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Absolute continuity

In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. We have the following chains of inclusions for functions over a compact subset of the real line:

 * absolutely continuous ⊆ uniformly continuous $$=$$ continuous

and, for a compact interval,

 * continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere.

Absolute continuity of functions
A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over $[0, π/2)$, x2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the Weierstrass function, which is not differentiable anywhere). Or it may be differentiable almost everywhere and its derivative f ′ may be Lebesgue integrable, but the integral of f ′ differs from the increment of f (how much f changes over an interval). This happens for example with the Cantor function.

Definition
Let $$I$$ be an interval in the real line $$\R$$. A function $$f\colon I \to \R$$ is absolutely continuous on $$I$$ if for every positive number $$\varepsilon$$, there is a positive number $$\delta$$ such that whenever a finite sequence of pairwise disjoint sub-intervals $$(x_k, y_k)$$ of $$I$$ with $$x_k < y_k \in I$$ satisfies
 * $$\sum_{k=1}^{N} (y_k - x_k) < \delta $$

then
 * $$ \sum_{k=1}^{N} | f(y_k) - f(x_k) | < \varepsilon.$$

The collection of all absolutely continuous functions on $$I$$ is denoted $$\operatorname{AC}(I)$$.

Equivalent definitions
The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:

 * 1) f is absolutely continuous;
 * 2) f has a derivative f ′ almost everywhere, the derivative is Lebesgue integrable, and $$ f(x) = f(a) + \int_a^x f'(t) \, dt $$ for all x on [a,b];
 * 3) there exists a Lebesgue integrable function g on [a,b] such that $$ f(x) = f(a) + \int_a^x g(t) \, dt $$ for all x in [a,b].

If these equivalent conditions are satisfied, then necessarily any function g as in condition 3. satisfies g = f ′ almost everywhere.

Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.

For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity.

Properties

 * The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous.
 * If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.
 * Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every (globally) Lipschitz-continuous function is absolutely continuous.
 * If f: [a,b] → R is absolutely continuous, then it is of bounded variation on [a,b].
 * If f: [a,b] → R is absolutely continuous, then it can be written as the difference of two monotonic nondecreasing absolutely continuous functions on [a,b].
 * If f: [a,b] → R is absolutely continuous, then it has the Luzin N property (that is, for any $$N \subseteq [a,b]$$ such that $$\lambda(N) = 0$$, it holds that $$\lambda(f(N)) = 0$$, where $$\lambda$$ stands for the Lebesgue measure on R).
 * f: I → R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property. This statement is also known as the Banach-Zareckiǐ theorem.
 * If f: I → R is absolutely continuous and g: R → R is globally Lipschitz-continuous, then the composition g ∘ f is absolutely continuous. Conversely, for every function g that is not globally Lipschitz continuous there exists an absolutely continuous function f such that g ∘ f is not absolutely continuous.

Examples
The following functions are uniformly continuous but not absolutely continuous: 0, & \text{if }x =0 \\ x \sin(1/x), & \text{if } x \neq 0 \end{cases} $$ on a finite interval containing the origin.
 * The Cantor function on [0, 1] (it is of bounded variation but not absolutely continuous);
 * The function:$$ f(x) = \begin{cases}

The following functions are absolutely continuous but not α-Hölder continuous:
 * The function f(x) = xβ on [0, c], for any 0 < β < α < 1

The following functions are absolutely continuous and α-Hölder continuous but not Lipschitz continuous:
 * The function f(x) = $\sqrt{x}$ on [0, c], for α ≤ 1/2.

Generalizations
Let (X, d) be a metric space and let I be an interval in the real line R. A function f: I → X is absolutely continuous on I if for every positive number $$\epsilon$$, there is a positive number $$\delta$$ such that whenever a finite sequence of pairwise disjoint sub-intervals [xk, yk] of I satisfies:

 * $$\sum_{k} \left| y_k - x_k \right| < \delta$$

then:

 * $$\sum_{k} d \left( f(y_k), f(x_k) \right) < \epsilon.$$

The collection of all absolutely continuous functions from I into X is denoted AC(I; X).

A further generalization is the space ACp(I; X) of curves f: I → X such that:

 * $$d \left( f(s), f(t) \right) \leq \int_s^t m(\tau) \,d\tau \text{ for all } [s, t] \subseteq I$$

for some m in the Lp space Lp(I).

Properties of these generalizations

 * Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.
 * If f: [a,b] → X is absolutely continuous, then it is of bounded variation on [a,b].
 * For f ∈ ACp(I; X), the metric derivative of f exists for λ-almost all times in I, and the metric derivative is the smallest m ∈ Lp(I; R) such that: $$d \left( f(s), f(t) \right) \leq \int_s^t m(\tau) \,d\tau \text{ for all } [s, t] \subseteq I.$$

Definition
A measure $$\mu$$ on Borel subsets of the real line is absolutely continuous with respect to the Lebesgue measure $$\lambda$$ if for every $$\lambda$$-measurable set $$A,$$ $$\lambda(A) = 0$$ implies $$\mu(A) = 0$$. Equivalently, $$\mu(A) > 0$$ implies $$\lambda(A) > 0$$. This condition is written as $$\mu \ll \lambda.$$ We say $$\mu$$ is dominated by $$\lambda.$$

In most applications, if a measure on the real line is simply said to be absolutely continuous — without specifying with respect to which other measure it is absolutely continuous — then absolute continuity with respect to the Lebesgue measure is meant.

The same principle holds for measures on Borel subsets of $$\mathbb{R}^n, n \geq 2.$$

Equivalent definitions
The following conditions on a finite measure $$\mu$$ on Borel subsets of the real line are equivalent:

 * 1) $$\mu$$ is absolutely continuous;
 * 2) For every positive number $$\varepsilon$$ there is a positive number $$\delta > 0$$ such that $$\mu(A) < \varepsilon$$ for all Borel sets $$A$$ of Lebesgue measure less than $$\delta;$$
 * 3) There exists a Lebesgue integrable function $$g$$ on the real line such that: $$\mu(A) = \int_A g \,d\lambda$$ for all Borel subsets $$A$$ of the real line.

For an equivalent definition in terms of functions see the section Relation between the two notions of absolute continuity.

Any other function satisfying (3) is equal to $$g$$ almost everywhere. Such a function is called Radon–Nikodym derivative, or density, of the absolutely continuous measure $$\mu.$$

Equivalence between (1), (2) and (3) holds also in $$\R^n$$ for all $$n = 1, 2, 3, \ldots.$$

Thus, the absolutely continuous measures on $$\R^n$$ are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have probability density functions.

Generalizations
If $$\mu$$ and $$\nu$$ are two measures on the same measurable space $$(X, \mathcal{A}),$$ $$\mu$$ is said to be  with respect to $$\nu$$ if $$\mu(A) = 0$$ for every set $$A$$ for which $$\nu(A) = 0.$$ This is written as "$$\mu\ll\nu$$". That is: $$\mu \ll \nu \qquad \text{ if and only if } \qquad \text{ for all } A\in\mathcal{A}, \quad (\nu(A) = 0\ \text{ implies } \ \mu (A) = 0).$$

When $$\mu\ll\nu,$$ then $$\nu$$ is said to be  $$\mu.$$

Absolute continuity of measures is reflexive and transitive, but is not antisymmetric, so it is a preorder rather than a partial order. Instead, if $$\mu \ll \nu$$ and $$\nu \ll \mu,$$ the measures $$\mu$$ and $$\nu$$ are said to be equivalent. Thus absolute continuity induces a partial ordering of such equivalence classes.

If $$\mu$$ is a signed or complex measure, it is said that $$\mu$$ is absolutely continuous with respect to $$\nu$$ if its variation $$|\mu|$$ satisfies $$|\mu| \ll \nu;$$ equivalently, if every set $$A$$ for which $$\nu(A) = 0$$ is $$\mu$$-null.

The Radon–Nikodym theorem states that if $$\mu$$ is absolutely continuous with respect to $$\nu,$$ and both measures are σ-finite, then $$\mu$$ has a density, or "Radon-Nikodym derivative", with respect to $$\nu,$$ which means that there exists a $$\nu$$-measurable function $$f$$ taking values in $$[0, +\infty),$$ denoted by $$f = d\mu / d\nu,$$ such that for any $$\nu$$-measurable set $$A$$ we have: $$\mu(A) = \int_A f \,d\nu.$$

Singular measures
Via Lebesgue's decomposition theorem, every σ-finite measure can be decomposed into the sum of an absolutely continuous measure and a singular measure with respect to another σ-finite measure. See singular measure for examples of measures that are not absolutely continuous.

Relation between the two notions of absolute continuity
A finite measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function:
 * $$F(x)=\mu((-\infty,x])$$

is an absolutely continuous real function. More generally, a function is locally (meaning on every bounded interval) absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.

If absolute continuity holds then the Radon–Nikodym derivative of μ is equal almost everywhere to the derivative of F.

More generally, the measure μ is assumed to be locally finite (rather than finite) and F(x) is defined as μ((0,x]) for x > 0, 0 for x = 0, and −μ((x,0]) for x < 0. In this case μ is the Lebesgue–Stieltjes measure generated by F. The relation between the two notions of absolute continuity still holds.

title: Absolute continuity


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Mikoyan MiG-31

The Mikoyan MiG-31 (Микоян МиГ-31; NATO reporting name: Foxhound) is a supersonic interceptor aircraft developed for the Soviet Air Forces by the Mikoyan design bureau as a replacement for the earlier MiG-25 "Foxbat"; the MiG-31 is based on and shares design elements with the MiG-25.

The MiG-31 is cited as the third fastest known operational combat aircraft in the world as of the 2020s. It continues to be operated by the Russian Aerospace Forces following the end of the Cold War and the collapse of the Soviet Union in 1991. The other operator, the Kazakh Air Defense Forces, retired the type in 2023. The Russian Defence Ministry expects the MiG-31 to remain in service until at least 2030; that was confirmed in 2020 when an announcement was made to extend the service lifetime of the existing airframes from 2,500 to 3,500 hours.

Background
The single-seat MiG-25 could achieve high speed, altitude and rate of climb; however, it lacked maneuverability at interception speeds and was difficult to fly at low altitudes. The MiG-25's speed was normally limited to Mach 2.83, but it could reach a maximum speed of at least Mach 3.2 with the risk of engine damage.

Development of the MiG-25's replacement began with the Ye-155MP (Е-155МП) prototype which first flew on 16 September 1975. Although it bore a superficial resemblance to the MiG-25, it had a longer fuselage to accommodate the radar operator's cockpit and was in many respects a new design. An important development was the MiG-31's advanced radar, capable of both look-up and look-down/shoot-down engagement, as well as multiple target tracking. This gave the Soviet Union an interceptor with the capability to engage the most likely Western intruders (low-flying cruise missiles and bombers) at long range. The MiG-31 replaced the Tu-128 as the Soviet Union's dedicated long-range interceptor, with far more advanced sensors and weapons, while its range is almost double that of the MiG-25.

Like that of its MiG-25 predecessor, the introduction of the MiG-31 was surrounded by early speculation and misinformation concerning its design and abilities. The West learned of the new interceptor from Lieutenant Viktor Belenko, a pilot who defected to Japan in 1976 with his MiG-25P. Belenko described an upcoming "Super Foxbat" with two seats and an ability to intercept cruise missiles. According to his testimony, the new interceptor was to have air intakes similar to the Mikoyan-Gurevich MiG-23, which the MiG-31 does not have, at least in production variants.

Production
Serial production of the MiG-31 began in 1979. The first production batch of 519 MiG-31s including 349 "baseline models" was produced at the Sokol plant between 1976 and 1988. The second batch of 101 MiG-31DZs was produced from 1989 to 1991. The final batch of 69 MiG-31B aircraft was produced between 1990 and 1994. From the final batch, 50 were retained by the Kazakhstan Air Force after the dissolution of the Soviet Union. Of the "baseline models", 40 airframes were upgraded to MiG-31BS standard.

Upgrades and replacement
Some upgrade programs have found their way into the MiG-31 fleet, like the MiG-31BM multirole version with upgraded avionics, new multimode radar, hands-on-throttle-and-stick (HOTAS) controls, liquid crystal (LCD) color multi-function displays (MFDs), ability to carry the R-77 missile and various Russian air-to-ground missiles (AGMs) such as the Kh-31 anti-radiation missile (ARM), a new and more powerful computer, and digital data links. A project to upgrade the Russian MiG-31 fleet to the MiG-31BM standard began in 2010; 100 aircraft were to be upgraded to the MiG-31BM standard by 2020. Russian Federation Defence Ministry chief Colonel Yuri Balyko has claimed that the upgrade will increase the combat effectiveness of the aircraft several times over. 18 MIG-31BMs were delivered in 2014. The Russian military was to receive more than 130 upgraded MiG-31BMs, with the first 24 aircraft already delivered according to Russian Deputy Defense Minister Yuri Borisov in his press briefing on 9 April 2015.

Russia planned to start development of a replacement for the MiG-31 by 2019. The aircraft would be called PAK-DP (ПАК ДП, Перспективный авиационный комплекс дальнего перехвата – Prospective Air Complex for Long-Range Interception). Development of the new aircraft, designated MiG-41, began in April 2013. Such development is favored over restarting MiG-31 production. In March 2014, Russian test pilot Anatoly Kvochur said that work began on a Mach 4 capable MiG-41 based on the MiG-31. Later reports stated that the development of the MiG-31 replacement was to begin in 2017, with the first aircraft to be delivered in 2020, and the replacement entering service in 2025.

Design

Like the MiG-25, the MiG-31 is a large twin-engine aircraft with side-mounted air intake ramps, a shoulder-mounted wing with an aspect ratio of 2.94, and twin vertical tailfins. Unlike the MiG-25, it has two seats, with the rear occupied by a dedicated weapon systems officer.

The MiG-31 was designed to fulfill the following mission objectives:
 * Intercept cruise missiles and their launch aircraft by reaching missile launch range in the shortest possible time after departing the loiter area;
 * Detect and destroy low flying cruise missiles, UAVs and helicopters;
 * Long range escort of strategic bombers;
 * Provide strategic air defense in areas not covered by ground-based air defense systems.

The MiG-31 is limited to five g when travelling at supersonic speeds. While flying under combat weight, its wing loading is marginal and its thrust-to-weight ratio is favorable. The MiG-31 is not designed for close combat or rapid turning.

The wings and airframe of the MiG-31 are stronger than those of the MiG-25, permitting supersonic flight at low altitudes. Like the MiG-25, its flight surfaces are built primarily of nickel-steel alloy, enabling the aircraft to tolerate kinetic heating at airspeeds approaching Mach 3. The MiG-31 airframe comprises 49% arc-welded nickel steel, 33% light metal alloy, 16% titanium and 2% composites. Its D30-F6 jet engines, each rated at 152 kN thrust, allow a maximum speed of Mach 1.23 at low altitude. High-altitude speed is temperature-redlined to Mach 2.83 – the thrust-to-drag ratio is sufficient for speeds in excess of Mach 3, but such speeds pose unacceptable hazards to engine and airframe life in routine use.

Electronics suite

The MiG-31 was among the first aircraft with a phased array radar.

The MiG-31 was the world's first operational fighter with a passive electronically scanned array (PESA) radar, the Zaslon S-800. Its maximum range against fighter-sized targets is approximately 200 km, and it can track up to 10 targets and simultaneously attack four of them with its Vympel R-33 missiles. The radar is matched with an infrared search and track (IRST) system in a retractable undernose fairing.

The MiG-31 was equipped with RK-RLDN and APD-518 digital secure datalinks. The RK-RLDN datalink is for communication with ground control centers. The APD-518 datalink enables a flight of four MiG-31 to automatically exchange radar-generated data within 200 km from each other. It also enables other aircraft with less sophisticated avionics, such as MiG-23s, 25s, and 29s and Su-15s and 27s, to be directed to targets spotted by MiG-31 (a maximum of four (long-range) for each MiG-31 aircraft). The A-50 AEW aircraft and MiG-31 can automatically exchange aerial and terrestrial radar target designation, as well as air defense. The MiG-31 is equipped with ECM of radar and infrared ranges.

The flight-navigation equipment of the MiG-31 includes a complex of automatic control system SAU-155МP and sighting-navigation complex KN-25 with two inertial systems and IP-1-72A with digital computer, electronic long range navigation system Radical NP (312) or A-331, electronic system of the long-range navigation A-723. Distant radio navigation is carried out by means of two systems: Chayka (similar to the system of Loran) and «Route» (similar to the system of Omega).

Similarly to the complex S-300 missile system, aircraft group with APD-518 can share data obtained by various radars from different directions (active or passive scanning of radiation) and summarize the data. The target can be detected passively (via its emissions of jamming or use of its radar) and/or actively simultaneously from many different directions (with the MiG-31 using its radar). Every aircraft with the APD-518 will have the exact data, even if it is not involved in the search. A group of four MiG-31 interceptors is able to control an area of air space across a total length of 800 to 900 km; its radar possessing a maximum detection range of 200 km in distance (radius) and the typical width of detection along the front of 225 km.
 * interacting with ground-based automated digital control system (ACS «Rubezh» Operating radius of 2000 km, can control multiple groups of planes), operating modes of remote aiming, semi-automated actions (coordinate support), singly, and also: to direct on the target missiles launched from the other aircraft.
 * Digital immune system provides the automatic exchange of tactical information in a group of four interceptors, remote one from another at a distance of 200 km and aiming at the target group of fighters with less-powerful avionics (in this case the aircraft performs the role of guidance point or repeater).

Radars

Adopted in 1981 RP-31 N007 backstop (Russian: Zaslon).
 * the range of detection of air targets with Zaslon-A: 200 km (for the purpose of a radar cross-section of 19 m2 on a collision angle with probability 0.5)
 * target detection distance with radar cross-section of 3 m2 in the rear within 35 km with a probability of 0.5
 * number of detected targets: 24 (was originally 10 )
 * number of targets for attack: 6 (was originally 4 )
 * range of automatic tracking: 120 km
 * detection of infrared signature targets: 56 km
 * Effective in the detection of cruise missiles and other targets against ground clutter
 * Until 2000, it was the world's only fighter in service equipped with phased array radar, when the Mitsubishi F-2 entered service with the J/APG-1 active phased array radar.
 * Able to intercept and destroy cruise missiles flying at extremely low altitudes.

Variant differences
The basic differences between other versions and the MiG-31BM are:
 * The onboard radar complex of the MiG-31BM can track 24 airborne targets at one time, six of which can be simultaneously attacked by R-33S missiles.
 * Modernized variants of the aircraft can be equipped with anti-radiation missiles Kh-31, Kh-25MR or MPU (up to six units), anti-ship Kh-31A (up to six), air-to-surface class missiles Kh-29 and Kh-59 (up to three) or Kh-59M (up to two units), up to six precision bombs KAB-1500 or eight KAB-500 with television or laser-guidance. Maximum mass of payload is 9000 kg.
 * The MiG-31M, MiG-31D, and MiG-31BM standard aircraft have an upgraded Zaslon-M radar, with larger antenna and greater detection range (said to be 400 km against AWACS-size targets) and the ability to attack multiple targets – air and ground – simultaneously. The Zaslon-M has a 1.4 m diameter (larger) antenna, with 50–100% better performance than Zaslon. In April 1994 it was used with an R-37 to hit a target at 300 km distance. It has a search range of 400 km for a 19–20 m2 RCS target and can track 24 targets at once, engaging six,  or 282 km for 5 m2. Relative target speed detection increased from Mach 5 to Mach 6, improving the probability of destroying fast-moving targets. The MiG-31BM is one of only a few aircraft able to intercept and destroy cruise missiles flying at extremely low altitude.

Cockpit

The aircraft is a two-seater with the rear seat occupant controlling the radar. Although cockpit controls are duplicated across cockpits, it is normal for the aircraft to be flown only from the front seat. The pilot flies the aircraft by means of a centre stick and left hand throttles. The rear cockpit has only two small vision ports on the sides of the canopy. The presence of the WSO (weapon systems operator) in the rear cockpit improves aircraft effectiveness since the WSO is entirely dedicated to radar operations and weapons deployment, thus decreasing the workload of the pilot and increasing efficiency. Both cockpits are fitted with zero/zero ejection seats which allow the crew to eject at any altitude and airspeed.

Armament

The MiG-31's main armament is four R-33 air-to-air missiles (NATO codename AA-9 'Amos') carried under the belly.
 * One GSh-6-23 23 mm cannon with 260 rounds. (The MiG-31's predecessor, the MiG-25 did not include a cannon.)
 * Fuselage recesses for four R-33 (AA-9 'Amos') or four R-37 (AA-13 'Arrow') (MiG-31BM only).
 * Four underwing pylons for a combination of six places for charging (two spaces to add removable fuel tanks )
 * Six R-37 (missile) long-range air-to-air missiles 280 km.
 * Four R-33 long-range missiles 300 km.
 * (?)× Kh-31 (AS-17 'Krypton') long-range air-to-ground missiles (200 km) for high-speed target (maneuvering with an overload of 8 g).
 * (?)× R-33 (AA-9 "Amos") (1981) 120 km, R-33S (1999) 160 km.
 * Two or four (superior limit) × R-40TD1 (AA-6 'Acrid') medium-range missiles (R-40 – 50-80 km), MiG-25P, 1970) launched at altitudes of 0.5 - 3 km (maneuvering with overload four g).
 * Four R-60 (AA-8 'Aphid') 2018 Moscow Victory Day Parade 66.jpg missile.]]
 * Four R-73 (AA-11 'Archer') short-range IR missiles,
 * Four R-77 (AA-12 'Adder') medium-range missiles (100 km) for high-speed target (maneuvering with overload of 12 g).
 * Some aircraft are equipped to launch the Kh-31P (AS-17 'Krypton') and Kh-58 (AS-11 'Kilter') anti-radiation missiles in the Suppression of Enemy Air Defenses (SEAD) role. Anti-ship missiles Kh-31A (up to six) and air-to-surface missiles Kh-59 and Kh-29T (up to three) or X-59M (up to two units), up to six air bombs KAB-1500, or up to eight KAB-500 with a television or laser-guidance. Maximum weight of the combat load is 9000 kg.
 * One Kh-47M2 Kinzhal high-precision ballistic missile with a range of about 2,000 km (including range of launching aircraft), Mach 10 speed. It can carry both conventional and nuclear warheads. This gave the MiG-31 long range strike capabilities for the first time, alongside its primary interceptor role. (Only MiG-31K variants equipped)

Operational history

Serial production of the MiG-31 began in 1979. The MiG-31 entered operational service with the Soviet Air Defence Forces (PVO) in 1981.

During the 2022 Russian invasion of Ukraine, MiG-31 aircraft have reportedly shot down several Ukrainian aircraft, mainly by using the long range R-37 air-to-air missile. By remaining at high speed and high altitude, MiG-31s have been able to operate virtually unopposed due to Ukrainian fighters lacking range, speed, or altitude necessary to engage the MiG-31.

The R-37M has, since October 2022, been the main threat against the Ukrainian Air Force. The Ukrainian Air Force has a significant lack of fire-and-forget missiles. They relied on the R-27 missiles, both the R-27ER and R-27ET; the R-27ER's range is 60 mi. A Ukrainian pilot must illuminate a Russian aircraft with radar to guide the missile to the target. Russian pilots firing the active-radar, fire-and-forget R-77 give them the ability to launch their missiles and then take evasive action. Ukrainian pilots were forced to "exploit ground clutter and terrain-masking to get close enough to fire before being engaged". During the first three days of the war both sides lost aircraft. The Ukrainians replaced them with older airframes that were made flyable. The Russian Aerospace Forces turned to the MiG-31 with the R-37M missile that has a range of 200 mi. Combined with the superior radar on the MiG-31, the Ukrainian Air Force has started losing more aircraft. A report by the Royal United Services Institute states that in October some six R-37Ms were being fired at the Ukrainian Air Force a day. Four MiG-31s were also deployed to Crimea. To avoid R-37M missiles, Ukraine has had to attempt to destroy MiG-31s while they are still on the ground, such as the attack on the Belbek airbase and an attempted drone attack in August.

On the early morning of 4 May 2023, a Kh-47M2 Kinzhal missile was intercepted by the Patriot air defense system according to the commander of the Ukrainian Air Force, general Mykola Oleschuk. The missile was fired from a MiG-31K in Russian airspace. Ukraine confirmed the intercept, saying it used the Patriot missile system protecting the Kyiv region.

On 15 May 2024, Russian-installed officials in Crimea claimed that a missile attack caused explosions and fires near the Belbek airfield. Some missiles (possibly ATACMS) struck dropping cluster munitions on the airfield, and according to photographic evidence and satellite imagery, at least two Russian ground-based air defense systems destroyed, likely destroying two MiG-31s, destroyed a fuel farm and damaged one Su-27.

On the morning of 16 May 2023, some six Kh-47M2 Kinzhal missiles were fired at Kyiv. Ukrainian Air Force spokesperson said that they were launched from six MiG-31Ks. Ukraine claims that all Kinzhals were intercepted but one may have damaged a Patriot battery launcher according to US officials, although there is limited evidence.

Export
In 1992 Russia offered the MiG-31 to Finland, which had a selection programme ongoing for a new fighter, but the offer was not submitted to the program, where Russia had submitted the MiG-29. Finland did not accept the offer, and chose the new fighter from the selection programme.

Syria ordered eight MiG-31E aircraft in 2007 for the Syrian Air Force. The order was suspended in May 2009 reportedly either due to Israeli pressure or lack of Syrian funds.

Variants

 * Ye-155MP (MiG-25MP): Prototype modification of the early MiG-31. First flight on 16 September 1975.

 * MiG-31: First variant which entered in serial production. 349 aircraft were built.

 * MiG-31M:Development of a more comprehensive advanced version, the MiG-31M, began in 1984 and first flew in 1985, but the dissolution of the Soviet Union prevented it from entering full production. One piece rounded windscreen, small side windows for rear cockpit, wider and deeper dorsal spine. Digital flight controls added, multifunction CRT cockpit displays, multi-mode phased array radar. No gun fitted in this model, refueling probe moved to starboard side of aircraft, fuselage weapon stations increased from 4 to 6 by adding two centre-line stations. Maximum TO weight increased to 52,000 kg using increased thrust D-30F6M engines instead of the D-30F6 engines. 1 prototype and 6 flyable pre-production units were produced.

 * MiG-31D: Two aircraft were designated as Type 31D and were manufactured as dedicated anti-satellite models with ballast in the nose instead of radars, flat fuselage undersurface (i.e. no recessed weapon system bays) and had large winglets above and below the wing-tips. Equipped with Vympel ASAT missiles. Two prototypes were built.

 * MiG-31LL: Special modification used as a flying laboratory for testing of ejection seats during flight.

 * MiG-31 01DZ: Two-seat all weather, all altitude interceptor. Designated as MiG-31 01DZ when fitted with air-to-air refueling probe. One hundred produced of DZ variant.

 * MiG-31B: Second production batch with upgraded avionics and in-flight refueling probe introduced in 1990. Its development was the result of the Soviet discovery that Phazotron radar division engineer Adolf Tolkachev had sold information on advanced radars to the West. A new version of the compromised radar was hastily developed. MiG-31B also have the improved ECM and EW equipment with integration of improved R-33S missiles. Long range navigation system compatible with Loran/Omega and Chaka ground stations added. This model replaced the 01DZ models in late 1990.

 * MiG-31E: Export version of the MiG-31B with simplified avionics. Never entered in serial production.

 * MiG-31BS: Designation applied to type 01DZ when converted to MiG-31B standard.

 * MiG-31BM: After passing state testing in 2008, this modernized variant of MiG-31B was approved for introduction into air force of Russia. 50 planes are modified to the MiG-31BM (Bolshaya Modernizatsiya/Big Modernization) standard, in accordance with the 2011 contract. The efficiency of the modernized MiG-31BM is 2.6 times greater than the basic MiG-31. The MiG-31BМ's maximum detection range for air targets was increased in the upgrade to 320 km. It had the ability to automatically track up to ten targets, and the latest units can track up to 24 targets and simultaneously engage up to eight targets. The on-board Argon-K is replaced with the new Baget 55-06 computer that selects four targets of the highest priority, which simultaneously are engaged by long-range R-33S air-to-air missiles. A new long range missile R-37 (missile) with a speed of Mach 6 and a range of up to 400 km is developed during the modernization process for use with the newly modernized MiG-31. The MiG-31BM has a multi-role capability, and is capable of using anti-radar, air-to-ship and air-to-ground missiles. Some of its avionics are unified with the MiG-29SMT. It also has a refueling probe. MiG-31BM set a sustained flight record for the model while spending seven hours and four minutes in the air while covering the distance of 8,000 km.

 * MiG-31BSM: An upgrade of the BS version, it is the latest modernization variant first time contracted in 2014 for modernization of 60 aircraft, it is very similar in some aspects to the BM standard. Unlike the BS standard, aircraft modernized into the BSM standard are equipped with air refueling probe. Improvements were made to the aircraft canopy, where new and better heat resistant glass was used, thus enabling the MiG-31BSM to fly with cruise speed of 3000 km/h at long distances without any damage. Furthermore, new faster central computer Baget-55-06 is used with addition of multi-functional displays, one for the pilot and three for the weapons operator-navigator. Also there is a new set of navigation equipment. The MiG-31BSM has multi-role capability with ability to use anti-radar, anti-ship and air-to-ground missiles. The main visible difference between the BS and BSM standards a rear-view periscope above the front cockpit canopy.

 * MiG-31K: Modified MiG-31BM variant able to carry the hypersonic Kh-47M2 Kinzhal ALBM. Ten aircraft were modified by May 2018. With this modification and with removed APU for air-to-air missiles, the aircraft gained a sole role of an attack aircraft.

 * MiG-31F: Planned fighter-bomber intended for use with TV, radar and laser-guided ASM weapon systems. Never entered serial production.

 * MiG-31FE: Planned export version of the MiG-31F.

 * MiG-31I (Ishim): Proposed modification for air launch to orbit of small spacecraft with a payload of 160 kg to 300 km altitude or 120 kg to 600 km altitude orbit.

 * MiG-31 (Izdeliye 08): MiG-31 modified into a launch-platform for the Izdeliye 293 Burevestnik anti-satellite missile. At least two prototypes are converted. Tests from September 2018.

 * MiG-31I: The latest version of the aircraft which reportedly adds an in-flight refueling capability and includes different engines, avionics and missiles, fly-by-wire and an electronic remote control system which allows to launch Kinzhal missiles at a predetermined moment. It is assumed that all MiG-31K aircraft will be upgraded to this standard. Completed flight tests in April 2024.

Operators

 * Russian Aerospace Forces
 * 4th Centre for Combat Application and Crew Training (Savasleyka)
 * 22nd Fighter Aviation Regiment (Tsentralnaya Uglovaya)
 * 98th Independent Composite Aviation Regiment (Monchegorsk)
 * 764th Fighter Aviation Regiment (Perm/Bolshoye Savino)
 * 712th Guards Fighter Aviation Regiment (Kansk)
 * 790th Fighter Aviation Regiment (Khotilovo/Borisovskiy)
 * 929th V.P. Chkalova State Flight Test Centre (Akhtubinsk)
 * From 85 to 131 MiG-31BM in inventory as of 2020. Deliveries of updated aircraft drawn from older models stocks continue as of 2024.           Ten jets have been modified to the MiG-31K version and carry the Kh-47M2 Kinzhal missile as of May 2018. With this modification and with removed APU for air-to-air missiles, the aircraft gained a sole role of an attack aircraft.
 * Russian Naval Aviation
 * 7060th Naval Aviation Air Base (Petropavlovsk-Kamchatskiy/Yelizovo)
 * 10 MiG-31B/BS and 22 MiG-31BM in inventory as of 2020.
 * 10 MiG-31B/BS and 22 MiG-31BM in inventory as of 2020.

Former operators

 * Soviet Air Forces aircraft passed to the Russian and Kazakh Air Forces after the dissolution of the Soviet Union.
 * Soviet Air Defence Forces
 * Soviet Air Defence Forces

 * Kazakh Air Defense Forces
 * 610th Air Base (Sary-Arka Airport)
 * Operated 31 aircraft. in inventory as of 2022. The type was retired in 2023 and put up for auction. The aircraft were reportedly sold to USA in April 2024,  however it remains unclear who actually bought them, with Kazakh officials claiming that only local companies were allowed to bid.
 * Operated 31 aircraft. in inventory as of 2022. The type was retired in 2023 and put up for auction. The aircraft were reportedly sold to USA in April 2024,  however it remains unclear who actually bought them, with Kazakh officials claiming that only local companies were allowed to bid.

Notable accidents
On 4 April 1984, a MiG-31 crashed while on a test flight, killing Mikoyan chief test pilot and Hero of the Soviet Union, Aleksandr Vasilyevich Fedotov and his navigator V. Zaitsev.

On 26 April 2017, a MiG-31 crashed after it was accidentally hit by "friendly fire" during a training session near the Telemba proving ground in Russian Far East. This occurred during a training exercise over the Telemba proving ground in Buryatia; both crew members ejected successfully. While Russian state media did not offer any details, independent investigators discovered from a leaked government document that the aircraft was shot down by an R-33 missile fired from another MiG-31 and that pilot error from both airplanes was at fault. The report also suggested problems with the Zaslon-AM radar and Baget-55 fire control system that might increase the risk of more accidental shootdowns.

On 16 April 2020, a MiG-31 interceptor of the Kazakh Air Force crashed in the country's Karaganda region.

On 8 April 2022, a MiG-31 of the Russian Aerospace Forces crashed in the Leningrad region.

On 2 December 2022, a MiG-31 of the Russian Aerospace Forces crashed during a training flight in the far eastern Primorsky Region.

On 26 April 2023, a MiG-31 of the Russian Aerospace Forces crashed during a training flight in northern Murmansk. The pilots reportedly ejected and survived.

On 4 July 2023, a MiG-31 of the Russian Aerospace Forces crashed during a training flight over Avacha Bay.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Mikoyan MiG-31


text:
Courtney Walsh

Courtney Andrew Walsh OJ (born 30 October 1962) is a Jamaican former cricketer who represented the West Indies from 1984 to 2001, captaining the West Indies in 22 Test matches. He is a fast bowler and considered one of the all-time greats, best known for a remarkable opening bowling partnership along with fellow West Indian Curtly Ambrose for several years. Walsh played 132 Tests and 205 ODIs for the West Indies and took 519 and 227 wickets respectively. He shared 421 Test wickets with Ambrose in 49 matches. He held the record of most Test wickets from 2000, after he broke the record of Kapil Dev. This record was later broken in 2004 by Shane Warne. He was the first bowler to reach 500 wickets in Test cricket. His autobiography is entitled "Heart of the Lion". Walsh was named one of the Wisden Cricketers of the Year in 1987. In October 2010, he was inducted into the ICC Cricket Hall of Fame. He was appointed as the Specialist Bowling Coach of Bangladesh Cricket Team in August 2016.

Early life and first-class career
Courtney Andrew Walsh was born on 30 October 1962 in Kingston, Jamaica. He played his early cricket there with the same cricket club for which Michael Holding also played cricket—the Melbourne club. Walsh first claim to fame came in 1979 when he took 10 wickets in an innings in school cricket and three years later made his first-class cricket debut. He played 427 matches of this format between 1981 and 2001, and took 1,807 wickets at the average of 21.71, including 104 five-wicket hauls and 20 ten-wicket hauls. Walsh played first-class cricket for Gloucestershire County Cricket Club (Gloucestershire CCC) from 1985 to 1998.

Walsh played cricket for the West Indies from 1984 to 2001, Gloucestershire County Cricket Club (Gloucestershire CCC) from 1984 to 1998, Jamaica cricket team from 1981–82 to 1999–00, Rest of the World XI in 1987 and West Indies A in 1991–92. He first played for Gloucestershire CCC in 1984 and was a mainstay of the side until 1998.

International career
Walsh made his Test debut against Australia in Perth in 1984, taking 2 wickets for 43 runs. He played six Test matches during the 1984–85 season, five against Australia in the 1984–85 series between the teams, and one Test against New Zealand in the home series. He took 16 wickets in the season conceding 507 runs. In the same season, Walsh also made his One Day International debut against Sri Lanka at Hobart during the World Series Cup. He took one wicket for 47 from 10 overs in the match. In the next two seasons, Walsh played a match at home against England, three matches against Pakistan and three matches against New Zealand, both outside the West Indies. He took 29 wickets from seven matches in these seasons, including a five-wicket haul against New Zealand. In 1987, Walsh was named as one of the Wisden Cricketers of the Year for his performance the previous year.

In the 1987–88 season, Walsh toured India and played four Test matches against them, taking 26 wickets at an average of 16.80. In the first Test of the series at Feroz Shah Kotla Ground, he took six wickets for 67 runs, including five wickets for 54 runs in India's second innings. In the first innings of the second Test at Wankhede Stadium, he repeated the same performance of five for 54. In that season, Pakistan cricket team visited the West Indies and played three Tests there. In the 1987 Cricket World Cup Walsh backed out to ball and run out Saleem Jaffar as he was backing up as Pakistan required two off the last ball to qualify for the semi-finals, Abdul Qadir eventually scored the winning runs and West Indies lost but Walsh was deservedly feted for his sportsmanship, he even received a hand-woven carpet from a local fan. Walsh underperformed in the series, taking only four wickets from three matches. He played four matches against England in 1988, and took 12 wickets at an average of 34.33. During the West Indies tour of Australia in 1988–89, Walsh played five Tests and took 17 wickets at 29.41. His best bowling in the series came the first innings of the first match, taking four wickets for 62 runs, including a hat-trick. He took an unusual hat-trick that covered two innings at the Brisbane Cricket Ground, dismissing Australia's Tony Dodemaide with the last ball of the first innings and Mike Veletta and Graeme Wood with his first two deliveries in the second. During that winter he also took 10 wickets in a Test match for the first time against India in Kingston.

In 1994, he was appointed captain of the West Indies for the tours of India and New Zealand after Richie Richardson was ordered to rest because of "acute fatigue syndrome". In 1995, he took 62 Test wickets at an average of 21.75 runs per wicket, a performance which he bettered in 2000 when he took 66 Test wickets at an average of 18.69, including 34 wickets in the Test series against England at an average of 12.82 runs per wicket. Coming close to the record for a West Indian bowler of 35 wickets in a Test series (set by Malcolm Marshall in 1988). In the 1990s, his partnership with Curtly Ambrose was one of the most feared bowling attacks in world cricket.

During the first part of his career, Walsh served as the "stock" bowler in an attack featuring Marshall, Joel Garner and later Ambrose, but after the retirement of Marshall and Garner took the role as opening bowler. His action lacked the elegance of those bowlers, but its economy and his natural athleticism ensured he was accurate and durable, even over very long spells and he used his height (about 198 cm, or six-foot-six) to extract vicious bounce. Even as he lost pace in the later stage of his career he continued to take wickets at an undiminished rate; teams tended to defend against him and Ambrose and attack the weaker third and fourth bowlers.

Walsh played his last ODI against New Zealand in 2000 and his last Test match against South Africa in his homeland, Jamaica, in 2001.

Walsh is one of only seven bowlers to have bowled over 5000 overs in Test cricket, the others being Muttiah Muralitharan of Sri Lanka, Shane Warne and Nathan Lyon of Australia, Anil Kumble of India and James Anderson and Stuart Broad of England. He is also one of only seven bowlers to have taken 500 or more Test wickets. In ODIs, Walsh was not as successful although he will be remembered for his best performance, 5 wickets for just 1 run against Sri Lanka in 1986. In first-class cricket, he took 5 wickets in an innings more than 100 times and 10 wickets in a match 20 times.

His highest score, coincidentally, was 30 in both forms of the game. By the end of his career, he had such a reputation for poor batting that the crowd would cheer every ball he faced. Whether this was in appreciation of his batting or an attempt to wind up the bowling side is open to interpretation. His most significant (if scoreless) innings came when he kept Brian Lara company during a last-wicket stand to ensure victory in the match (and thus a draw in the series) against Australia in the Third Test in 1999 (Walsh also took seven wickets in the match). A trademark of his batting was an elaborate windmilling leave-alone. Walsh is a much loved and respected cricketer and the West Indies have yet to find a fast bowler with anything approaching his talents.

Walsh is also famous for his sportsmanlike gesture of not mankading last man Saleem Jaffar of Pakistan in a World Cup match in 1987, which cost the West Indies the match and a place in the semi-finals.

He is currently a regular feature of the Lashings World XI alongside other cricket legends including Sachin Tendulkar and fellow West Indian Richie Richardson. In May 2004, Walsh was chosen one of the Jamaica's five greatest cricketers of all time.

He has a dubious distinction of being one of the two players to play more than 100 test matches and not make a half-century, the other being Nathan Lyon.

Records
Walsh took 5 wickets in an ODI match conceding only one run against Sri Lanka in December 1986, a match which the West Indies won at the Sharjah Cricket Association Stadium, Sharjah by 193 runs. This was his solitary five-wicket haul in the format. In February 1998, he played his 100th Test, against England in Georgetown, Guyana. In 2000, Walsh became the highest wicket-taker in Test cricket, breaking a six-year-old record of Kapil Dev's 434 wickets. He achieved the feat in his 114th match, which are 17 matches less than Kapil Dev. Walsh became the first bowler to reach 500 wickets in the history of Test cricket. He achieved the feat against South Africa in 2001, getting Jacques Kallis leg before wicket in Port-of-Spain, Trinidad. Throughout his Test career, Walsh produced one of the greatest opening partnership with Curtly Ambrose as a bowler, and shared 421 wickets from 49 matches with the latter. Walsh's 519 wickets in Test cricket was a record at that time, which was surpassed by Sri Lankan Muttiah Muralitharan in 2004.

Walsh's feats with the bat are rather less flattering, as indicated by an average of seven in both Test cricket and ODIs. He also holds the record for the most Test ducks (43), and also held the record for the most "not outs" – 61 times – until passed by James Anderson in 2017.

Walsh played 132 Tests and 205 ODI matches, and took 519 and 227 wickets respectively. He took 22 five wicket hauls in Tests—of which five fifers came in the first 63 appearances and 17 in the later 69 appearances—and one in ODIs.

Walsh holds the record for the best bowling figures (13 for 55) for a captain in a Test match.

Retirement
During his Test career, spanned over seventeen years, Walsh bowled 5004.1 overs, captured 519 wickets at an average of 24.45 runs and at a strike rate of 57.55 in 132 Test matches. Cricket critics considered him that he was "one of the most admired cricketers of recent times and will long be remembered as one of the game's most revered players." He played last time in Test cricket against South Africa in April 2001, a match West Indies won by 130 runs at Sabina Park, Kingston, Jamaica. He took six wickets for 103 runs in the match. In ODIs, he took 227 wickets from 205 matches at an average of 30.47. His last ODI came in January 2000, against New Zealand at Jade Stadium, Christchurch in which he took one wicket for 70 runs.

Former West Indian captain Clive Lloyd attributed Walsh: "I don't think you'll find another Courtney Walsh around and if I was a young fast bowlers, I'd want to emulate him." Former West Indian all-rounder Garry Sobers said about him that the "young crop of fast bowlers can take from him his dedication to West Indies and his ability to be always there, trying and giving 100 per cent in difficult conditions."

Post-retirement
After his career as a selector for the West Indies national cricket team, Walsh signed as bowling coach of Bangladesh in August, 2016. His contract expired after the 2019 Cricket World Cup and he left his position as Bangladesh bowling coach along with then Bangladesh head coach Steve Rhodes.

Courtney Walsh is the son of Joan Wollaston and also owns a restaurant in Jamaica called Cuddyz. In November 2019, he was appointed as the assistant coach to the head coach Gus Logie for the West Indies women's cricket team.

Coaching career
He was appointed as talent scout and fast bowling coach for Kings XI Punjab.

Walsh was appointed bowling coach of the Bangladesh national cricket team in September 2016 on a three-year contract. He was appointed interim head coach in February 2018 following the resignation of Chandika Hathurusingha, with his first tournament in charge being the 2018 Nidahas Trophy.

In October 2020, he was appointed as head coach of West Indies women's national cricket team. After the 2023 T20 World Cup, Cricket West Indies decided not to renew his contract.

Awards and honours
Walsh was named one of the Wisden Cricketers of the Year in 1987, and the cricket almanack Wisden noted his "three distinct speeds, all delivered with the same action", and his "sparing use of the bouncer, his shorter deliveries generally threatening the batsman's rib-cage, a tactic which, allied to change of pace, produced many catches in the short-leg area off splice or glove." Walsh was selected as one of the West Indian Cricket Cricketer of the Year in 1988. He was named one Jamaica's greatest cricketers of all time in 2004.

In October 2010, he was inducted into the ICC Cricket Hall of Fame along with Joel Garner, joining the other fifteen West Indian players.

title: Courtney Walsh