurrence relation with initial term a 0 = 0. {\displaystyle a_{0}=0.} A linear recurrence with constant coefficients is a recurrence relation of the form where c 0 , … , c k {\displaystyle c_{0},\dots ,c_{k}} are constants . There

858-400: A sequence are discussed after the examples. The prime numbers are the natural numbers greater than 1 that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics , particularly in number theory where many results related to them exist. The Fibonacci numbers comprise

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936-440: A sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence ( a n ) {\displaystyle (a_{n})} is normally denoted lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} . If ( a n ) {\displaystyle (a_{n})}

1014-404: A sequence is defined as the number of terms in the sequence. A sequence of a finite length n is also called an n -tuple . Finite sequences include the empty sequence  ( ) that has no elements. Normally, the term infinite sequence refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence

1092-463: A sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting

1170-409: A sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways. Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be

1248-450: A sequence of sequences: ( ( a m , n ) n ∈ N ) m ∈ N {\textstyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }} denotes a sequence whose m th term is the sequence ( a m , n ) n ∈ N {\textstyle (a_{m,n})_{n\in \mathbb {N} }} . An alternative to writing

1326-565: Is monotonically decreasing if each consecutive term is less than or equal to the previous one, and is strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function . The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing , respectively. If

1404-593: Is semi-volatile . The term is used to describe a memory that has some limited non-volatile duration after power is removed, but then data is ultimately lost. A typical goal when using a semi-volatile memory is to provide the high performance and durability associated with volatile memories while providing some benefits of non-volatile memory. For example, some non-volatile memory types experience wear when written. A worn cell has increased volatility but otherwise continues to work. Data locations which are written frequently can thus be directed to use worn circuits. As long as

1482-474: Is a divergent sequence, then the expression lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} is meaningless. A sequence of real numbers ( a n ) {\displaystyle (a_{n})} converges to a real number L {\displaystyle L} if, for all ε > 0 {\displaystyle \varepsilon >0} , there exists

1560-468: Is a general method for expressing the general term a n {\displaystyle a_{n}} of such a sequence as a function of n ; see Linear recurrence . In the case of the Fibonacci sequence, one has c 0 = 0 , c 1 = c 2 = 1 , {\displaystyle c_{0}=0,c_{1}=c_{2}=1,} and the resulting function of n

1638-411: Is a simple classical example, defined by the recurrence relation with initial terms a 0 = 0 {\displaystyle a_{0}=0} and a 1 = 1 {\displaystyle a_{1}=1} . From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. A complicated example of a sequence defined by

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1716-401: Is a strictly increasing sequence of positive integers. Some other types of sequences that are easy to define include: An important property of a sequence is convergence . If a sequence converges, it converges to a particular value known as the limit . If a sequence converges to some limit, then it is convergent . A sequence that does not converge is divergent . Informally, a sequence has

1794-464: Is a system where each program is given an area of memory to use and is prevented from going outside that range. If the operating system detects that a program has tried to alter memory that does not belong to it, the program is terminated (or otherwise restricted or redirected). This way, only the offending program crashes, and other programs are not affected by the misbehavior (whether accidental or intentional). Use of protected memory greatly enhances both

1872-409: Is also used to describe semi-volatile behavior constructed from other memory types, such as nvSRAM , which combines SRAM and a non-volatile memory on the same chip , where an external signal copies data from the volatile memory to the non-volatile memory, but if power is removed before the copy occurs, the data is lost. Another example is battery-backed RAM , which uses an external battery to power

1950-464: Is bi-infinite. This sequence could be denoted ( 2 n ) n = − ∞ ∞ {\textstyle {(2n)}_{n=-\infty }^{\infty }} . A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }}

2028-409: Is called a lower bound . If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded . A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of

2106-465: Is called a singly infinite sequence or a one-sided infinite sequence when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence , two-way infinite sequence , or doubly infinite sequence . A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ),

2184-418: Is called an index , and the set of values that it can take is called the index set . It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , which denotes a sequence whose n th element

2262-470: Is computer memory that requires power to maintain the stored information. Most modern semiconductor volatile memory is either static RAM (SRAM) or dynamic RAM (DRAM). DRAM dominates for desktop system memory. SRAM is used for CPU cache . SRAM is also found in small embedded systems requiring little memory. SRAM retains its contents as long as the power is connected and may use a simpler interface, but commonly uses six transistors per bit . Dynamic RAM

2340-416: Is easily discernible by inspection. Other examples are sequences of functions , whose elements are functions instead of numbers. The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences. Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of π . One such notation

2418-406: Is given by Binet's formula . A holonomic sequence is a sequence defined by a recurrence relation of the form where c 1 , … , c k {\displaystyle c_{1},\dots ,c_{k}} are polynomials in n . For most holonomic sequences, there is no explicit formula for expressing a n {\displaystyle a_{n}} as

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2496-503: Is given by the variable a n {\displaystyle a_{n}} . For example: One can consider multiple sequences at the same time by using different variables; e.g. ( b n ) n ∈ N {\textstyle (b_{n})_{n\in \mathbb {N} }} could be a different sequence than ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} . One can even consider

2574-431: Is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation. The Fibonacci sequence

2652-404: Is monotonically increasing if and only if a n + 1 ≥ a n {\textstyle a_{n+1}\geq a_{n}} for all n ∈ N . {\displaystyle n\in \mathbf {N} .} If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing . A sequence

2730-918: Is more complicated for interfacing and control, needing regular refresh cycles to prevent losing its contents, but uses only one transistor and one capacitor per bit, allowing it to reach much higher densities and much cheaper per-bit costs. Non-volatile memory can retain the stored information even when not powered. Examples of non-volatile memory include read-only memory , flash memory , most types of magnetic computer storage devices (e.g. hard disk drives , floppy disks and magnetic tape ), optical discs , and early computer storage methods such as magnetic drum , paper tape and punched cards . Non-volatile memory technologies under development include ferroelectric RAM , programmable metallization cell , Spin-transfer torque magnetic RAM , SONOS , resistive random-access memory , racetrack memory , Nano-RAM , 3D XPoint , and millipede memory . A third category of memory

2808-411: Is organized into memory cells each storing one bit (0 or 1). Flash memory organization includes both one bit per memory cell and a multi-level cell capable of storing multiple bits per cell. The memory cells are grouped into words of fixed word length , for example, 1, 2, 4, 8, 16, 32, 64 or 128 bits. Each word can be accessed by a binary address of N bits, making it possible to store 2 words in

2886-482: Is physically stored or whether the user's computer will have enough memory. The operating system will place actively used data in RAM, which is much faster than hard disks. When the amount of RAM is not sufficient to run all the current programs, it can result in a situation where the computer spends more time moving data from RAM to disk and back than it does accomplishing tasks; this is known as thrashing . Protected memory

2964-500: Is replaced by the expression dist ⁡ ( a n , L ) {\displaystyle \operatorname {dist} (a_{n},L)} , which denotes the distance between a n {\displaystyle a_{n}} and L {\displaystyle L} . If ( a n ) {\displaystyle (a_{n})} and ( b n ) {\displaystyle (b_{n})} are convergent sequences, then

3042-568: Is to write down a general formula for computing the n th term as a function of n , enclose it in parentheses, and include a subscript indicating the set of values that n can take. For example, in this notation the sequence of even numbers could be written as ( 2 n ) n ∈ N {\textstyle (2n)_{n\in \mathbb {N} }} . The sequence of squares could be written as ( n 2 ) n ∈ N {\textstyle (n^{2})_{n\in \mathbb {N} }} . The variable n

3120-501: The Electrotechnical Laboratory in 1972. Flash memory was invented by Fujio Masuoka at Toshiba in the early 1980s. Masuoka and colleagues presented the invention of NOR flash in 1984, and then NAND flash in 1987. Toshiba commercialized NAND flash memory in 1987. Developments in technology and economies of scale have made possible so-called very large memory (VLM) computers. Volatile memory

3198-464: The Fibonacci sequence F {\displaystyle F} is generally denoted as F n {\displaystyle F_{n}} . In computing and computer science , finite sequences are usually called strings , words or lists , with the specific technical term chosen depending on the type of object the sequence enumerates and the different ways to represent

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3276-499: The Royal Radar Establishment proposed digital storage systems that use CMOS (complementary MOS) memory cells, in addition to MOSFET power devices for the power supply , switched cross-coupling, switches and delay-line storage . The development of silicon-gate MOS integrated circuit (MOS IC) technology by Federico Faggin at Fairchild in 1968 enabled the production of MOS memory chips . NMOS memory

3354-558: The System/360 Model 95 . Toshiba introduced bipolar DRAM memory cells for its Toscal BC-1411 electronic calculator in 1965. While it offered improved performance, bipolar DRAM could not compete with the lower price of the then dominant magnetic-core memory. MOS technology is the basis for modern DRAM. In 1966, Robert H. Dennard at the IBM Thomas J. Watson Research Center was working on MOS memory. While examining

3432-550: The Whirlwind I computer in 1953. Magnetic-core memory was the dominant form of memory until the development of MOS semiconductor memory in the 1960s. The first semiconductor memory was implemented as a flip-flop circuit in the early 1960s using bipolar transistors . Semiconductor memory made from discrete devices was first shipped by Texas Instruments to the United States Air Force in 1961. In

3510-441: The codomain of the sequence is fixed by context, for example by requiring it to be the set R of real numbers, the set C of complex numbers, or a topological space . Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, a n rather than a ( n ) . There are terminological differences as well:

3588-427: The convergence properties of sequences. In particular, sequences are the basis for series , which are important in differential equations and analysis . Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers . There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify

3666-411: The limit of a sequence of rational numbers (e.g. via its decimal expansion , also see completeness of the real numbers ). As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π , that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that

3744-420: The natural numbers . In the second and third bullets, there is a well-defined sequence ( a k ) k = 1 ∞ {\textstyle {(a_{k})}_{k=1}^{\infty }} , but it is not the same as the sequence denoted by the expression. Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion . This

3822-561: The Arma Division of the American Bosch Arma Corporation. In 1967, Dawon Kahng and Simon Sze of Bell Labs proposed that the floating gate of a MOS semiconductor device could be used for the cell of a reprogrammable ROM, which led to Dov Frohman of Intel inventing EPROM (erasable PROM) in 1971. EEPROM (electrically erasable PROM) was developed by Yasuo Tarui, Yutaka Hayashi and Kiyoko Naga at

3900-490: The characteristics of MOS technology, he found it was possible to build capacitors , and that storing a charge or no charge on the MOS capacitor could represent the 1 and 0 of a bit, while the MOS transistor could control writing the charge to the capacitor. This led to his development of a single-transistor DRAM memory cell. In 1967, Dennard filed a patent for a single-transistor DRAM memory cell based on MOS technology. This led to

3978-441: The complex modulus, i.e. | z | = z ∗ z {\displaystyle |z|={\sqrt {z^{*}z}}} . If ( a n ) {\displaystyle (a_{n})} is a sequence of points in a metric space , then the formula can be used to define convergence, if the expression | a n − L | {\displaystyle |a_{n}-L|}

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4056-714: The computer memory can be transferred to storage; a common way of doing this is through a memory management technique called virtual memory . Modern computer memory is implemented as semiconductor memory , where data is stored within memory cells built from MOS transistors and other components on an integrated circuit . There are two main kinds of semiconductor memory: volatile and non-volatile . Examples of non-volatile memory are flash memory and ROM , PROM , EPROM , and EEPROM memory. Examples of volatile memory are dynamic random-access memory (DRAM) used for primary storage and static random-access memory (SRAM) used mainly for CPU cache . Most semiconductor memory

4134-417: The context or a specific convention. In mathematical analysis , a sequence is often denoted by letters in the form of a n {\displaystyle a_{n}} , b n {\displaystyle b_{n}} and c n {\displaystyle c_{n}} , where the subscript n refers to the n th element of the sequence; for example, the n th element of

4212-432: The definitions and notations introduced below. In this article, a sequence is formally defined as a function whose domain is an interval of integers . This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring

4290-597: The delay line, the Williams tube and Selectron tube , originated in 1946, both using electron beams in glass tubes as means of storage. Using cathode-ray tubes , Fred Williams invented the Williams tube, which was the first random-access computer memory . The Williams tube was able to store more information than the Selectron tube (the Selectron was limited to 256 bits, while the Williams tube could store thousands) and

4368-697: The domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation ( k 2 ) ) k = 1 10 {\textstyle (k^{2}){\vphantom {)}}_{k=1}^{10}} denotes the ten-term sequence of squares ( 1 , 4 , 9 , … , 100 ) {\displaystyle (1,4,9,\ldots ,100)} . The limits ∞ {\displaystyle \infty } and − ∞ {\displaystyle -\infty } are allowed, but they do not represent valid values for

4446-433: The domain of a sequence to be the set of natural numbers . This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition,

4524-401: The early 1940s. Through the construction of a glass tube filled with mercury and plugged at each end with a quartz crystal, delay lines could store bits of information in the form of sound waves propagating through the mercury, with the quartz crystals acting as transducers to read and write bits. Delay-line memory was limited to a capacity of up to a few thousand bits. Two alternatives to

4602-594: The first commercial DRAM IC chip, the Intel 1103 in October 1970. Synchronous dynamic random-access memory (SDRAM) later debuted with the Samsung KM48SL2000 chip in 1992. The term memory is also often used to refer to non-volatile memory including read-only memory (ROM) through modern flash memory . Programmable read-only memory (PROM) was invented by Wen Tsing Chow in 1956, while working for

4680-412: The following limits exist, and can be computed as follows: Computer memory Computer memory stores information, such as data and programs, for immediate use in the computer . The term memory is often synonymous with the terms RAM , main memory , or primary storage . Archaic synonyms for main memory include core (for magnetic core memory) and store . Main memory operates at

4758-518: The following types: Virtual memory is a system where physical memory is managed by the operating system typically with assistance from a memory management unit , which is part of many modern CPUs . It allows multiple types of memory to be used. For example, some data can be stored in RAM while other data is stored on a hard drive (e.g. in a swapfile ), functioning as an extension of the cache hierarchy . This offers several advantages. Computer programmers no longer need to worry about where their data

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4836-474: The index, only the supremum or infimum of such values, respectively. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }} is the same as the sequence ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , and does not contain an additional term "at infinity". The sequence (

4914-434: The integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). Other examples of sequences include those made up of rational numbers , real numbers and complex numbers . The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as

4992-449: The late 1960s. The invention of the metal–oxide–semiconductor field-effect transistor ( MOSFET ) enabled the practical use of metal–oxide–semiconductor (MOS) transistors as memory cell storage elements. MOS memory was developed by John Schmidt at Fairchild Semiconductor in 1964. In addition to higher performance, MOS semiconductor memory was cheaper and consumed less power than magnetic core memory. In 1965, J. Wood and R. Ball of

5070-447: The location is updated within some known retention time, the data stays valid. After a period of time without update, the value is copied to a less-worn circuit with longer retention. Writing first to the worn area allows a high write rate while avoiding wear on the not-worn circuits. As a second example, an STT-RAM can be made non-volatile by building large cells, but doing so raises the cost per bit and power requirements and reduces

5148-457: The memory device in case of external power loss. If power is off for an extended period of time, the battery may run out, resulting in data loss. Proper management of memory is vital for a computer system to operate properly. Modern operating systems have complex systems to properly manage memory. Failure to do so can lead to bugs or slow performance. Improper management of memory is a common cause of bugs and security vulnerabilities, including

5226-478: The memory. In the early 1940s, memory technology often permitted a capacity of a few bytes. The first electronic programmable digital computer , the ENIAC , using thousands of vacuum tubes , could perform simple calculations involving 20 numbers of ten decimal digits stored in the vacuum tubes. The next significant advance in computer memory came with acoustic delay-line memory , developed by J. Presper Eckert in

5304-628: The positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved. Formally, a subsequence of the sequence ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} is any sequence of the form ( a n k ) k ∈ N {\textstyle (a_{n_{k}})_{k\in \mathbb {N} }} , where ( n k ) k ∈ N {\displaystyle (n_{k})_{k\in \mathbb {N} }}

5382-409: The reliability and security of a computer system. Without protected memory, it is possible that a bug in one program will alter the memory used by another program. This will cause that other program to run off of corrupted memory with unpredictable results. If the operating system's memory is corrupted, the entire computer system may crash and need to be rebooted . At times programs intentionally alter

5460-479: The same year, the concept of solid-state memory on an integrated circuit (IC) chip was proposed by applications engineer Bob Norman at Fairchild Semiconductor . The first bipolar semiconductor memory IC chip was the SP95 introduced by IBM in 1965. While semiconductor memory offered improved performance over magnetic-core memory, it remained larger and more expensive and did not displace magnetic-core memory until

5538-427: The sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite , as in these examples, or infinite , such as the sequence of all even positive integers (2, 4, 6, ...). The position of an element in a sequence is its rank or index ; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on

5616-415: The sequence in computer memory . Infinite sequences are called streams . The empty sequence ( ) is included in most notions of sequence. It may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions , spaces , and other mathematical structures using

5694-426: The sequence of real numbers ( a n ) is such that all the terms are less than some real number M , then the sequence is said to be bounded from above . In other words, this means that there exists M such that for all n , a n ≤ M . Any such M is called an upper bound . Likewise, if, for some real m , a n ≥ m for all n greater than some N , then the sequence is bounded from below and any such m

5772-460: The set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes ( a k ) {\textstyle (a_{k})} for an arbitrary sequence. Often, the index k is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in In some cases, the elements of the sequence are related naturally to

5850-546: The value of a sequence at the lowest input (often 1) is called the "first element" of the sequence, the value at the second smallest input (often 2) is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. f , a sequence abstracted from its input is usually written by a notation such as ( a n ) n ∈ A {\textstyle (a_{n})_{n\in A}} , or just as (

5928-434: The write speed. Using small cells improves cost, power, and speed, but leads to semi-volatile behavior. In some applications, the increased volatility can be managed to provide many benefits of a non-volatile memory, for example by removing power but forcing a wake-up before data is lost; or by caching read-only data and discarding the cached data if the power-off time exceeds the non-volatile threshold. The term semi-volatile

6006-638: Was commercialized by IBM in the early 1970s. MOS memory overtook magnetic core memory as the dominant memory technology in the early 1970s. The two main types of volatile random-access memory (RAM) are static random-access memory (SRAM) and dynamic random-access memory (DRAM). Bipolar SRAM was invented by Robert Norman at Fairchild Semiconductor in 1963, followed by the development of MOS SRAM by John Schmidt at Fairchild in 1964. SRAM became an alternative to magnetic-core memory, but requires six transistors for each bit of data. Commercial use of SRAM began in 1965, when IBM introduced their SP95 SRAM chip for

6084-421: Was less expensive. The Williams tube was nevertheless frustratingly sensitive to environmental disturbances. Efforts began in the late 1940s to find non-volatile memory . Magnetic-core memory allowed for memory recall after power loss. It was developed by Frederick W. Viehe and An Wang in the late 1940s, and improved by Jay Forrester and Jan A. Rajchman in the early 1950s, before being commercialized with

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