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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
dc071e90b17f62e9ff7848734c4de1263633c62b	the slabs viewpoint makes it easy to define	the slabs viewpoint makes it easy to define the lebesgue integral, in terms of basic calculus. suppose that f {\displaystyle f} is a (lebesgue measurable) function, taking non-negative values (possibly including + ∞ {\displaystyle +\infty }). define the distribution function of f {\displaystyle f} as the "width of a sl...	wikipedia
1a244e347cdeafe64a483d540b5a723bdfdedf45	one can think of the lebesgue integral either	one can think of the lebesgue integral either in terms of slabs or simple functions. intuitively, the area under a simple function can be partitioned into slabs based on the (finite) collection of values in the range of a simple function (a real interval). conversely, the (finite) collection of slabs in the undergraph ...	wikipedia
c4b77fe2a3aa62e3065e944879cfe4ca4efb4f72	an equivalent way to introduce the lebesgue integral	an equivalent way to introduce the lebesgue integral is to use so-called simple functions, which generalize the step functions of riemann integration. consider, for example, determining the cumulative covid-19 case count from a graph of smoothed cases each day (right).	wikipedia
f871fbfa46217a77fd3bf3b99e44e089abf7c6e9	for the lebesgue integral, the range is partitioned	for the lebesgue integral, the range is partitioned into intervals, and so the region under the graph is partitioned into horizontal "slabs" (which may not be connected sets). the area of a small horizontal "slab" under the graph of f, of height dy, is equal to the measure of the slab's width times dy: μ ({ x ∣ f (x) >...	wikipedia
2041f8b581b79b505ed94ff2284e10875a873b8b	for the riemann integral, the domain is partitioned	for the riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph. the areas of these bars are added together, and this approximates the integral, in effect by summing areas of the form f (x) dx where f (x) is the height of a rectangle and dx is its width.	wikipedia
e0e83083ca59ac67de8cd9363bb169cd0759037d	folland (1999) summarizes the difference between the riemann	folland (1999) summarizes the difference between the riemann and lebesgue approaches thus: "to compute the riemann integral of f, one partitions the domain into subintervals", while in the lebesgue integral, "one is in effect partitioning the range of f."	wikipedia
71ed8a9a6b878c47bddbdf5d6f71f2ec25e3cd7f	the insight is that one should be able	the insight is that one should be able to rearrange the values of a function freely, while preserving the value of the integral. this process of rearrangement can convert a very pathological function into one that is "nice" from the point of view of integration, and thus let such pathological functions be integrated.	wikipedia
9ea79a0cb3025888c13a72f266ccd4521307d0b5	i have to pay a certain sum, which	i have to pay a certain sum, which i have collected in my pocket. i take the bills and coins out of my pocket and give them to the creditor in the order i find them until i have reached the total sum. this is the riemann integral. but i can proceed differently. after i have taken all the money out of my pocket i order ...	wikipedia
12e04f865870ef1a9e6544ffaa47b3811f49de6c	while the riemann integral considers the area under	while the riemann integral considers the area under a curve as made out of vertical rectangles, the lebesgue definition considers horizontal slabs that are not necessarily just rectangles, and so it is more flexible. for this reason, the lebesgue definition makes it possible to calculate integrals for a broader class o...	wikipedia
81f8a4794c6f265b2b5b463af0034f200ed1cf30	however, riemann integration does not interact well with	however, riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. this is important, for instance, in the study of fourier series, fourier transforms, and other topics. the lebesgue integral describes better how and when it is possible...	wikipedia
b3d0dfdacdb8fc37815d3322aaec7db423d35708	as part of a general movement toward rigor	as part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. the riemann integral —proposed by bernhard riemann (1826–1866)—is a broadly successful attempt to provide such a foundation. riemann's definition starts with the c...	wikipedia
0374d2ba0a1a07397b329513e3eb2954367292e9	the integral of a positive real function f	the integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f, between a and b. this notion of area fits some functions, mainly piecewise continuous functions, including elementary functions, for example polynomials. however, the graphs of other functions, for...	wikipedia
87e50a5df68e775654c83f9d6c1b1fa55798ce2d	the term lebesgue integration can mean either the	the term lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the lebesgue measure.	wikipedia
557f8515ae34ff50ccf4bc7c7ebbc61f287e50f3	the lebesgue integral is more general than the	the lebesgue integral is more general than the riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. it can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the riemann integral. the lebesgue in...	wikipedia
75b53794517d25e9e86b0519c27882ede682264e	in mathematics, the integral of a non-negative function	in mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x axis. the lebesgue integral, named after french mathematician henri lebesgue, is one way to make this concept rigorous and to extend it to more gen...	wikipedia
a76dc9700097d413828329fff0d37e840d7dc9c3	transistor–transistor logic (ttl) is a logic family built	transistor–transistor logic (ttl) is a logic family built from bipolar junction transistors. its name signifies that transistors perform both the logic function and the amplifying function, as opposed to earlier resistor–transistor logic (rtl) and diode–transistor logic (dtl). ttl integrated circuits (ics) were widely ...	wikipedia
73f7e7d289f622ad8a369d04ad6e496ca471cdc5	ttl serial refers to single-ended serial communication using	ttl serial refers to single-ended serial communication using raw transistor voltage levels: "low" for 0 and "high" for 1. uart over ttl serial is a common debug interface for embedded devices. handheld devices such as graphing calculators and nmea 0183 -compliant gps receivers and fishfinders also commonly use uart wit...	wikipedia
0ea40c331ffac55fe20dd1423422a6824369b4eb	a ttl gate may operate inadvertently as an	a ttl gate may operate inadvertently as an analog amplifier if the input is connected to a slowly changing input signal that traverses the unspecified region from 0.8 v to 2 v. the output can be erratic when the input is in this range. a slowly changing input like this can also cause excess power dissipation in the out...	wikipedia
18ac2ed06464cc6fb617cc1d754b695cc03e25dc	while originally designed to handle logic-level digital signals,	while originally designed to handle logic-level digital signals, a ttl inverter can be biased as an analog amplifier. connecting a resistor between the output and the input biases the ttl element as a negative feedback amplifier. such amplifiers may be useful to convert analog signals to the digital domain but would no...	wikipedia
c1d30c5763dbb2b0f2e9293447a5f243eb594ac8	before the advent of vlsi devices, ttl integrated	before the advent of vlsi devices, ttl integrated circuits were a standard method of construction for the processors of minicomputer and midrange mainframe computers, such as the dec vax and data general eclipse; however some computer families were based on proprietary components (e.g. fairchild ctl) while supercompute...	wikipedia
964e40f06f538664beaf19b77d52b2a627942c4d	most manufacturers offer commercial and extended temperature ranges:	most manufacturers offer commercial and extended temperature ranges: for example texas instruments 7400 series parts are rated from 0 to 70 °c, and 5400 series devices over the military-specification temperature range of −55 to +125 °c.	wikipedia
571788a7a1a64bfc824370c2e585993c42812bbd	variations of and successors to the basic ttl	variations of and successors to the basic ttl family, which has a typical gate propagation delay of 10ns and a power dissipation of 10 mw per gate, for a power–delay product (pdp) or switching energy of about 100 pj, include:	wikipedia
5d48801cb7cace4eafa19d8e288115032c12a4a0	successive generations of technology produced compatible parts with	successive generations of technology produced compatible parts with improved power consumption or switching speed, or both. although vendors uniformly marketed these various product lines as ttl with schottky diodes, some of the underlying circuits, such as used in the ls family, could rather be considered dtl.	wikipedia
819e5659c1cf5190667eb0ea9dd28d484fa58e8a	since the mid 1980s, several manufacturers supply cmos	since the mid 1980s, several manufacturers supply cmos logic equivalents with ttl-compatible input and output levels, usually bearing part numbers similar to the equivalent ttl component and with the same pinouts. for example, the 74hct00 series provides many drop-in replacements for bipolar 7400 series parts, but uses...	wikipedia
19f4408301aaeb8fc9140a401610ffcb41abb9ae	the ttl "totem-pole" output structure often has a	the ttl "totem-pole" output structure often has a momentary overlap when both the upper and lower transistors are conducting, resulting in a substantial pulse of current drawn from the power supply. these pulses can couple in unexpected ways between multiple integrated circuit packages, resulting in reduced noise margi...	wikipedia
62d7dbd0e62ca37a5b26a11e2a5a67fc670ec1fe	due to the output structure of ttl devices,	due to the output structure of ttl devices, the output impedance is asymmetrical between the high and low state, making them unsuitable for driving transmission lines. this drawback is usually overcome by buffering the outputs with special line-driver devices where signals need to be sent through cables. ecl, by virtue...	wikipedia
0150a7cf97159fca61aff4cef8df4ba16d95b21f	ttl devices consume substantially more power than equivalent	ttl devices consume substantially more power than equivalent cmos devices at rest, but power consumption does not increase with clock speed as rapidly as for cmos devices. compared to contemporary ecl circuits, ttl uses less power and has easier design rules but is substantially slower. designers can combine ecl and tt...	wikipedia
ba9134975c53c3191bb67b1311f781483b5b05b0	ttl is particularly well suited to bipolar integrated	ttl is particularly well suited to bipolar integrated circuits because additional inputs to a gate merely required additional emitters on a shared base region of the input transistor. if individually packaged transistors were used, the cost of all the transistors would discourage one from using such an input structure....	wikipedia
a8512bdcceff15742cadafdae8b32cd3ecd9acd6	beam-lead chip dies without packages were made for	beam-lead chip dies without packages were made for assembly into larger arrays as hybrid integrated circuits. parts for military and aerospace applications were packaged in flatpacks, a form of surface-mount package, with leads suitable for welding or soldering to printed circuit boards. today, many ttl-compatible devi...	wikipedia
63ac4500803128b3e5d9951d4882ad2f80ebfaac	like most integrated circuits of the period 1963–1990,	like most integrated circuits of the period 1963–1990, commercial ttl devices are usually packaged in dual in-line packages (dips), usually with 14 to 24 pins, for through-hole or socket mounting. epoxy plastic (pdip) packages were often used for commercial temperature range components, while ceramic packages (cdip) we...	wikipedia
dd789a8e1d4d7993b2ec2571e0669e295d86e5dc	in some cases (e.g., when the output of	in some cases (e.g., when the output of a ttl logic gate needs to be used for driving the input of a cmos gate), the voltage level of the "totem-pole" output stage at output logical "1" can be increased closer to v by connecting an external resistor between the v4 collector and the positive rail. it pulls up the v cath...	wikipedia
96bdd86cb07905532fea5afd976915d69febd902	standard ttl circuits operate with a 5- volt	standard ttl circuits operate with a 5- volt power supply. a ttl input signal is defined as "low" when between 0 v and 0.8 v with respect to the ground terminal, and "high" when between 2 v and v (5 v), and if a voltage signal ranging between 0.8 v and 2.0 v is sent into the input of a ttl gate, there is no certain res...	wikipedia
dee86c1f2fcbe166d2ac8f866e0e1d7275f5e697	like dtl, ttl is a current-sinking logic since	like dtl, ttl is a current-sinking logic since a current must be drawn from inputs to bring them to a logic 0 voltage level. the driving stage must absorb up to 1.6 ma from a standard ttl input while not allowing the voltage to rise to more than 0.4 volts. the output stage of the most common ttl gates is specified to f...	wikipedia
baa726135ce97219c954ad8d68dfd8c2e9c24bce	the main advantage of ttl with a "totem-pole"	the main advantage of ttl with a "totem-pole" output stage is the low output resistance at output logical "1". it is determined by the upper output transistor v operating in active region as an emitter follower. the resistor r does not increase the output resistance since it is connected in the v collector and its infl...	wikipedia
4c5092f8029d8eaf7979c8a709e8caeabaaa5e3c	in the middle of the transition, the resistor	in the middle of the transition, the resistor r limits the current flowing directly through the series connected transistor v, diode v and transistor v that are all conducting. it also limits the output current in the case of output logical "1" and short connection to the ground. the strength of the gate may be increas...	wikipedia
597e8d550e33467646f87b18d2a34a88e9e987e0	when v is "on", it activates v, driving	when v is "on", it activates v, driving low voltage (logical "0") to the output. again there is a current-steering effect: the series combination of v 's c-e junction and v 's b-e junction is in parallel with the series of v b-e, v 's anode-cathode junction, and v c-e. the second series combination has the higher thres...	wikipedia
3239921237a469024d11e8a89702f2505bc7db1a	to solve the problem with the high output	to solve the problem with the high output resistance of the simple output stage the second schematic adds to this a "totem-pole" (" push–pull ") output. it consists of the two n-p-n transistors v and v, the "lifting" diode v and the current-limiting resistor r (see the figure on the right). it is driven by applying the...	wikipedia
3fd528751712f3a8e1015690ec996b82bba29c19	a common variation omits the collector resistor of	a common variation omits the collector resistor of the output transistor, making an open-collector output. this allows the designer to fabricate wired logic by connecting the open-collector outputs of several logic gates together and providing a single external pull-up resistor. if any of the logic gates becomes logic ...	wikipedia
332334f3c8d48f61ef5ad9c33d9e34891f35a405	the main disadvantage of ttl with a simple	the main disadvantage of ttl with a simple output stage is the relatively high output resistance at output logical "1" that is completely determined by the output collector resistor. it limits the number of inputs that can be connected (the fanout). some advantage of the simple output stage is the high voltage level (u...	wikipedia
e5d5d4c4e8152cd9074a1084af2a353524b9d986	an input logical zero. note that the base–collector	an input logical zero. note that the base–collector junction of the multiple-emitter transistor and the base–emitter junction of the output transistor are in series between the bottom of the resistor and ground. if one input voltage becomes zero, the corresponding base–emitter junction of the multiple-emitter transisto...	wikipedia